diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedOrdinaryTransfer.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedOrdinaryTransfer.lean index 44419d0d6685b447cdc310d5d8e0f504b1565ca8..b846d3077fadc8fa7269feeef713593528da0757 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedOrdinaryTransfer.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedOrdinaryTransfer.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Gauge.PrescribedAngleMain -import OAI.Probability.SLE.Geometry.ActualExplicitTransfer - -namespace OAI - -/-! Transfer of the exact prescribed-gauge criterion from a regular -Brownian presentation to the original ordinary SLE probability space. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -local instance prescribedTransferMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance prescribedTransferBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem RegularSLEPresentation.prescribed_main_of_canonical_angle_bounds - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0<κ) (hκ8 : κ<8) (hγ : ∀ t, Measurable (fun ω => γ ω t)) - (hgrowth : PublishedDiskUnivalentGrowth) - (htrace : PublishedTraceGeneration - ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) - κ (fun t p => p t)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) - (δ : ℝ) (hδ : 0<δ) - (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → - ∀ θ∈Ioo (0:ℝ) Real.pi, - (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ - canonicalAngleAverage - ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) - κ v f θ ∧ - canonicalAngleAverage - ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) - κ v f θ ≤ 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) : - MainConclusion P (Target.gauge κ W) γ := by - let Q := P.comap (Subtype.val : a.samples → Ω) - let : IsProbabilityMeasure Q := a.probability - have hBm : Measurable (Function.uncurry a.B) := - measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B - obtain ⟨C⟩ := Multipoint.exists_canonicalMass a.brownian hBm a.continuous_B a.zero_B - hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth - have hmain := C.prescribed_main_of_canonical_angle_bounds a.brownian a.continuous_B - a.zero_B hκ hκ8 a.F a.witness (fun t => (hγ t).comp measurable_subtype_coe) - hgrowth htrace W hW δ hδ hangle - have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := - MeasurableEmbedding.subtype_coe a.measurable_samples - constructor - · rw [← a.measurePreserving_val.map_eq,hemb.ae_map_iff] - exact hmain.1 - · intro L hL - obtain ⟨hm,M,hM⟩ := hmain.2 L hL - refine ⟨(a.measurePreserving_val.aemeasurable_comp_iff hemb).mp hm,M,?_⟩ - have he := a.measurePreserving_val.lintegral_comp_emb hemb - (fun ω => hausdorffGauge (Target.gauge κ W) (wholeTrace (γ ω)∩Metric.closedBall 0 L)) - rw [← he] - exact hM - -theorem ordinary_prescribed_main_of_canonical_angle_bounds - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hgrowth : PublishedDiskUnivalentGrowth) - (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) - (δ : ℝ) (hδ : 0<δ) - (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → - ∀ θ∈Ioo (0:ℝ) Real.pi, - (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ - canonicalAngleAverage (ordinaryRegularDriverLaw hSLE) κ v f θ ∧ - canonicalAngleAverage (ordinaryRegularDriverLaw hSLE) κ v f θ ≤ - 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) : - MainConclusion P (Target.gauge κ W) γ := - (ordinaryRegularPresentation hSLE).prescribed_main_of_canonical_angle_bounds - hκ hκ8 hSLE.1 hgrowth htrace W hW δ hδ hangle - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedAngleMain +import OAI.Probability.SLE.Geometry.ActualExplicitTransfer + +namespace OAI + +/-! Transfer of the exact prescribed-gauge criterion from a regular +Brownian presentation to the original ordinary SLE probability space. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +local instance prescribedTransferMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance prescribedTransferBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem RegularSLEPresentation.prescribed_main_of_canonical_angle_bounds + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0<κ) (hκ8 : κ<8) (hγ : ∀ t, Measurable (fun ω => γ ω t)) + (hgrowth : PublishedDiskUnivalentGrowth) + (htrace : PublishedTraceGeneration + ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) + κ (fun t p => p t)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) + (δ : ℝ) (hδ : 0<δ) + (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → + ∀ θ∈Ioo (0:ℝ) Real.pi, + (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ + canonicalAngleAverage + ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) + κ v f θ ∧ + canonicalAngleAverage + ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) + κ v f θ ≤ 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) : + MainConclusion P (Target.gauge κ W) γ := by + let Q := P.comap (Subtype.val : a.samples → Ω) + let : IsProbabilityMeasure Q := a.probability + have hBm : Measurable (Function.uncurry a.B) := + measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B + obtain ⟨C⟩ := Multipoint.exists_canonicalMass a.brownian hBm a.continuous_B a.zero_B + hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth + have hmain := C.prescribed_main_of_canonical_angle_bounds a.brownian a.continuous_B + a.zero_B hκ hκ8 a.F a.witness (fun t => (hγ t).comp measurable_subtype_coe) + hgrowth htrace W hW δ hδ hangle + have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := + MeasurableEmbedding.subtype_coe a.measurable_samples + constructor + · rw [← a.measurePreserving_val.map_eq,hemb.ae_map_iff] + exact hmain.1 + · intro L hL + obtain ⟨hm,M,hM⟩ := hmain.2 L hL + refine ⟨(a.measurePreserving_val.aemeasurable_comp_iff hemb).mp hm,M,?_⟩ + have he := a.measurePreserving_val.lintegral_comp_emb hemb + (fun ω => hausdorffGauge (Target.gauge κ W) (wholeTrace (γ ω)∩Metric.closedBall 0 L)) + rw [← he] + exact hM + +theorem ordinary_prescribed_main_of_canonical_angle_bounds + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hgrowth : PublishedDiskUnivalentGrowth) + (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) + (δ : ℝ) (hδ : 0<δ) + (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → + ∀ θ∈Ioo (0:ℝ) Real.pi, + (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ + canonicalAngleAverage (ordinaryRegularDriverLaw hSLE) κ v f θ ∧ + canonicalAngleAverage (ordinaryRegularDriverLaw hSLE) κ v f θ ≤ + 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) : + MainConclusion P (Target.gauge κ W) γ := + (ordinaryRegularPresentation hSLE).prescribed_main_of_canonical_angle_bounds + hκ hκ8 hSLE.1 hgrowth htrace W hW δ hδ hangle + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedPublishedMain.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedPublishedMain.lean index 6a037a90bd3e73058c4642406ef906f2612a912f..585b75b4dfbca308365205c22ad1a4df0acfe0c6 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedPublishedMain.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedPublishedMain.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Angle.CanonicalAngleMixing -import OAI.Probability.SLE.Gauge.PrescribedOrdinaryTransfer - -namespace OAI - -/-! The first paper's prescribed moment gauge, from precisely named -published inputs. The angular mixing estimate is proved internally. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -local instance prescribedPublishedMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance prescribedPublishedBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -/-- The exact prescribed gauge gives positive finite mass to every positive-time -compact trace segment and finite expected mass of the whole trace in bounded disks. -Published inputs: Duren (1983), section 2.3 (the disk univalent-growth estimate), -and Rohde--Schramm (2005), Theorem 5.1 (trace generation under the driver law). -/ -theorem RegularSLEPresentation.prescribed_main_of_published_inputs - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0<κ) (hκ8 : κ<8) (hγ : ∀ t, Measurable (fun ω => γ ω t)) - (hgrowth : PublishedDiskUnivalentGrowth) - (htrace : PublishedTraceGeneration - ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) - κ (fun t p => p t)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) : - MainConclusion P (Target.gauge κ W) γ := by - have : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability - have hBm : Measurable (Function.uncurry a.B) := - measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B - apply a.prescribed_main_of_canonical_angle_bounds hκ hκ8 hγ hgrowth htrace W hW - (angleSpectralMixingTime κ) (angleSpectralMixingTime_pos hκ) - exact canonicalAngleAverage_uniform_bounds a.brownian hBm a.continuous_B a.zero_B - hκ hκ8 (fun ω => γ ω) a.G a.F a.witness - -/-- The first paper's full prescribed-gauge Hausdorff conclusion for ordinary -chordal SLE, with only the cited published conformal and trace inputs. -/ -theorem ordinary_prescribed_mainConclusion_of_published_inputs - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hgrowth : PublishedDiskUnivalentGrowth) - (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) : - MainConclusion P (Target.gauge κ W) γ := - (ordinaryRegularPresentation hSLE).prescribed_main_of_published_inputs - hκ hκ8 hSLE.1 hgrowth htrace W hW - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Angle.CanonicalAngleMixing +import OAI.Probability.SLE.Gauge.PrescribedOrdinaryTransfer + +namespace OAI + +/-! The first paper's prescribed moment gauge, from precisely named +published inputs. The angular mixing estimate is proved internally. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +local instance prescribedPublishedMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance prescribedPublishedBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +/-- The exact prescribed gauge gives positive finite mass to every positive-time +compact trace segment and finite expected mass of the whole trace in bounded disks. +Published inputs: Duren (1983), section 2.3 (the disk univalent-growth estimate), +and Rohde--Schramm (2005), Theorem 5.1 (trace generation under the driver law). -/ +theorem RegularSLEPresentation.prescribed_main_of_published_inputs + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0<κ) (hκ8 : κ<8) (hγ : ∀ t, Measurable (fun ω => γ ω t)) + (hgrowth : PublishedDiskUnivalentGrowth) + (htrace : PublishedTraceGeneration + ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) + κ (fun t p => p t)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) : + MainConclusion P (Target.gauge κ W) γ := by + have : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability + have hBm : Measurable (Function.uncurry a.B) := + measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B + apply a.prescribed_main_of_canonical_angle_bounds hκ hκ8 hγ hgrowth htrace W hW + (angleSpectralMixingTime κ) (angleSpectralMixingTime_pos hκ) + exact canonicalAngleAverage_uniform_bounds a.brownian hBm a.continuous_B a.zero_B + hκ hκ8 (fun ω => γ ω) a.G a.F a.witness + +/-- The first paper's full prescribed-gauge Hausdorff conclusion for ordinary +chordal SLE, with only the cited published conformal and trace inputs. -/ +theorem ordinary_prescribed_mainConclusion_of_published_inputs + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hgrowth : PublishedDiskUnivalentGrowth) + (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) : + MainConclusion P (Target.gauge κ W) γ := + (ordinaryRegularPresentation hSLE).prescribed_main_of_published_inputs + hκ hκ8 hSLE.1 hgrowth htrace W hW + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusFinalLaw.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusFinalLaw.lean index d8eeb7aef0b90dbe70a11955a9078f298fc1d96a..698e370ef559578fbaae1ffde9b0ca1030b86f0f 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusFinalLaw.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusFinalLaw.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Gauge.PrescribedRadiusTest - -namespace OAI - -/-! A stopped disk test has the same kernel bound under every later finite -centered law. The terminal cover is not used in this change of measure. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance finalTestMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance finalTestBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.prescribed_disk_test_under_final_law - {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b V : ℝ) - (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r) - (hR : (2*z.im)*Real.exp (-u)=(2048*Real.pi)*r) (hV : u+v ≤ V) : - ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB - (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0 - κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A → - (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v - (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω) - ∂finiteCenteredLaw P κ B γ G z V) ≤ - finiteCenteredLaw P κ B γ G z V - (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) - z ((4096*Real.pi)*r) - (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) - let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u - let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) - have hσ : IsStoppingTime ℱ σ := - actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu - have huv : 0 ≤ u+v := add_nonneg hu hv.le - have hT : IsStoppingTime ℱ T := - actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u+v) huv - have hσT (ω : Ω) : σ ω ≤ T ω := - radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z hu (by linarith) - let f := fun ω => R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v - (extendedMarkedAngle κ B γ G σ z ω) - let test := C.prescribedBallTest T z ((4096*Real.pi)*r) - (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) - let Q := finiteCenteredLaw P κ B γ G z (u+v) - let QV := finiteCenteredLaw P κ B γ G z V - let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - have hfm : Measurable[hσ.measurableSpace] f := - (R.angleDiskTail_measurable _ κ b v).comp - (extendedMarkedAngle_measurable B hm hc κ γ G F hw σ hσ z) - intro A hA - have hleft : (∫⁻ ω in A, f ω ∂QV)=(∫⁻ ω in A, f ω ∂Q) := by - have h := @lintegral_eq_of_stopped_events Ω mΩ QV Q hσ.measurableSpace - hσ.measurableSpace_le (fun E hE => - finiteCenteredLaw_consistent_at_extended_stop hB hBm hc h0 κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz (u+v) V huv hV σ hσ - (Eventually.of_forall hσT) E hE) (A.indicator f) (hfm.indicator hA) - simpa only [lintegral_indicator (hσ.measurableSpace_le _ hA)] using h - have hAm : MeasurableSet[hT.measurableSpace] A := - hσ.measurableSpace_mono hT hσT A hA - have htest : MeasurableSet[hT.measurableSpace] test := - C.prescribedBallTest_measurable T hT _ _ _ - have hright : QV (A∩test)=Q (A∩test) := - finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz (u+v) V huv hV T hT - (Eventually.of_forall fun _ => rfl) (A∩test) (hAm.inter htest) - change (∫⁻ ω in A, f ω ∂QV) ≤ QV (A∩test) - rw [hleft,hright] - exact C.prescribed_disk_test_at_radius hB hc h0 hκ hκ8 F hw R he D hD hDf hDp - hgrowth hpublished z hz u v r b hu hv hr hR A hA - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedRadiusTest + +namespace OAI + +/-! A stopped disk test has the same kernel bound under every later finite +centered law. The terminal cover is not used in this change of measure. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance finalTestMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance finalTestBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.prescribed_disk_test_under_final_law + {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b V : ℝ) + (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r) + (hR : (2*z.im)*Real.exp (-u)=(2048*Real.pi)*r) (hV : u+v ≤ V) : + ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB + (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0 + κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A → + (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v + (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω) + ∂finiteCenteredLaw P κ B γ G z V) ≤ + finiteCenteredLaw P κ B γ G z V + (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) + z ((4096*Real.pi)*r) + (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) + let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u + let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) + have hσ : IsStoppingTime ℱ σ := + actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu + have huv : 0 ≤ u+v := add_nonneg hu hv.le + have hT : IsStoppingTime ℱ T := + actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u+v) huv + have hσT (ω : Ω) : σ ω ≤ T ω := + radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z hu (by linarith) + let f := fun ω => R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v + (extendedMarkedAngle κ B γ G σ z ω) + let test := C.prescribedBallTest T z ((4096*Real.pi)*r) + (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) + let Q := finiteCenteredLaw P κ B γ G z (u+v) + let QV := finiteCenteredLaw P κ B γ G z V + let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + have hfm : Measurable[hσ.measurableSpace] f := + (R.angleDiskTail_measurable _ κ b v).comp + (extendedMarkedAngle_measurable B hm hc κ γ G F hw σ hσ z) + intro A hA + have hleft : (∫⁻ ω in A, f ω ∂QV)=(∫⁻ ω in A, f ω ∂Q) := by + have h := @lintegral_eq_of_stopped_events Ω mΩ QV Q hσ.measurableSpace + hσ.measurableSpace_le (fun E hE => + finiteCenteredLaw_consistent_at_extended_stop hB hBm hc h0 κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz (u+v) V huv hV σ hσ + (Eventually.of_forall hσT) E hE) (A.indicator f) (hfm.indicator hA) + simpa only [lintegral_indicator (hσ.measurableSpace_le _ hA)] using h + have hAm : MeasurableSet[hT.measurableSpace] A := + hσ.measurableSpace_mono hT hσT A hA + have htest : MeasurableSet[hT.measurableSpace] test := + C.prescribedBallTest_measurable T hT _ _ _ + have hright : QV (A∩test)=Q (A∩test) := + finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz (u+v) V huv hV T hT + (Eventually.of_forall fun _ => rfl) (A∩test) (hAm.inter htest) + change (∫⁻ ω in A, f ω ∂QV) ≤ QV (A∩test) + rw [hleft,hright] + exact C.prescribed_disk_test_at_radius hB hc h0 hκ hκ8 F hw R he D hD hDf hDp + hgrowth hpublished z hz u v r b hu hv hr hR A hA + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusLevels.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusLevels.lean index 4b9e4334fdecb9dbc2b54f126e778ba1f6972fc4..377744a84b28d8b885ed250dfafd2cffab082000 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusLevels.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedRadiusLevels.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Gauge.BatchRadiusGaps -import OAI.Probability.SLE.Gauge.GaugeEntrySmallness - -namespace OAI - -/-! Physical logarithmic levels for the exact prescribed radii. Uniformity - includes moving centers down to height exp(-n). -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge.Target - -def entryPhysicalLevel (κ : ℝ) (W : WienerLaw) (e : Entry) (A : ℝ) (z : ℂ) : ℝ := - Real.log ((2*z.im)/A) + (e.1:ℝ)^2 + - 2*a κ*(entryPosition κ W e:ℝ)*((batchLevel κ W e.1:ℝ)+(e.1:ℝ)) - -lemma entryPhysicalLevel_radius (κ : ℝ) (W : WienerLaw) (e : Entry) - {A : ℝ} (hA : 0 B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ) - (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r) - (hR : (2*z.im)*Real.exp (-u)=(2048*Real.pi)*r) : - ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB - (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0 - κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A → - (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v - (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω) - ∂finiteCenteredLaw P κ B γ G z (u+v)) ≤ - finiteCenteredLaw P κ B γ G z (u+v) - (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) - z ((4096*Real.pi)*r) - (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by - classical - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) - let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u - have hσ : IsStoppingTime ℱ σ := - actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu - let Q := finiteCenteredLaw P κ B γ G z (u+v) - let f := fun ω => R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v - (extendedMarkedAngle κ B γ G σ z ω) - let test := C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) - z ((4096*Real.pi)*r) (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) - intro A hA - let E := fun n : ℕ => A∩{ω | σ ω≤((n:ℝ≥0):WithTop ℝ≥0)} - have hEm (n : ℕ) : MeasurableSet (E n) := - hσ.measurableSpace_le _ (hA.inter (hσ.measurableSet_le' n)) - have hmono : Monotone E := by - intro n k hnk ω hω - change ω∈A ∧ σ ω≤((n:ℝ≥0):WithTop ℝ≥0) at hω - change ω∈A ∧ σ ω≤((k:ℝ≥0):WithTop ℝ≥0) - exact ⟨hω.1,hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))⟩ - have hfinite : ∀ᵐ ω ∂Q, σ ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw) - z hz u (u+v) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl) - have hbound (n : ℕ) : (∫⁻ ω in E n, f ω ∂Q) ≤ Q (A∩test) := by - let τ := cutoffStop σ n - have hτ := cutoffStop_stopping ℱ σ hσ (n:ℝ≥0) - have hEn : MeasurableSet[hτ.measurableSpace] (E n) := - cutoffStop_success_measurable ℱ σ hσ n A hA - have hstate (ω : Ω) (hω : ω∈E n) : - z ∈ survivingDomain (γ ω) (τ ω) ∧ - fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u) := by - rw [show τ ω=(σ ω).untopD 0 from cutoffStop_eq_on_success σ n ω hω.2] - exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u - (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2) - have hh := C.prescribed_disk_conditional_bound hB hc h0 hκ hκ8 F hw R he - D hD hDf hDp hgrowth hpublished τ hτ z hz u v r b hu hv hr (E n) hEn - (fun ω hω => (hstate ω hω).1) (fun ω hω => (hstate ω hω).2) - (fun ω hω => (hstate ω hω).2.trans hR) univ MeasurableSet.univ - simp only [univ_inter] at hh - have hid (ω : Ω) (hω : ω∈E n) : f ω=R.angleDiskTail - (P.map (sourceDriverPath B hc)) κ b v (stoppedMarkedCenter κ B γ G τ z ω).arg := by - dsimp only [f,extendedMarkedAngle] - rw [ite_eq_right (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)] - congr 2 - exact (stoppedMarkedCenter_time_congr κ B γ G τ - (fun ω => (σ ω).untopD 0) z ω (cutoffStop_eq_on_success σ n ω hω.2)).symm - rw [setLIntegral_congr_fun (hEm n) hid] - exact hh.trans (measure_mono (inter_subset_inter inter_subset_left subset_rfl)) - have hcover : A =ᵐ[Q] ⋃ n, E n := by - filter_upwards [hfinite] with ω hω - apply propext - constructor - · intro hAω - lift σ ω to ℝ≥0 using hω with t ht - obtain ⟨n,hn⟩ := exists_nat_ge (t:ℝ) - exact mem_iUnion.mpr ⟨n,hAω,by - change σ ω≤((n:ℝ≥0):WithTop ℝ≥0) - rw [← ht] - exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)⟩ - · intro hωE - obtain ⟨n,hn⟩ := mem_iUnion.mp hωE - exact hn.1 - have habs : (Q.withDensity f).AbsolutelyContinuous Q := withDensity_absolutelyContinuous Q f - calc - (∫⁻ ω in A, f ω ∂Q) = Q.withDensity f A := (withDensity_apply _ (hσ.measurableSpace_le _ hA)).symm - _ = Q.withDensity f (⋃ n, E n) := measure_congr (habs.ae_le hcover) - _ = ⨆ n, Q.withDensity f (E n) := hmono.measure_iUnion - _ ≤ Q (A∩test) := iSup_le fun n => by - rw [withDensity_apply _ (hEm n)] - exact hbound n - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedDiskConditional +import OAI.Probability.SLE.Geometry.ExtendedDiskAngle + +namespace OAI + +/-! The prescribed disk success kernel at the literal radius stop. Finite +cutoffs exhaust that stop under the actual finite centered law. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance radiusTestMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance radiusTestBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.prescribed_disk_test_at_radius + {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ) + (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r) + (hR : (2*z.im)*Real.exp (-u)=(2048*Real.pi)*r) : + ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB + (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0 + κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A → + (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v + (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω) + ∂finiteCenteredLaw P κ B γ G z (u+v)) ≤ + finiteCenteredLaw P κ B γ G z (u+v) + (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) + z ((4096*Real.pi)*r) + (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by + classical + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) + let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u + have hσ : IsStoppingTime ℱ σ := + actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu + let Q := finiteCenteredLaw P κ B γ G z (u+v) + let f := fun ω => R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v + (extendedMarkedAngle κ B γ G σ z ω) + let test := C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) + z ((4096*Real.pi)*r) (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) + intro A hA + let E := fun n : ℕ => A∩{ω | σ ω≤((n:ℝ≥0):WithTop ℝ≥0)} + have hEm (n : ℕ) : MeasurableSet (E n) := + hσ.measurableSpace_le _ (hA.inter (hσ.measurableSet_le' n)) + have hmono : Monotone E := by + intro n k hnk ω hω + change ω∈A ∧ σ ω≤((n:ℝ≥0):WithTop ℝ≥0) at hω + change ω∈A ∧ σ ω≤((k:ℝ≥0):WithTop ℝ≥0) + exact ⟨hω.1,hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))⟩ + have hfinite : ∀ᵐ ω ∂Q, σ ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw) + z hz u (u+v) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl) + have hbound (n : ℕ) : (∫⁻ ω in E n, f ω ∂Q) ≤ Q (A∩test) := by + let τ := cutoffStop σ n + have hτ := cutoffStop_stopping ℱ σ hσ (n:ℝ≥0) + have hEn : MeasurableSet[hτ.measurableSpace] (E n) := + cutoffStop_success_measurable ℱ σ hσ n A hA + have hstate (ω : Ω) (hω : ω∈E n) : + z ∈ survivingDomain (γ ω) (τ ω) ∧ + fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u) := by + rw [show τ ω=(σ ω).untopD 0 from cutoffStop_eq_on_success σ n ω hω.2] + exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u + (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2) + have hh := C.prescribed_disk_conditional_bound hB hc h0 hκ hκ8 F hw R he + D hD hDf hDp hgrowth hpublished τ hτ z hz u v r b hu hv hr (E n) hEn + (fun ω hω => (hstate ω hω).1) (fun ω hω => (hstate ω hω).2) + (fun ω hω => (hstate ω hω).2.trans hR) univ MeasurableSet.univ + simp only [univ_inter] at hh + have hid (ω : Ω) (hω : ω∈E n) : f ω=R.angleDiskTail + (P.map (sourceDriverPath B hc)) κ b v (stoppedMarkedCenter κ B γ G τ z ω).arg := by + dsimp only [f,extendedMarkedAngle] + rw [ite_eq_right (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)] + congr 2 + exact (stoppedMarkedCenter_time_congr κ B γ G τ + (fun ω => (σ ω).untopD 0) z ω (cutoffStop_eq_on_success σ n ω hω.2)).symm + rw [setLIntegral_congr_fun (hEm n) hid] + exact hh.trans (measure_mono (inter_subset_inter inter_subset_left subset_rfl)) + have hcover : A =ᵐ[Q] ⋃ n, E n := by + filter_upwards [hfinite] with ω hω + apply propext + constructor + · intro hAω + lift σ ω to ℝ≥0 using hω with t ht + obtain ⟨n,hn⟩ := exists_nat_ge (t:ℝ) + exact mem_iUnion.mpr ⟨n,hAω,by + change σ ω≤((n:ℝ≥0):WithTop ℝ≥0) + rw [← ht] + exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)⟩ + · intro hωE + obtain ⟨n,hn⟩ := mem_iUnion.mp hωE + exact hn.1 + have habs : (Q.withDensity f).AbsolutelyContinuous Q := withDensity_absolutelyContinuous Q f + calc + (∫⁻ ω in A, f ω ∂Q) = Q.withDensity f A := (withDensity_apply _ (hσ.measurableSpace_le _ hA)).symm + _ = Q.withDensity f (⋃ n, E n) := measure_congr (habs.ae_le hcover) + _ = ⨆ n, Q.withDensity f (E n) := hmono.measure_iUnion + _ ≤ Q (A∩test) := iSup_le fun n => by + rw [withDensity_apply _ (hEm n)] + exact hbound n + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedRowSum.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedRowSum.lean index cf432b92c363f97df12ea235ee2aa33d0eaa6964..acd5d7733dc74d218c3f0f70a18810834e7a6870 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedRowSum.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedRowSum.lean @@ -1,58 +1,58 @@ -import OAI.Probability.SLE.SimpleTails -import Mathlib.Analysis.SumIntegralComparisons - -namespace OAI - -/-! The half-integer row heights have an integrable boundary singularity. -This is the row sum used in the prescribed cover's missed-cell estimate. -/ -noncomputable section -open Set Filter MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -lemma half_integer_row_sum_bound (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1) (N : ℕ) : - (∑ j∈Finset.range N, ((j:ℝ)+1/2)^(-α)) ≤ - (2:ℝ)^α/(1-α)*(N:ℝ)^(1-α) := by - have hq : 0<1-α := by linarith - have hfi : IntegrableOn (fun x : ℝ => x^(-α)) (Icc 0 (N:ℝ)) := by - have ha := (integrableOn_abs_heightPower α (N:ℝ) hα1).mono_set - (show Icc (0:ℝ) N ⊆ Icc (-(N:ℝ)) N by - intro x hx - exact ⟨(neg_nonpos.mpr (Nat.cast_nonneg N)).trans hx.1,hx.2⟩) - apply ha.congr_fun _ measurableSet_Icc - intro x hx - dsimp only - rw [abs_of_nonneg hx.1] - have hanti : AntitoneOn (fun x : ℝ => x^(-α)) (Ioc 0 (N:ℝ)) := by - intro x hx y _ hxy - exact Real.rpow_le_rpow_of_nonpos hx.1 hxy (neg_nonpos.mpr hα0) - have hsum : (∑ j∈Finset.range N, ((j:ℝ)+1)^(-α)) ≤ - ∫ x in (0:ℝ)..(N:ℝ), x^(-α) := by - have ha : AntitoneOn (fun x : ℝ => x^(-α)) (Ioc 0 (0+(N:ℝ))) := by - simpa only [zero_add] using hanti - simpa only [zero_add,Nat.cast_add,Nat.cast_one] using - ha.sum_le_integral_of_integrableOn (x₀ := 0) (a := N) (by simpa only [zero_add] using hfi) - have hint : (∫ x in (0:ℝ)..(N:ℝ), x^(-α)) = (N:ℝ)^(1-α)/(1-α) := by - rw [integral_rpow (Or.inl (by linarith : -1 < -α))] - rw [show -α+1=1-α by ring,Real.zero_rpow hq.ne',sub_zero] - have hhalf (j : ℕ) : ((j:ℝ)+1/2)^(-α) ≤ (2:ℝ)^α*((j:ℝ)+1)^(-α) := by - have h := Real.rpow_le_rpow_of_nonpos - (show (0:ℝ)<((j:ℝ)+1)/2 by positivity) - (show ((j:ℝ)+1)/2 ≤ (j:ℝ)+1/2 by nlinarith [Nat.cast_nonneg (α := ℝ) j]) - (neg_nonpos.mpr hα0) - refine h.trans_eq ?_ - rw [Real.div_rpow (by positivity) (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_inv_eq_mul] - ring - calc - _ ≤ ∑ j∈Finset.range N, (2:ℝ)^α*((j:ℝ)+1)^(-α) := Finset.sum_le_sum (fun j _ => hhalf j) - _ = (2:ℝ)^α*∑ j∈Finset.range N, ((j:ℝ)+1)^(-α) := (Finset.mul_sum ..).symm - _ ≤ (2:ℝ)^α*(∫ x in (0:ℝ)..(N:ℝ), x^(-α)) := - mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (by norm_num) _) - _ = _ := by rw [hint]; ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.SimpleTails +import Mathlib.Analysis.SumIntegralComparisons + +namespace OAI + +/-! The half-integer row heights have an integrable boundary singularity. +This is the row sum used in the prescribed cover's missed-cell estimate. -/ +noncomputable section +open Set Filter MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +lemma half_integer_row_sum_bound (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1) (N : ℕ) : + (∑ j∈Finset.range N, ((j:ℝ)+1/2)^(-α)) ≤ + (2:ℝ)^α/(1-α)*(N:ℝ)^(1-α) := by + have hq : 0<1-α := by linarith + have hfi : IntegrableOn (fun x : ℝ => x^(-α)) (Icc 0 (N:ℝ)) := by + have ha := (integrableOn_abs_heightPower α (N:ℝ) hα1).mono_set + (show Icc (0:ℝ) N ⊆ Icc (-(N:ℝ)) N by + intro x hx + exact ⟨(neg_nonpos.mpr (Nat.cast_nonneg N)).trans hx.1,hx.2⟩) + apply ha.congr_fun _ measurableSet_Icc + intro x hx + dsimp only + rw [abs_of_nonneg hx.1] + have hanti : AntitoneOn (fun x : ℝ => x^(-α)) (Ioc 0 (N:ℝ)) := by + intro x hx y _ hxy + exact Real.rpow_le_rpow_of_nonpos hx.1 hxy (neg_nonpos.mpr hα0) + have hsum : (∑ j∈Finset.range N, ((j:ℝ)+1)^(-α)) ≤ + ∫ x in (0:ℝ)..(N:ℝ), x^(-α) := by + have ha : AntitoneOn (fun x : ℝ => x^(-α)) (Ioc 0 (0+(N:ℝ))) := by + simpa only [zero_add] using hanti + simpa only [zero_add,Nat.cast_add,Nat.cast_one] using + ha.sum_le_integral_of_integrableOn (x₀ := 0) (a := N) (by simpa only [zero_add] using hfi) + have hint : (∫ x in (0:ℝ)..(N:ℝ), x^(-α)) = (N:ℝ)^(1-α)/(1-α) := by + rw [integral_rpow (Or.inl (by linarith : -1 < -α))] + rw [show -α+1=1-α by ring,Real.zero_rpow hq.ne',sub_zero] + have hhalf (j : ℕ) : ((j:ℝ)+1/2)^(-α) ≤ (2:ℝ)^α*((j:ℝ)+1)^(-α) := by + have h := Real.rpow_le_rpow_of_nonpos + (show (0:ℝ)<((j:ℝ)+1)/2 by positivity) + (show ((j:ℝ)+1)/2 ≤ (j:ℝ)+1/2 by nlinarith [Nat.cast_nonneg (α := ℝ) j]) + (neg_nonpos.mpr hα0) + refine h.trans_eq ?_ + rw [Real.div_rpow (by positivity) (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_inv_eq_mul] + ring + calc + _ ≤ ∑ j∈Finset.range N, (2:ℝ)^α*((j:ℝ)+1)^(-α) := Finset.sum_le_sum (fun j _ => hhalf j) + _ = (2:ℝ)^α*∑ j∈Finset.range N, ((j:ℝ)+1)^(-α) := (Finset.mul_sum ..).symm + _ ≤ (2:ℝ)^α*(∫ x in (0:ℝ)..(N:ℝ), x^(-α)) := + mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (by norm_num) _) + _ = _ := by rw [hint]; ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedTestPast.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedTestPast.lean index 351f6bb17374dc7a7cc3125607d34db705b0ea20..893ea5b459b90a790c14e50d536ea0b0a98b843b 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedTestPast.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedTestPast.lean @@ -1,93 +1,93 @@ -import OAI.Probability.SLE.Gauge.PrescribedBatchClock - -namespace OAI - -/-! The history before each test is taken at the preceding test endpoint. -The intervening radius gap is reserved for angular mixing. -/ -noncomputable section -open Set Filter - -namespace SLEExactGauge.Target - -def prescribedTestPastLevel {J : ℕ} (κ : ℝ) (W : WienerLaw) (n : ℕ) - (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : ℕ) : ℝ := - if i=0 then 0 else prescribedBatchLevel κ W n e A z (i-1)+ - 2*a κ*(batchLevel κ W n:ℝ) - -lemma prescribedTestPastLevel_nonneg {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) - (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) - (hl : ∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) A z) (i : ℕ) : - 0 ≤ prescribedTestPastLevel κ W n e A z i := by - have ha := internal_a_pos hκ - unfold prescribedTestPastLevel prescribedBatchLevel - split_ifs - · exact le_rfl - · exact add_nonneg (hl _) (by positivity) - · exact add_nonneg (hl _) (by positivity) - -lemma prescribedTestPastLevel_succ {κ : ℝ} (W : WienerLaw) (n : ℕ) - {J : ℕ} (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : Fin J) : - prescribedTestPastLevel κ W n e A z (i.val+1)= - entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ) := by - simp only [prescribedTestPastLevel,Nat.add_one_ne_zero,ite_false,Nat.add_sub_cancel, - prescribedBatchLevel,dite_eq_left i.isLt,Fin.eta] - -lemma prescribedTestPastLevel_monotone {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) - {n J : ℕ} (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1) - (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ) - (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1) - (horder : StrictMono (fun i => entryPosition κ W (n,some (e i)))) - (A : ℝ) (z : ℂ) (hl : ∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) A z) : - Monotone (prescribedTestPastLevel κ W n e A z) := by - intro i j hij - by_cases hi : i=0 - · subst i - simpa only [prescribedTestPastLevel,ite_true] using - prescribedTestPastLevel_nonneg hκ W n e A z hl j - · have hj : j ≠ 0 := by omega - simp only [prescribedTestPastLevel,ite_eq_right hi,ite_eq_right hj] - exact add_le_add - (prescribedBatchLevel_monotone hκ W hn hm1 hJ e he horder A z - (Nat.sub_le_sub_right hij 1)) le_rfl - -lemma prescribedTestPastLevel_gap {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) - {n J : ℕ} (hn : 1 ≤ n) (e : Fin J → ℤ×ℕ) - (horder : StrictMono (fun i => entryPosition κ W (n,some (e i)))) - (A : ℝ) (z : ℂ) (δ : ℝ) - (hfirst : ∀ i, δ ≤ entryPhysicalLevel κ W (n,some (e i)) A z) - (hδ : δ ≤ 2*a κ*(n:ℝ)) (i : Fin J) : - prescribedTestPastLevel κ W n e A z i.val+δ ≤ - entryPhysicalLevel κ W (n,some (e i)) A z := by - by_cases hi : i.val=0 - · simpa only [prescribedTestPastLevel,hi,ite_true,zero_add] using hfirst i - · let j : Fin J := ⟨i.val-1,by omega⟩ - have heq : i.val=j.val+1 := by dsimp only [j]; omega - rw [heq,prescribedTestPastLevel_succ W n e A z j] - have hpos := horder (show j < i from by - dsimp only [j] - exact Nat.sub_lt (Nat.pos_of_ne_zero hi) (by decide)) - have hh := entryPhysicalLevel_gap hκ W hn hpos A z - linarith - -lemma eventually_prescribedTestPastLevel_gap {κ : ℝ} (hκ : 0 < κ) - (W : WienerLaw) {A : ℝ} (hA : 0 < A) (δ : ℝ) : - ∀ᶠ n : ℕ in Filter.atTop, ∀ (J : ℕ) (e : Fin J → ℤ×ℕ), - StrictMono (fun i => entryPosition κ W (n,some (e i))) → - ∀ z : ℂ, Real.exp (-(n:ℝ)) ≤ z.im → ∀ i : Fin J, - prescribedTestPastLevel κ W n e A z i.val+δ ≤ - entryPhysicalLevel κ W (n,some (e i)) A z := by - have ha : 0 < 2*a κ := mul_pos (by norm_num) (internal_a_pos hκ) - have hn : ∀ᶠ n : ℕ in Filter.atTop, δ/(2*a κ) ≤ (n:ℝ) := - tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _) - filter_upwards [eventually_entryPhysicalLevel_large hκ W hA δ,hn, - Filter.eventually_ge_atTop 1] with n hlevel hδ hn1 - intro J e horder z hz i - apply prescribedTestPastLevel_gap hκ W hn1 e horder A z δ - (fun j => (hlevel (some (e j)) z hz).le) _ i - simpa only [mul_comm] using (div_le_iff₀ ha).mp hδ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedBatchClock + +namespace OAI + +/-! The history before each test is taken at the preceding test endpoint. +The intervening radius gap is reserved for angular mixing. -/ +noncomputable section +open Set Filter + +namespace SLEExactGauge.Target + +def prescribedTestPastLevel {J : ℕ} (κ : ℝ) (W : WienerLaw) (n : ℕ) + (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : ℕ) : ℝ := + if i=0 then 0 else prescribedBatchLevel κ W n e A z (i-1)+ + 2*a κ*(batchLevel κ W n:ℝ) + +lemma prescribedTestPastLevel_nonneg {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) + (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) + (hl : ∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) A z) (i : ℕ) : + 0 ≤ prescribedTestPastLevel κ W n e A z i := by + have ha := internal_a_pos hκ + unfold prescribedTestPastLevel prescribedBatchLevel + split_ifs + · exact le_rfl + · exact add_nonneg (hl _) (by positivity) + · exact add_nonneg (hl _) (by positivity) + +lemma prescribedTestPastLevel_succ {κ : ℝ} (W : WienerLaw) (n : ℕ) + {J : ℕ} (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : Fin J) : + prescribedTestPastLevel κ W n e A z (i.val+1)= + entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ) := by + simp only [prescribedTestPastLevel,Nat.add_one_ne_zero,ite_false,Nat.add_sub_cancel, + prescribedBatchLevel,dite_eq_left i.isLt,Fin.eta] + +lemma prescribedTestPastLevel_monotone {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) + {n J : ℕ} (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1) + (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ) + (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1) + (horder : StrictMono (fun i => entryPosition κ W (n,some (e i)))) + (A : ℝ) (z : ℂ) (hl : ∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) A z) : + Monotone (prescribedTestPastLevel κ W n e A z) := by + intro i j hij + by_cases hi : i=0 + · subst i + simpa only [prescribedTestPastLevel,ite_true] using + prescribedTestPastLevel_nonneg hκ W n e A z hl j + · have hj : j ≠ 0 := by omega + simp only [prescribedTestPastLevel,ite_eq_right hi,ite_eq_right hj] + exact add_le_add + (prescribedBatchLevel_monotone hκ W hn hm1 hJ e he horder A z + (Nat.sub_le_sub_right hij 1)) le_rfl + +lemma prescribedTestPastLevel_gap {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) + {n J : ℕ} (hn : 1 ≤ n) (e : Fin J → ℤ×ℕ) + (horder : StrictMono (fun i => entryPosition κ W (n,some (e i)))) + (A : ℝ) (z : ℂ) (δ : ℝ) + (hfirst : ∀ i, δ ≤ entryPhysicalLevel κ W (n,some (e i)) A z) + (hδ : δ ≤ 2*a κ*(n:ℝ)) (i : Fin J) : + prescribedTestPastLevel κ W n e A z i.val+δ ≤ + entryPhysicalLevel κ W (n,some (e i)) A z := by + by_cases hi : i.val=0 + · simpa only [prescribedTestPastLevel,hi,ite_true,zero_add] using hfirst i + · let j : Fin J := ⟨i.val-1,by omega⟩ + have heq : i.val=j.val+1 := by dsimp only [j]; omega + rw [heq,prescribedTestPastLevel_succ W n e A z j] + have hpos := horder (show j < i from by + dsimp only [j] + exact Nat.sub_lt (Nat.pos_of_ne_zero hi) (by decide)) + have hh := entryPhysicalLevel_gap hκ W hn hpos A z + linarith + +lemma eventually_prescribedTestPastLevel_gap {κ : ℝ} (hκ : 0 < κ) + (W : WienerLaw) {A : ℝ} (hA : 0 < A) (δ : ℝ) : + ∀ᶠ n : ℕ in Filter.atTop, ∀ (J : ℕ) (e : Fin J → ℤ×ℕ), + StrictMono (fun i => entryPosition κ W (n,some (e i))) → + ∀ z : ℂ, Real.exp (-(n:ℝ)) ≤ z.im → ∀ i : Fin J, + prescribedTestPastLevel κ W n e A z i.val+δ ≤ + entryPhysicalLevel κ W (n,some (e i)) A z := by + have ha : 0 < 2*a κ := mul_pos (by norm_num) (internal_a_pos hκ) + have hn : ∀ᶠ n : ℕ in Filter.atTop, δ/(2*a κ) ≤ (n:ℝ) := + tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _) + filter_upwards [eventually_entryPhysicalLevel_large hκ W hA δ,hn, + Filter.eventually_ge_atTop 1] with n hlevel hδ hn1 + intro J e horder z hz i + apply prescribedTestPastLevel_gap hκ W hn1 e horder A z δ + (fun j => (hlevel (some (e j)) z hz).le) _ i + simpa only [mul_comm] using (div_le_iff₀ ha).mp hδ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedUniformGeometry.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedUniformGeometry.lean index 4fa3429b610768ae6d75c51386230e6d7f343288..3c928d8e8f91c3f4fbffed6f46738207ff35da0d 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedUniformGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedUniformGeometry.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Gauge.PrescribedBatchClock - -namespace OAI - -/-! One late-batch cutoff supplies all clock and visit geometry, uniformly - over every center above the shrinking boundary strip. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge.Target - -theorem eventually_prescribed_uniform_geometry {κ : ℝ} (hκ : 0<κ) (W : WienerLaw) - (hm1 : 0 < moment κ W 1) - (hJ : ∀ n, 1≤n → moment κ W 1^n ≤ moment κ W n) : - ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, Real.exp (-(n:ℝ)) ≤ z.im → - (∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) (2048*Real.pi) z) ∧ - 4*entryRadius κ W (n,none) ≤ z.im ∧ - ∀ (k : ℤ) (j : ℕ), j < multiplicity κ W n k → - entryPhysicalLevel κ W (n,some (k,j)) (2048*Real.pi) z+ - 2*a κ*(batchLevel κ W n:ℝ)+1 ≤ - entryPhysicalLevel κ W (n,none) (2048*Real.pi) z := by - have ha : 0<2*a κ := mul_pos (by norm_num) (internal_a_pos hκ) - have hlarge : ∀ᶠ n : ℕ in atTop, 1/(2*a κ) ≤ (n:ℝ) := - tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _) - filter_upwards [eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 0, - hlarge,eventually_ge_atTop 3] with n hl hn hn3 - intro z hz - have hn1 : 1≤n := by omega - have hnR : (3:ℝ) ≤ n := by exact_mod_cast hn3 - have hunit : 1 ≤ 2*a κ*(n:ℝ) := by - have hh := (div_le_iff₀ ha).mp hn - linarith - refine ⟨fun o => (hl o z hz).le,?_,?_⟩ - · apply le_trans _ hz - calc - _ ≤ 4*Real.exp (-(n:ℝ)^2) := - mul_le_mul_of_nonneg_left (entryRadius_le_batch_exp hκ W (n,none)) (by norm_num) - _ ≤ Real.exp (-(n:ℝ)) := by - rw [show (4:ℝ)=Real.exp (Real.log 4) by rw [Real.exp_log (by norm_num)],← Real.exp_add] - apply Real.exp_le_exp.mpr - have hh : Real.log 4 ≤ 3 := by - have hl4 := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4) - linarith - nlinarith - · intro k j hj - have hh := entryPhysicalLevel_cleanup_gap hκ W hn1 hm1 (hJ n hn1) hj (2048*Real.pi) z - linarith - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedBatchClock + +namespace OAI + +/-! One late-batch cutoff supplies all clock and visit geometry, uniformly + over every center above the shrinking boundary strip. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge.Target + +theorem eventually_prescribed_uniform_geometry {κ : ℝ} (hκ : 0<κ) (W : WienerLaw) + (hm1 : 0 < moment κ W 1) + (hJ : ∀ n, 1≤n → moment κ W 1^n ≤ moment κ W n) : + ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, Real.exp (-(n:ℝ)) ≤ z.im → + (∀ o, 0 ≤ entryPhysicalLevel κ W (n,o) (2048*Real.pi) z) ∧ + 4*entryRadius κ W (n,none) ≤ z.im ∧ + ∀ (k : ℤ) (j : ℕ), j < multiplicity κ W n k → + entryPhysicalLevel κ W (n,some (k,j)) (2048*Real.pi) z+ + 2*a κ*(batchLevel κ W n:ℝ)+1 ≤ + entryPhysicalLevel κ W (n,none) (2048*Real.pi) z := by + have ha : 0<2*a κ := mul_pos (by norm_num) (internal_a_pos hκ) + have hlarge : ∀ᶠ n : ℕ in atTop, 1/(2*a κ) ≤ (n:ℝ) := + tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _) + filter_upwards [eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 0, + hlarge,eventually_ge_atTop 3] with n hl hn hn3 + intro z hz + have hn1 : 1≤n := by omega + have hnR : (3:ℝ) ≤ n := by exact_mod_cast hn3 + have hunit : 1 ≤ 2*a κ*(n:ℝ) := by + have hh := (div_le_iff₀ ha).mp hn + linarith + refine ⟨fun o => (hl o z hz).le,?_,?_⟩ + · apply le_trans _ hz + calc + _ ≤ 4*Real.exp (-(n:ℝ)^2) := + mul_le_mul_of_nonneg_left (entryRadius_le_batch_exp hκ W (n,none)) (by norm_num) + _ ≤ Real.exp (-(n:ℝ)) := by + rw [show (4:ℝ)=Real.exp (Real.log 4) by rw [Real.exp_log (by norm_num)],← Real.exp_add] + apply Real.exp_le_exp.mpr + have hh : Real.log 4 ≤ 3 := by + have hl4 := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4) + linarith + nlinarith + · intro k j hj + have hh := entryPhysicalLevel_cleanup_gap hκ W hn1 hm1 (hJ n hn1) hj (2048*Real.pi) z + linarith + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedUpperAngle.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedUpperAngle.lean index cb42c7b8b2df1805284e026de65c736ec9f2ed36..1cdfefa787e51a555451dc2a69d5e62e5dff5fbb 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedUpperAngle.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedUpperAngle.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Gauge.PrescribedCanonicalMixing -import OAI.Probability.SLE.Gauge.CanonicalPrescribedUpper - -namespace OAI - -/-! A uniform canonical angle transition bound implies the prescribed-gauge -upper cover estimate. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance upperAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance upperAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.prescribed_expectation_of_canonical_angle_bound - {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) - (hγ : ∀ t, Measurable (fun ω => γ ω t)) - (δ : ℝ) (hδ : 0<δ) - (hangle : ∀ (v : ℝ), δ≤v → ∀ (f : ℝ → ℝ≥0∞), Measurable f → - ∀ θ∈Ioo (0:ℝ) Real.pi, (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ - canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ) - (L : ℝ) (hL : 0≤L) (K : Set ℂ) (hK : IsClosed K) - (hKL : K⊆closedBall 0 L) (hH : ∀ z∈K, 0≤z.im) : - (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (γ ω)∩K) ∂P)<∞ := by - classical - have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1 - have hJn (n : ℕ) := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n - have hdata (n : ℕ) : ∃ (J : ℕ) (e : Fin J → ℤ×ℕ), - (∀ i, (e i).2 Target.entryPosition κ W (n,some (e i)))) := by - by_cases hn : 3≤n - · obtain ⟨J,e,hi,he,ho⟩ := Target.ordered_repeated_batch κ W (by omega) hm1 (hJn n) - exact ⟨J,e,fun i => (he _).mp ⟨i,rfl⟩,fun _ => ⟨hi,he,ho⟩⟩ - · refine ⟨0,Fin.elim0,?_,?_⟩ - · exact fun i => Fin.elim0 i - · exact fun h => False.elim (hn h) - choose J e hentries horder using hdata - apply C.prescribed_expectation_of_finite_failures hB hc h0 hκ hκ8 F hw hγ hpublished - hgrowth W hW J e hentries (4096*Real.pi) ((128*Real.pi)^dimension κ) L - (by have hp := Real.pi_gt_three; linarith) (Real.rpow_pos_of_pos (by positivity) _) hL K hK hKL hH - filter_upwards [Target.eventually_prescribedTestPastLevel_gap hκ W - (by positivity : 0<2048*Real.pi) δ, - Target.eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 0, - eventually_ge_atTop 3] with n hgap hlevel hn - intro ij _hij hy - let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij - have hz : z∈upperHalfPlane := lt_of_lt_of_le (Real.exp_pos _) hy - obtain ⟨hi,he',ho⟩ := horder n hn - exact C.prescribed_batch_failure_of_canonical_angle_bound hB hc h0 hκ hκ8 F hw R he - D hD hDf hDp hgrowth hpublished W hW hn (e n) hi he' ho z hz - (fun o => (hlevel o z hy).le) δ hδ (hgap (J n) (e n) ho z hy) hangle - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedCanonicalMixing +import OAI.Probability.SLE.Gauge.CanonicalPrescribedUpper + +namespace OAI + +/-! A uniform canonical angle transition bound implies the prescribed-gauge +upper cover estimate. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance upperAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance upperAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.prescribed_expectation_of_canonical_angle_bound + {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) + (hγ : ∀ t, Measurable (fun ω => γ ω t)) + (δ : ℝ) (hδ : 0<δ) + (hangle : ∀ (v : ℝ), δ≤v → ∀ (f : ℝ → ℝ≥0∞), Measurable f → + ∀ θ∈Ioo (0:ℝ) Real.pi, (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ + canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ) + (L : ℝ) (hL : 0≤L) (K : Set ℂ) (hK : IsClosed K) + (hKL : K⊆closedBall 0 L) (hH : ∀ z∈K, 0≤z.im) : + (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (γ ω)∩K) ∂P)<∞ := by + classical + have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1 + have hJn (n : ℕ) := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n + have hdata (n : ℕ) : ∃ (J : ℕ) (e : Fin J → ℤ×ℕ), + (∀ i, (e i).2 Target.entryPosition κ W (n,some (e i)))) := by + by_cases hn : 3≤n + · obtain ⟨J,e,hi,he,ho⟩ := Target.ordered_repeated_batch κ W (by omega) hm1 (hJn n) + exact ⟨J,e,fun i => (he _).mp ⟨i,rfl⟩,fun _ => ⟨hi,he,ho⟩⟩ + · refine ⟨0,Fin.elim0,?_,?_⟩ + · exact fun i => Fin.elim0 i + · exact fun h => False.elim (hn h) + choose J e hentries horder using hdata + apply C.prescribed_expectation_of_finite_failures hB hc h0 hκ hκ8 F hw hγ hpublished + hgrowth W hW J e hentries (4096*Real.pi) ((128*Real.pi)^dimension κ) L + (by have hp := Real.pi_gt_three; linarith) (Real.rpow_pos_of_pos (by positivity) _) hL K hK hKL hH + filter_upwards [Target.eventually_prescribedTestPastLevel_gap hκ W + (by positivity : 0<2048*Real.pi) δ, + Target.eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 0, + eventually_ge_atTop 3] with n hgap hlevel hn + intro ij _hij hy + let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij + have hz : z∈upperHalfPlane := lt_of_lt_of_le (Real.exp_pos _) hy + obtain ⟨hi,he',ho⟩ := horder n hn + exact C.prescribed_batch_failure_of_canonical_angle_bound hB hc h0 hκ hκ8 F hw R he + D hD hDf hDp hgrowth hpublished W hW hn (e n) hi he' ho z hz + (fun o => (hlevel o z hy).le) δ hδ (hgap (J n) (e n) ho z hy) hangle + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedVisitBound.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedVisitBound.lean index e2ef474a6c358708d8285ec6c7ee8682dd45f516..f0d0ee0f52b282015dbf02109cc16b1523d78a43 100644 --- a/lean/OAI/Probability/SLE/Gauge/PrescribedVisitBound.lean +++ b/lean/OAI/Probability/SLE/Gauge/PrescribedVisitBound.lean @@ -1,57 +1,57 @@ -import OAI.Probability.SLE.Gauge.PrescribedCandidateCover -import OAI.Probability.SLE.SausagePointBound - -namespace OAI - -/-! The ordinary one-point estimate bounds every actual missed cell, - including the bottom half-row of the shifted grid. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem prescribed_missed_cell_visit_bound - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (ν : Ω → Measure ℂ) (W : Target.WienerLaw) (n : ℕ) - {J : ℕ} (e : Fin J → ℤ×ℕ) (C c R : ℝ) (K : Set ℂ) - (ij : ℤ×ℤ) (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K) : - P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω)} ≤ - ENNReal.ofReal ((sausagePointConstant κ*(4:ℝ)^codimension κ)* - (Target.entryRadius κ W (n,none))^codimension κ* - (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)) := by - classical - let r := Target.entryRadius κ W (n,none) - let z := prescribedGridCenter r ij - have hr : 0 Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (ν : Ω → Measure ℂ) (W : Target.WienerLaw) (n : ℕ) + {J : ℕ} (e : Fin J → ℤ×ℕ) (C c R : ℝ) (K : Set ℂ) + (ij : ℤ×ℤ) (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K) : + P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω)} ≤ + ENNReal.ofReal ((sausagePointConstant κ*(4:ℝ)^codimension κ)* + (Target.entryRadius κ W (n,none))^codimension κ* + (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)) := by + classical + let r := Target.entryRadius κ W (n,none) + let z := prescribedGridCenter r ij + have hr : 0 P z) := by - have hc : Continuous (fun z : closedRibbon => P z) := hP.domRestrict - have hinj : Function.Injective (fun z : closedRibbon => P z) := by - intro z w he - exact Subtype.ext (hi z.2 w.2 he) - have hproper : IsProperMap (fun z : closedRibbon => P z) := by - apply isProperMap_iff_isCompact_preimage.mpr - refine ⟨hc, fun K hK => ?_⟩ - obtain ⟨R, hR⟩ := hK.isBounded.exists_norm_le - have hclosed : IsClosed (closedRibbon ∩ P ⁻¹' K) := - hP.preimage_isClosed_of_isClosed closedRibbon_isClosed hK.isClosed - have hbounded : Bornology.IsBounded (closedRibbon ∩ P ⁻¹' K) := by - apply isBounded_iff_forall_norm_le.mpr - refine ⟨T + |a.im| + R + 2, fun z hz => ?_⟩ - exact (ribbon_coordinate_norm_bound P σ T hT a htail hwidth z hz.1).trans - (by linarith [hR (P z) hz.2]) - apply Subtype.isCompact_iff.mpr - have he : (Subtype.val '' ((fun z : closedRibbon => P z) ⁻¹' K)) = - closedRibbon ∩ P ⁻¹' K := by - ext z - simp only [mem_image, mem_preimage, mem_inter_iff] - constructor - · rintro ⟨w, hw, rfl⟩ - exact ⟨w.2, hw⟩ - · rintro ⟨hz, hPz⟩ - exact ⟨⟨z, hz⟩, hPz, rfl⟩ - rw [he] - exact (isCompact_iff_isClosed_bounded).mpr ⟨hclosed, hbounded⟩ - exact .of_continuous_injective_isClosedMap hc hinj hproper.isClosedMap - -lemma coordinateInverse_continuousOn_of_embedding - (P : ℂ → ℂ) (he : Topology.IsEmbedding (fun z : closedRibbon => P z)) : - ContinuousOn (coordinateInverse P closedRibbon) (P '' closedRibbon) := by - let q : (P '' closedRibbon) → closedRibbon := fun z => - ⟨coordinateInverse P closedRibbon z, coordinateInverse_mem z.2⟩ - have hq : Continuous q := by - apply he.continuous_iff.mpr - have hf : (fun z : closedRibbon => P z) ∘ q = Subtype.val := by - funext z - exact coordinateInverse_right z.2 - rw [hf] - exact continuous_subtype_val - exact continuousOn_iff_continuous_domRestrict.mpr (continuous_subtype_val.comp hq) - -theorem exists_global_ribbon_progress - (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) : - ∃ g : C(ℂ, ℝ), ∀ z ∈ closedRibbon, g (P z) = z.re := by - have hclosed : IsClosed (P '' closedRibbon) := by - have hr : range (fun z : closedRibbon => P z) = P '' closedRibbon := by - ext z - simp - exact hr ▸ he.isClosed_range - have hp : ContinuousOn (longitudinalCoordinate P closedRibbon 0) (P '' closedRibbon) := by - exact (Complex.continuous_re.comp_continuousOn - (coordinateInverse_continuousOn_of_embedding P he.isEmbedding)).sub continuousOn_const - obtain ⟨g, hg⟩ := FrontSuccessMeasurability.exists_continuous_progress_extension - (longitudinalCoordinate P closedRibbon 0) (P '' closedRibbon) hclosed hp - have hi : InjOn P closedRibbon := by - intro z hz w hw heq - exact congrArg Subtype.val (he.injective (show P (⟨z, hz⟩ : closedRibbon) = - P (⟨w, hw⟩ : closedRibbon) from heq)) - refine ⟨g, fun z hz => ?_⟩ - rw [← hg (mem_image_of_mem P hz), longitudinalCoordinate, coordinateInverse_left hi hz, sub_zero] - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonHistoryAdvance +import Mathlib.Topology.Maps.Proper.CompactlyGenerated + +namespace OAI + +/-! An escaping terminal straight piece makes the complete normal ribbon +proper. Its inverse coordinates therefore extend continuously to the +ambient plane, supplying one actual adapted front process for iteration. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RibbonCoordinates + +lemma closedRibbon_isClosed : IsClosed closedRibbon := + (isClosed_le continuous_const Complex.continuous_re).inter + (isClosed_le Complex.continuous_im.abs continuous_const) + +lemma ribbon_coordinate_norm_bound + (P : ℂ → ℂ) (σ : ℝ → ℂ) (T : ℝ) (_hT : 0 ≤ T) (a : ℂ) + (htail : ∀ t : ℝ, T ≤ t → σ t = a + ((t - T : ℝ) : ℂ) * Complex.I) + (hwidth : ∀ z ∈ closedRibbon, ‖P z - σ z.re‖ ≤ 1) + (z : ℂ) (hz : z ∈ closedRibbon) : + ‖z‖ ≤ T + |a.im| + ‖P z‖ + 2 := by + have hre : z.re ≤ T + |a.im| + ‖P z‖ + 1 := by + by_cases hzt : z.re ≤ T + · linarith [abs_nonneg a.im, norm_nonneg (P z)] + · have him : z.re - T + a.im = (σ z.re).im := by + rw [htail z.re (le_of_not_ge hzt)] + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_im, mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] + ring + have hσ : ‖σ z.re‖ ≤ ‖P z‖ + 1 := by + calc + _ ≤ ‖P z‖ + ‖σ z.re - P z‖ := norm_le_insert' _ _ + _ ≤ _ := add_le_add le_rfl (by simpa only [norm_sub_rev] using hwidth z hz) + have hh := Complex.im_le_norm (σ z.re) + rw [← him] at hh + linarith [neg_abs_le a.im] + have hn := Complex.norm_le_abs_re_add_abs_im z + rw [abs_of_nonneg hz.1] at hn + linarith [hz.2, abs_nonneg a.im, norm_nonneg (P z)] + +theorem ribbon_closedEmbedding_of_linear_tail + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (σ : ℝ → ℂ) (T : ℝ) (hT : 0 ≤ T) (a : ℂ) + (htail : ∀ t : ℝ, T ≤ t → σ t = a + ((t - T : ℝ) : ℂ) * Complex.I) + (hwidth : ∀ z ∈ closedRibbon, ‖P z - σ z.re‖ ≤ 1) : + Topology.IsClosedEmbedding (fun z : closedRibbon => P z) := by + have hc : Continuous (fun z : closedRibbon => P z) := hP.domRestrict + have hinj : Function.Injective (fun z : closedRibbon => P z) := by + intro z w he + exact Subtype.ext (hi z.2 w.2 he) + have hproper : IsProperMap (fun z : closedRibbon => P z) := by + apply isProperMap_iff_isCompact_preimage.mpr + refine ⟨hc, fun K hK => ?_⟩ + obtain ⟨R, hR⟩ := hK.isBounded.exists_norm_le + have hclosed : IsClosed (closedRibbon ∩ P ⁻¹' K) := + hP.preimage_isClosed_of_isClosed closedRibbon_isClosed hK.isClosed + have hbounded : Bornology.IsBounded (closedRibbon ∩ P ⁻¹' K) := by + apply isBounded_iff_forall_norm_le.mpr + refine ⟨T + |a.im| + R + 2, fun z hz => ?_⟩ + exact (ribbon_coordinate_norm_bound P σ T hT a htail hwidth z hz.1).trans + (by linarith [hR (P z) hz.2]) + apply Subtype.isCompact_iff.mpr + have he : (Subtype.val '' ((fun z : closedRibbon => P z) ⁻¹' K)) = + closedRibbon ∩ P ⁻¹' K := by + ext z + simp only [mem_image, mem_preimage, mem_inter_iff] + constructor + · rintro ⟨w, hw, rfl⟩ + exact ⟨w.2, hw⟩ + · rintro ⟨hz, hPz⟩ + exact ⟨⟨z, hz⟩, hPz, rfl⟩ + rw [he] + exact (isCompact_iff_isClosed_bounded).mpr ⟨hclosed, hbounded⟩ + exact .of_continuous_injective_isClosedMap hc hinj hproper.isClosedMap + +lemma coordinateInverse_continuousOn_of_embedding + (P : ℂ → ℂ) (he : Topology.IsEmbedding (fun z : closedRibbon => P z)) : + ContinuousOn (coordinateInverse P closedRibbon) (P '' closedRibbon) := by + let q : (P '' closedRibbon) → closedRibbon := fun z => + ⟨coordinateInverse P closedRibbon z, coordinateInverse_mem z.2⟩ + have hq : Continuous q := by + apply he.continuous_iff.mpr + have hf : (fun z : closedRibbon => P z) ∘ q = Subtype.val := by + funext z + exact coordinateInverse_right z.2 + rw [hf] + exact continuous_subtype_val + exact continuousOn_iff_continuous_domRestrict.mpr (continuous_subtype_val.comp hq) + +theorem exists_global_ribbon_progress + (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) : + ∃ g : C(ℂ, ℝ), ∀ z ∈ closedRibbon, g (P z) = z.re := by + have hclosed : IsClosed (P '' closedRibbon) := by + have hr : range (fun z : closedRibbon => P z) = P '' closedRibbon := by + ext z + simp + exact hr ▸ he.isClosed_range + have hp : ContinuousOn (longitudinalCoordinate P closedRibbon 0) (P '' closedRibbon) := by + exact (Complex.continuous_re.comp_continuousOn + (coordinateInverse_continuousOn_of_embedding P he.isEmbedding)).sub continuousOn_const + obtain ⟨g, hg⟩ := FrontSuccessMeasurability.exists_continuous_progress_extension + (longitudinalCoordinate P closedRibbon 0) (P '' closedRibbon) hclosed hp + have hi : InjOn P closedRibbon := by + intro z hz w hw heq + exact congrArg Subtype.val (he.injective (show P (⟨z, hz⟩ : closedRibbon) = + P (⟨w, hw⟩ : closedRibbon) from heq)) + refine ⟨g, fun z hz => ?_⟩ + rw [← hg (mem_image_of_mem P hz), longitudinalCoordinate, coordinateInverse_left hi hz, sub_zero] + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/PublishedRibbonIteration.lean b/lean/OAI/Probability/SLE/Gauge/PublishedRibbonIteration.lean index 5de83062a64e066630e2b1328ab2e15206b60d86..0d4deecb66ea7606ecf75be39123d7a2f13be6a5 100644 --- a/lean/OAI/Probability/SLE/Gauge/PublishedRibbonIteration.lean +++ b/lean/OAI/Probability/SLE/Gauge/PublishedRibbonIteration.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Gauge.ActualRibbonIteration - -namespace OAI - -/-! Uniform iteration of actual front-and-mass passages from explicit -analytic inputs, for any chart and number of steps satisfying the -deterministic ribbon bounds. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates Target - -local instance publishedIterationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance publishedIterationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem uniform_actual_ribbon_iteration - {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport μ γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ q c v₀ : ℝ, 0 < q ∧ q ≤ 1 ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (P : ℂ → ℂ), - Topology.IsClosedEmbedding (fun z : closedRibbon => P z) → - ContinuousOn P closedRibbon → InjOn P closedRibbon → P 0 = 0 → - (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → - Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop → - (∀ k : ℝ, LipschitzOnWith C P (ribbonStepPatch k)) → - (∀ k : ℝ, ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, - dist z w ≤ L * dist (P z) (P w)) → - ∀ (g : C(ℂ, ℝ)), (∀ z ∈ closedRibbon, g (P z) = z.re) → ∀ N : ℕ, - (ENNReal.ofReal q) ^ N * μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε) ≤ - μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩ - ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) := by - obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ := - uniform_conditional_actual_ribbon_step_extended μ κ hκ hκ8 B hB hm hc γ G F hw - hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW - C L hC hL ε hε hεsmall - refine ⟨min q 1, c, v₀, lt_min hq (by norm_num), min_le_right _ _, hcm, hv₀, ?_⟩ - intro v hv hvcut P hemb hP hi hP0 hH hescape hu hl g hg N - let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) - have ha : Adapted ℱ (fun t ω => γ ω t) := brownian_capacity_trace_adapted B hm hc κ γ G F hw - have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 - have hg0 (ω : Ω) : g (γ ω 0) = 0 := by - rw [(hw ω).2.1, ← hP0] - exact hg 0 (by constructor <;> norm_num) - apply le_trans (mul_le_mul' (pow_le_pow_left' - (ENNReal.ofReal_le_ofReal (min_le_left q 1)) N) le_rfl) - apply actual_ribbon_successes_lower_bound μ ℱ γ ha hγc g hg0 P hemb ε κ v c - (ENNReal.ofReal q) ?_ N - intro n A hA hAE - let k : ℝ := (n : ℝ) + 1 - have hk : 1 ≤ k := by dsimp only [k]; linarith [Nat.cast_nonneg (α := ℝ) n] - have hτ := ribbonFrontStop_stopping ℱ g γ ha hγc k - have hS := ribbonPathStep_measurable P C L ε k κ v c hplanar hεsmall hk hemb hP (hl k) - have hb := actual_ribbon_step_on_good_prefix μ ℱ (fun ω t => Real.sqrt κ * B t ω) - γ G F (fun ω => continuous_const.mul (hc ω)) hw C L hL ε κ v c hεsmall - (ENNReal.ofReal q) P hemb hP hi hH hescape g hg hg0 k hk - (fun j _ => hu j) (fun j _ => hl j) hτ - (hstep v hv hvcut _ hτ _ hS) A hA hAE - simpa only [actualRibbonSuccess, ribbonIntegerFront, k, Nat.cast_add, Nat.cast_one] using hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ActualRibbonIteration + +namespace OAI + +/-! Uniform iteration of actual front-and-mass passages from explicit +analytic inputs, for any chart and number of steps satisfying the +deterministic ribbon bounds. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates Target + +local instance publishedIterationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance publishedIterationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem uniform_actual_ribbon_iteration + {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport μ γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ q c v₀ : ℝ, 0 < q ∧ q ≤ 1 ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (P : ℂ → ℂ), + Topology.IsClosedEmbedding (fun z : closedRibbon => P z) → + ContinuousOn P closedRibbon → InjOn P closedRibbon → P 0 = 0 → + (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → + Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop → + (∀ k : ℝ, LipschitzOnWith C P (ribbonStepPatch k)) → + (∀ k : ℝ, ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, + dist z w ≤ L * dist (P z) (P w)) → + ∀ (g : C(ℂ, ℝ)), (∀ z ∈ closedRibbon, g (P z) = z.re) → ∀ N : ℕ, + (ENNReal.ofReal q) ^ N * μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε) ≤ + μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩ + ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) := by + obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ := + uniform_conditional_actual_ribbon_step_extended μ κ hκ hκ8 B hB hm hc γ G F hw + hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW + C L hC hL ε hε hεsmall + refine ⟨min q 1, c, v₀, lt_min hq (by norm_num), min_le_right _ _, hcm, hv₀, ?_⟩ + intro v hv hvcut P hemb hP hi hP0 hH hescape hu hl g hg N + let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable) + have ha : Adapted ℱ (fun t ω => γ ω t) := brownian_capacity_trace_adapted B hm hc κ γ G F hw + have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 + have hg0 (ω : Ω) : g (γ ω 0) = 0 := by + rw [(hw ω).2.1, ← hP0] + exact hg 0 (by constructor <;> norm_num) + apply le_trans (mul_le_mul' (pow_le_pow_left' + (ENNReal.ofReal_le_ofReal (min_le_left q 1)) N) le_rfl) + apply actual_ribbon_successes_lower_bound μ ℱ γ ha hγc g hg0 P hemb ε κ v c + (ENNReal.ofReal q) ?_ N + intro n A hA hAE + let k : ℝ := (n : ℝ) + 1 + have hk : 1 ≤ k := by dsimp only [k]; linarith [Nat.cast_nonneg (α := ℝ) n] + have hτ := ribbonFrontStop_stopping ℱ g γ ha hγc k + have hS := ribbonPathStep_measurable P C L ε k κ v c hplanar hεsmall hk hemb hP (hl k) + have hb := actual_ribbon_step_on_good_prefix μ ℱ (fun ω t => Real.sqrt κ * B t ω) + γ G F (fun ω => continuous_const.mul (hc ω)) hw C L hL ε κ v c hεsmall + (ENNReal.ofReal q) P hemb hP hi hH hescape g hg hg0 k hk + (fun j _ => hu j) (fun j _ => hl j) hτ + (hstep v hv hvcut _ hτ _ hS) A hA hAE + simpa only [actualRibbonSuccess, ribbonIntegerFront, k, Nat.cast_add, Nat.cast_one] using hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RandomBadGrid.lean b/lean/OAI/Probability/SLE/Gauge/RandomBadGrid.lean index f497c498eb39c2fb0e6d2f34ea29446a4e8fc423..305ca0b9a6f1992d3ce74231c200411fb79cff51 100644 --- a/lean/OAI/Probability/SLE/Gauge/RandomBadGrid.lean +++ b/lean/OAI/Probability/SLE/Gauge/RandomBadGrid.lean @@ -1,121 +1,121 @@ -import OAI.Probability.SLE.Gauge.GoodGridRestriction - -namespace OAI - -/-! Measurability and almost-sure finiteness of the bad-grid mass sum. -Only expectations of the actual cell masses occur in the criterion. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma measurableSet_random_badGridCells - {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure ℂ) (hμ : Measurable μ) - (r H : ℝ) : MeasurableSet {p : Ω × ℂ | p.2 ∈ badGridCells (μ p.1) r H} := by - classical - let Q := fun i : ℤ × ℤ => closedBall (coverGridCenter r i) (2*r) - have he : {p : Ω × ℂ | p.2 ∈ badGridCells (μ p.1) r H} = - ⋃ i : ℤ × ℤ, {ω | ENNReal.ofReal H < μ ω (Q i)} ×ˢ Q i := by - ext p - constructor - · intro hp - obtain ⟨i,hi⟩ := mem_iUnion.mp hp - change p.2 ∈ if ENNReal.ofReal H < μ p.1 (Q i) then Q i else ∅ at hi - by_cases h : ENNReal.ofReal H < μ p.1 (Q i) - · rw [ite_eq_left h] at hi - exact mem_iUnion.mpr ⟨i,h,hi⟩ - · rw [ite_eq_right h] at hi - exact hi.elim - · intro hp - obtain ⟨i,hi⟩ := mem_iUnion.mp hp - apply mem_iUnion.mpr - refine ⟨i,?_⟩ - change p.2 ∈ if ENNReal.ofReal H < μ p.1 (Q i) then Q i else ∅ - rw [ite_eq_left (show ENNReal.ofReal H < μ p.1 (Q i) from hi.1)] - exact hi.2 - rw [he] - apply MeasurableSet.iUnion - intro i - exact (measurableSet_lt measurable_const - ((Measure.measurable_coe measurableSet_closedBall).comp hμ)).prod measurableSet_closedBall - -lemma random_badGrid_mass_measurable - {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure ℂ) (hμ : Measurable μ) - (hfin : ∀ ω, IsFiniteMeasure (μ ω)) (r H : ℝ) : - Measurable (fun ω => μ ω (badGridCells (μ ω) r H)) := by - let K : Kernel Ω ℂ := ⟨μ,hμ⟩ - exact Kernel.measurable_kernel_prodMk_left_of_finite (κ:=K) - (measurableSet_random_badGridCells μ hμ r H) hfin - -lemma badGrid_mass_le_sum (μ : Measure ℂ) (r H : ℝ) : - μ (badGridCells μ r H) ≤ ∑' i : ℤ × ℤ, - if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then - μ (closedBall (coverGridCenter r i) (2*r)) else 0 := by - classical - calc - _ ≤ ∑' i : ℤ × ℤ, μ (if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then - closedBall (coverGridCenter r i) (2*r) else ∅) := measure_iUnion_le _ - _ = _ := by - apply tsum_congr - intro i - split_ifs <;> simp - -lemma expected_badGrid_mass_le_sum - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure ℂ) (hμ : Measurable μ) (r H : ℝ) : - (∫⁻ ω, μ ω (badGridCells (μ ω) r H) ∂P) ≤ ∑' i : ℤ × ℤ, - ∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then - μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P := by - classical - calc - _ ≤ ∫⁻ ω, ∑' i : ℤ × ℤ, - if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then - μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P := - lintegral_mono (fun ω => badGrid_mass_le_sum (μ ω) r H) - _ = _ := by - apply lintegral_tsum - intro i - exact (Measurable.ite (measurableSet_lt measurable_const - ((Measure.measurable_coe measurableSet_closedBall).comp hμ)) - ((Measure.measurable_coe measurableSet_closedBall).comp hμ) measurable_const).aemeasurable - -lemma ae_prescribedBadGrid_sum_finite - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω)) - (κ : ℝ) (W : Target.WienerLaw) (D : ℝ) - (hsum : (∑' e : Target.GaugeIndex κ W, - ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞) : - ∀ᵐ ω ∂P, (∑' e : Target.GaugeIndex κ W, μ ω (prescribedBadGrid (μ ω) κ W D e)) ≠ ∞ := by - have hm (e : Target.GaugeIndex κ W) : - Measurable (fun ω => μ ω (prescribedBadGrid (μ ω) κ W D e)) := - random_badGrid_mass_measurable μ hμ hfin _ _ - have hi : (∫⁻ ω, ∑' e : Target.GaugeIndex κ W, - μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞ := by - rw [lintegral_tsum (fun e => (hm e).aemeasurable)] - exact hsum - exact (ae_lt_top (Measurable.tsum hm) hi).mono (fun _ h => h.ne) - -lemma ae_positive_prescribed_hausdorff_of_bad_mass - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω)) - (κ : ℝ) (W : Target.WienerLaw) - (hg : Target.IsHausdorffGauge (Target.gauge κ W)) - (hm1 : 0 < Target.moment κ W 1) - (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n) - (D : ℝ) (hD : 0 ≤ D) - (hsum : (∑' e : Target.GaugeIndex κ W, - ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞) : - ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → 0 < μ ω E → - 0 < Target.hausdorff (Target.gauge κ W) E := by - filter_upwards [ae_prescribedBadGrid_sum_finite P μ hμ hfin κ W D hsum] with ω hω - let := hfin ω - exact fun E hE hp => positive_prescribed_hausdorff_of_summable_bad_mass - (μ ω) κ W hg hm1 hJ D hD hω E hE hp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.GoodGridRestriction + +namespace OAI + +/-! Measurability and almost-sure finiteness of the bad-grid mass sum. +Only expectations of the actual cell masses occur in the criterion. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma measurableSet_random_badGridCells + {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure ℂ) (hμ : Measurable μ) + (r H : ℝ) : MeasurableSet {p : Ω × ℂ | p.2 ∈ badGridCells (μ p.1) r H} := by + classical + let Q := fun i : ℤ × ℤ => closedBall (coverGridCenter r i) (2*r) + have he : {p : Ω × ℂ | p.2 ∈ badGridCells (μ p.1) r H} = + ⋃ i : ℤ × ℤ, {ω | ENNReal.ofReal H < μ ω (Q i)} ×ˢ Q i := by + ext p + constructor + · intro hp + obtain ⟨i,hi⟩ := mem_iUnion.mp hp + change p.2 ∈ if ENNReal.ofReal H < μ p.1 (Q i) then Q i else ∅ at hi + by_cases h : ENNReal.ofReal H < μ p.1 (Q i) + · rw [ite_eq_left h] at hi + exact mem_iUnion.mpr ⟨i,h,hi⟩ + · rw [ite_eq_right h] at hi + exact hi.elim + · intro hp + obtain ⟨i,hi⟩ := mem_iUnion.mp hp + apply mem_iUnion.mpr + refine ⟨i,?_⟩ + change p.2 ∈ if ENNReal.ofReal H < μ p.1 (Q i) then Q i else ∅ + rw [ite_eq_left (show ENNReal.ofReal H < μ p.1 (Q i) from hi.1)] + exact hi.2 + rw [he] + apply MeasurableSet.iUnion + intro i + exact (measurableSet_lt measurable_const + ((Measure.measurable_coe measurableSet_closedBall).comp hμ)).prod measurableSet_closedBall + +lemma random_badGrid_mass_measurable + {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure ℂ) (hμ : Measurable μ) + (hfin : ∀ ω, IsFiniteMeasure (μ ω)) (r H : ℝ) : + Measurable (fun ω => μ ω (badGridCells (μ ω) r H)) := by + let K : Kernel Ω ℂ := ⟨μ,hμ⟩ + exact Kernel.measurable_kernel_prodMk_left_of_finite (κ:=K) + (measurableSet_random_badGridCells μ hμ r H) hfin + +lemma badGrid_mass_le_sum (μ : Measure ℂ) (r H : ℝ) : + μ (badGridCells μ r H) ≤ ∑' i : ℤ × ℤ, + if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then + μ (closedBall (coverGridCenter r i) (2*r)) else 0 := by + classical + calc + _ ≤ ∑' i : ℤ × ℤ, μ (if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then + closedBall (coverGridCenter r i) (2*r) else ∅) := measure_iUnion_le _ + _ = _ := by + apply tsum_congr + intro i + split_ifs <;> simp + +lemma expected_badGrid_mass_le_sum + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure ℂ) (hμ : Measurable μ) (r H : ℝ) : + (∫⁻ ω, μ ω (badGridCells (μ ω) r H) ∂P) ≤ ∑' i : ℤ × ℤ, + ∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then + μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P := by + classical + calc + _ ≤ ∫⁻ ω, ∑' i : ℤ × ℤ, + if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then + μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P := + lintegral_mono (fun ω => badGrid_mass_le_sum (μ ω) r H) + _ = _ := by + apply lintegral_tsum + intro i + exact (Measurable.ite (measurableSet_lt measurable_const + ((Measure.measurable_coe measurableSet_closedBall).comp hμ)) + ((Measure.measurable_coe measurableSet_closedBall).comp hμ) measurable_const).aemeasurable + +lemma ae_prescribedBadGrid_sum_finite + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω)) + (κ : ℝ) (W : Target.WienerLaw) (D : ℝ) + (hsum : (∑' e : Target.GaugeIndex κ W, + ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞) : + ∀ᵐ ω ∂P, (∑' e : Target.GaugeIndex κ W, μ ω (prescribedBadGrid (μ ω) κ W D e)) ≠ ∞ := by + have hm (e : Target.GaugeIndex κ W) : + Measurable (fun ω => μ ω (prescribedBadGrid (μ ω) κ W D e)) := + random_badGrid_mass_measurable μ hμ hfin _ _ + have hi : (∫⁻ ω, ∑' e : Target.GaugeIndex κ W, + μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞ := by + rw [lintegral_tsum (fun e => (hm e).aemeasurable)] + exact hsum + exact (ae_lt_top (Measurable.tsum hm) hi).mono (fun _ h => h.ne) + +lemma ae_positive_prescribed_hausdorff_of_bad_mass + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω)) + (κ : ℝ) (W : Target.WienerLaw) + (hg : Target.IsHausdorffGauge (Target.gauge κ W)) + (hm1 : 0 < Target.moment κ W 1) + (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n) + (D : ℝ) (hD : 0 ≤ D) + (hsum : (∑' e : Target.GaugeIndex κ W, + ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≠ ∞) : + ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → 0 < μ ω E → + 0 < Target.hausdorff (Target.gauge κ W) E := by + filter_upwards [ae_prescribedBadGrid_sum_finite P μ hμ hfin κ W D hsum] with ω hω + let := hfin ω + exact fun E hE hp => positive_prescribed_hausdorff_of_summable_bad_mass + (μ ω) κ W hg hm1 hJ D hD hω E hE hp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RandomPassageBounds.lean b/lean/OAI/Probability/SLE/Gauge/RandomPassageBounds.lean index d1fe62c55e9d60f13e75dcea6b0197c4c8b7cca2..b115449f63688af2bdce2cadcd5f8bda3231f396 100644 --- a/lean/OAI/Probability/SLE/Gauge/RandomPassageBounds.lean +++ b/lean/OAI/Probability/SLE/Gauge/RandomPassageBounds.lean @@ -1,136 +1,136 @@ -import OAI.Probability.SLE.Geometry.FiniteBoundaryRestart -import OAI.Probability.SLE.Geometry.BoundaryScaleLaw -import OAI.Probability.SLE.Gauge.CoverChangeOfLaw -import Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner - -namespace OAI - -/-! Quantitative passage bounds for measurable random stopped charts and -finite stopped changes of measure. These are integration identities, with -the actual section estimates supplied by the ribbon proof. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open BoundaryScaleLaw - -local instance : LocallyCompactSpace closedHalfPlane := - isClosed_closedHalfPlane.locallyCompactSpace -local instance randomPassagePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance randomPassagePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ -local instance randomPassageClosedMeasurable : MeasurableSpace C(ℝ≥0, closedHalfPlane) := borel _ -local instance randomPassageClosedBorel : BorelSpace C(ℝ≥0, closedHalfPlane) := ⟨rfl⟩ - -lemma measurable_random_finiteBoundaryPath - {Ω : Type*} [MeasurableSpace Ω] - (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb) : - Measurable (fun x : Ω × C(ℝ≥0, ℂ) => finiteBoundaryPath (Fb x.1) x.2) := by - have hc : Continuous (fun x : C(ℝ≥0, closedHalfPlane) × C(closedHalfPlane, ℂ) => - x.2.comp x.1) := ContinuousMap.continuous_comp' - exact hc.measurable.comp - ((measurable_liftClosedPath.comp measurable_snd).prodMk (hFb.comp measurable_fst)) - -lemma measurableSet_random_finite_frontMassEvent - {Ω : Type*} [MeasurableSpace Ω] - (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb) - (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) - (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : - MeasurableSet {x : Ω × C(ℝ≥0, ℂ) | - finiteBoundaryPath (Fb x.1) x.2 ∈ frontMassEvent p V a ε κ v c R} := - (measurableSet_frontMassEvent p V a ε κ v c R K hK hp hV hVK).preimage - (measurable_random_finiteBoundaryPath Fb hFb) - -lemma restart_condLExp_event_sections - {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] - (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Y : Ω → X) (hY : Measurable[mΩ] Y) - (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → - @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) - (C : Set (Ω × X)) (hC : @MeasurableSet (Ω × X) (m.prod mX) C) : - (fun ω => ν {x | (ω, x) ∈ C}) =ᵐ[P] - P⁻[({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) | m] := by - have hf : @Measurable (Ω × X) ℝ≥0∞ (m.prod mX) _ (C.indicator (fun _ => 1)) := by - let : MeasurableSpace (Ω × X) := m.prod mX - exact measurable_const.indicator hC - let : MeasurableSpace Ω := mΩ - have he := @restart_condLExp_random_test Ω X mΩ mX P inferInstance ν inferInstance m hm Y hY hlaw - (C.indicator (fun _ => 1)) hf - have hs (ω : Ω) : (∫⁻ x, C.indicator (fun _ => (1 : ℝ≥0∞)) (ω, x) ∂ν) = - ν {x | (ω, x) ∈ C} := by - change (∫⁻ x, ({x | (ω, x) ∈ C}).indicator (fun _ => (1 : ℝ≥0∞)) x ∂ν) = _ - have hsec : MeasurableSet {x | (ω, x) ∈ C} := - hC.preimage measurable_prodMk_left - rw [lintegral_indicator hsec, setLIntegral_const, one_mul] - have hind : (fun ω => C.indicator (fun _ => (1 : ℝ≥0∞)) (ω, Y ω)) = - ({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) := by - funext ω - by_cases h : (ω, Y ω) ∈ C <;> simp [h] - simpa only [hs, hind] using he - -/-- Uniform section bounds integrate on every event in the stopped past. -No countable list of possible pasts and no independence of steps is used. -/ -theorem restart_event_lower_bound - {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] - (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Y : Ω → X) (hY : Measurable[mΩ] Y) - (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → - @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) - (C : Set (Ω × X)) (hC : @MeasurableSet (Ω × X) (m.prod mX) C) - (E : Set Ω) (q : ℝ≥0∞) - (hsections : ∀ᵐ ω ∂P, ω ∈ E → q ≤ ν {x | (ω, x) ∈ C}) - (A : Set Ω) (hA : MeasurableSet[m] A) (hAE : A ⊆ E) : - q * P A ≤ P (A ∩ {ω | (ω, Y ω) ∈ C}) := by - let : MeasurableSpace Ω := mΩ - have he := @restart_condLExp_event_sections Ω X mΩ mX P inferInstance ν inferInstance m hm Y hY hlaw C hC - have hactual : MeasurableSet[mΩ] {ω | (ω, Y ω) ∈ C} := - hC.preimage ((measurable_id'' hm).prodMk hY) - calc - _ = ∫⁻ ω in A, q ∂P := (setLIntegral_const A q).symm - _ ≤ ∫⁻ ω in A, P⁻[({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) | m] ω ∂P := by - apply lintegral_mono_ae - apply (ae_restrict_iff' (hm A hA)).mpr - filter_upwards [he, hsections] with ω hω hsec hωA - exact hω ▸ hsec (hAE hωA) - _ = ∫⁻ ω in A, ({ω | (ω, Y ω) ∈ C}).indicator (fun _ => (1 : ℝ≥0∞)) ω ∂P := - setLIntegral_condLExp hm P _ hA - _ = P (A ∩ {ω | (ω, Y ω) ∈ C}) := by - rw [setLIntegral_indicator hactual, setLIntegral_const, one_mul, inter_comm] - -/-- A lower bound on the density at a finite stop transfers any event -observable there to the terminal tilted law. -/ -theorem normalizedDensity_event_lower_bound_of_condExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Z M : Ω → ℝ) (hZ : Integrable Z P) (hZpos : 0 ≤ᵐ[P] Z) - (hM : M =ᵐ[P] P[Z | m]) - (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c) - (E : Set Ω) (hE : MeasurableSet[m] E) - (hbound : ∀ᵐ ω ∂P, ω ∈ E → c * a ≤ M ω) : - ENNReal.ofReal c * P E ≤ - P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E := by - let : MeasurableSpace Ω := mΩ - have hcond := condLExp_ofReal m hZ hZpos - have hmass : ENNReal.ofReal (c * a) * P E ≤ ∫⁻ ω in E, ENNReal.ofReal (Z ω) ∂P := by - calc - _ = ∫⁻ ω in E, ENNReal.ofReal (c * a) ∂P := (setLIntegral_const E _).symm - _ ≤ ∫⁻ ω in E, P⁻[(fun ω => ENNReal.ofReal (Z ω)) | m] ω ∂P := by - apply lintegral_mono_ae - apply (ae_restrict_iff' (hm E hE)).mpr - filter_upwards [hcond, hM, hbound] with ω hω hMω hb hωE - rw [hω, ← hMω] - exact ENNReal.ofReal_le_ofReal (hb hωE) - _ = _ := setLIntegral_condLExp hm P _ hE - rw [setLIntegral_eq_mul_normalizedDensity P Z ha (hm E hE), - ENNReal.ofReal_mul hc, mul_comm (ENNReal.ofReal c), mul_assoc] at hmass - exact (ENNReal.mul_le_mul_iff_right (ENNReal.ofReal_ne_zero_iff.mpr ha) - ENNReal.ofReal_ne_top).mp (by simpa only [mul_comm (ENNReal.ofReal a)] using hmass) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteBoundaryRestart +import OAI.Probability.SLE.Geometry.BoundaryScaleLaw +import OAI.Probability.SLE.Gauge.CoverChangeOfLaw +import Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner + +namespace OAI + +/-! Quantitative passage bounds for measurable random stopped charts and +finite stopped changes of measure. These are integration identities, with +the actual section estimates supplied by the ribbon proof. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open BoundaryScaleLaw + +local instance : LocallyCompactSpace closedHalfPlane := + isClosed_closedHalfPlane.locallyCompactSpace +local instance randomPassagePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance randomPassagePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ +local instance randomPassageClosedMeasurable : MeasurableSpace C(ℝ≥0, closedHalfPlane) := borel _ +local instance randomPassageClosedBorel : BorelSpace C(ℝ≥0, closedHalfPlane) := ⟨rfl⟩ + +lemma measurable_random_finiteBoundaryPath + {Ω : Type*} [MeasurableSpace Ω] + (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb) : + Measurable (fun x : Ω × C(ℝ≥0, ℂ) => finiteBoundaryPath (Fb x.1) x.2) := by + have hc : Continuous (fun x : C(ℝ≥0, closedHalfPlane) × C(closedHalfPlane, ℂ) => + x.2.comp x.1) := ContinuousMap.continuous_comp' + exact hc.measurable.comp + ((measurable_liftClosedPath.comp measurable_snd).prodMk (hFb.comp measurable_fst)) + +lemma measurableSet_random_finite_frontMassEvent + {Ω : Type*} [MeasurableSpace Ω] + (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb) + (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) + (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : + MeasurableSet {x : Ω × C(ℝ≥0, ℂ) | + finiteBoundaryPath (Fb x.1) x.2 ∈ frontMassEvent p V a ε κ v c R} := + (measurableSet_frontMassEvent p V a ε κ v c R K hK hp hV hVK).preimage + (measurable_random_finiteBoundaryPath Fb hFb) + +lemma restart_condLExp_event_sections + {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] + (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Y : Ω → X) (hY : Measurable[mΩ] Y) + (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → + @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) + (C : Set (Ω × X)) (hC : @MeasurableSet (Ω × X) (m.prod mX) C) : + (fun ω => ν {x | (ω, x) ∈ C}) =ᵐ[P] + P⁻[({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) | m] := by + have hf : @Measurable (Ω × X) ℝ≥0∞ (m.prod mX) _ (C.indicator (fun _ => 1)) := by + let : MeasurableSpace (Ω × X) := m.prod mX + exact measurable_const.indicator hC + let : MeasurableSpace Ω := mΩ + have he := @restart_condLExp_random_test Ω X mΩ mX P inferInstance ν inferInstance m hm Y hY hlaw + (C.indicator (fun _ => 1)) hf + have hs (ω : Ω) : (∫⁻ x, C.indicator (fun _ => (1 : ℝ≥0∞)) (ω, x) ∂ν) = + ν {x | (ω, x) ∈ C} := by + change (∫⁻ x, ({x | (ω, x) ∈ C}).indicator (fun _ => (1 : ℝ≥0∞)) x ∂ν) = _ + have hsec : MeasurableSet {x | (ω, x) ∈ C} := + hC.preimage measurable_prodMk_left + rw [lintegral_indicator hsec, setLIntegral_const, one_mul] + have hind : (fun ω => C.indicator (fun _ => (1 : ℝ≥0∞)) (ω, Y ω)) = + ({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) := by + funext ω + by_cases h : (ω, Y ω) ∈ C <;> simp [h] + simpa only [hs, hind] using he + +/-- Uniform section bounds integrate on every event in the stopped past. +No countable list of possible pasts and no independence of steps is used. -/ +theorem restart_event_lower_bound + {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] + (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Y : Ω → X) (hY : Measurable[mΩ] Y) + (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → + @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) + (C : Set (Ω × X)) (hC : @MeasurableSet (Ω × X) (m.prod mX) C) + (E : Set Ω) (q : ℝ≥0∞) + (hsections : ∀ᵐ ω ∂P, ω ∈ E → q ≤ ν {x | (ω, x) ∈ C}) + (A : Set Ω) (hA : MeasurableSet[m] A) (hAE : A ⊆ E) : + q * P A ≤ P (A ∩ {ω | (ω, Y ω) ∈ C}) := by + let : MeasurableSpace Ω := mΩ + have he := @restart_condLExp_event_sections Ω X mΩ mX P inferInstance ν inferInstance m hm Y hY hlaw C hC + have hactual : MeasurableSet[mΩ] {ω | (ω, Y ω) ∈ C} := + hC.preimage ((measurable_id'' hm).prodMk hY) + calc + _ = ∫⁻ ω in A, q ∂P := (setLIntegral_const A q).symm + _ ≤ ∫⁻ ω in A, P⁻[({ω | (ω, Y ω) ∈ C}).indicator (fun _ => 1) | m] ω ∂P := by + apply lintegral_mono_ae + apply (ae_restrict_iff' (hm A hA)).mpr + filter_upwards [he, hsections] with ω hω hsec hωA + exact hω ▸ hsec (hAE hωA) + _ = ∫⁻ ω in A, ({ω | (ω, Y ω) ∈ C}).indicator (fun _ => (1 : ℝ≥0∞)) ω ∂P := + setLIntegral_condLExp hm P _ hA + _ = P (A ∩ {ω | (ω, Y ω) ∈ C}) := by + rw [setLIntegral_indicator hactual, setLIntegral_const, one_mul, inter_comm] + +/-- A lower bound on the density at a finite stop transfers any event +observable there to the terminal tilted law. -/ +theorem normalizedDensity_event_lower_bound_of_condExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Z M : Ω → ℝ) (hZ : Integrable Z P) (hZpos : 0 ≤ᵐ[P] Z) + (hM : M =ᵐ[P] P[Z | m]) + (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c) + (E : Set Ω) (hE : MeasurableSet[m] E) + (hbound : ∀ᵐ ω ∂P, ω ∈ E → c * a ≤ M ω) : + ENNReal.ofReal c * P E ≤ + P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E := by + let : MeasurableSpace Ω := mΩ + have hcond := condLExp_ofReal m hZ hZpos + have hmass : ENNReal.ofReal (c * a) * P E ≤ ∫⁻ ω in E, ENNReal.ofReal (Z ω) ∂P := by + calc + _ = ∫⁻ ω in E, ENNReal.ofReal (c * a) ∂P := (setLIntegral_const E _).symm + _ ≤ ∫⁻ ω in E, P⁻[(fun ω => ENNReal.ofReal (Z ω)) | m] ω ∂P := by + apply lintegral_mono_ae + apply (ae_restrict_iff' (hm E hE)).mpr + filter_upwards [hcond, hM, hbound] with ω hω hMω hb hωE + rw [hω, ← hMω] + exact ENNReal.ofReal_le_ofReal (hb hωE) + _ = _ := setLIntegral_condLExp hm P _ hE + rw [setLIntegral_eq_mul_normalizedDensity P Z ha (hm E hE), + ENNReal.ofReal_mul hc, mul_comm (ENNReal.ofReal c), mul_assoc] at hmass + exact (ENNReal.mul_le_mul_iff_right (ENNReal.ofReal_ne_zero_iff.mpr ha) + ENNReal.ofReal_ne_top).mp (by simpa only [mul_comm (ENNReal.ofReal a)] using hmass) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RecenteredRibbonPassage.lean b/lean/OAI/Probability/SLE/Gauge/RecenteredRibbonPassage.lean index 9c9adaf5307d3a82518740c3c5ea97de7e1fe046..e2510e66d899cd5946b0fb0366fc037d3546a13d 100644 --- a/lean/OAI/Probability/SLE/Gauge/RecenteredRibbonPassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/RecenteredRibbonPassage.lean @@ -1,193 +1,193 @@ -import OAI.Probability.SLE.Geometry.RecenteredGuideCompactness -import OAI.Probability.SLE.Stochastic.PlanarCoordinatePassage -import OAI.Probability.SLE.Gauge.RibbonSequenceCompactness - -namespace OAI - -/-! Actual passage along a recentered conformal subsequence, with its -finite physical realization constructed from the boundary chart. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates SequencePassage PassageSuccess FirstFrontPassage - -theorem recentered_ribbon_subsequence_passage - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) - (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (Gbase ω) (Fbase ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) - (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) - (hU : ∀ n, Continuous (U n)) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) - (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) - (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) - (αₙ : ℕ → ℝ → ℂ) (Cg : ℝ≥0) (s c : ℝ) - (hCg : 0 < (Cg : ℝ)) (hs : 0 < s) (hcpos : 0 < c) - (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) - (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) - (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → - c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) - (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) - (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) - (Qₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) - (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C Lq : ℝ≥0) (Rq : ℝ) - (hC : 0 < C) (hLq : 0 < Lq) - (hQupper : ∀ n, LipschitzOnWith C (Qₙ n) K) - (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) - (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) - (xₙ : ℕ → ℝ) (ε : ℝ) (hε : 0 < ε) - (hxₙ : ∀ n, |xₙ n| ≤ ε / 4) - (hcore : {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ interior K) - (hshape : ∀ n, EqOn (fun t => αₙ n t - γₙ n (T n)) - (fun t => Qₙ n (guideCoordinate 0 (xₙ n) t)) (Icc 0 2)) - (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / - ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) : - ∃ (φ : ℕ → ℕ) (b : ℕ → ℝ) (Fb : ℕ → C(closedHalfPlane, ℂ)), - StrictMono φ ∧ (∀ n, 0 < b n) ∧ - (∀ n (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb n z = - normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b n) z - - γₙ (φ n) (T (φ n))) ∧ - let η := fun n ω => (finiteBoundaryPath (Fb n) - (sourceTracePath γ (fun ω => (hwbase ω).1) ω) : ℝ≥0 → ℂ) - ∃ c₁ q : ℝ, 0 < c₁ ∧ 0 < q ∧ ∀ᶠ n in atTop, - ENNReal.ofReal q ≤ P {ω | - ω ∈ FrontSuccess η (fun n => longitudinalCoordinate (Qₙ (φ n)) K 0) - (fun _ => 0) (fun n => physicalPassageRegion (Qₙ (φ n)) K C ε (ε / Lq)) - (fun n => Qₙ (φ n) 1) (ε / Lq) n ∧ - ENNReal.ofReal c₁ ≤ segmentSausageCharge κ (η n ω) 0 - (frontTime (fun t => longitudinalCoordinate (Qₙ (φ n)) K 0 (η n ω t)) 1) - (v (φ n)) (physicalMiddleRegion (Qₙ (φ n)) K 0 ε)} := by - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - have hadapt := brownian_capacity_trace_adapted B₀ hm hc κ γ Gbase Fbase hwbase - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := - (hadapt t).mono (ℱ.le t) le_rfl - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨Gbase ω, Fbase ω, hwbase ω⟩)⟩ - have hxin : ∀ n, xₙ n ∈ Icc (-ε / 4) (ε / 4) := by - intro n - have hh := abs_le.mp (hxₙ n) - constructor <;> linarith [hh.1, hh.2] - obtain ⟨x, hx, ψ, hψ, hxlim⟩ := isCompact_Icc.tendsto_subseq hxin - have hxabs : |x| ≤ ε / 4 := abs_le.mpr ⟨by linarith [hx.1], hx.2⟩ - obtain ⟨Q, χ, hχ, hQconv, hQu, hQl, _⟩ := exists_bilip_uniform_subsequence - (fun n => Qₙ (ψ n)) K hK x₀ hx₀ C Lq Rq (fun n => hQupper (ψ n)) - (fun n => hQlower (ψ n)) (fun n => hQtip (ψ n)) - let ρ := ψ ∘ χ - obtain ⟨b, Fb, βₙ, θ, F, α, β, hb, hext, hβc, hβ0, hβHn, hbarα, - hθ, hF, hiF, hFne, homit, hFconv, hαconv, hβconv, _, _, hαclim, _, - hβlim0, hβlimc, hβlimi, hβlimH, hrel⟩ := - recentered_actual_guide_compactness P κ hκ hκ8 γ hSLE hcompact hboundary - (fun n => U (ρ n)) (fun n => γₙ (ρ n)) (fun n => G (ρ n)) (fun n => F₀ (ρ n)) - (fun n => hU (ρ n)) (fun n => hw (ρ n)) (fun n => T (ρ n)) (fun n => hT (ρ n)) - (fun n => hfresh (ρ n)) (fun n => αₙ (ρ n)) Cg s c hCg hs hcpos - (fun n => hαL (ρ n)) (fun n => hα0 (ρ n)) - (fun n => hchord (ρ n)) (fun n => hαD (ρ n)) - (fun δ hδ hδ2 => by - obtain ⟨r, hr, hh⟩ := hclear δ hδ hδ2 - exact ⟨r, hr, fun n => hh (ρ n)⟩) - (fun n => hangle (ρ n)) - let φ := ρ ∘ θ - let b' := fun n => b (θ n) - let Fb' := fun n => Fb (θ n) - let Fn := fun n z => normalizedInverse (F₀ (φ n) (T (φ n))) - (U (φ n) (T (φ n))) (b' n) z - γₙ (φ n) (T (φ n)) - let α' := fun n t => αₙ (φ n) t - γₙ (φ n) (T (φ n)) - have hφ : StrictMono φ := (hψ.comp hχ).comp hθ - have hQconv' : TendstoUniformlyOn (fun n => Qₙ (φ n)) Q atTop K := by - intro u hu - exact hθ.tendsto_atTop.eventually (hQconv u hu) - have hguideK n t (ht : t ∈ Icc (0 : ℝ) 2) : guideCoordinate 0 (xₙ (φ n)) t ∈ K := by - apply interior_subset - apply hcore - exact ⟨by simpa using ht.1, by simpa using ht.2, - (guideCoordinate_im_bound ht.1).trans (hxₙ (φ n))⟩ - have hshapeLim : EqOn α (fun t => Q (guideCoordinate 0 x t)) (Icc 0 2) := - coordinate_guide_limit_identity (fun n => Qₙ (φ n)) Q K hK.isClosed hQu.continuousOn - hQconv' (fun n => xₙ (φ n)) x - (hxlim.comp (hχ.comp hθ).tendsto_atTop) α' α hαconv hguideK - (fun n => hshape (φ n)) - have hmem t (ht : t ∈ Icc (0 : ℝ) (3 / 2)) : guideCoordinate 0 x t ∈ interior K := by - apply hcore - exact ⟨by simpa using ht.1, by simpa using ht.2.trans (by norm_num : (3 / 2 : ℝ) ≤ 2), - (guideCoordinate_im_bound ht.1).trans hxabs⟩ - have hbase : (1 : ℂ) ∈ K := by - apply interior_subset - apply hcore - refine ⟨by norm_num, by norm_num, ?_⟩ - simpa only [Complex.one_im, abs_zero] using (show 0 ≤ ε / 4 by positivity) - let Fbar := fun n z => if hz : z ∈ closedHalfPlane then (Fb' n ⟨z, hz⟩ : OnePoint ℂ) - else OnePoint.infty - have hbar n : ContinuousOn (Fbar n) closedHalfPlane := by - rw [continuousOn_iff_continuous_domRestrict] - convert OnePoint.continuous_coe.comp (Fb' n).continuous using 1 - funext z - change (if hz : (z : ℂ) ∈ closedHalfPlane then (Fb' n ⟨z, hz⟩ : OnePoint ℂ) - else OnePoint.infty) = (Fb' n z : OnePoint ℂ) - rw [dite_eq_left z.2] - have hbarext n z (hz : z ∈ upperHalfPlane) : Fbar n z = (Fn n z : OnePoint ℂ) := by - have hz' : z ∈ closedHalfPlane := (show 0 < z.im from hz).le - dsimp only [Fbar] - rw [dite_eq_left hz'] - exact congrArg (fun z : ℂ => (z : OnePoint ℂ)) (hext (θ n) ⟨z, hz'⟩ hz) - have hbarα' n t (ht : t ∈ Icc (0 : ℝ) 2) : Fbar n (βₙ (θ n) t) = (α' n t : OnePoint ℂ) := by - dsimp only [Fbar] - rw [dite_eq_left (show βₙ (θ n) t ∈ closedHalfPlane from hβHn (θ n) t ht)] - exact congrArg (fun z : ℂ => (z : OnePoint ℂ)) (hbarα (θ n) t ht _) - let gp := sourceTracePath γ (fun ω => (hwbase ω).1) - let η := fun n ω => (finiteBoundaryPath (Fb' n) (gp ω) : ℝ≥0 → ℂ) - have hηc n : ∀ᵐ ω ∂P, Continuous (η n ω) := - Eventually.of_forall (fun ω => (finiteBoundaryPath (Fb' n) (gp ω)).continuous) - have hηbar n : ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ) := by - apply Eventually.of_forall - intro ω t - have hh : ClosedPathProperty (gp ω) := fun t => (hwbase ω).2.2.1 t - dsimp only [η] - rw [finiteBoundaryPath_apply _ _ hh] - dsimp only [Fbar] - rw [dite_eq_left (show γ ω t ∈ closedHalfPlane from (hwbase ω).2.2.1 t)] - rfl - have hαL' n : LipschitzOnWith Cg (α' n) (Icc 0 2) := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro u hu w hw' - simpa only [α', dist_sub_right] using (hαL (φ n)).dist_le_mul u hu w hw' - have hnorm n := normalized_capacity_charts (hw (φ n)) (T (φ n)) (b' n) (hb (θ n)) - have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hnorm n).1.sub_const _ - have hiFn n : InjOn (Fn n) upperHalfPlane := by - intro z hz w hw' he - exact (hnorm n).2.1 hz hw' (sub_left_inj.mp he) - obtain ⟨B, _, Tc, _, c₁, q, hc₁, hq, hprob⟩ := planar_coordinate_limit_passage - κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase hwbase hboundary.to_onePoint hsupport hplanar - (fun n => Qₙ (φ n)) Q K C Lq hC hLq (fun n => hQupper (φ n)) - (fun n => hQlower (φ n)) hQconv' hQu.continuousOn hbase - Fn Fbar F hFn hiFn hbar hbarext hF hiF hFne hFconv - α' (fun n => βₙ (θ n)) α β Cg hαL' hαconv hαclim - (fun n => hβc (θ n)) (fun n => hβ0 (θ n)) (fun n => hβHn (θ n)) - hbarα' hβconv hβlimc hβlim0 hβlimi hβlimH hrel homit x ε (ε / Lq) - (by linarith) (div_pos hε hLq) hshapeLim hmem - (fun n => v (φ n)) (fun n => hv (φ n)) (hlim.comp hφ.tendsto_atTop) η hηc hηbar - refine ⟨φ, b', Fb', hφ, (fun n => hb (θ n)), (fun n => hext (θ n)), - c₁, q, hc₁, hq, ?_⟩ - filter_upwards [hprob] with n hn - apply hn.trans (measure_mono ?_) - rintro ω ⟨⟨hfinite, hhit, hbefore, hpath, hterminal⟩, hmass⟩ - exact ⟨⟨hfinite, hhit, bot_le.trans_lt hbefore, hpath, hterminal⟩, - hmass.trans (segmentSausageCharge_mono κ (η n ω) bot_le le_rfl (v (φ n)) Subset.rfl)⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.RecenteredGuideCompactness +import OAI.Probability.SLE.Stochastic.PlanarCoordinatePassage +import OAI.Probability.SLE.Gauge.RibbonSequenceCompactness + +namespace OAI + +/-! Actual passage along a recentered conformal subsequence, with its +finite physical realization constructed from the boundary chart. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates SequencePassage PassageSuccess FirstFrontPassage + +theorem recentered_ribbon_subsequence_passage + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) + (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (Gbase ω) (Fbase ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) + (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) + (hU : ∀ n, Continuous (U n)) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) + (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) + (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) + (αₙ : ℕ → ℝ → ℂ) (Cg : ℝ≥0) (s c : ℝ) + (hCg : 0 < (Cg : ℝ)) (hs : 0 < s) (hcpos : 0 < c) + (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) + (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) + (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → + c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) + (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) + (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) + (Qₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) + (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C Lq : ℝ≥0) (Rq : ℝ) + (hC : 0 < C) (hLq : 0 < Lq) + (hQupper : ∀ n, LipschitzOnWith C (Qₙ n) K) + (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) + (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) + (xₙ : ℕ → ℝ) (ε : ℝ) (hε : 0 < ε) + (hxₙ : ∀ n, |xₙ n| ≤ ε / 4) + (hcore : {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ interior K) + (hshape : ∀ n, EqOn (fun t => αₙ n t - γₙ n (T n)) + (fun t => Qₙ n (guideCoordinate 0 (xₙ n) t)) (Icc 0 2)) + (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / + ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) : + ∃ (φ : ℕ → ℕ) (b : ℕ → ℝ) (Fb : ℕ → C(closedHalfPlane, ℂ)), + StrictMono φ ∧ (∀ n, 0 < b n) ∧ + (∀ n (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb n z = + normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b n) z - + γₙ (φ n) (T (φ n))) ∧ + let η := fun n ω => (finiteBoundaryPath (Fb n) + (sourceTracePath γ (fun ω => (hwbase ω).1) ω) : ℝ≥0 → ℂ) + ∃ c₁ q : ℝ, 0 < c₁ ∧ 0 < q ∧ ∀ᶠ n in atTop, + ENNReal.ofReal q ≤ P {ω | + ω ∈ FrontSuccess η (fun n => longitudinalCoordinate (Qₙ (φ n)) K 0) + (fun _ => 0) (fun n => physicalPassageRegion (Qₙ (φ n)) K C ε (ε / Lq)) + (fun n => Qₙ (φ n) 1) (ε / Lq) n ∧ + ENNReal.ofReal c₁ ≤ segmentSausageCharge κ (η n ω) 0 + (frontTime (fun t => longitudinalCoordinate (Qₙ (φ n)) K 0 (η n ω t)) 1) + (v (φ n)) (physicalMiddleRegion (Qₙ (φ n)) K 0 ε)} := by + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + have hadapt := brownian_capacity_trace_adapted B₀ hm hc κ γ Gbase Fbase hwbase + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := + (hadapt t).mono (ℱ.le t) le_rfl + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨Gbase ω, Fbase ω, hwbase ω⟩)⟩ + have hxin : ∀ n, xₙ n ∈ Icc (-ε / 4) (ε / 4) := by + intro n + have hh := abs_le.mp (hxₙ n) + constructor <;> linarith [hh.1, hh.2] + obtain ⟨x, hx, ψ, hψ, hxlim⟩ := isCompact_Icc.tendsto_subseq hxin + have hxabs : |x| ≤ ε / 4 := abs_le.mpr ⟨by linarith [hx.1], hx.2⟩ + obtain ⟨Q, χ, hχ, hQconv, hQu, hQl, _⟩ := exists_bilip_uniform_subsequence + (fun n => Qₙ (ψ n)) K hK x₀ hx₀ C Lq Rq (fun n => hQupper (ψ n)) + (fun n => hQlower (ψ n)) (fun n => hQtip (ψ n)) + let ρ := ψ ∘ χ + obtain ⟨b, Fb, βₙ, θ, F, α, β, hb, hext, hβc, hβ0, hβHn, hbarα, + hθ, hF, hiF, hFne, homit, hFconv, hαconv, hβconv, _, _, hαclim, _, + hβlim0, hβlimc, hβlimi, hβlimH, hrel⟩ := + recentered_actual_guide_compactness P κ hκ hκ8 γ hSLE hcompact hboundary + (fun n => U (ρ n)) (fun n => γₙ (ρ n)) (fun n => G (ρ n)) (fun n => F₀ (ρ n)) + (fun n => hU (ρ n)) (fun n => hw (ρ n)) (fun n => T (ρ n)) (fun n => hT (ρ n)) + (fun n => hfresh (ρ n)) (fun n => αₙ (ρ n)) Cg s c hCg hs hcpos + (fun n => hαL (ρ n)) (fun n => hα0 (ρ n)) + (fun n => hchord (ρ n)) (fun n => hαD (ρ n)) + (fun δ hδ hδ2 => by + obtain ⟨r, hr, hh⟩ := hclear δ hδ hδ2 + exact ⟨r, hr, fun n => hh (ρ n)⟩) + (fun n => hangle (ρ n)) + let φ := ρ ∘ θ + let b' := fun n => b (θ n) + let Fb' := fun n => Fb (θ n) + let Fn := fun n z => normalizedInverse (F₀ (φ n) (T (φ n))) + (U (φ n) (T (φ n))) (b' n) z - γₙ (φ n) (T (φ n)) + let α' := fun n t => αₙ (φ n) t - γₙ (φ n) (T (φ n)) + have hφ : StrictMono φ := (hψ.comp hχ).comp hθ + have hQconv' : TendstoUniformlyOn (fun n => Qₙ (φ n)) Q atTop K := by + intro u hu + exact hθ.tendsto_atTop.eventually (hQconv u hu) + have hguideK n t (ht : t ∈ Icc (0 : ℝ) 2) : guideCoordinate 0 (xₙ (φ n)) t ∈ K := by + apply interior_subset + apply hcore + exact ⟨by simpa using ht.1, by simpa using ht.2, + (guideCoordinate_im_bound ht.1).trans (hxₙ (φ n))⟩ + have hshapeLim : EqOn α (fun t => Q (guideCoordinate 0 x t)) (Icc 0 2) := + coordinate_guide_limit_identity (fun n => Qₙ (φ n)) Q K hK.isClosed hQu.continuousOn + hQconv' (fun n => xₙ (φ n)) x + (hxlim.comp (hχ.comp hθ).tendsto_atTop) α' α hαconv hguideK + (fun n => hshape (φ n)) + have hmem t (ht : t ∈ Icc (0 : ℝ) (3 / 2)) : guideCoordinate 0 x t ∈ interior K := by + apply hcore + exact ⟨by simpa using ht.1, by simpa using ht.2.trans (by norm_num : (3 / 2 : ℝ) ≤ 2), + (guideCoordinate_im_bound ht.1).trans hxabs⟩ + have hbase : (1 : ℂ) ∈ K := by + apply interior_subset + apply hcore + refine ⟨by norm_num, by norm_num, ?_⟩ + simpa only [Complex.one_im, abs_zero] using (show 0 ≤ ε / 4 by positivity) + let Fbar := fun n z => if hz : z ∈ closedHalfPlane then (Fb' n ⟨z, hz⟩ : OnePoint ℂ) + else OnePoint.infty + have hbar n : ContinuousOn (Fbar n) closedHalfPlane := by + rw [continuousOn_iff_continuous_domRestrict] + convert OnePoint.continuous_coe.comp (Fb' n).continuous using 1 + funext z + change (if hz : (z : ℂ) ∈ closedHalfPlane then (Fb' n ⟨z, hz⟩ : OnePoint ℂ) + else OnePoint.infty) = (Fb' n z : OnePoint ℂ) + rw [dite_eq_left z.2] + have hbarext n z (hz : z ∈ upperHalfPlane) : Fbar n z = (Fn n z : OnePoint ℂ) := by + have hz' : z ∈ closedHalfPlane := (show 0 < z.im from hz).le + dsimp only [Fbar] + rw [dite_eq_left hz'] + exact congrArg (fun z : ℂ => (z : OnePoint ℂ)) (hext (θ n) ⟨z, hz'⟩ hz) + have hbarα' n t (ht : t ∈ Icc (0 : ℝ) 2) : Fbar n (βₙ (θ n) t) = (α' n t : OnePoint ℂ) := by + dsimp only [Fbar] + rw [dite_eq_left (show βₙ (θ n) t ∈ closedHalfPlane from hβHn (θ n) t ht)] + exact congrArg (fun z : ℂ => (z : OnePoint ℂ)) (hbarα (θ n) t ht _) + let gp := sourceTracePath γ (fun ω => (hwbase ω).1) + let η := fun n ω => (finiteBoundaryPath (Fb' n) (gp ω) : ℝ≥0 → ℂ) + have hηc n : ∀ᵐ ω ∂P, Continuous (η n ω) := + Eventually.of_forall (fun ω => (finiteBoundaryPath (Fb' n) (gp ω)).continuous) + have hηbar n : ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ) := by + apply Eventually.of_forall + intro ω t + have hh : ClosedPathProperty (gp ω) := fun t => (hwbase ω).2.2.1 t + dsimp only [η] + rw [finiteBoundaryPath_apply _ _ hh] + dsimp only [Fbar] + rw [dite_eq_left (show γ ω t ∈ closedHalfPlane from (hwbase ω).2.2.1 t)] + rfl + have hαL' n : LipschitzOnWith Cg (α' n) (Icc 0 2) := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro u hu w hw' + simpa only [α', dist_sub_right] using (hαL (φ n)).dist_le_mul u hu w hw' + have hnorm n := normalized_capacity_charts (hw (φ n)) (T (φ n)) (b' n) (hb (θ n)) + have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hnorm n).1.sub_const _ + have hiFn n : InjOn (Fn n) upperHalfPlane := by + intro z hz w hw' he + exact (hnorm n).2.1 hz hw' (sub_left_inj.mp he) + obtain ⟨B, _, Tc, _, c₁, q, hc₁, hq, hprob⟩ := planar_coordinate_limit_passage + κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase hwbase hboundary.to_onePoint hsupport hplanar + (fun n => Qₙ (φ n)) Q K C Lq hC hLq (fun n => hQupper (φ n)) + (fun n => hQlower (φ n)) hQconv' hQu.continuousOn hbase + Fn Fbar F hFn hiFn hbar hbarext hF hiF hFne hFconv + α' (fun n => βₙ (θ n)) α β Cg hαL' hαconv hαclim + (fun n => hβc (θ n)) (fun n => hβ0 (θ n)) (fun n => hβHn (θ n)) + hbarα' hβconv hβlimc hβlim0 hβlimi hβlimH hrel homit x ε (ε / Lq) + (by linarith) (div_pos hε hLq) hshapeLim hmem + (fun n => v (φ n)) (fun n => hv (φ n)) (hlim.comp hφ.tendsto_atTop) η hηc hηbar + refine ⟨φ, b', Fb', hφ, (fun n => hb (θ n)), (fun n => hext (θ n)), + c₁, q, hc₁, hq, ?_⟩ + filter_upwards [hprob] with n hn + apply hn.trans (measure_mono ?_) + rintro ω ⟨⟨hfinite, hhit, hbefore, hpath, hterminal⟩, hmass⟩ + exact ⟨⟨hfinite, hhit, bot_le.trans_lt hbefore, hpath, hterminal⟩, + hmass.trans (segmentSausageCharge_mono κ (η n ω) bot_le le_rfl (v (φ n)) Subset.rfl)⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RestartDenseWeight.lean b/lean/OAI/Probability/SLE/Gauge/RestartDenseWeight.lean index ff847252e80cc7d32d0b3fd3e0b1c654a60e9ac1..839d0e24685595dfa0b9d67fb9641df8c55e71e1 100644 --- a/lean/OAI/Probability/SLE/Gauge/RestartDenseWeight.lean +++ b/lean/OAI/Probability/SLE/Gauge/RestartDenseWeight.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Mass.RestartStoppedMass -import OAI.Probability.SLE.Gauge.CanonicalDenseSections - -namespace OAI - -/-! A successful canonical dense section, multiplied by its exact centered -weight, contributes to the actual physical stopped-mass event. This proves -the pointwise density domination needed in conditional section integration. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open FrontSuccessMeasurability - -local instance denseWeightRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance denseWeightRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance denseWeightComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance denseWeightComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -lemma IsCapacityTwoWitness.normalized_restart_dense_weight_le - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a) - (κ : ℝ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (p : C(ℝ≥0, ℝ)) - (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) (Φ p) G' F') - (hdriver : (fun t => Real.sqrt κ * p t) = normalizedRestartDriver U T a) - (w : ℂ) (hwH : w ∈ upperHalfPlane) (u q : ℝ) (hq : 0 ≤ q) - (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) = - (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u)) - (Fb : C(closedHalfPlane, ℂ)) - (hpath : ∀ t, finiteBoundaryPath Fb (Φ p) t = γ (T + a ^ 2 * t)) - (b r v : ℝ) (m : ℕ) : - (canonicalDenseEvent κ Φ Fb q w b r m v (F T ((a : ℂ) * w + (U T : ℂ)))).indicator - (fun p => ENNReal.ofReal - (greenDensity κ U γ G T (F T ((a : ℂ) * w + (U T : ℂ))) / - initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ)))) * - canonicalFiniteDensity κ (q, w, p)) p ≤ - if ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ - stoppedWholeSausageCharge κ ⟨γ, hw.1⟩ - (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q)) v - (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) then - ENNReal.ofReal (stoppedGreen κ U γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q) / - initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ)))) - else 0 := by - classical - have hb' : IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) G' F' := hdriver ▸ hb - have hzH : (a : ℂ) * w + (U T : ℂ) ∈ upperHalfPlane := by - change 0 < ((a : ℂ) * w + (U T : ℂ)).im - simpa only [Complex.add_im, Complex.ofReal_im, add_zero, Complex.mul_im, - Complex.ofReal_re, zero_mul] using mul_pos (show 0 < (a : ℝ) from ha) hwH - have hphysH := (hw.2.2.2.2.2.2.1 T hzH).1.1 - by_cases hC : p ∈ canonicalDenseEvent κ Φ Fb q w b r m v (F T ((a : ℂ) * w + (U T : ℂ))) - · rw [indicator_of_mem hC, canonicalFiniteDensity_eq κ p hb' q hq w] - by_cases hfinite : radiusLevelTime (Φ p) G' w q < ∞ - · have hmass := hw.normalized_restart_stopped_mass_le hU T a ha hb w u q hbase - (finiteBoundaryPath Fb (Φ p)) hpath hfinite κ v - (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) - have hcanon := canonicalRadiusLevel_eq κ p hb' q hq w - have hC' : ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ - stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ p)) - (radiusLevelTime (Φ p) G' w q) v (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) := by - simpa only [canonicalDenseEvent, mem_ofPred_eq, hcanon] using hC - rw [ite_eq_left (hC'.trans hmass)] - rw [hdriver, ← ENNReal.ofReal_mul - (div_nonneg (greenDensity_nonneg hw κ T _) (initialGreen_pos κ _ hphysH).le)] - apply ENNReal.ofReal_le_ofReal - have hbound := hw.normalized_restart_stopped_green_lower hU T a ha hb κ w hwH u q hbase - calc - _ = (greenDensity κ U γ G T (F T ((a : ℂ) * w + (U T : ℂ))) * - (stoppedGreen κ (normalizedRestartDriver U T a) (Φ p) G' w q / initialGreen κ w)) / - initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ))) := by ring - _ ≤ _ := div_le_div_of_nonneg_right hbound (initialGreen_pos κ _ hphysH).le - · rw [hdriver, stoppedGreen, ite_eq_right hfinite, zero_div, ENNReal.ofReal_zero, mul_zero] - exact bot_le - · rw [indicator_of_notMem hC] - exact bot_le - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RestartStoppedMass +import OAI.Probability.SLE.Gauge.CanonicalDenseSections + +namespace OAI + +/-! A successful canonical dense section, multiplied by its exact centered +weight, contributes to the actual physical stopped-mass event. This proves +the pointwise density domination needed in conditional section integration. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open FrontSuccessMeasurability + +local instance denseWeightRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance denseWeightRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance denseWeightComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance denseWeightComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +lemma IsCapacityTwoWitness.normalized_restart_dense_weight_le + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a) + (κ : ℝ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (p : C(ℝ≥0, ℝ)) + (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) (Φ p) G' F') + (hdriver : (fun t => Real.sqrt κ * p t) = normalizedRestartDriver U T a) + (w : ℂ) (hwH : w ∈ upperHalfPlane) (u q : ℝ) (hq : 0 ≤ q) + (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) = + (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u)) + (Fb : C(closedHalfPlane, ℂ)) + (hpath : ∀ t, finiteBoundaryPath Fb (Φ p) t = γ (T + a ^ 2 * t)) + (b r v : ℝ) (m : ℕ) : + (canonicalDenseEvent κ Φ Fb q w b r m v (F T ((a : ℂ) * w + (U T : ℂ)))).indicator + (fun p => ENNReal.ofReal + (greenDensity κ U γ G T (F T ((a : ℂ) * w + (U T : ℂ))) / + initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ)))) * + canonicalFiniteDensity κ (q, w, p)) p ≤ + if ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ + stoppedWholeSausageCharge κ ⟨γ, hw.1⟩ + (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q)) v + (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) then + ENNReal.ofReal (stoppedGreen κ U γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q) / + initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ)))) + else 0 := by + classical + have hb' : IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) G' F' := hdriver ▸ hb + have hzH : (a : ℂ) * w + (U T : ℂ) ∈ upperHalfPlane := by + change 0 < ((a : ℂ) * w + (U T : ℂ)).im + simpa only [Complex.add_im, Complex.ofReal_im, add_zero, Complex.mul_im, + Complex.ofReal_re, zero_mul] using mul_pos (show 0 < (a : ℝ) from ha) hwH + have hphysH := (hw.2.2.2.2.2.2.1 T hzH).1.1 + by_cases hC : p ∈ canonicalDenseEvent κ Φ Fb q w b r m v (F T ((a : ℂ) * w + (U T : ℂ))) + · rw [indicator_of_mem hC, canonicalFiniteDensity_eq κ p hb' q hq w] + by_cases hfinite : radiusLevelTime (Φ p) G' w q < ∞ + · have hmass := hw.normalized_restart_stopped_mass_le hU T a ha hb w u q hbase + (finiteBoundaryPath Fb (Φ p)) hpath hfinite κ v + (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) + have hcanon := canonicalRadiusLevel_eq κ p hb' q hq w + have hC' : ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ + stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ p)) + (radiusLevelTime (Φ p) G' w q) v (closedBall (F T ((a : ℂ) * w + (U T : ℂ))) r) := by + simpa only [canonicalDenseEvent, mem_ofPred_eq, hcanon] using hC + rw [ite_eq_left (hC'.trans hmass)] + rw [hdriver, ← ENNReal.ofReal_mul + (div_nonneg (greenDensity_nonneg hw κ T _) (initialGreen_pos κ _ hphysH).le)] + apply ENNReal.ofReal_le_ofReal + have hbound := hw.normalized_restart_stopped_green_lower hU T a ha hb κ w hwH u q hbase + calc + _ = (greenDensity κ U γ G T (F T ((a : ℂ) * w + (U T : ℂ))) * + (stoppedGreen κ (normalizedRestartDriver U T a) (Φ p) G' w q / initialGreen κ w)) / + initialGreen κ (F T ((a : ℂ) * w + (U T : ℂ))) := by ring + _ ≤ _ := div_le_div_of_nonneg_right hbound (initialGreen_pos κ _ hphysH).le + · rw [hdriver, stoppedGreen, ite_eq_right hfinite, zero_div, ENNReal.ofReal_zero, mul_zero] + exact bot_le + · rw [indicator_of_notMem hC] + exact bot_le + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RestrictedConformalSausage.lean b/lean/OAI/Probability/SLE/Gauge/RestrictedConformalSausage.lean index 672da497095f45eff150815d8ce8be1331151be8..0dc0e35e1c70ccabefed74d7e44c311f24b7ef00 100644 --- a/lean/OAI/Probability/SLE/Gauge/RestrictedConformalSausage.lean +++ b/lean/OAI/Probability/SLE/Gauge/RestrictedConformalSausage.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Gauge.ConformalSausageTransfer -import OAI.Probability.SLE.Mass.StoppedConformalPassageMass - -namespace OAI - -/-! Area transport of the part of a stopped sausage in a prescribed -region. Agreement of the physical trace with the conformal image is only -needed at the nearby interior witnesses. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -theorem restricted_segment_sausage_conformal_transfer - (κ : ℝ) (F F' : ℂ → ℂ) (γ η : ℝ≥0 → ℂ) (s t : ℝ≥0) - (D K R : Set ℂ) (hD : MeasurableSet D) (hDK : D ⊆ K) (hK : Convex ℝ K) - (hinj : InjOn F K) (hF : ∀ z ∈ K, HasDerivAt F (F' z) z) - {c v ell : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell) - (hlo : ∀ z ∈ D, c ≤ ‖F' z‖) - (C : ℝ≥0) (hup : ∀ z ∈ K, ‖F' z‖₊ ≤ C) - (hscale : (C : ℝ) * ell ≤ 1) - (hnear : ∀ z ∈ D, ∀ y : ℂ, dist z y < ell * v → y ∈ K) - (heq : ∀ u ∈ Icc s t, γ u ∈ K → η u = F (γ u)) - (himage : F '' D ⊆ R) : - ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) * - segmentSausageCharge κ γ s t (ell * v) D ≤ - segmentSausageCharge κ η s t v R := by - have hLip : LipschitzOnWith C F K := - hK.lipschitzOnWith_of_nnnorm_hasDerivWithin_le - (fun z hz => (hF z hz).hasDerivWithinAt) hup - apply normalized_conformal_area_lower - (hD.inter (source_sausage_isOpen γ s t (ell * v)).measurableSet) - (hinj.mono (inter_subset_left.trans hDK)) - (fun z hz => hF z (hDK hz.1)) hc hv hell (fun z hz => hlo z hz.1) - rintro _ ⟨z, ⟨hzD, u, hu, hd⟩, rfl⟩ - refine ⟨himage (mem_image_of_mem F hzD), u, hu, ?_⟩ - have huK : γ u ∈ K := hnear z hzD (γ u) hd - rw [heq u hu huK] - have hdist := hLip.dist_le_mul z (hDK hzD) (γ u) huK - by_cases hC : C = 0 - · have hdist0 : dist (F z) (F (γ u)) ≤ 0 := by - simpa only [hC, NNReal.coe_zero, zero_mul] using hdist - exact hdist0.trans_lt hv - · have hCp : 0 < (C : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hC) - calc - dist (F z) (F (γ u)) ≤ (C : ℝ) * dist z (γ u) := hdist - _ < (C : ℝ) * (ell * v) := mul_lt_mul_of_pos_left hd hCp - _ ≤ v := by nlinarith [mul_le_mul_of_nonneg_right hscale hv.le] - -lemma conformal_transfer_coefficient - (κ : ℝ) {c C a : ℝ} (hC : 0 < C) (ha : 0 < a) : - (c * a) ^ 2 * (1 / (C * a)) ^ codimension κ = - (c ^ 2 * (1 / C) ^ codimension κ) * a ^ dimension κ := by - have he : (1 / (C * a)) = (1 / C) * (1 / a) := by ring - rw [he, Real.mul_rpow (by positivity) (by positivity), - Real.div_rpow zero_le_one ha.le, Real.one_rpow] - calc - _ = (c ^ 2 * (1 / C) ^ codimension κ) * (a ^ 2 / a ^ codimension κ) := by ring - _ = _ := by - congr 1 - rw [← Real.rpow_two, ← Real.rpow_sub ha] - congr 1 - unfold codimension - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ConformalSausageTransfer +import OAI.Probability.SLE.Mass.StoppedConformalPassageMass + +namespace OAI + +/-! Area transport of the part of a stopped sausage in a prescribed +region. Agreement of the physical trace with the conformal image is only +needed at the nearby interior witnesses. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +theorem restricted_segment_sausage_conformal_transfer + (κ : ℝ) (F F' : ℂ → ℂ) (γ η : ℝ≥0 → ℂ) (s t : ℝ≥0) + (D K R : Set ℂ) (hD : MeasurableSet D) (hDK : D ⊆ K) (hK : Convex ℝ K) + (hinj : InjOn F K) (hF : ∀ z ∈ K, HasDerivAt F (F' z) z) + {c v ell : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell) + (hlo : ∀ z ∈ D, c ≤ ‖F' z‖) + (C : ℝ≥0) (hup : ∀ z ∈ K, ‖F' z‖₊ ≤ C) + (hscale : (C : ℝ) * ell ≤ 1) + (hnear : ∀ z ∈ D, ∀ y : ℂ, dist z y < ell * v → y ∈ K) + (heq : ∀ u ∈ Icc s t, γ u ∈ K → η u = F (γ u)) + (himage : F '' D ⊆ R) : + ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) * + segmentSausageCharge κ γ s t (ell * v) D ≤ + segmentSausageCharge κ η s t v R := by + have hLip : LipschitzOnWith C F K := + hK.lipschitzOnWith_of_nnnorm_hasDerivWithin_le + (fun z hz => (hF z hz).hasDerivWithinAt) hup + apply normalized_conformal_area_lower + (hD.inter (source_sausage_isOpen γ s t (ell * v)).measurableSet) + (hinj.mono (inter_subset_left.trans hDK)) + (fun z hz => hF z (hDK hz.1)) hc hv hell (fun z hz => hlo z hz.1) + rintro _ ⟨z, ⟨hzD, u, hu, hd⟩, rfl⟩ + refine ⟨himage (mem_image_of_mem F hzD), u, hu, ?_⟩ + have huK : γ u ∈ K := hnear z hzD (γ u) hd + rw [heq u hu huK] + have hdist := hLip.dist_le_mul z (hDK hzD) (γ u) huK + by_cases hC : C = 0 + · have hdist0 : dist (F z) (F (γ u)) ≤ 0 := by + simpa only [hC, NNReal.coe_zero, zero_mul] using hdist + exact hdist0.trans_lt hv + · have hCp : 0 < (C : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hC) + calc + dist (F z) (F (γ u)) ≤ (C : ℝ) * dist z (γ u) := hdist + _ < (C : ℝ) * (ell * v) := mul_lt_mul_of_pos_left hd hCp + _ ≤ v := by nlinarith [mul_le_mul_of_nonneg_right hscale hv.le] + +lemma conformal_transfer_coefficient + (κ : ℝ) {c C a : ℝ} (hC : 0 < C) (ha : 0 < a) : + (c * a) ^ 2 * (1 / (C * a)) ^ codimension κ = + (c ^ 2 * (1 / C) ^ codimension κ) * a ^ dimension κ := by + have he : (1 / (C * a)) = (1 / C) * (1 / a) := by ring + rw [he, Real.mul_rpow (by positivity) (by positivity), + Real.div_rpow zero_le_one ha.le, Real.one_rpow] + calc + _ = (c ^ 2 * (1 / C) ^ codimension κ) * (a ^ 2 / a ^ codimension κ) := by ring + _ = _ := by + congr 1 + rw [← Real.rpow_two, ← Real.rpow_sub ha] + congr 1 + unfold codimension + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RetainedPassageExtraction.lean b/lean/OAI/Probability/SLE/Gauge/RetainedPassageExtraction.lean index c13c2397d15899f738e34f405fa96e1edfeb1cfa..bd6432e8a6442a27728b2071038e1c5863f4616a 100644 --- a/lean/OAI/Probability/SLE/Gauge/RetainedPassageExtraction.lean +++ b/lean/OAI/Probability/SLE/Gauge/RetainedPassageExtraction.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Gauge.RibbonPassageMass -import OAI.Probability.SLE.Gauge.ConformalCompactSausage - -namespace OAI - -/-! Countable-to-finite retention in the ribbon passage argument. The loss -is bounded for every varying early-containment event by the same discarded -set. Sausage exceptions are subtracted unconditionally. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -theorem finite_retention_of_countable_cover - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (E : Set Ω) (D : ℕ → Set Ω) - (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (ε : ℝ≥0∞) (hε : 0 < ε) : - ∃ N : ℕ, P (E \ ⋃ i ∈ Finset.range N, D i) < ε := by - let S : ℕ → Set Ω := fun n => E \ ⋃ i ∈ Finset.range n, D i - have hSm : ∀ n, MeasurableSet (S n) := fun n => - hE.diff ((Finset.range n).measurableSet_biUnion (fun i _ => hD i)) - have hSa : Antitone S := by - intro n k hnk ω hω - refine ⟨hω.1, ?_⟩ - intro hn - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hn - exact hω.2 (mem_iUnion₂.mpr ⟨i, Finset.range_mono hnk hi, hDi⟩) - have hi : ⋂ n, S n = E \ ⋃ i, D i := by - ext ω - constructor - · intro h - refine ⟨(mem_iInter.mp h 0).1, ?_⟩ - intro hω - obtain ⟨i, hi⟩ := mem_iUnion.mp hω - exact (mem_iInter.mp h (i + 1)).2 - (mem_iUnion₂.mpr ⟨i, Finset.mem_range.mpr (Nat.lt_succ_self i), hi⟩) - · rintro ⟨hEω, hω⟩ - apply mem_iInter.mpr - intro n - refine ⟨hEω, ?_⟩ - intro hn - obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hn - exact hω (mem_iUnion.mpr ⟨i, hi⟩) - have ht : Tendsto (fun n => P (S n)) atTop (𝓝 0) := by - simpa only [Function.comp_def, hi, hcover] using - tendsto_measure_iInter_atTop (fun n => (hSm n).nullMeasurableSet) - hSa ⟨0, measure_ne_top P _⟩ - exact (ht.eventually (gt_mem_nhds hε)).exists - -/-- Discarding one small set preserves a fixed fraction of every member -of a varying family; no independence is involved. -/ -theorem varying_events_after_finite_retention - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (E : Set Ω) (D : ℕ → Set Ω) (G : ℕ → Set Ω) - (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) : - ∃ N : ℕ, ∀ n, ENNReal.ofReal (q / 2) ≤ - P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by - obtain ⟨N, hN⟩ := finite_retention_of_countable_cover P E D hE hD hcover - (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) - refine ⟨N, fun n => ?_⟩ - let R := ⋃ i ∈ Finset.range N, D i - have hsub : G n ∩ E ⊆ (G n ∩ E ∩ R) ∪ (E \ R) := by - intro ω hω - by_cases hR : ω ∈ R - · exact Or.inl ⟨hω, hR⟩ - · exact Or.inr ⟨hω.2, hR⟩ - have hbound : ENNReal.ofReal q ≤ P (G n ∩ E ∩ R) + P (E \ R) := - (hG n).trans ((measure_mono hsub).trans (measure_union_le _ _)) - by_contra h - have hhalf : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = - ENNReal.ofReal q := by - rw [← ENNReal.ofReal_add (by positivity) (by positivity)] - congr 1 - ring - exact (not_lt_of_ge hbound) (hhalf ▸ ENNReal.add_lt_add (lt_of_not_ge h) hN) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonPassageMass +import OAI.Probability.SLE.Gauge.ConformalCompactSausage + +namespace OAI + +/-! Countable-to-finite retention in the ribbon passage argument. The loss +is bounded for every varying early-containment event by the same discarded +set. Sausage exceptions are subtracted unconditionally. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +theorem finite_retention_of_countable_cover + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (E : Set Ω) (D : ℕ → Set Ω) + (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (ε : ℝ≥0∞) (hε : 0 < ε) : + ∃ N : ℕ, P (E \ ⋃ i ∈ Finset.range N, D i) < ε := by + let S : ℕ → Set Ω := fun n => E \ ⋃ i ∈ Finset.range n, D i + have hSm : ∀ n, MeasurableSet (S n) := fun n => + hE.diff ((Finset.range n).measurableSet_biUnion (fun i _ => hD i)) + have hSa : Antitone S := by + intro n k hnk ω hω + refine ⟨hω.1, ?_⟩ + intro hn + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hn + exact hω.2 (mem_iUnion₂.mpr ⟨i, Finset.range_mono hnk hi, hDi⟩) + have hi : ⋂ n, S n = E \ ⋃ i, D i := by + ext ω + constructor + · intro h + refine ⟨(mem_iInter.mp h 0).1, ?_⟩ + intro hω + obtain ⟨i, hi⟩ := mem_iUnion.mp hω + exact (mem_iInter.mp h (i + 1)).2 + (mem_iUnion₂.mpr ⟨i, Finset.mem_range.mpr (Nat.lt_succ_self i), hi⟩) + · rintro ⟨hEω, hω⟩ + apply mem_iInter.mpr + intro n + refine ⟨hEω, ?_⟩ + intro hn + obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hn + exact hω (mem_iUnion.mpr ⟨i, hi⟩) + have ht : Tendsto (fun n => P (S n)) atTop (𝓝 0) := by + simpa only [Function.comp_def, hi, hcover] using + tendsto_measure_iInter_atTop (fun n => (hSm n).nullMeasurableSet) + hSa ⟨0, measure_ne_top P _⟩ + exact (ht.eventually (gt_mem_nhds hε)).exists + +/-- Discarding one small set preserves a fixed fraction of every member +of a varying family; no independence is involved. -/ +theorem varying_events_after_finite_retention + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (E : Set Ω) (D : ℕ → Set Ω) (G : ℕ → Set Ω) + (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) : + ∃ N : ℕ, ∀ n, ENNReal.ofReal (q / 2) ≤ + P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by + obtain ⟨N, hN⟩ := finite_retention_of_countable_cover P E D hE hD hcover + (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) + refine ⟨N, fun n => ?_⟩ + let R := ⋃ i ∈ Finset.range N, D i + have hsub : G n ∩ E ⊆ (G n ∩ E ∩ R) ∪ (E \ R) := by + intro ω hω + by_cases hR : ω ∈ R + · exact Or.inl ⟨hω, hR⟩ + · exact Or.inr ⟨hω.2, hR⟩ + have hbound : ENNReal.ofReal q ≤ P (G n ∩ E ∩ R) + P (E \ R) := + (hG n).trans ((measure_mono hsub).trans (measure_union_le _ _)) + by_contra h + have hhalf : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = + ENNReal.ofReal q := by + rw [← ENNReal.ofReal_add (by positivity) (by positivity)] + congr 1 + ring + exact (not_lt_of_ge hbound) (hhalf ▸ ENNReal.add_lt_add (lt_of_not_ge h) hN) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RetainedRegionMargins.lean b/lean/OAI/Probability/SLE/Gauge/RetainedRegionMargins.lean index a10dd5cfafc44762dccb01cff41b7439a814696b..9b1a0ffe3830200c2c7e0fdf2a56b08d8f0cb249 100644 --- a/lean/OAI/Probability/SLE/Gauge/RetainedRegionMargins.lean +++ b/lean/OAI/Probability/SLE/Gauge/RetainedRegionMargins.lean @@ -1,191 +1,191 @@ -import OAI.Probability.SLE.Gauge.RetainedTubeSausage - -namespace OAI - -/-! Physical middle-strip margins are chosen separately for each retained -compact set. A finite probability retention then permits their finite union; -no inclusion of the entire limiting open strip in every moving strip is used. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RetainedTubeSausage - -/-- Each compact subset of the limiting middle strip has one open physical -neighborhood lying in every sufficiently late moving middle strip. -/ -theorem eventual_middle_neighborhood - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (K : Set ℂ) (hK : IsCompact K) - (hband : K ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3}) : - ∃ R : Set ℂ, IsOpen R ∧ K ⊆ R ∧ - ∀ᶠ n in atTop, R ⊆ U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} := by - let V := U ∩ p ⁻¹' Ioo (1 / 3 : ℝ) (2 / 3) - have hV : IsOpen V := hp.isOpen_inter_preimage hU isOpen_Ioo - obtain ⟨δ, hδ, hthick⟩ := hK.exists_cthickening_subset_open hV hband - let C := cthickening δ K - let R := thickening δ K - have hC : IsCompact C := hK.cthickening - have hCU : C ⊆ U := hthick.trans inter_subset_left - have hCR : R ⊆ C := thickening_subset_cthickening δ K - have hpC : IsCompact (p '' C) := hC.image_of_continuousOn (hp.mono hCU) - have hpband : p '' C ⊆ Ioo (1 / 3 : ℝ) (2 / 3) := by - rintro _ ⟨z, hz, rfl⟩ - exact (hthick hz).2 - obtain ⟨ε, hε, heps⟩ := hpC.exists_thickening_subset_open isOpen_Ioo hpband - have hu : TendstoUniformlyOn pₙ p atTop C := - (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hC).mp (hconv.mono hCU) - refine ⟨R, isOpen_thickening, self_subset_thickening hδ K, ?_⟩ - filter_upwards [hu _ (Metric.dist_mem_uniformity hε)] with n hn - intro z hz - have hzC := hCR hz - refine ⟨hCU hzC, heps ?_⟩ - exact mem_thickening_iff.mpr ⟨p z, mem_image_of_mem p hzC, - by simpa only [dist_comm] using hn z hzC⟩ - -/-- Finite retention permits compact-dependent physical neighborhoods. -/ -theorem countable_stopped_mass_ae_window_compact_regions - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) - (R : ℕ → Set ℂ) (hR : ∀ i, IsOpen (R i)) (himage : ∀ i, F '' K i ⊆ R i) - (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) - (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ i, ∀ᶠ n in atTop, R i ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by - classical - obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover - (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) - have hN : ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ - P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by - filter_upwards [hG] with n hn - let A := ⋃ i ∈ Finset.range N, D i - have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by - intro ω hω - by_cases hA : ω ∈ A - · exact Or.inl ⟨hω, hA⟩ - · exact Or.inr ⟨hω.2, hA⟩ - have hb := hn.trans ((measure_mono hsub).trans (measure_union_le _ _)) - by_contra h - have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = - ENNReal.ofReal q := by - rw [← ENNReal.ofReal_add (by positivity) (by positivity)] - congr 1 - ring - exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall) - let A := ⋃ i ∈ Finset.range N, D i - let Q := ⋃ i ∈ Finset.range N, K i - let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A - let R₀ : Set ℂ := ⋃ i ∈ Finset.range N, R i - have hR₀ : IsOpen R₀ := isOpen_iUnion fun i => isOpen_iUnion fun _ => hR i - have hregion₀ : ∀ᶠ n in atTop, R₀ ⊆ Rₙ n := by - have hf := (Finset.eventually_all (Finset.range N)).mpr (fun i _ => hregion i) - filter_upwards [hf] with n hn - exact iUnion₂_subset (fun i hi => hn i hi) - have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) - have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) - have hQR : F '' Q ⊆ R₀ := by - rintro w ⟨z, hz, rfl⟩ - obtain ⟨i, hiN, hi⟩ := mem_iUnion₂.mp hz - exact mem_iUnion₂.mpr ⟨i, hiN, himage i (mem_image_of_mem F hi)⟩ - have hQn : Q.Nonempty := by - obtain ⟨n, hn⟩ := hN.exists - have hHpos : 0 < P (H n) := - (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le hn - obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 - exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, - hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ - let I := (Finset.range N).image J - have hI : ∀ L ∈ I, L.1 < L.2 := by - intro L hL - obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL - exact hJ i - have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := - (Finset.eventually_all _).mpr (fun i _ => hwindow i) - have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → - ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by - filter_upwards [hwin] with n hn - filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω - intro hH - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 - exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, - (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), - hω i hi ⟨hH.1, hDi⟩⟩ - obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R₀ hR₀ hQR I hI v hv hlim - η s t Rₙ H (q / 2) (by positivity) hN hregion₀ hr - refine ⟨a, ha, ?_⟩ - filter_upwards [hmass] with n hn - have hqq : q / 2 / 2 = q / 4 := by ring - rw [hqq] at hn - exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) - -/-- Local coordinate convergence supplies the compact-dependent physical -margins needed by the retained stopped-mass theorem. -/ -theorem countable_stopped_mass_ae_window_middle_regions - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (himage : ∀ i, F '' K i ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3}) - (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) - (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by - have hneighborhood (i : ℕ) := eventual_middle_neighborhood pₙ p U hU hp hpconv - (F '' K i) ((hK i).image_of_continuousOn (hF.continuousOn.mono (hKH i))) (himage i) - choose R hR himageR hRlate using hneighborhood - apply countable_stopped_mass_ae_window_compact_regions P κ hκ hκ8 γ hSLE - Fₙ F hFn hinj hF hne hconv K hK hKH R hR himageR J hJ E D hE hD hcover hretain - v hv hlim η s t Rₙ ?_ G q hq hG hwindow - intro i - filter_upwards [hRlate i, hregion] with n hn htarget - exact hn.trans htarget - -end SLEExactGauge.RetainedTubeSausage - -end - -end OAI +import OAI.Probability.SLE.Gauge.RetainedTubeSausage + +namespace OAI + +/-! Physical middle-strip margins are chosen separately for each retained +compact set. A finite probability retention then permits their finite union; +no inclusion of the entire limiting open strip in every moving strip is used. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RetainedTubeSausage + +/-- Each compact subset of the limiting middle strip has one open physical +neighborhood lying in every sufficiently late moving middle strip. -/ +theorem eventual_middle_neighborhood + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (K : Set ℂ) (hK : IsCompact K) + (hband : K ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3}) : + ∃ R : Set ℂ, IsOpen R ∧ K ⊆ R ∧ + ∀ᶠ n in atTop, R ⊆ U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} := by + let V := U ∩ p ⁻¹' Ioo (1 / 3 : ℝ) (2 / 3) + have hV : IsOpen V := hp.isOpen_inter_preimage hU isOpen_Ioo + obtain ⟨δ, hδ, hthick⟩ := hK.exists_cthickening_subset_open hV hband + let C := cthickening δ K + let R := thickening δ K + have hC : IsCompact C := hK.cthickening + have hCU : C ⊆ U := hthick.trans inter_subset_left + have hCR : R ⊆ C := thickening_subset_cthickening δ K + have hpC : IsCompact (p '' C) := hC.image_of_continuousOn (hp.mono hCU) + have hpband : p '' C ⊆ Ioo (1 / 3 : ℝ) (2 / 3) := by + rintro _ ⟨z, hz, rfl⟩ + exact (hthick hz).2 + obtain ⟨ε, hε, heps⟩ := hpC.exists_thickening_subset_open isOpen_Ioo hpband + have hu : TendstoUniformlyOn pₙ p atTop C := + (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hC).mp (hconv.mono hCU) + refine ⟨R, isOpen_thickening, self_subset_thickening hδ K, ?_⟩ + filter_upwards [hu _ (Metric.dist_mem_uniformity hε)] with n hn + intro z hz + have hzC := hCR hz + refine ⟨hCU hzC, heps ?_⟩ + exact mem_thickening_iff.mpr ⟨p z, mem_image_of_mem p hzC, + by simpa only [dist_comm] using hn z hzC⟩ + +/-- Finite retention permits compact-dependent physical neighborhoods. -/ +theorem countable_stopped_mass_ae_window_compact_regions + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) + (R : ℕ → Set ℂ) (hR : ∀ i, IsOpen (R i)) (himage : ∀ i, F '' K i ⊆ R i) + (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) + (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ i, ∀ᶠ n in atTop, R i ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by + classical + obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover + (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) + have hN : ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ + P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by + filter_upwards [hG] with n hn + let A := ⋃ i ∈ Finset.range N, D i + have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by + intro ω hω + by_cases hA : ω ∈ A + · exact Or.inl ⟨hω, hA⟩ + · exact Or.inr ⟨hω.2, hA⟩ + have hb := hn.trans ((measure_mono hsub).trans (measure_union_le _ _)) + by_contra h + have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = + ENNReal.ofReal q := by + rw [← ENNReal.ofReal_add (by positivity) (by positivity)] + congr 1 + ring + exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall) + let A := ⋃ i ∈ Finset.range N, D i + let Q := ⋃ i ∈ Finset.range N, K i + let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A + let R₀ : Set ℂ := ⋃ i ∈ Finset.range N, R i + have hR₀ : IsOpen R₀ := isOpen_iUnion fun i => isOpen_iUnion fun _ => hR i + have hregion₀ : ∀ᶠ n in atTop, R₀ ⊆ Rₙ n := by + have hf := (Finset.eventually_all (Finset.range N)).mpr (fun i _ => hregion i) + filter_upwards [hf] with n hn + exact iUnion₂_subset (fun i hi => hn i hi) + have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) + have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) + have hQR : F '' Q ⊆ R₀ := by + rintro w ⟨z, hz, rfl⟩ + obtain ⟨i, hiN, hi⟩ := mem_iUnion₂.mp hz + exact mem_iUnion₂.mpr ⟨i, hiN, himage i (mem_image_of_mem F hi)⟩ + have hQn : Q.Nonempty := by + obtain ⟨n, hn⟩ := hN.exists + have hHpos : 0 < P (H n) := + (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le hn + obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 + exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, + hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ + let I := (Finset.range N).image J + have hI : ∀ L ∈ I, L.1 < L.2 := by + intro L hL + obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL + exact hJ i + have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := + (Finset.eventually_all _).mpr (fun i _ => hwindow i) + have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → + ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by + filter_upwards [hwin] with n hn + filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω + intro hH + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 + exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, + (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), + hω i hi ⟨hH.1, hDi⟩⟩ + obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R₀ hR₀ hQR I hI v hv hlim + η s t Rₙ H (q / 2) (by positivity) hN hregion₀ hr + refine ⟨a, ha, ?_⟩ + filter_upwards [hmass] with n hn + have hqq : q / 2 / 2 = q / 4 := by ring + rw [hqq] at hn + exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) + +/-- Local coordinate convergence supplies the compact-dependent physical +margins needed by the retained stopped-mass theorem. -/ +theorem countable_stopped_mass_ae_window_middle_regions + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (himage : ∀ i, F '' K i ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3}) + (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) + (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by + have hneighborhood (i : ℕ) := eventual_middle_neighborhood pₙ p U hU hp hpconv + (F '' K i) ((hK i).image_of_continuousOn (hF.continuousOn.mono (hKH i))) (himage i) + choose R hR himageR hRlate using hneighborhood + apply countable_stopped_mass_ae_window_compact_regions P κ hκ hκ8 γ hSLE + Fₙ F hFn hinj hF hne hconv K hK hKH R hR himageR J hJ E D hE hD hcover hretain + v hv hlim η s t Rₙ ?_ G q hq hG hwindow + intro i + filter_upwards [hRlate i, hregion] with n hn htarget + exact hn.trans htarget + +end SLEExactGauge.RetainedTubeSausage + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RetainedTubeSausage.lean b/lean/OAI/Probability/SLE/Gauge/RetainedTubeSausage.lean index d8ca4c3840e74178819f2bd20fc1df3a491b3125..3b90867b6ed067f6700c7f4774caaa0f96cc73ef 100644 --- a/lean/OAI/Probability/SLE/Gauge/RetainedTubeSausage.lean +++ b/lean/OAI/Probability/SLE/Gauge/RetainedTubeSausage.lean @@ -1,416 +1,416 @@ -import OAI.Probability.SLE.Gauge.PassageRetention -import OAI.Probability.SLE.Mass.StoppedConformalPassageMass - -namespace OAI - -/-! Actual open-tube passage certificates imply a positive stopped sausage -mass. The retained time window is derived from the front coordinate, rather -than supplied as a separate geometric hypothesis. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RetainedTubeSausage -open PassageRetention - -/-- Uniformly convergent physical coordinates preserve a strict front bound -on each compact retained parameter set. -/ -theorem eventually_front_lt_on_compact - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hF : ContinuousOn F upperHalfPlane) - (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) - (hKU : F '' K ⊆ U) (q : ℝ) (hbound : ∀ z ∈ K, p (F z) < q) : - ∀ᶠ n in atTop, ∀ z ∈ K, pₙ n (Fₙ n z) < q := by - by_cases hKn : K.Nonempty - · let V := U ∩ p ⁻¹' Iio q - have hV : IsOpen V := hp.isOpen_inter_preimage hU isOpen_Iio - have hFK : IsCompact (F '' K) := hK.image_of_continuousOn (hF.mono hKH) - have hFKV : F '' K ⊆ V := by - rintro w ⟨z, hz, rfl⟩ - exact ⟨hKU (mem_image_of_mem F hz), hbound z hz⟩ - obtain ⟨δ, hδ, hthick⟩ := hFK.exists_cthickening_subset_open hV hFKV - let L := cthickening (δ / 2) (F '' K) - have hL : IsCompact L := hFK.cthickening - have hLV : L ⊆ V := (cthickening_mono (by linarith) (F '' K)).trans hthick - have hLn : L.Nonempty := (hKn.image F).mono (self_subset_cthickening _) - have hpL : ContinuousOn p L := hp.mono (hLV.trans inter_subset_left) - obtain ⟨z₀, hz₀, hmax⟩ := hL.exists_isMaxOn hLn hpL - have hz₀q : p z₀ < q := (hLV hz₀).2 - have huF : TendstoUniformlyOn Fₙ F atTop K := - (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hFconv.mono hKH) - have huP : TendstoUniformlyOn pₙ p atTop L := - (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hL).mp - (hpconv.mono (hLV.trans inter_subset_left)) - filter_upwards [huF _ (Metric.dist_mem_uniformity (by positivity : 0 < δ / 2)), - huP _ (Metric.dist_mem_uniformity (sub_pos.mpr hz₀q))] with n hnF hnP - intro z hz - have hzL : Fₙ n z ∈ L := - mem_cthickening_of_dist_le _ (F z) (δ / 2) (F '' K) - (mem_image_of_mem F hz) (by simpa only [dist_comm] using (hnF z hz).le) - have hdiff := hnP (Fₙ n z) hzL - have hupper : p (Fₙ n z) ≤ p z₀ := hmax hzL - rw [Real.dist_eq] at hdiff - have hh := (abs_lt.mp hdiff).1 - linarith - · have hKe : K = ∅ := not_nonempty_iff_eq_empty.mp hKn - simp only [hKe, mem_empty_iff_false, IsEmpty.forall_iff, implies_true, - eventually_true] - -/-- The countable stopped-mass reduction also permits almost-sure window -and chart identities, as supplied by the actual boundary restart. -/ -theorem countable_stopped_mass_ae_window - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) - (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) - (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) - (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by - classical - obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG - let A := ⋃ i ∈ Finset.range N, D i - let Q := ⋃ i ∈ Finset.range N, K i - let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A - have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) - have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) - have hQR : F '' Q ⊆ R := by - rintro w ⟨z, hz, rfl⟩ - obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz - exact himage i (mem_image_of_mem F hi) - have hQn : Q.Nonempty := by - have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0) - obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 - exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, - hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ - let I := (Finset.range N).image J - have hI : ∀ L ∈ I, L.1 < L.2 := by - intro L hL - obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL - exact hJ i - have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := - (Finset.eventually_all _).mpr (fun i _ => hwindow i) - have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → - ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by - filter_upwards [hwin] with n hn - filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω - intro hH - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 - exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, - (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), - hω i hi ⟨hH.1, hDi⟩⟩ - obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim - η s t Rₙ H (q / 2) (by positivity) (Eventually.of_forall hN) hregion hr - refine ⟨a, ha, ?_⟩ - filter_upwards [hmass] with n hn - have hqq : q / 2 / 2 = q / 4 := by ring - rw [hqq] at hn - exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) - -/-- The countable stopped-mass reduction also permits almost-sure window -and chart identities, as supplied by the actual boundary restart. -/ -theorem countable_stopped_mass_ae_window_eventually - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) - (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) - (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) - (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by - classical - obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover - (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) - have hN : ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ - P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by - filter_upwards [hG] with n hn - let A := ⋃ i ∈ Finset.range N, D i - have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by - intro ω hω - by_cases hA : ω ∈ A - · exact Or.inl ⟨hω, hA⟩ - · exact Or.inr ⟨hω.2, hA⟩ - have hb := hn.trans ((measure_mono hsub).trans (measure_union_le _ _)) - by_contra h - have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = - ENNReal.ofReal q := by - rw [← ENNReal.ofReal_add (by positivity) (by positivity)] - congr 1 - ring - exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall) - let A := ⋃ i ∈ Finset.range N, D i - let Q := ⋃ i ∈ Finset.range N, K i - let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A - have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) - have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) - have hQR : F '' Q ⊆ R := by - rintro w ⟨z, hz, rfl⟩ - obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz - exact himage i (mem_image_of_mem F hi) - have hQn : Q.Nonempty := by - obtain ⟨n, hn⟩ := hN.exists - have hHpos : 0 < P (H n) := - (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le hn - obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 - exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, - hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ - let I := (Finset.range N).image J - have hI : ∀ L ∈ I, L.1 < L.2 := by - intro L hL - obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL - exact hJ i - have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := - (Finset.eventually_all _).mpr (fun i _ => hwindow i) - have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → - ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by - filter_upwards [hwin] with n hn - filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω - intro hH - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 - exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, - (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), - hω i hi ⟨hH.1, hDi⟩⟩ - obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim - η s t Rₙ H (q / 2) (by positivity) hN hregion hr - refine ⟨a, ha, ?_⟩ - filter_upwards [hmass] with n hn - have hqq : q / 2 / 2 = q / 4 := by ring - rw [hqq] at hn - exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) - -/-- Actual tube passages produce retained sausage mass before the varying -front stop. Compact intervals and stop windows are both derived here. -/ -theorem retained_tube_stopped_sausage_mass - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) - (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P) - (σ : Ω → ℝ≥0) (hσ : Measurable σ) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)}) - (E : Set Ω) (hE : MeasurableSet E) - (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W) - (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3) - (R : Set ℂ) (hR : IsOpen R) - (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (τ : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane → - η n ω t = Fₙ n (f ω t)) - (hτ : ∀ n, ∀ᵐ ω ∂P, ω ∈ G n ∩ E → - σ ω ≤ τ n ω ∧ 2 / 3 ≤ pₙ n (η n ω (τ n ω))) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (σ ω) (τ n ω) (v n) (Rₙ n)} := by - let progress := fun z => p (F z) - have hprogress : ContinuousOn progress O := - hp.comp (hF.continuousOn.mono hOH) hFOU - obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ := - stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart - let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N - (beyondFrontRegion O progress) (enumerateRetainedPassage i) - let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1 - let J : ℕ → ℝ≥0 × ℝ≥0 := fun i => - (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1) - have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1 - have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1 - have hQR : ∀ i, F '' Q i ⊆ R := by - rintro i _ ⟨z, hz, rfl⟩ - have hh := (hM _).2 hz - exact hmiddle z hh.1 hh.2.1 hh.2.2 - have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _ - have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _ - have hnull : P (E \ ⋃ i, D i) = 0 := by - rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty] - have hretain : ∀ i ω, ω ∈ D i → - (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by - intro i ω hω - exact hω.1.2.2.2.1.image_subset - have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - σ ω ≤ (J i).1 ∧ (J i).2 ≤ τ n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by - intro i - let j := enumerateRetainedPassage i - have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1 - have hLU : F '' L j.1.2.2.2 ⊆ U := by - rintro _ ⟨z, hz, rfl⟩ - exact hFOU ((hL _).2 hz).1 - have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv - (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2) - filter_upwards [hlt] with n hn - filter_upwards [hη n, hτ n] with ω hηω hτω - intro hω - have hDi : ω ∈ retainedPassageCertificate f σ K M L N - (beyondFrontRegion O progress) j := hω.2 - obtain ⟨⟨hσs, _, _, hmid, hlow⟩, _⟩ := hDi - have htau := hτω hω.1 - have hbefore : intervalEnd j.1 ≤ τ n ω := by - by_contra hbad - have hmem : f ω (τ n ω) ∈ L j.1.2.2.2 := - hlow.2 ⟨htau.1, (lt_of_not_ge hbad).le⟩ - have hh := hn _ hmem - rw [← hηω (τ n ω) htau.1 (hLH hmem)] at hh - exact (not_le_of_gt hh) htau.2 - refine ⟨hσs.le, hbefore, ?_⟩ - intro t ht - exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1) - exact countable_stopped_mass_ae_window P κ hκ hκ8 (fun ω => f ω) hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain - v hv hlim η (fun _ => σ) τ Rₙ hregion G q hq hG hwindow -/-- Actual tube passages produce retained sausage mass before the varying -front stop. Compact intervals and stop windows are both derived here. -/ -theorem retained_tube_stopped_sausage_mass_eventually - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) - (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P) - (σ : Ω → ℝ≥0) (hσ : Measurable σ) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)}) - (E : Set Ω) (hE : MeasurableSet E) - (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W) - (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3) - (R : Set ℂ) (hR : IsOpen R) - (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (τ : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) - (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) - (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane → - η n ω t = Fₙ n (f ω t)) - (hτ : ∀ n, ∀ᵐ ω ∂P, ω ∈ G n ∩ E → - σ ω ≤ τ n ω ∧ 2 / 3 ≤ pₙ n (η n ω (τ n ω))) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - segmentSausageCharge κ (η n ω) (σ ω) (τ n ω) (v n) (Rₙ n)} := by - let progress := fun z => p (F z) - have hprogress : ContinuousOn progress O := - hp.comp (hF.continuousOn.mono hOH) hFOU - obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ := - stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart - let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N - (beyondFrontRegion O progress) (enumerateRetainedPassage i) - let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1 - let J : ℕ → ℝ≥0 × ℝ≥0 := fun i => - (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1) - have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1 - have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1 - have hQR : ∀ i, F '' Q i ⊆ R := by - rintro i _ ⟨z, hz, rfl⟩ - have hh := (hM _).2 hz - exact hmiddle z hh.1 hh.2.1 hh.2.2 - have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _ - have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _ - have hnull : P (E \ ⋃ i, D i) = 0 := by - rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty] - have hretain : ∀ i ω, ω ∈ D i → - (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by - intro i ω hω - exact hω.1.2.2.2.1.image_subset - have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → - σ ω ≤ (J i).1 ∧ (J i).2 ≤ τ n ω ∧ - EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by - intro i - let j := enumerateRetainedPassage i - have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1 - have hLU : F '' L j.1.2.2.2 ⊆ U := by - rintro _ ⟨z, hz, rfl⟩ - exact hFOU ((hL _).2 hz).1 - have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv - (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2) - filter_upwards [hlt] with n hn - filter_upwards [hη n, hτ n] with ω hηω hτω - intro hω - have hDi : ω ∈ retainedPassageCertificate f σ K M L N - (beyondFrontRegion O progress) j := hω.2 - obtain ⟨⟨hσs, _, _, hmid, hlow⟩, _⟩ := hDi - have htau := hτω hω.1 - have hbefore : intervalEnd j.1 ≤ τ n ω := by - by_contra hbad - have hmem : f ω (τ n ω) ∈ L j.1.2.2.2 := - hlow.2 ⟨htau.1, (lt_of_not_ge hbad).le⟩ - have hh := hn _ hmem - rw [← hηω (τ n ω) htau.1 (hLH hmem)] at hh - exact (not_le_of_gt hh) htau.2 - refine ⟨hσs.le, hbefore, ?_⟩ - intro t ht - exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1) - exact countable_stopped_mass_ae_window_eventually P κ hκ hκ8 (fun ω => f ω) hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain - v hv hlim η (fun _ => σ) τ Rₙ hregion G q hq hG hwindow - -end SLEExactGauge.RetainedTubeSausage - -end - -end OAI +import OAI.Probability.SLE.Gauge.PassageRetention +import OAI.Probability.SLE.Mass.StoppedConformalPassageMass + +namespace OAI + +/-! Actual open-tube passage certificates imply a positive stopped sausage +mass. The retained time window is derived from the front coordinate, rather +than supplied as a separate geometric hypothesis. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RetainedTubeSausage +open PassageRetention + +/-- Uniformly convergent physical coordinates preserve a strict front bound +on each compact retained parameter set. -/ +theorem eventually_front_lt_on_compact + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hF : ContinuousOn F upperHalfPlane) + (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) + (hKU : F '' K ⊆ U) (q : ℝ) (hbound : ∀ z ∈ K, p (F z) < q) : + ∀ᶠ n in atTop, ∀ z ∈ K, pₙ n (Fₙ n z) < q := by + by_cases hKn : K.Nonempty + · let V := U ∩ p ⁻¹' Iio q + have hV : IsOpen V := hp.isOpen_inter_preimage hU isOpen_Iio + have hFK : IsCompact (F '' K) := hK.image_of_continuousOn (hF.mono hKH) + have hFKV : F '' K ⊆ V := by + rintro w ⟨z, hz, rfl⟩ + exact ⟨hKU (mem_image_of_mem F hz), hbound z hz⟩ + obtain ⟨δ, hδ, hthick⟩ := hFK.exists_cthickening_subset_open hV hFKV + let L := cthickening (δ / 2) (F '' K) + have hL : IsCompact L := hFK.cthickening + have hLV : L ⊆ V := (cthickening_mono (by linarith) (F '' K)).trans hthick + have hLn : L.Nonempty := (hKn.image F).mono (self_subset_cthickening _) + have hpL : ContinuousOn p L := hp.mono (hLV.trans inter_subset_left) + obtain ⟨z₀, hz₀, hmax⟩ := hL.exists_isMaxOn hLn hpL + have hz₀q : p z₀ < q := (hLV hz₀).2 + have huF : TendstoUniformlyOn Fₙ F atTop K := + (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hFconv.mono hKH) + have huP : TendstoUniformlyOn pₙ p atTop L := + (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hL).mp + (hpconv.mono (hLV.trans inter_subset_left)) + filter_upwards [huF _ (Metric.dist_mem_uniformity (by positivity : 0 < δ / 2)), + huP _ (Metric.dist_mem_uniformity (sub_pos.mpr hz₀q))] with n hnF hnP + intro z hz + have hzL : Fₙ n z ∈ L := + mem_cthickening_of_dist_le _ (F z) (δ / 2) (F '' K) + (mem_image_of_mem F hz) (by simpa only [dist_comm] using (hnF z hz).le) + have hdiff := hnP (Fₙ n z) hzL + have hupper : p (Fₙ n z) ≤ p z₀ := hmax hzL + rw [Real.dist_eq] at hdiff + have hh := (abs_lt.mp hdiff).1 + linarith + · have hKe : K = ∅ := not_nonempty_iff_eq_empty.mp hKn + simp only [hKe, mem_empty_iff_false, IsEmpty.forall_iff, implies_true, + eventually_true] + +/-- The countable stopped-mass reduction also permits almost-sure window +and chart identities, as supplied by the actual boundary restart. -/ +theorem countable_stopped_mass_ae_window + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) + (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) + (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) + (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by + classical + obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG + let A := ⋃ i ∈ Finset.range N, D i + let Q := ⋃ i ∈ Finset.range N, K i + let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A + have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) + have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) + have hQR : F '' Q ⊆ R := by + rintro w ⟨z, hz, rfl⟩ + obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz + exact himage i (mem_image_of_mem F hi) + have hQn : Q.Nonempty := by + have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0) + obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 + exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, + hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ + let I := (Finset.range N).image J + have hI : ∀ L ∈ I, L.1 < L.2 := by + intro L hL + obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL + exact hJ i + have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := + (Finset.eventually_all _).mpr (fun i _ => hwindow i) + have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → + ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by + filter_upwards [hwin] with n hn + filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω + intro hH + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 + exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, + (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), + hω i hi ⟨hH.1, hDi⟩⟩ + obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim + η s t Rₙ H (q / 2) (by positivity) (Eventually.of_forall hN) hregion hr + refine ⟨a, ha, ?_⟩ + filter_upwards [hmass] with n hn + have hqq : q / 2 / 2 = q / 4 := by ring + rw [hqq] at hn + exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) + +/-- The countable stopped-mass reduction also permits almost-sure window +and chart identities, as supplied by the actual boundary restart. -/ +theorem countable_stopped_mass_ae_window_eventually + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane) + (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) + (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) + (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by + classical + obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover + (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity)) + have hN : ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ + P (G n ∩ E ∩ ⋃ i ∈ Finset.range N, D i) := by + filter_upwards [hG] with n hn + let A := ⋃ i ∈ Finset.range N, D i + have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by + intro ω hω + by_cases hA : ω ∈ A + · exact Or.inl ⟨hω, hA⟩ + · exact Or.inr ⟨hω.2, hA⟩ + have hb := hn.trans ((measure_mono hsub).trans (measure_union_le _ _)) + by_contra h + have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = + ENNReal.ofReal q := by + rw [← ENNReal.ofReal_add (by positivity) (by positivity)] + congr 1 + ring + exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall) + let A := ⋃ i ∈ Finset.range N, D i + let Q := ⋃ i ∈ Finset.range N, K i + let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A + have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) + have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) + have hQR : F '' Q ⊆ R := by + rintro w ⟨z, hz, rfl⟩ + obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz + exact himage i (mem_image_of_mem F hi) + have hQn : Q.Nonempty := by + obtain ⟨n, hn⟩ := hN.exists + have hHpos : 0 < P (H n) := + (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le hn + obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 + exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, + hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩ + let I := (Finset.range N).image J + have hI : ∀ L ∈ I, L.1 < L.2 := by + intro L hL + obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL + exact hJ i + have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) := + (Finset.eventually_all _).mpr (fun i _ => hwindow i) + have hr : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n → + ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by + filter_upwards [hwin] with n hn + filter_upwards [(Finset.eventually_all _).mpr hn] with ω hω + intro hH + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hH.2 + exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, + (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl), + hω i hi ⟨hH.1, hDi⟩⟩ + obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass_ae P κ hκ hκ8 γ hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim + η s t Rₙ H (q / 2) (by positivity) hN hregion hr + refine ⟨a, ha, ?_⟩ + filter_upwards [hmass] with n hn + have hqq : q / 2 / 2 = q / 4 := by ring + rw [hqq] at hn + exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) + +/-- Actual tube passages produce retained sausage mass before the varying +front stop. Compact intervals and stop windows are both derived here. -/ +theorem retained_tube_stopped_sausage_mass + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) + (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P) + (σ : Ω → ℝ≥0) (hσ : Measurable σ) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)}) + (E : Set Ω) (hE : MeasurableSet E) + (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W) + (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3) + (R : Set ℂ) (hR : IsOpen R) + (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (τ : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane → + η n ω t = Fₙ n (f ω t)) + (hτ : ∀ n, ∀ᵐ ω ∂P, ω ∈ G n ∩ E → + σ ω ≤ τ n ω ∧ 2 / 3 ≤ pₙ n (η n ω (τ n ω))) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (σ ω) (τ n ω) (v n) (Rₙ n)} := by + let progress := fun z => p (F z) + have hprogress : ContinuousOn progress O := + hp.comp (hF.continuousOn.mono hOH) hFOU + obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ := + stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart + let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N + (beyondFrontRegion O progress) (enumerateRetainedPassage i) + let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1 + let J : ℕ → ℝ≥0 × ℝ≥0 := fun i => + (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1) + have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1 + have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1 + have hQR : ∀ i, F '' Q i ⊆ R := by + rintro i _ ⟨z, hz, rfl⟩ + have hh := (hM _).2 hz + exact hmiddle z hh.1 hh.2.1 hh.2.2 + have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _ + have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _ + have hnull : P (E \ ⋃ i, D i) = 0 := by + rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty] + have hretain : ∀ i ω, ω ∈ D i → + (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by + intro i ω hω + exact hω.1.2.2.2.1.image_subset + have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + σ ω ≤ (J i).1 ∧ (J i).2 ≤ τ n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by + intro i + let j := enumerateRetainedPassage i + have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1 + have hLU : F '' L j.1.2.2.2 ⊆ U := by + rintro _ ⟨z, hz, rfl⟩ + exact hFOU ((hL _).2 hz).1 + have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv + (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2) + filter_upwards [hlt] with n hn + filter_upwards [hη n, hτ n] with ω hηω hτω + intro hω + have hDi : ω ∈ retainedPassageCertificate f σ K M L N + (beyondFrontRegion O progress) j := hω.2 + obtain ⟨⟨hσs, _, _, hmid, hlow⟩, _⟩ := hDi + have htau := hτω hω.1 + have hbefore : intervalEnd j.1 ≤ τ n ω := by + by_contra hbad + have hmem : f ω (τ n ω) ∈ L j.1.2.2.2 := + hlow.2 ⟨htau.1, (lt_of_not_ge hbad).le⟩ + have hh := hn _ hmem + rw [← hηω (τ n ω) htau.1 (hLH hmem)] at hh + exact (not_le_of_gt hh) htau.2 + refine ⟨hσs.le, hbefore, ?_⟩ + intro t ht + exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1) + exact countable_stopped_mass_ae_window P κ hκ hκ8 (fun ω => f ω) hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain + v hv hlim η (fun _ => σ) τ Rₙ hregion G q hq hG hwindow +/-- Actual tube passages produce retained sausage mass before the varying +front stop. Compact intervals and stop windows are both derived here. -/ +theorem retained_tube_stopped_sausage_mass_eventually + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) + (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P) + (σ : Ω → ℝ≥0) (hσ : Measurable σ) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)}) + (E : Set Ω) (hE : MeasurableSet E) + (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W) + (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3) + (R : Set ℂ) (hR : IsOpen R) + (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (τ : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ) + (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E)) + (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane → + η n ω t = Fₙ n (f ω t)) + (hτ : ∀ n, ∀ᵐ ω ∂P, ω ∈ G n ∩ E → + σ ω ≤ τ n ω ∧ 2 / 3 ≤ pₙ n (η n ω (τ n ω))) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + segmentSausageCharge κ (η n ω) (σ ω) (τ n ω) (v n) (Rₙ n)} := by + let progress := fun z => p (F z) + have hprogress : ContinuousOn progress O := + hp.comp (hF.continuousOn.mono hOH) hFOU + obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ := + stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart + let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N + (beyondFrontRegion O progress) (enumerateRetainedPassage i) + let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1 + let J : ℕ → ℝ≥0 × ℝ≥0 := fun i => + (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1) + have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1 + have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1 + have hQR : ∀ i, F '' Q i ⊆ R := by + rintro i _ ⟨z, hz, rfl⟩ + have hh := (hM _).2 hz + exact hmiddle z hh.1 hh.2.1 hh.2.2 + have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _ + have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _ + have hnull : P (E \ ⋃ i, D i) = 0 := by + rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty] + have hretain : ∀ i ω, ω ∈ D i → + (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by + intro i ω hω + exact hω.1.2.2.2.1.image_subset + have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → + σ ω ≤ (J i).1 ∧ (J i).2 ≤ τ n ω ∧ + EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by + intro i + let j := enumerateRetainedPassage i + have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1 + have hLU : F '' L j.1.2.2.2 ⊆ U := by + rintro _ ⟨z, hz, rfl⟩ + exact hFOU ((hL _).2 hz).1 + have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv + (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2) + filter_upwards [hlt] with n hn + filter_upwards [hη n, hτ n] with ω hηω hτω + intro hω + have hDi : ω ∈ retainedPassageCertificate f σ K M L N + (beyondFrontRegion O progress) j := hω.2 + obtain ⟨⟨hσs, _, _, hmid, hlow⟩, _⟩ := hDi + have htau := hτω hω.1 + have hbefore : intervalEnd j.1 ≤ τ n ω := by + by_contra hbad + have hmem : f ω (τ n ω) ∈ L j.1.2.2.2 := + hlow.2 ⟨htau.1, (lt_of_not_ge hbad).le⟩ + have hh := hn _ hmem + rw [← hηω (τ n ω) htau.1 (hLH hmem)] at hh + exact (not_le_of_gt hh) htau.2 + refine ⟨hσs.le, hbefore, ?_⟩ + intro t ht + exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1) + exact countable_stopped_mass_ae_window_eventually P κ hκ hκ8 (fun ω => f ω) hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain + v hv hlim η (fun _ => σ) τ Rₙ hregion G q hq hG hwindow + +end SLEExactGauge.RetainedTubeSausage + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonAccessGuide.lean b/lean/OAI/Probability/SLE/Gauge/RibbonAccessGuide.lean index df95f853216a535576fccdc669d518cd7d443a1a..fc2fcbd0932aab440502644b826a030adfc44ac1 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonAccessGuide.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonAccessGuide.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Gauge.RibbonFrontFreshness - -namespace OAI - -/-! The physical ribbon guide supplies surviving-domain and compactness -geometry under a two-sided waypoint angle hypothesis. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -theorem ribbon_access_guide_geometry - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) - (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) (hxp : ContinuousOn xp (Icc 0 T)) - (hxp0 : (xp 0).re = 0) (hcoords : MapsTo xp (Icc 0 T) closedRibbon) - (hγ : EqOn γ (P ∘ xp) (Icc 0 T)) - (k x : ℝ) (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) - (htip : xp T = (k : ℂ) + (x : ℂ) * Complex.I) - (hfront : ∀ t : ℝ≥0, t < T → (xp t).re < k) - (C : ℝ≥0) (K : ℝ) (hK : 0 < K) - (hL : LipschitzOnWith C P (guidePatch k)) - (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, - dist u v ≤ K * dist (P u) (P v)) : - let α := fun t => P (guideCoordinate k x t) - ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ α 0 = γ T ∧ - MapsTo α (Ioc 0 2) (survivingDomain γ T) ∧ - (∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) ∧ - LipschitzOnWith (2 * C) α (Icc 0 2) ∧ - (∀ u ∈ Icc (0 : ℝ) 2, ∀ v ∈ Icc (0 : ℝ) 2, - (1 / K) * |u - v| ≤ dist (α u) (α v)) ∧ - (∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ t ∈ Icc δ 2, ball (α t) r ⊆ survivingDomain γ T) := by - dsimp only - have htipre : (xp T).re = k := by simp [htip] - have htipim : (xp T).im = x := by simp [htip] - have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by - rintro z ⟨t, ht, rfl⟩ - refine ⟨xp t, ⟨hcoords ht, ?_⟩, (hγ ht).symm⟩ - by_cases he : t = T - · simpa only [he, htipre] using le_rfl (a := k) - · exact (hfront t (lt_of_le_of_ne ht.2 he)).le - have hunbounded := futureRibbon_unbounded_of_centerline_escape P k hescape - have hguidePatch : MapsTo (guideCoordinate k x) (Icc (0 : ℝ) 2) (guidePatch k) := - fun _ ht => guideCoordinate_mem_patch hx ht - have hguideClosed := hguidePatch.mono_right (guidePatch_subset_closedRibbon hk) - have hαcont : ContinuousOn (fun t => P (guideCoordinate k x t)) (Icc 0 2) := - hP.comp (guideCoordinate_continuous k x).continuousOn hguideClosed - have hαinj : InjOn (fun t => P (guideCoordinate k x t)) (Icc 0 2) := by - intro u hu v hv he - have hc := hi (hguideClosed hu) (hguideClosed hv) he - have hre := congrArg Complex.re hc - simp only [guideCoordinate_re] at hre - linarith - have hα0 : P (guideCoordinate k x 0) = γ T := by - rw [guideCoordinate_zero, ← htip] - exact (hγ ⟨bot_le, le_rfl⟩).symm - have hαD : MapsTo (fun t => P (guideCoordinate k x t)) (Ioc 0 2) - (survivingDomain γ T) := by - intro t ht - apply futureRibbon_survives P hP hi hk hH γ T hpast hunbounded - apply mem_image_of_mem - change k < (guideCoordinate k x t).re ∧ |(guideCoordinate k x t).im| < 1 - exact ⟨by rw [guideCoordinate_re]; linarith [ht.1], - (guideCoordinate_im_bound ht.1.le).trans_lt (hx.trans_lt (by norm_num))⟩ - have hfresh := ribbon_front_tip_survives_prior P hP hi hH hescape γ xp T hxp hxp0 hcoords hγ - (fun t ht => by rw [htipre]; exact hfront t ht) - (by rw [htipim]; exact hx.trans_lt (by norm_num)) - exact ⟨hαcont, hαinj, hα0, hαD, hfresh, physical_guide_lipschitz P C hx hL, - physical_guide_lower_chord P hx hK hlower, - fun δ hδ hδ2 => physical_guide_clearance_away hpublished P hP hi hk hx hK - hH γ T hpast hunbounded hlower hδ hδ2⟩ - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonFrontFreshness + +namespace OAI + +/-! The physical ribbon guide supplies surviving-domain and compactness +geometry under a two-sided waypoint angle hypothesis. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +theorem ribbon_access_guide_geometry + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) + (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) (hxp : ContinuousOn xp (Icc 0 T)) + (hxp0 : (xp 0).re = 0) (hcoords : MapsTo xp (Icc 0 T) closedRibbon) + (hγ : EqOn γ (P ∘ xp) (Icc 0 T)) + (k x : ℝ) (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) + (htip : xp T = (k : ℂ) + (x : ℂ) * Complex.I) + (hfront : ∀ t : ℝ≥0, t < T → (xp t).re < k) + (C : ℝ≥0) (K : ℝ) (hK : 0 < K) + (hL : LipschitzOnWith C P (guidePatch k)) + (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, + dist u v ≤ K * dist (P u) (P v)) : + let α := fun t => P (guideCoordinate k x t) + ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ α 0 = γ T ∧ + MapsTo α (Ioc 0 2) (survivingDomain γ T) ∧ + (∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) ∧ + LipschitzOnWith (2 * C) α (Icc 0 2) ∧ + (∀ u ∈ Icc (0 : ℝ) 2, ∀ v ∈ Icc (0 : ℝ) 2, + (1 / K) * |u - v| ≤ dist (α u) (α v)) ∧ + (∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ t ∈ Icc δ 2, ball (α t) r ⊆ survivingDomain γ T) := by + dsimp only + have htipre : (xp T).re = k := by simp [htip] + have htipim : (xp T).im = x := by simp [htip] + have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by + rintro z ⟨t, ht, rfl⟩ + refine ⟨xp t, ⟨hcoords ht, ?_⟩, (hγ ht).symm⟩ + by_cases he : t = T + · simpa only [he, htipre] using le_rfl (a := k) + · exact (hfront t (lt_of_le_of_ne ht.2 he)).le + have hunbounded := futureRibbon_unbounded_of_centerline_escape P k hescape + have hguidePatch : MapsTo (guideCoordinate k x) (Icc (0 : ℝ) 2) (guidePatch k) := + fun _ ht => guideCoordinate_mem_patch hx ht + have hguideClosed := hguidePatch.mono_right (guidePatch_subset_closedRibbon hk) + have hαcont : ContinuousOn (fun t => P (guideCoordinate k x t)) (Icc 0 2) := + hP.comp (guideCoordinate_continuous k x).continuousOn hguideClosed + have hαinj : InjOn (fun t => P (guideCoordinate k x t)) (Icc 0 2) := by + intro u hu v hv he + have hc := hi (hguideClosed hu) (hguideClosed hv) he + have hre := congrArg Complex.re hc + simp only [guideCoordinate_re] at hre + linarith + have hα0 : P (guideCoordinate k x 0) = γ T := by + rw [guideCoordinate_zero, ← htip] + exact (hγ ⟨bot_le, le_rfl⟩).symm + have hαD : MapsTo (fun t => P (guideCoordinate k x t)) (Ioc 0 2) + (survivingDomain γ T) := by + intro t ht + apply futureRibbon_survives P hP hi hk hH γ T hpast hunbounded + apply mem_image_of_mem + change k < (guideCoordinate k x t).re ∧ |(guideCoordinate k x t).im| < 1 + exact ⟨by rw [guideCoordinate_re]; linarith [ht.1], + (guideCoordinate_im_bound ht.1.le).trans_lt (hx.trans_lt (by norm_num))⟩ + have hfresh := ribbon_front_tip_survives_prior P hP hi hH hescape γ xp T hxp hxp0 hcoords hγ + (fun t ht => by rw [htipre]; exact hfront t ht) + (by rw [htipim]; exact hx.trans_lt (by norm_num)) + exact ⟨hαcont, hαinj, hα0, hαD, hfresh, physical_guide_lipschitz P C hx hL, + physical_guide_lower_chord P hx hK hlower, + fun δ hδ hδ2 => physical_guide_clearance_away hpublished P hP hi hk hx hK + hH γ T hpast hunbounded hlower hδ hδ2⟩ + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonConcatenation.lean b/lean/OAI/Probability/SLE/Gauge/RibbonConcatenation.lean index 5c624c6b8484b8ede4cbfb2bfd806a4cc74e61f3..608bc1b6d36382e0b724e1bdf6d7ba521512d085 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonConcatenation.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonConcatenation.lean @@ -1,101 +1,101 @@ -import OAI.Probability.SLE.Gauge.DenseCoverCore - -namespace OAI - -/-! Adapted concatenation and the mass/probability scaling of a serpentine -ribbon with a quadratic number of narrow steps. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -/-- Success probabilities multiply as lower bounds under adapted restarts. -The successive steps need not be independent. -/ -theorem adapted_successes_lower_bound - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) - (F : Filtration ℕ mΩ) (success : ℕ → Set Ω) - (hmeas : ∀ n, MeasurableSet[F (n + 1)] (success n)) - (q : ℝ≥0∞) - (hstep : ∀ n (B : Set Ω), MeasurableSet[F n] B → - q * P B ≤ P (B ∩ success n)) : - ∀ N : ℕ, q ^ N * P univ ≤ P (⋂ n ∈ Finset.range N, success n) := by - intro N - induction N with - | zero => simp - | succ N ih => - let B : Set Ω := ⋂ n ∈ Finset.range N, success n - have hB : MeasurableSet[F N] B := by - apply MeasurableSet.iInter - intro n - apply MeasurableSet.iInter - intro hn - exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n) - have he : (⋂ n ∈ Finset.range (N + 1), success n) = B ∩ success N := by - ext ω - simp only [mem_iInter, Finset.mem_range, mem_inter_iff, B] - constructor - · intro hω - exact ⟨fun n hn => hω n (Nat.lt_succ_of_lt hn), hω N (Nat.lt_succ_self N)⟩ - · rintro ⟨hω, hN⟩ n hn - rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl - · exact hω n hn - · exact hN - rw [he, pow_succ] - calc - q ^ N * q * P univ = q * (q ^ N * P univ) := by ac_rfl - _ ≤ q * P B := mul_le_mul' le_rfl ih - _ ≤ P (B ∩ success N) := hstep N B hB - -lemma ribbon_mass_power_cancellation (κ : ℝ) {m : ℝ} (hm : 0 < m) : - m ^ 2 * (1 / m) ^ dimension κ = m ^ codimension κ := by - rw [one_div, Real.inv_rpow hm.le, ← Real.rpow_neg hm.le, - ← Real.rpow_two, ← Real.rpow_add hm] - rfl - -/-- The disjoint middle strips of `m²` steps of width `1/m` carry mass of -order `m^(2-d)`, with no dependence assumption in this deterministic sum. -/ -theorem ribbon_m_squared_mass_lower {ι : Type*} (I : Finset ι) - (ν : Measure ℂ) (D : ι → Set ℂ) - (hdisj : (I : Set ι).PairwiseDisjoint D) - (hmeas : ∀ i ∈ I, MeasurableSet (D i)) - (κ : ℝ) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) - (b : ℝ) (_hb : 0 ≤ b) - (hmass : ∀ i ∈ I, ENNReal.ofReal (b * (1 / (m : ℝ)) ^ dimension κ) ≤ ν (D i)) : - ENNReal.ofReal (b * (m : ℝ) ^ codimension κ) ≤ ν (⋃ i ∈ I, D i) := by - have hmR : (0 : ℝ) < m := by exact_mod_cast hm - rw [measure_biUnion_finset hdisj hmeas] - calc - _ = ∑ _i ∈ I, ENNReal.ofReal (b * (1 / (m : ℝ)) ^ dimension κ) := by - rw [Finset.sum_const, nsmul_eq_mul, hcard, - ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)] - congr 1 - rw [Nat.cast_pow] - calc - b * (m : ℝ) ^ codimension κ = b * - ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by - rw [ribbon_mass_power_cancellation κ hmR] - _ = _ := by ring - _ ≤ _ := Finset.sum_le_sum hmass - -/-- A quadratic number of fixed-probability passages has the exponential -cost used by the dense visit. -/ -theorem ribbon_quadratic_probability {q : ℝ} (hq : 0 < q) (hq1 : q ≤ 1) : - ∃ C : ℝ, 0 < C ∧ ∀ m : ℕ, - Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ (m ^ 2) := by - have hlog : Real.log q ≤ 0 := Real.log_nonpos hq.le hq1 - refine ⟨1 - Real.log q, by linarith, ?_⟩ - intro m - calc - _ ≤ Real.exp ((m ^ 2 : ℕ) * Real.log q) := by - apply Real.exp_le_exp.mpr - rw [Nat.cast_pow] - nlinarith [sq_nonneg (m : ℝ)] - _ = q ^ (m ^ 2) := by rw [Real.exp_nat_mul, Real.exp_log hq] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.DenseCoverCore + +namespace OAI + +/-! Adapted concatenation and the mass/probability scaling of a serpentine +ribbon with a quadratic number of narrow steps. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +/-- Success probabilities multiply as lower bounds under adapted restarts. +The successive steps need not be independent. -/ +theorem adapted_successes_lower_bound + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) + (F : Filtration ℕ mΩ) (success : ℕ → Set Ω) + (hmeas : ∀ n, MeasurableSet[F (n + 1)] (success n)) + (q : ℝ≥0∞) + (hstep : ∀ n (B : Set Ω), MeasurableSet[F n] B → + q * P B ≤ P (B ∩ success n)) : + ∀ N : ℕ, q ^ N * P univ ≤ P (⋂ n ∈ Finset.range N, success n) := by + intro N + induction N with + | zero => simp + | succ N ih => + let B : Set Ω := ⋂ n ∈ Finset.range N, success n + have hB : MeasurableSet[F N] B := by + apply MeasurableSet.iInter + intro n + apply MeasurableSet.iInter + intro hn + exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n) + have he : (⋂ n ∈ Finset.range (N + 1), success n) = B ∩ success N := by + ext ω + simp only [mem_iInter, Finset.mem_range, mem_inter_iff, B] + constructor + · intro hω + exact ⟨fun n hn => hω n (Nat.lt_succ_of_lt hn), hω N (Nat.lt_succ_self N)⟩ + · rintro ⟨hω, hN⟩ n hn + rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl + · exact hω n hn + · exact hN + rw [he, pow_succ] + calc + q ^ N * q * P univ = q * (q ^ N * P univ) := by ac_rfl + _ ≤ q * P B := mul_le_mul' le_rfl ih + _ ≤ P (B ∩ success N) := hstep N B hB + +lemma ribbon_mass_power_cancellation (κ : ℝ) {m : ℝ} (hm : 0 < m) : + m ^ 2 * (1 / m) ^ dimension κ = m ^ codimension κ := by + rw [one_div, Real.inv_rpow hm.le, ← Real.rpow_neg hm.le, + ← Real.rpow_two, ← Real.rpow_add hm] + rfl + +/-- The disjoint middle strips of `m²` steps of width `1/m` carry mass of +order `m^(2-d)`, with no dependence assumption in this deterministic sum. -/ +theorem ribbon_m_squared_mass_lower {ι : Type*} (I : Finset ι) + (ν : Measure ℂ) (D : ι → Set ℂ) + (hdisj : (I : Set ι).PairwiseDisjoint D) + (hmeas : ∀ i ∈ I, MeasurableSet (D i)) + (κ : ℝ) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) + (b : ℝ) (_hb : 0 ≤ b) + (hmass : ∀ i ∈ I, ENNReal.ofReal (b * (1 / (m : ℝ)) ^ dimension κ) ≤ ν (D i)) : + ENNReal.ofReal (b * (m : ℝ) ^ codimension κ) ≤ ν (⋃ i ∈ I, D i) := by + have hmR : (0 : ℝ) < m := by exact_mod_cast hm + rw [measure_biUnion_finset hdisj hmeas] + calc + _ = ∑ _i ∈ I, ENNReal.ofReal (b * (1 / (m : ℝ)) ^ dimension κ) := by + rw [Finset.sum_const, nsmul_eq_mul, hcard, + ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)] + congr 1 + rw [Nat.cast_pow] + calc + b * (m : ℝ) ^ codimension κ = b * + ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by + rw [ribbon_mass_power_cancellation κ hmR] + _ = _ := by ring + _ ≤ _ := Finset.sum_le_sum hmass + +/-- A quadratic number of fixed-probability passages has the exponential +cost used by the dense visit. -/ +theorem ribbon_quadratic_probability {q : ℝ} (hq : 0 < q) (hq1 : q ≤ 1) : + ∃ C : ℝ, 0 < C ∧ ∀ m : ℕ, + Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ (m ^ 2) := by + have hlog : Real.log q ≤ 0 := Real.log_nonpos hq.le hq1 + refine ⟨1 - Real.log q, by linarith, ?_⟩ + intro m + calc + _ ≤ Real.exp ((m ^ 2 : ℕ) * Real.log q) := by + apply Real.exp_le_exp.mpr + rw [Nat.cast_pow] + nlinarith [sq_nonneg (m : ℝ)] + _ = q ^ (m ^ 2) := by rw [Real.exp_nat_mul, Real.exp_log hq] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateCompactness.lean b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateCompactness.lean index fb25cac223905e024b6d0d109cb74347aa09dc43..9033a41d48c1605d672fe268c9fb367e19d4eb9a 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateCompactness.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateCompactness.lean @@ -1,350 +1,350 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry -import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli - -namespace OAI - -/-! Common physical domains and convergence of inverse ribbon coordinates -on a fixed compact coordinate patch. The uniform lower Lipschitz bound -controls the inverse without requiring derivative convergence. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal BoundedContinuousFunction - -namespace SLEExactGauge.RibbonCoordinates - -def coordinateInverse (P : ℂ → ℂ) (K : Set ℂ) : ℂ → ℂ := Function.invFunOn P K - -theorem coordinateInverse_mem {P : ℂ → ℂ} {K : Set ℂ} {z : ℂ} - (hz : z ∈ P '' K) : coordinateInverse P K z ∈ K := - Function.invFunOn_mem hz - -theorem coordinateInverse_right {P : ℂ → ℂ} {K : Set ℂ} {z : ℂ} - (hz : z ∈ P '' K) : P (coordinateInverse P K z) = z := - Function.invFunOn_eq hz - -theorem coordinateInverse_left {P : ℂ → ℂ} {K : Set ℂ} - (hi : InjOn P K) {z : ℂ} (hz : z ∈ K) : - coordinateInverse P K (P z) = z := - hi.leftInvOn_invFunOn hz - -/-- Physical coordinate regions agree with inverse-coordinate membership. -/ -theorem coordinate_image_region_iff {P : ℂ → ℂ} {K A : Set ℂ} - (hi : InjOn P K) (hA : A ⊆ K) {z : ℂ} : - z ∈ P '' A ↔ z ∈ P '' K ∧ coordinateInverse P K z ∈ A := by - constructor - · rintro ⟨x, hx, rfl⟩ - exact ⟨mem_image_of_mem P (hA hx), by simpa only [coordinateInverse_left hi (hA hx)] using hx⟩ - · rintro ⟨hz, hAq⟩ - exact ⟨coordinateInverse P K z, hAq, coordinateInverse_right hz⟩ - -theorem coordinateInverse_lipschitz - (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) - (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : - LipschitzOnWith L (coordinateInverse P K) (P '' K) := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro x hx y hy - have hh := hlower _ (coordinateInverse_mem hx) _ (coordinateInverse_mem hy) - simpa only [coordinateInverse_right hx, coordinateInverse_right hy] using hh - -theorem lower_lipschitz_uniform_limit - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : - ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y) := by - intro x hx y hy - exact ge_of_tendsto (tendsto_const_nhds.mul ((hconv.tendsto_at hx).dist (hconv.tendsto_at hy))) - (Eventually.of_forall fun n => hlower n x hx y hy) - -/-- Common image containment converts uniform forward convergence into -uniform inverse convergence on any retained physical set. -/ -theorem coordinateInverse_uniform_on - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S : Set ℂ) (L : ℝ≥0) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hSP : S ⊆ P '' K) (hcover : ∀ᶠ n in atTop, S ⊆ Pₙ n '' K) : - TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) (coordinateInverse P K) atTop S := by - apply Metric.tendstoUniformlyOn_iff.mpr - intro ε hε - have hL : 0 < (L : ℝ) + 1 := by positivity - have hη : 0 < ε / ((L : ℝ) + 1) := div_pos hε hL - filter_upwards [hcover, Metric.tendstoUniformlyOn_iff.mp hconv _ hη] with n hn hclose - intro z hz - have hq : coordinateInverse P K z ∈ K := coordinateInverse_mem (hSP hz) - have hqn : coordinateInverse (Pₙ n) K z ∈ K := coordinateInverse_mem (hn hz) - have he := hlower n _ hqn _ hq - rw [coordinateInverse_right (hn hz)] at he - have hd := hclose _ hq - rw [coordinateInverse_right (hSP hz)] at hd - have he' : dist (coordinateInverse P K z) (coordinateInverse (Pₙ n) K z) ≤ - (L : ℝ) * dist z (Pₙ n (coordinateInverse P K z)) := by - simpa only [dist_comm] using he - calc - _ ≤ (L : ℝ) * dist z (Pₙ n (coordinateInverse P K z)) := he' - _ ≤ ((L : ℝ) + 1) * dist z (Pₙ n (coordinateInverse P K z)) := - mul_le_mul_of_nonneg_right (by linarith) dist_nonneg - _ < ((L : ℝ) + 1) * (ε / ((L : ℝ) + 1)) := mul_lt_mul_of_pos_left hd hL - _ = ε := mul_div_cancel₀ ε hL.ne' - -/-- The uniform inverse bound makes every coordinate chart injective. -/ -theorem lower_lipschitz_injOn - (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) - (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : - InjOn P K := by - intro x hx y hy he - have hh := hlower x hx y hy - rw [he, dist_self, mul_zero] at hh - exact dist_eq_zero.mp (le_antisymm hh dist_nonneg) - -/-- The limiting coordinate image has an open interior physical domain. -/ -theorem isOpen_image_interior - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (K : Set ℂ) - (hP : ContinuousOn P K) (hi : InjOn P K) : - IsOpen (P '' interior K) := by - apply isOpen_iff_mem_nhds.mpr - rintro z ⟨x, hx, rfl⟩ - obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp isOpen_interior x hx - have hsub : ball x r ⊆ K := hball.trans interior_subset - have hopen := hpublished P x r hr (hP.mono hsub) (hi.mono hsub) - exact mem_of_superset - (hopen.mem_nhds (mem_image_of_mem P (mem_ball_self hr))) - (image_mono hball) - -/-- Compact physical subsets of the limiting interior image belong to -all sufficiently late moving interior images. -/ -theorem eventual_compact_image_interior - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S : Set ℂ) (L : ℝ≥0) - (hL : 0 < L) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hS : IsCompact S) (hSP : S ⊆ P '' interior K) : - ∀ᶠ n in atTop, S ⊆ Pₙ n '' interior K := by - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hi := lower_lipschitz_injOn P K L hlim - have hSK : S ⊆ P '' K := hSP.trans (image_mono interior_subset) - let Q := coordinateInverse P K '' S - have hQ : IsCompact Q := hS.image_of_continuousOn - ((coordinateInverse_lipschitz P K L hlim).continuousOn.mono hSK) - have hQK : Q ⊆ interior K := by - rintro _ ⟨z, hz, rfl⟩ - obtain ⟨x, hx, he⟩ := hSP hz - have hqx : coordinateInverse P K z = x := hi - (coordinateInverse_mem (hSK hz)) (interior_subset hx) - ((coordinateInverse_right (hSK hz)).trans he.symm) - simpa only [hqx] using hx - obtain ⟨ρ, hρ, hthick⟩ := hQ.exists_cthickening_subset_open isOpen_interior hQK - have hLr : 0 < (L : ℝ) := hL - filter_upwards [Metric.tendstoUniformlyOn_iff.mp hconv (ρ / L) (div_pos hρ hLr)] with n hn - intro z hz - let x := coordinateInverse P K z - have hxQ : x ∈ Q := mem_image_of_mem _ hz - have hxK : x ∈ K := interior_subset (hQK hxQ) - have hball : closedBall x ρ ⊆ interior K := by - intro y hy - exact hthick (mem_cthickening_of_dist_le y x ρ Q hxQ hy) - have hballK := hball.trans interior_subset - have hiₙ := lower_lipschitz_injOn (Pₙ n) K L (hlower n) - have hopen := hpublished (Pₙ n) x ρ hρ - ((hPn n).mono (ball_subset_closedBall.trans hballK)) - (hiₙ.mono (ball_subset_closedBall.trans hballK)) - have hcontains := ball_subset_image_ball (Pₙ n) x hρ hLr - ((hPn n).mono hballK) hopen - (fun y hy => hlower n y (hballK hy) x hxK) - apply (image_mono (ball_subset_closedBall.trans hball)) - apply hcontains - have hh := hn x hxK - rw [coordinateInverse_right (hSK hz)] at hh - simpa only [mem_ball, dist_comm] using hh - -/-- The inverse ribbon charts converge locally uniformly on the full -limiting interior image. Only compact subsets are eventually common. -/ -theorem coordinateInverse_locally_uniform - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) - (hL : 0 < L) (hP : ContinuousOn P K) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : - TendstoLocallyUniformlyOn (fun n => coordinateInverse (Pₙ n) K) - (coordinateInverse P K) atTop (P '' interior K) := by - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hopen := isOpen_image_interior hpublished P K hP - (lower_lipschitz_injOn P K L hlim) - apply (tendstoLocallyUniformlyOn_iff_forall_isCompact hopen).mpr - intro S hSP hS - exact coordinateInverse_uniform_on Pₙ P K S L hconv hlower - (hSP.trans (image_mono interior_subset)) - ((eventual_compact_image_interior hpublished Pₙ P K S L hL hPn hconv hlower hS hSP).mono - (fun _ h => h.trans (image_mono interior_subset))) - -/-- A compact retained physical patch admits one common open neighborhood -contained in every sufficiently late coordinate image. -/ -theorem common_physical_neighborhood - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A : Set ℂ) (L : ℝ≥0) - (hL : 0 < L) (hP : ContinuousOn P K) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hA : IsCompact A) (hAP : A ⊆ P '' interior K) : - ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ - closure U ⊆ P '' interior K ∧ - (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' interior K) ∧ - TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) - (coordinateInverse P K) atTop (closure U) := by - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hopen := isOpen_image_interior hpublished P K hP - (lower_lipschitz_injOn P K L hlim) - obtain ⟨ρ, hρ, hthick⟩ := hA.exists_cthickening_subset_open hopen hAP - let U := thickening ρ A - have hclosure : closure U ⊆ cthickening ρ A := closure_thickening_subset_cthickening ρ A - have hUc : IsCompact (closure U) := (hA.cthickening).of_isClosed_subset isClosed_closure hclosure - have hUP : closure U ⊆ P '' interior K := hclosure.trans hthick - have hlate := eventual_compact_image_interior hpublished Pₙ P K (closure U) L hL hPn - hconv hlower hUc hUP - exact ⟨U, isOpen_thickening, self_subset_thickening hρ A, hUc, hUP, hlate, - coordinateInverse_uniform_on Pₙ P K (closure U) L hconv hlower - (hUP.trans (image_mono interior_subset)) - (hlate.mono (fun _ h => h.trans (image_mono interior_subset)))⟩ - -/-- The source longitudinal coordinate is the real part of the inverse -ribbon chart, translated by the current front coordinate. -/ -def longitudinalCoordinate (P : ℂ → ℂ) (K : Set ℂ) (k : ℝ) (z : ℂ) : ℝ := - (coordinateInverse P K z).re - k - -theorem longitudinalCoordinate_locally_uniform - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k : ℝ) - (hL : 0 < L) (hP : ContinuousOn P K) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : - TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K k) - (longitudinalCoordinate P K k) atTop (P '' interior K) := by - have hh := coordinateInverse_locally_uniform hpublished Pₙ P K L hL hP hPn hconv hlower - have hu : UniformContinuous (fun z : ℂ => z.re - k) := - Complex.uniformContinuous_re.sub uniformContinuous_const - exact hu.comp_tendstoLocallyUniformlyOn hh - -theorem longitudinalCoordinate_continuousOn - (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k : ℝ) - (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : - ContinuousOn (longitudinalCoordinate P K k) (P '' K) := by - exact (Complex.continuous_re.comp_continuousOn - (coordinateInverse_lipschitz P K L hlower).continuousOn).sub continuousOn_const - -/-- A retained compact part of a ribbon has a common open physical chart -on which the actual longitudinal coordinates are eventually continuous -and converge locally uniformly. -/ -theorem common_longitudinal_chart - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A : Set ℂ) (L : ℝ≥0) (k : ℝ) - (hL : 0 < L) (hP : ContinuousOn P K) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hA : IsCompact A) (hAP : A ⊆ P '' interior K) : - ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ - closure U ⊆ P '' interior K ∧ - (∀ᶠ n in atTop, U ⊆ Pₙ n '' interior K) ∧ - ContinuousOn (longitudinalCoordinate P K k) U ∧ - (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K k) U) ∧ - TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K k) - (longitudinalCoordinate P K k) atTop U := by - obtain ⟨U, hU, hAU, hUc, hUP, hlate, _⟩ := common_physical_neighborhood - hpublished Pₙ P K A L hL hP hPn hconv hlower hA hAP - have hUImage : U ⊆ P '' interior K := subset_closure.trans hUP - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - refine ⟨U, hU, hAU, hUc, hUP, hlate.mono (fun _ h => subset_closure.trans h), - (longitudinalCoordinate_continuousOn P K L k hlim).mono - (hUImage.trans (image_mono interior_subset)), ?_, ?_⟩ - · filter_upwards [hlate] with n hn - exact (longitudinalCoordinate_continuousOn (Pₙ n) K L k (hlower n)).mono - (subset_closure.trans (hn.trans (image_mono interior_subset))) - · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L k hL hP hPn - hconv hlower).mono hUImage - -/-- A compact set with limiting coordinates in an open coordinate region -has its moving coordinates in that same region for all sufficiently late -charts. This includes simultaneous longitudinal and transverse margins. -/ -theorem eventual_compact_coordinate_region - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S A : Set ℂ) (L : ℝ≥0) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hS : IsCompact S) (hSP : S ⊆ P '' K) - (hcover : ∀ᶠ n in atTop, S ⊆ Pₙ n '' K) - (hA : IsOpen A) (hSA : coordinateInverse P K '' S ⊆ A) : - ∀ᶠ n in atTop, S ⊆ Pₙ n '' (K ∩ A) := by - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hcompact : IsCompact (coordinateInverse P K '' S) := - hS.image_of_continuousOn ((coordinateInverse_lipschitz P K L hlim).continuousOn.mono hSP) - obtain ⟨ε, hε, heps⟩ := hcompact.exists_thickening_subset_open hA hSA - have hu := coordinateInverse_uniform_on Pₙ P K S L hconv hlower hSP hcover - filter_upwards [hcover, Metric.tendstoUniformlyOn_iff.mp hu ε hε] with n hn hclose - intro z hz - refine ⟨coordinateInverse (Pₙ n) K z, ⟨coordinateInverse_mem (hn hz), ?_⟩, - coordinateInverse_right (hn hz)⟩ - exact heps (mem_thickening_iff.mpr ⟨coordinateInverse P K z, mem_image_of_mem _ hz, - by simpa only [dist_comm] using hclose z hz⟩) - -/-- The admissible two-point patch bounds and a bounded marked point give -an actual uniformly convergent subsequence of the coordinate maps. Both -Lipschitz bounds pass to the limit, retaining an embedded limiting patch. -/ -theorem exists_bilip_uniform_subsequence - (Pₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) - (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C L : ℝ≥0) (R : ℝ) - (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (htip : ∀ n, ‖Pₙ n x₀‖ ≤ R) : - ∃ P : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧ - TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K ∧ - LipschitzOnWith C P K ∧ - (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) ∧ - InjOn P K := by - classical - let : CompactSpace K := isCompact_iff_compactSpace.mp hK - let f : ℕ → K →ᵇ ℂ := fun n => BoundedContinuousFunction.mkOfCompact - ⟨K.domRestrict (Pₙ n), (hupper n).continuousOn.domRestrict⟩ - obtain ⟨M, hM⟩ := hK.isBounded.subset_closedBall x₀ - have hbound n (x : K) : ‖f n x‖ ≤ R + C * M := by - have hd := (hupper n).dist_le_mul x x.property x₀ hx₀ - have hh := norm_le_norm_add_const_of_dist_le hd - exact hh.trans (add_le_add (htip n) - (mul_le_mul_of_nonneg_left (hM x.property) C.coe_nonneg)) - have heq : Equicontinuous ((↑) : Set.range f → K → ℂ) := by - apply (LipschitzWith.uniformEquicontinuous _ C ?_).equicontinuous - rintro ⟨g, n, rfl⟩ - exact (hupper n).to_restrict - have hcompact : IsCompact (closure (Set.range f)) := - BoundedContinuousFunction.arzela_ascoli (closedBall (0 : ℂ) (R + C * M)) - (isCompact_closedBall _ _) (Set.range f) - (by rintro g x ⟨n, rfl⟩; simpa only [mem_closedBall, dist_zero_right] using hbound n x) - heq - obtain ⟨b, _, φ, hφ, hlim⟩ := hcompact.tendsto_subseq - (fun n => subset_closure (mem_range_self n)) - have hb : TendstoUniformly (fun n => f (φ n)) b atTop := - BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hlim - let P : ℂ → ℂ := fun z => if hz : z ∈ K then b ⟨z, hz⟩ else 0 - have hu : TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K := by - apply Metric.tendstoUniformlyOn_iff.mpr - intro ε hε - filter_upwards [Metric.tendstoUniformly_iff.mp hb ε hε] with n hn - intro z hz - simpa [P, f, hz] using hn ⟨z, hz⟩ - have hPu : LipschitzOnWith C P K := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro x hx y hy - exact le_of_tendsto ((hu.tendsto_at hx).dist (hu.tendsto_at hy)) - (Eventually.of_forall fun n => (hupper (φ n)).dist_le_mul x hx y hy) - have hPl := lower_lipschitz_uniform_limit (fun n => Pₙ (φ n)) P K L hu - (fun n => hlower (φ n)) - exact ⟨P, φ, hφ, hu, hPu, hPl, lower_lipschitz_injOn P K L hPl⟩ - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry +import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli + +namespace OAI + +/-! Common physical domains and convergence of inverse ribbon coordinates +on a fixed compact coordinate patch. The uniform lower Lipschitz bound +controls the inverse without requiring derivative convergence. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal BoundedContinuousFunction + +namespace SLEExactGauge.RibbonCoordinates + +def coordinateInverse (P : ℂ → ℂ) (K : Set ℂ) : ℂ → ℂ := Function.invFunOn P K + +theorem coordinateInverse_mem {P : ℂ → ℂ} {K : Set ℂ} {z : ℂ} + (hz : z ∈ P '' K) : coordinateInverse P K z ∈ K := + Function.invFunOn_mem hz + +theorem coordinateInverse_right {P : ℂ → ℂ} {K : Set ℂ} {z : ℂ} + (hz : z ∈ P '' K) : P (coordinateInverse P K z) = z := + Function.invFunOn_eq hz + +theorem coordinateInverse_left {P : ℂ → ℂ} {K : Set ℂ} + (hi : InjOn P K) {z : ℂ} (hz : z ∈ K) : + coordinateInverse P K (P z) = z := + hi.leftInvOn_invFunOn hz + +/-- Physical coordinate regions agree with inverse-coordinate membership. -/ +theorem coordinate_image_region_iff {P : ℂ → ℂ} {K A : Set ℂ} + (hi : InjOn P K) (hA : A ⊆ K) {z : ℂ} : + z ∈ P '' A ↔ z ∈ P '' K ∧ coordinateInverse P K z ∈ A := by + constructor + · rintro ⟨x, hx, rfl⟩ + exact ⟨mem_image_of_mem P (hA hx), by simpa only [coordinateInverse_left hi (hA hx)] using hx⟩ + · rintro ⟨hz, hAq⟩ + exact ⟨coordinateInverse P K z, hAq, coordinateInverse_right hz⟩ + +theorem coordinateInverse_lipschitz + (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) + (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : + LipschitzOnWith L (coordinateInverse P K) (P '' K) := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro x hx y hy + have hh := hlower _ (coordinateInverse_mem hx) _ (coordinateInverse_mem hy) + simpa only [coordinateInverse_right hx, coordinateInverse_right hy] using hh + +theorem lower_lipschitz_uniform_limit + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : + ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y) := by + intro x hx y hy + exact ge_of_tendsto (tendsto_const_nhds.mul ((hconv.tendsto_at hx).dist (hconv.tendsto_at hy))) + (Eventually.of_forall fun n => hlower n x hx y hy) + +/-- Common image containment converts uniform forward convergence into +uniform inverse convergence on any retained physical set. -/ +theorem coordinateInverse_uniform_on + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S : Set ℂ) (L : ℝ≥0) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hSP : S ⊆ P '' K) (hcover : ∀ᶠ n in atTop, S ⊆ Pₙ n '' K) : + TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) (coordinateInverse P K) atTop S := by + apply Metric.tendstoUniformlyOn_iff.mpr + intro ε hε + have hL : 0 < (L : ℝ) + 1 := by positivity + have hη : 0 < ε / ((L : ℝ) + 1) := div_pos hε hL + filter_upwards [hcover, Metric.tendstoUniformlyOn_iff.mp hconv _ hη] with n hn hclose + intro z hz + have hq : coordinateInverse P K z ∈ K := coordinateInverse_mem (hSP hz) + have hqn : coordinateInverse (Pₙ n) K z ∈ K := coordinateInverse_mem (hn hz) + have he := hlower n _ hqn _ hq + rw [coordinateInverse_right (hn hz)] at he + have hd := hclose _ hq + rw [coordinateInverse_right (hSP hz)] at hd + have he' : dist (coordinateInverse P K z) (coordinateInverse (Pₙ n) K z) ≤ + (L : ℝ) * dist z (Pₙ n (coordinateInverse P K z)) := by + simpa only [dist_comm] using he + calc + _ ≤ (L : ℝ) * dist z (Pₙ n (coordinateInverse P K z)) := he' + _ ≤ ((L : ℝ) + 1) * dist z (Pₙ n (coordinateInverse P K z)) := + mul_le_mul_of_nonneg_right (by linarith) dist_nonneg + _ < ((L : ℝ) + 1) * (ε / ((L : ℝ) + 1)) := mul_lt_mul_of_pos_left hd hL + _ = ε := mul_div_cancel₀ ε hL.ne' + +/-- The uniform inverse bound makes every coordinate chart injective. -/ +theorem lower_lipschitz_injOn + (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) + (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : + InjOn P K := by + intro x hx y hy he + have hh := hlower x hx y hy + rw [he, dist_self, mul_zero] at hh + exact dist_eq_zero.mp (le_antisymm hh dist_nonneg) + +/-- The limiting coordinate image has an open interior physical domain. -/ +theorem isOpen_image_interior + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (K : Set ℂ) + (hP : ContinuousOn P K) (hi : InjOn P K) : + IsOpen (P '' interior K) := by + apply isOpen_iff_mem_nhds.mpr + rintro z ⟨x, hx, rfl⟩ + obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp isOpen_interior x hx + have hsub : ball x r ⊆ K := hball.trans interior_subset + have hopen := hpublished P x r hr (hP.mono hsub) (hi.mono hsub) + exact mem_of_superset + (hopen.mem_nhds (mem_image_of_mem P (mem_ball_self hr))) + (image_mono hball) + +/-- Compact physical subsets of the limiting interior image belong to +all sufficiently late moving interior images. -/ +theorem eventual_compact_image_interior + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S : Set ℂ) (L : ℝ≥0) + (hL : 0 < L) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hS : IsCompact S) (hSP : S ⊆ P '' interior K) : + ∀ᶠ n in atTop, S ⊆ Pₙ n '' interior K := by + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hi := lower_lipschitz_injOn P K L hlim + have hSK : S ⊆ P '' K := hSP.trans (image_mono interior_subset) + let Q := coordinateInverse P K '' S + have hQ : IsCompact Q := hS.image_of_continuousOn + ((coordinateInverse_lipschitz P K L hlim).continuousOn.mono hSK) + have hQK : Q ⊆ interior K := by + rintro _ ⟨z, hz, rfl⟩ + obtain ⟨x, hx, he⟩ := hSP hz + have hqx : coordinateInverse P K z = x := hi + (coordinateInverse_mem (hSK hz)) (interior_subset hx) + ((coordinateInverse_right (hSK hz)).trans he.symm) + simpa only [hqx] using hx + obtain ⟨ρ, hρ, hthick⟩ := hQ.exists_cthickening_subset_open isOpen_interior hQK + have hLr : 0 < (L : ℝ) := hL + filter_upwards [Metric.tendstoUniformlyOn_iff.mp hconv (ρ / L) (div_pos hρ hLr)] with n hn + intro z hz + let x := coordinateInverse P K z + have hxQ : x ∈ Q := mem_image_of_mem _ hz + have hxK : x ∈ K := interior_subset (hQK hxQ) + have hball : closedBall x ρ ⊆ interior K := by + intro y hy + exact hthick (mem_cthickening_of_dist_le y x ρ Q hxQ hy) + have hballK := hball.trans interior_subset + have hiₙ := lower_lipschitz_injOn (Pₙ n) K L (hlower n) + have hopen := hpublished (Pₙ n) x ρ hρ + ((hPn n).mono (ball_subset_closedBall.trans hballK)) + (hiₙ.mono (ball_subset_closedBall.trans hballK)) + have hcontains := ball_subset_image_ball (Pₙ n) x hρ hLr + ((hPn n).mono hballK) hopen + (fun y hy => hlower n y (hballK hy) x hxK) + apply (image_mono (ball_subset_closedBall.trans hball)) + apply hcontains + have hh := hn x hxK + rw [coordinateInverse_right (hSK hz)] at hh + simpa only [mem_ball, dist_comm] using hh + +/-- The inverse ribbon charts converge locally uniformly on the full +limiting interior image. Only compact subsets are eventually common. -/ +theorem coordinateInverse_locally_uniform + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) + (hL : 0 < L) (hP : ContinuousOn P K) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : + TendstoLocallyUniformlyOn (fun n => coordinateInverse (Pₙ n) K) + (coordinateInverse P K) atTop (P '' interior K) := by + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hopen := isOpen_image_interior hpublished P K hP + (lower_lipschitz_injOn P K L hlim) + apply (tendstoLocallyUniformlyOn_iff_forall_isCompact hopen).mpr + intro S hSP hS + exact coordinateInverse_uniform_on Pₙ P K S L hconv hlower + (hSP.trans (image_mono interior_subset)) + ((eventual_compact_image_interior hpublished Pₙ P K S L hL hPn hconv hlower hS hSP).mono + (fun _ h => h.trans (image_mono interior_subset))) + +/-- A compact retained physical patch admits one common open neighborhood +contained in every sufficiently late coordinate image. -/ +theorem common_physical_neighborhood + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A : Set ℂ) (L : ℝ≥0) + (hL : 0 < L) (hP : ContinuousOn P K) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hA : IsCompact A) (hAP : A ⊆ P '' interior K) : + ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ + closure U ⊆ P '' interior K ∧ + (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' interior K) ∧ + TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) + (coordinateInverse P K) atTop (closure U) := by + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hopen := isOpen_image_interior hpublished P K hP + (lower_lipschitz_injOn P K L hlim) + obtain ⟨ρ, hρ, hthick⟩ := hA.exists_cthickening_subset_open hopen hAP + let U := thickening ρ A + have hclosure : closure U ⊆ cthickening ρ A := closure_thickening_subset_cthickening ρ A + have hUc : IsCompact (closure U) := (hA.cthickening).of_isClosed_subset isClosed_closure hclosure + have hUP : closure U ⊆ P '' interior K := hclosure.trans hthick + have hlate := eventual_compact_image_interior hpublished Pₙ P K (closure U) L hL hPn + hconv hlower hUc hUP + exact ⟨U, isOpen_thickening, self_subset_thickening hρ A, hUc, hUP, hlate, + coordinateInverse_uniform_on Pₙ P K (closure U) L hconv hlower + (hUP.trans (image_mono interior_subset)) + (hlate.mono (fun _ h => h.trans (image_mono interior_subset)))⟩ + +/-- The source longitudinal coordinate is the real part of the inverse +ribbon chart, translated by the current front coordinate. -/ +def longitudinalCoordinate (P : ℂ → ℂ) (K : Set ℂ) (k : ℝ) (z : ℂ) : ℝ := + (coordinateInverse P K z).re - k + +theorem longitudinalCoordinate_locally_uniform + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k : ℝ) + (hL : 0 < L) (hP : ContinuousOn P K) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) : + TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K k) + (longitudinalCoordinate P K k) atTop (P '' interior K) := by + have hh := coordinateInverse_locally_uniform hpublished Pₙ P K L hL hP hPn hconv hlower + have hu : UniformContinuous (fun z : ℂ => z.re - k) := + Complex.uniformContinuous_re.sub uniformContinuous_const + exact hu.comp_tendstoLocallyUniformlyOn hh + +theorem longitudinalCoordinate_continuousOn + (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k : ℝ) + (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) : + ContinuousOn (longitudinalCoordinate P K k) (P '' K) := by + exact (Complex.continuous_re.comp_continuousOn + (coordinateInverse_lipschitz P K L hlower).continuousOn).sub continuousOn_const + +/-- A retained compact part of a ribbon has a common open physical chart +on which the actual longitudinal coordinates are eventually continuous +and converge locally uniformly. -/ +theorem common_longitudinal_chart + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A : Set ℂ) (L : ℝ≥0) (k : ℝ) + (hL : 0 < L) (hP : ContinuousOn P K) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hA : IsCompact A) (hAP : A ⊆ P '' interior K) : + ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ + closure U ⊆ P '' interior K ∧ + (∀ᶠ n in atTop, U ⊆ Pₙ n '' interior K) ∧ + ContinuousOn (longitudinalCoordinate P K k) U ∧ + (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K k) U) ∧ + TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K k) + (longitudinalCoordinate P K k) atTop U := by + obtain ⟨U, hU, hAU, hUc, hUP, hlate, _⟩ := common_physical_neighborhood + hpublished Pₙ P K A L hL hP hPn hconv hlower hA hAP + have hUImage : U ⊆ P '' interior K := subset_closure.trans hUP + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + refine ⟨U, hU, hAU, hUc, hUP, hlate.mono (fun _ h => subset_closure.trans h), + (longitudinalCoordinate_continuousOn P K L k hlim).mono + (hUImage.trans (image_mono interior_subset)), ?_, ?_⟩ + · filter_upwards [hlate] with n hn + exact (longitudinalCoordinate_continuousOn (Pₙ n) K L k (hlower n)).mono + (subset_closure.trans (hn.trans (image_mono interior_subset))) + · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L k hL hP hPn + hconv hlower).mono hUImage + +/-- A compact set with limiting coordinates in an open coordinate region +has its moving coordinates in that same region for all sufficiently late +charts. This includes simultaneous longitudinal and transverse margins. -/ +theorem eventual_compact_coordinate_region + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K S A : Set ℂ) (L : ℝ≥0) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hS : IsCompact S) (hSP : S ⊆ P '' K) + (hcover : ∀ᶠ n in atTop, S ⊆ Pₙ n '' K) + (hA : IsOpen A) (hSA : coordinateInverse P K '' S ⊆ A) : + ∀ᶠ n in atTop, S ⊆ Pₙ n '' (K ∩ A) := by + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hcompact : IsCompact (coordinateInverse P K '' S) := + hS.image_of_continuousOn ((coordinateInverse_lipschitz P K L hlim).continuousOn.mono hSP) + obtain ⟨ε, hε, heps⟩ := hcompact.exists_thickening_subset_open hA hSA + have hu := coordinateInverse_uniform_on Pₙ P K S L hconv hlower hSP hcover + filter_upwards [hcover, Metric.tendstoUniformlyOn_iff.mp hu ε hε] with n hn hclose + intro z hz + refine ⟨coordinateInverse (Pₙ n) K z, ⟨coordinateInverse_mem (hn hz), ?_⟩, + coordinateInverse_right (hn hz)⟩ + exact heps (mem_thickening_iff.mpr ⟨coordinateInverse P K z, mem_image_of_mem _ hz, + by simpa only [dist_comm] using hclose z hz⟩) + +/-- The admissible two-point patch bounds and a bounded marked point give +an actual uniformly convergent subsequence of the coordinate maps. Both +Lipschitz bounds pass to the limit, retaining an embedded limiting patch. -/ +theorem exists_bilip_uniform_subsequence + (Pₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) + (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C L : ℝ≥0) (R : ℝ) + (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (htip : ∀ n, ‖Pₙ n x₀‖ ≤ R) : + ∃ P : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧ + TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K ∧ + LipschitzOnWith C P K ∧ + (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) ∧ + InjOn P K := by + classical + let : CompactSpace K := isCompact_iff_compactSpace.mp hK + let f : ℕ → K →ᵇ ℂ := fun n => BoundedContinuousFunction.mkOfCompact + ⟨K.domRestrict (Pₙ n), (hupper n).continuousOn.domRestrict⟩ + obtain ⟨M, hM⟩ := hK.isBounded.subset_closedBall x₀ + have hbound n (x : K) : ‖f n x‖ ≤ R + C * M := by + have hd := (hupper n).dist_le_mul x x.property x₀ hx₀ + have hh := norm_le_norm_add_const_of_dist_le hd + exact hh.trans (add_le_add (htip n) + (mul_le_mul_of_nonneg_left (hM x.property) C.coe_nonneg)) + have heq : Equicontinuous ((↑) : Set.range f → K → ℂ) := by + apply (LipschitzWith.uniformEquicontinuous _ C ?_).equicontinuous + rintro ⟨g, n, rfl⟩ + exact (hupper n).to_restrict + have hcompact : IsCompact (closure (Set.range f)) := + BoundedContinuousFunction.arzela_ascoli (closedBall (0 : ℂ) (R + C * M)) + (isCompact_closedBall _ _) (Set.range f) + (by rintro g x ⟨n, rfl⟩; simpa only [mem_closedBall, dist_zero_right] using hbound n x) + heq + obtain ⟨b, _, φ, hφ, hlim⟩ := hcompact.tendsto_subseq + (fun n => subset_closure (mem_range_self n)) + have hb : TendstoUniformly (fun n => f (φ n)) b atTop := + BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hlim + let P : ℂ → ℂ := fun z => if hz : z ∈ K then b ⟨z, hz⟩ else 0 + have hu : TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K := by + apply Metric.tendstoUniformlyOn_iff.mpr + intro ε hε + filter_upwards [Metric.tendstoUniformly_iff.mp hb ε hε] with n hn + intro z hz + simpa [P, f, hz] using hn ⟨z, hz⟩ + have hPu : LipschitzOnWith C P K := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro x hx y hy + exact le_of_tendsto ((hu.tendsto_at hx).dist (hu.tendsto_at hy)) + (Eventually.of_forall fun n => (hupper (φ n)).dist_le_mul x hx y hy) + have hPl := lower_lipschitz_uniform_limit (fun n => Pₙ (φ n)) P K L hu + (fun n => hlower (φ n)) + exact ⟨P, φ, hφ, hu, hPu, hPl, lower_lipschitz_injOn P K L hPl⟩ + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateGeometry.lean b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateGeometry.lean index 03dcd12e0523fcfe404b13ca805bcc5aff84679b..bf75dccc3152273f621f008cfb3ea4474ded124d 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateGeometry.lean @@ -1,348 +1,348 @@ -import OAI.Probability.SLE.Gauge.RibbonSurvival -import OAI.Probability.SLE.Geometry.PublishedPlanarDiskOpenness -import Mathlib.Analysis.Normed.Module.Connected - -namespace OAI - -/-! Coordinate balls in an embedded ribbon. These deterministic lemmas -turn the local lower Lipschitz bound into physical clearance, and identify -the future ribbon with part of the actual unbounded surviving component. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -/-- An open coordinate disk whose inverse has Lipschitz constant `K` -contains the physical disk of radius `r / K`. Compactness of the closed -coordinate disk prevents an interior physical point from escaping its image. -/ -theorem ball_subset_image_ball - (P : ℂ → ℂ) (x : ℂ) {r K : ℝ} (hr : 0 < r) (hK : 0 < K) - (hP : ContinuousOn P (closedBall x r)) - (hopen : IsOpen (P '' ball x r)) - (hlower : ∀ y ∈ closedBall x r, dist y x ≤ K * dist (P y) (P x)) : - ball (P x) (r / K) ⊆ P '' ball x r := by - let A := P '' ball x r - let Q := P '' closedBall x r - have hQ : IsCompact Q := (isCompact_closedBall x r).image_of_continuousOn hP - have hAQ : A ⊆ Q := image_mono ball_subset_closedBall - have hsub : ball (P x) (r / K) ⊆ A ∪ Qᶜ := by - intro z hz - by_cases hzQ : z ∈ Q - · left - obtain ⟨y, hy, rfl⟩ := hzQ - refine ⟨y, ?_, rfl⟩ - have hd := hlower y hy - have hz' : dist (P y) (P x) < r / K := hz - have hh := (lt_div_iff₀ hK).mp hz' - exact lt_of_le_of_lt hd (by nlinarith [hh]) - · exact Or.inr hzQ - apply IsPreconnected.subset_left_of_subset_union hopen hQ.isClosed.isOpen_compl - (disjoint_compl_right.mono_left hAQ) hsub _ isPreconnected_ball - exact ⟨P x, mem_ball_self (div_pos hr hK), mem_image_of_mem P (mem_ball_self hr)⟩ - -def closedRibbon : Set ℂ := {z | 0 ≤ z.re ∧ |z.im| ≤ 1} -def futureRibbon (k : ℝ) : Set ℂ := {z | k < z.re ∧ |z.im| < 1} - -theorem futureRibbon_subset_closedRibbon {k : ℝ} (hk : 0 ≤ k) : - futureRibbon k ⊆ closedRibbon := by - intro z hz - exact ⟨hk.trans hz.1.le, hz.2.le⟩ - -theorem futureRibbon_convex (k : ℝ) : Convex ℝ (futureRibbon k) := by - have hre : Convex ℝ {z : ℂ | k < z.re} := - convex_halfSpace_gt Complex.reLm.isLinear k - have him : Convex ℝ {z : ℂ | z.im ∈ Ioo (-1 : ℝ) 1} := - (convex_Ioo (-1 : ℝ) 1).is_linear_preimage Complex.imLm.isLinear - simpa only [futureRibbon, abs_lt, mem_Ioo, ofPred_and] using hre.inter him - -/-- Any unbounded connected free set belongs to the unbounded component -used in the actual Loewner surviving domain. -/ -theorem free_preconnected_subset_surviving - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (S : Set ℂ) - (hS : IsPreconnected S) (hfree : S ⊆ upperHalfPlane \ (γ '' Icc 0 T)) - (hunbounded : ¬ Bornology.IsBounded S) : S ⊆ survivingDomain γ T := by - intro z hz - refine ⟨hfree hz, ?_⟩ - intro hbounded - exact hunbounded (hbounded.subset (hS.subset_connectedComponentIn hz hfree)) - -/-- Embedding excludes all nonlocal pieces of the stopped past from the -future ribbon. Escape to infinity places the whole future ribbon in the -surviving component, without assuming a filled-hull containment. -/ -theorem futureRibbon_survives - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) - (hi : InjOn P closedRibbon) {k : ℝ} (hk : 0 ≤ k) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) : - P '' futureRibbon k ⊆ survivingDomain γ T := by - have hsub := futureRibbon_subset_closedRibbon hk - apply free_preconnected_subset_surviving γ T _ - ((futureRibbon_convex k).isPreconnected.image P (hP.mono hsub)) _ hunbounded - rintro z ⟨y, hy, rfl⟩ - refine ⟨hH y (hsub hy) (hk.trans_lt hy.1), ?_⟩ - intro hyPast - obtain ⟨x, hx, he⟩ := hpast hyPast - have hxy : x = y := hi hx.1 (hsub hy) he - exact (not_le_of_gt hy.1) (hxy ▸ hx.2) - -/-- A coordinate disk strictly ahead of the front and away from the two -ribbon edges gives a physical disk in the actual surviving domain. -/ -theorem coordinate_ball_clearance - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) - (hi : InjOn P closedRibbon) {k : ℝ} (hk : 0 ≤ k) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) - (x : ℂ) {r K : ℝ} (hr : 0 < r) (hK : 0 < K) - (hfront : k + r < x.re) (hedge : |x.im| + r < 1) - (hopen : IsOpen (P '' ball x r)) - (hlower : ∀ y ∈ closedBall x r, dist y x ≤ K * dist (P y) (P x)) : - ball (P x) (r / K) ⊆ survivingDomain γ T := by - have hball : closedBall x r ⊆ futureRibbon k := by - intro y hy - have hd : ‖y - x‖ ≤ r := by simpa only [mem_closedBall, dist_eq_norm] using hy - have hre := (Complex.abs_re_le_norm (y - x)).trans hd - have him := (Complex.abs_im_le_norm (y - x)).trans hd - simp only [Complex.sub_re, Complex.sub_im] at hre him - refine ⟨by linarith [(abs_le.mp hre).1], ?_⟩ - have htri : |y.im| ≤ |y.im - x.im| + |x.im| := by - simpa only [sub_add_cancel] using abs_add_le (y.im - x.im) x.im - linarith - exact (ball_subset_image_ball P x hr hK - (hP.mono (hball.trans (futureRibbon_subset_closedRibbon hk))) hopen hlower).trans - ((image_mono (ball_subset_closedBall.trans hball)).trans - (futureRibbon_survives P hP hi hk hH γ T hpast hunbounded)) - -/-- The paper's access guide, before applying the ribbon coordinate map. -/ -def guideCoordinate (k x t : ℝ) : ℂ := - ((k + t : ℝ) : ℂ) + ((x * max (1 - 4 * t) 0 : ℝ) : ℂ) * Complex.I - -@[simp] theorem guideCoordinate_re (k x t : ℝ) : - (guideCoordinate k x t).re = k + t := by simp [guideCoordinate] - -@[simp] theorem guideCoordinate_im (k x t : ℝ) : - (guideCoordinate k x t).im = x * max (1 - 4 * t) 0 := by simp [guideCoordinate] - -theorem guideCoordinate_continuous (k x : ℝ) : Continuous (guideCoordinate k x) := by - unfold guideCoordinate - fun_prop - -theorem guideCoordinate_im_bound {k x t : ℝ} (ht : 0 ≤ t) : - |(guideCoordinate k x t).im| ≤ |x| := by - rw [guideCoordinate_im, abs_mul, abs_of_nonneg (le_max_right (1 - 4 * t) 0)] - exact mul_le_of_le_one_right (abs_nonneg x) (max_le (by linarith) (by norm_num)) - -/-- The longitudinal coordinate gives the lower chord bound directly. -/ -theorem guideCoordinate_lower_chord (k x u v : ℝ) : - |u - v| ≤ dist (guideCoordinate k x u) (guideCoordinate k x v) := by - have hh := Complex.abs_re_le_norm (guideCoordinate k x u - guideCoordinate k x v) - simpa only [Complex.sub_re, guideCoordinate_re, add_sub_add_left_eq_sub, - dist_eq_norm] using hh - -theorem guideCoordinate_upper_chord (k x u v : ℝ) : - dist (guideCoordinate k x u) (guideCoordinate k x v) ≤ - (1 + 4 * |x|) * |u - v| := by - have hmax := abs_max_sub_max_le_abs (1 - 4 * u) (1 - 4 * v) 0 - have he : (1 - 4 * u) - (1 - 4 * v) = -4 * (u - v) := by ring - rw [he, abs_mul] at hmax - norm_num at hmax - have him : |x * max (1 - 4 * u) 0 - x * max (1 - 4 * v) 0| ≤ - |x| * (4 * |u - v|) := by - rw [← mul_sub, abs_mul] - exact mul_le_mul_of_nonneg_left hmax (abs_nonneg x) - have hh := Complex.norm_le_abs_re_add_abs_im - (guideCoordinate k x u - guideCoordinate k x v) - simp only [Complex.sub_re, Complex.sub_im, guideCoordinate_re, guideCoordinate_im, - add_sub_add_left_eq_sub] at hh - rw [dist_eq_norm] - nlinarith - -def guideRadius (t : ℝ) : ℝ := min t 1 / 4 - -theorem guideRadius_pos {t : ℝ} (ht : 0 < t) : 0 < guideRadius t := - div_pos (lt_min ht zero_lt_one) (by norm_num) - -/-- The guide is at most a quarter of the ribbon half-width from the -centerline; its coordinate disks have radius proportional to `min(t,1)`. -/ -theorem guideCoordinate_disk_margins - {k x t : ℝ} (hx : |x| ≤ 1 / 4) (ht : 0 < t) : - k + guideRadius t < (guideCoordinate k x t).re ∧ - |(guideCoordinate k x t).im| + guideRadius t < 1 := by - have hrt : guideRadius t ≤ t / 4 := div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num) - have hr1 : guideRadius t ≤ 1 / 4 := div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num) - have hi := (guideCoordinate_im_bound (k := k) (x := x) ht.le).trans hx - rw [guideCoordinate_re] - constructor <;> linarith - -/-- Applying the coordinate disk lemma proves the paper's guide clearance -for every positive guide time, with the actual surviving domain. -/ -theorem physical_guide_clearance - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) - (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) - (hx : |x| ≤ 1 / 4) (hK : 0 < K) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) - (hopen : ∀ t ∈ Ioc (0 : ℝ) 2, - IsOpen (P '' ball (guideCoordinate k x t) (guideRadius t))) - (hlower : ∀ t ∈ Ioc (0 : ℝ) 2, - ∀ y ∈ closedBall (guideCoordinate k x t) (guideRadius t), - dist y (guideCoordinate k x t) ≤ K * dist (P y) (P (guideCoordinate k x t))) : - ∀ t ∈ Ioc (0 : ℝ) 2, - ball (P (guideCoordinate k x t)) (min t 1 / (4 * K)) ⊆ survivingDomain γ T := by - intro t ht - obtain ⟨hfront, hedge⟩ := guideCoordinate_disk_margins (k := k) hx ht.1 - have hh := coordinate_ball_clearance P hP hi hk hH γ T hpast hunbounded - (guideCoordinate k x t) (guideRadius_pos ht.1) hK hfront hedge (hopen t ht) (hlower t ht) - simpa only [guideRadius, div_div] using hh - -/-- The fixed length-four patch appearing in the admissible ribbon class. -/ -def guidePatch (k : ℝ) : Set ℂ := {z | k ≤ z.re ∧ z.re ≤ k + 3 ∧ |z.im| ≤ 1} - -theorem guideCoordinate_disk_subset_patch - {k x t : ℝ} (hx : |x| ≤ 1 / 4) (ht : t ∈ Ioc (0 : ℝ) 2) : - closedBall (guideCoordinate k x t) (guideRadius t) ⊆ guidePatch k := by - intro y hy - have hd : ‖y - guideCoordinate k x t‖ ≤ guideRadius t := by - simpa only [mem_closedBall, dist_eq_norm] using hy - have hre := (Complex.abs_re_le_norm (y - guideCoordinate k x t)).trans hd - have him := (Complex.abs_im_le_norm (y - guideCoordinate k x t)).trans hd - simp only [Complex.sub_re, Complex.sub_im, guideCoordinate_re, guideCoordinate_im] at hre him - obtain ⟨hfront, hedge⟩ := guideCoordinate_disk_margins (k := k) hx ht.1 - have hr1 : guideRadius t ≤ 1 / 4 := - div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num) - have hi := (guideCoordinate_im_bound (k := k) (x := x) ht.1.le).trans hx - have htri : |y.im| ≤ |y.im - (guideCoordinate k x t).im| + - |(guideCoordinate k x t).im| := by - simpa only [sub_add_cancel] using - abs_add_le (y.im - (guideCoordinate k x t).im) (guideCoordinate k x t).im - change k ≤ y.re ∧ y.re ≤ k + 3 ∧ |y.im| ≤ 1 - simp only [guideCoordinate_re] at hfront - simp only [guideCoordinate_im] at htri hi - exact ⟨by linarith [(abs_le.mp hre).1], by linarith [(abs_le.mp hre).2, ht.2], - by linarith⟩ - -theorem guidePatch_subset_closedRibbon {k : ℝ} (hk : 0 ≤ k) : - guidePatch k ⊆ closedRibbon := by - intro z hz - exact ⟨hk.trans hz.1, hz.2.2⟩ - -/-- The actual fixed patch inverse-Lipschitz bound suffices for all guide -disks. The only topological parameter is the cited planar disk theorem. -/ -theorem physical_guide_clearance_of_patch - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) - (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) - (hx : |x| ≤ 1 / 4) (hK : 0 < K) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) - (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, - dist u v ≤ K * dist (P u) (P v)) : - ∀ t ∈ Ioc (0 : ℝ) 2, - ball (P (guideCoordinate k x t)) (min t 1 / (4 * K)) ⊆ survivingDomain γ T := by - apply physical_guide_clearance P hP hi hk hx hK hH γ T hpast hunbounded - · intro t ht - have hsub := (ball_subset_closedBall.trans (guideCoordinate_disk_subset_patch hx ht)).trans - (guidePatch_subset_closedRibbon hk) - exact hpublished P _ _ (guideRadius_pos ht.1) (hP.mono hsub) (hi.mono hsub) - · intro t ht y hy - have hsub := guideCoordinate_disk_subset_patch (k := k) hx ht - exact hlower y (hsub hy) _ (hsub (mem_closedBall_self (guideRadius_pos ht.1).le)) - -/-- Uniform clearance on every closed interval away from the tip is the -form used by joint physical/conformal guide compactness. -/ -theorem physical_guide_clearance_away - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) - (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) - (hx : |x| ≤ 1 / 4) (hK : 0 < K) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) - (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, - dist u v ≤ K * dist (P u) (P v)) - {δ : ℝ} (hδ : 0 < δ) (_hδ2 : δ ≤ 2) : - ∃ r : ℝ, 0 < r ∧ ∀ t ∈ Icc δ 2, - ball (P (guideCoordinate k x t)) r ⊆ survivingDomain γ T := by - refine ⟨min δ 1 / (4 * K), div_pos (lt_min hδ zero_lt_one) (by positivity), ?_⟩ - intro t ht - apply (ball_subset_ball ?_).trans (physical_guide_clearance_of_patch hpublished P hP hi - hk hx hK hH γ T hpast hunbounded hlower t ⟨hδ.trans_le ht.1, ht.2⟩) - exact div_le_div_of_nonneg_right (min_le_min_right 1 ht.1) (by positivity) - -theorem guideCoordinate_mem_patch {k x t : ℝ} - (hx : |x| ≤ 1 / 4) (ht : t ∈ Icc (0 : ℝ) 2) : - guideCoordinate k x t ∈ guidePatch k := by - change k ≤ (guideCoordinate k x t).re ∧ - (guideCoordinate k x t).re ≤ k + 3 ∧ |(guideCoordinate k x t).im| ≤ 1 - rw [guideCoordinate_re] - exact ⟨by linarith [ht.1], by linarith [ht.2], - (guideCoordinate_im_bound ht.1).trans (hx.trans (by norm_num))⟩ - -/-- The fixed coordinate patch speed bound gives the physical guide speed -bound required by the compactness and access-localization arguments. -/ -theorem physical_guide_lipschitz - (P : ℂ → ℂ) (C : ℝ≥0) {k x : ℝ} (hx : |x| ≤ 1 / 4) - (hL : LipschitzOnWith C P (guidePatch k)) : - LipschitzOnWith (2 * C) (fun t => P (guideCoordinate k x t)) (Icc (0 : ℝ) 2) := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro u hu v hv - have hp := hL.dist_le_mul _ (guideCoordinate_mem_patch hx hu) _ - (guideCoordinate_mem_patch hx hv) - have hq := guideCoordinate_upper_chord k x u v - have hcq : dist (guideCoordinate k x u) (guideCoordinate k x v) ≤ 2 * |u - v| := by - exact hq.trans (mul_le_mul_of_nonneg_right (by linarith) (abs_nonneg _)) - have hh := mul_le_mul_of_nonneg_left hcq C.coe_nonneg - simp only [NNReal.coe_mul, NNReal.coe_ofNat, Real.dist_eq] - nlinarith - -/-- The lower patch Lipschitz bound makes the physical guide uniformly -simple, including its initial boundary point. -/ -theorem physical_guide_lower_chord - (P : ℂ → ℂ) {k x K : ℝ} (hx : |x| ≤ 1 / 4) (hK : 0 < K) - (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, - dist u v ≤ K * dist (P u) (P v)) : - ∀ u ∈ Icc (0 : ℝ) 2, ∀ v ∈ Icc (0 : ℝ) 2, - (1 / K) * |u - v| ≤ dist (P (guideCoordinate k x u)) (P (guideCoordinate k x v)) := by - intro u hu v hv - have hh := (guideCoordinate_lower_chord k x u v).trans - (hlower _ (guideCoordinate_mem_patch hx hu) _ (guideCoordinate_mem_patch hx hv)) - rw [one_div_mul_eq_div, div_le_iff₀ hK] - simpa only [mul_comm] using hh - -@[simp] theorem guideCoordinate_zero (k x : ℝ) : - guideCoordinate k x 0 = (k : ℂ) + (x : ℂ) * Complex.I := by simp [guideCoordinate] - -@[simp] theorem guideCoordinate_waypoint (k x : ℝ) : - guideCoordinate k x (1 / 2) = ((k + 1 / 2 : ℝ) : ℂ) := by - norm_num [guideCoordinate] - -/-- The paper's escape condition on the centerline supplies the -unboundedness used to select the actual surviving component. -/ -theorem futureRibbon_unbounded_of_centerline_escape - (P : ℂ → ℂ) (k : ℝ) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : - ¬ Bornology.IsBounded (P '' futureRibbon k) := by - intro hb - obtain ⟨R, _, hR⟩ := hb.exists_pos_norm_lt - obtain ⟨s, hs, hsR⟩ := ((eventually_gt_atTop k).and - (hescape.eventually (eventually_gt_atTop R))).exists - have hz : (s : ℂ) ∈ futureRibbon k := by - change k < (s : ℂ).re ∧ |(s : ℂ).im| < 1 - simpa using And.intro hs (by norm_num : |(0 : ℝ)| < 1) - exact (not_lt_of_ge hsR.le) (hR _ (mem_image_of_mem P hz)) - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonSurvival +import OAI.Probability.SLE.Geometry.PublishedPlanarDiskOpenness +import Mathlib.Analysis.Normed.Module.Connected + +namespace OAI + +/-! Coordinate balls in an embedded ribbon. These deterministic lemmas +turn the local lower Lipschitz bound into physical clearance, and identify +the future ribbon with part of the actual unbounded surviving component. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +/-- An open coordinate disk whose inverse has Lipschitz constant `K` +contains the physical disk of radius `r / K`. Compactness of the closed +coordinate disk prevents an interior physical point from escaping its image. -/ +theorem ball_subset_image_ball + (P : ℂ → ℂ) (x : ℂ) {r K : ℝ} (hr : 0 < r) (hK : 0 < K) + (hP : ContinuousOn P (closedBall x r)) + (hopen : IsOpen (P '' ball x r)) + (hlower : ∀ y ∈ closedBall x r, dist y x ≤ K * dist (P y) (P x)) : + ball (P x) (r / K) ⊆ P '' ball x r := by + let A := P '' ball x r + let Q := P '' closedBall x r + have hQ : IsCompact Q := (isCompact_closedBall x r).image_of_continuousOn hP + have hAQ : A ⊆ Q := image_mono ball_subset_closedBall + have hsub : ball (P x) (r / K) ⊆ A ∪ Qᶜ := by + intro z hz + by_cases hzQ : z ∈ Q + · left + obtain ⟨y, hy, rfl⟩ := hzQ + refine ⟨y, ?_, rfl⟩ + have hd := hlower y hy + have hz' : dist (P y) (P x) < r / K := hz + have hh := (lt_div_iff₀ hK).mp hz' + exact lt_of_le_of_lt hd (by nlinarith [hh]) + · exact Or.inr hzQ + apply IsPreconnected.subset_left_of_subset_union hopen hQ.isClosed.isOpen_compl + (disjoint_compl_right.mono_left hAQ) hsub _ isPreconnected_ball + exact ⟨P x, mem_ball_self (div_pos hr hK), mem_image_of_mem P (mem_ball_self hr)⟩ + +def closedRibbon : Set ℂ := {z | 0 ≤ z.re ∧ |z.im| ≤ 1} +def futureRibbon (k : ℝ) : Set ℂ := {z | k < z.re ∧ |z.im| < 1} + +theorem futureRibbon_subset_closedRibbon {k : ℝ} (hk : 0 ≤ k) : + futureRibbon k ⊆ closedRibbon := by + intro z hz + exact ⟨hk.trans hz.1.le, hz.2.le⟩ + +theorem futureRibbon_convex (k : ℝ) : Convex ℝ (futureRibbon k) := by + have hre : Convex ℝ {z : ℂ | k < z.re} := + convex_halfSpace_gt Complex.reLm.isLinear k + have him : Convex ℝ {z : ℂ | z.im ∈ Ioo (-1 : ℝ) 1} := + (convex_Ioo (-1 : ℝ) 1).is_linear_preimage Complex.imLm.isLinear + simpa only [futureRibbon, abs_lt, mem_Ioo, ofPred_and] using hre.inter him + +/-- Any unbounded connected free set belongs to the unbounded component +used in the actual Loewner surviving domain. -/ +theorem free_preconnected_subset_surviving + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (S : Set ℂ) + (hS : IsPreconnected S) (hfree : S ⊆ upperHalfPlane \ (γ '' Icc 0 T)) + (hunbounded : ¬ Bornology.IsBounded S) : S ⊆ survivingDomain γ T := by + intro z hz + refine ⟨hfree hz, ?_⟩ + intro hbounded + exact hunbounded (hbounded.subset (hS.subset_connectedComponentIn hz hfree)) + +/-- Embedding excludes all nonlocal pieces of the stopped past from the +future ribbon. Escape to infinity places the whole future ribbon in the +surviving component, without assuming a filled-hull containment. -/ +theorem futureRibbon_survives + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) + (hi : InjOn P closedRibbon) {k : ℝ} (hk : 0 ≤ k) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) : + P '' futureRibbon k ⊆ survivingDomain γ T := by + have hsub := futureRibbon_subset_closedRibbon hk + apply free_preconnected_subset_surviving γ T _ + ((futureRibbon_convex k).isPreconnected.image P (hP.mono hsub)) _ hunbounded + rintro z ⟨y, hy, rfl⟩ + refine ⟨hH y (hsub hy) (hk.trans_lt hy.1), ?_⟩ + intro hyPast + obtain ⟨x, hx, he⟩ := hpast hyPast + have hxy : x = y := hi hx.1 (hsub hy) he + exact (not_le_of_gt hy.1) (hxy ▸ hx.2) + +/-- A coordinate disk strictly ahead of the front and away from the two +ribbon edges gives a physical disk in the actual surviving domain. -/ +theorem coordinate_ball_clearance + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) + (hi : InjOn P closedRibbon) {k : ℝ} (hk : 0 ≤ k) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) + (x : ℂ) {r K : ℝ} (hr : 0 < r) (hK : 0 < K) + (hfront : k + r < x.re) (hedge : |x.im| + r < 1) + (hopen : IsOpen (P '' ball x r)) + (hlower : ∀ y ∈ closedBall x r, dist y x ≤ K * dist (P y) (P x)) : + ball (P x) (r / K) ⊆ survivingDomain γ T := by + have hball : closedBall x r ⊆ futureRibbon k := by + intro y hy + have hd : ‖y - x‖ ≤ r := by simpa only [mem_closedBall, dist_eq_norm] using hy + have hre := (Complex.abs_re_le_norm (y - x)).trans hd + have him := (Complex.abs_im_le_norm (y - x)).trans hd + simp only [Complex.sub_re, Complex.sub_im] at hre him + refine ⟨by linarith [(abs_le.mp hre).1], ?_⟩ + have htri : |y.im| ≤ |y.im - x.im| + |x.im| := by + simpa only [sub_add_cancel] using abs_add_le (y.im - x.im) x.im + linarith + exact (ball_subset_image_ball P x hr hK + (hP.mono (hball.trans (futureRibbon_subset_closedRibbon hk))) hopen hlower).trans + ((image_mono (ball_subset_closedBall.trans hball)).trans + (futureRibbon_survives P hP hi hk hH γ T hpast hunbounded)) + +/-- The paper's access guide, before applying the ribbon coordinate map. -/ +def guideCoordinate (k x t : ℝ) : ℂ := + ((k + t : ℝ) : ℂ) + ((x * max (1 - 4 * t) 0 : ℝ) : ℂ) * Complex.I + +@[simp] theorem guideCoordinate_re (k x t : ℝ) : + (guideCoordinate k x t).re = k + t := by simp [guideCoordinate] + +@[simp] theorem guideCoordinate_im (k x t : ℝ) : + (guideCoordinate k x t).im = x * max (1 - 4 * t) 0 := by simp [guideCoordinate] + +theorem guideCoordinate_continuous (k x : ℝ) : Continuous (guideCoordinate k x) := by + unfold guideCoordinate + fun_prop + +theorem guideCoordinate_im_bound {k x t : ℝ} (ht : 0 ≤ t) : + |(guideCoordinate k x t).im| ≤ |x| := by + rw [guideCoordinate_im, abs_mul, abs_of_nonneg (le_max_right (1 - 4 * t) 0)] + exact mul_le_of_le_one_right (abs_nonneg x) (max_le (by linarith) (by norm_num)) + +/-- The longitudinal coordinate gives the lower chord bound directly. -/ +theorem guideCoordinate_lower_chord (k x u v : ℝ) : + |u - v| ≤ dist (guideCoordinate k x u) (guideCoordinate k x v) := by + have hh := Complex.abs_re_le_norm (guideCoordinate k x u - guideCoordinate k x v) + simpa only [Complex.sub_re, guideCoordinate_re, add_sub_add_left_eq_sub, + dist_eq_norm] using hh + +theorem guideCoordinate_upper_chord (k x u v : ℝ) : + dist (guideCoordinate k x u) (guideCoordinate k x v) ≤ + (1 + 4 * |x|) * |u - v| := by + have hmax := abs_max_sub_max_le_abs (1 - 4 * u) (1 - 4 * v) 0 + have he : (1 - 4 * u) - (1 - 4 * v) = -4 * (u - v) := by ring + rw [he, abs_mul] at hmax + norm_num at hmax + have him : |x * max (1 - 4 * u) 0 - x * max (1 - 4 * v) 0| ≤ + |x| * (4 * |u - v|) := by + rw [← mul_sub, abs_mul] + exact mul_le_mul_of_nonneg_left hmax (abs_nonneg x) + have hh := Complex.norm_le_abs_re_add_abs_im + (guideCoordinate k x u - guideCoordinate k x v) + simp only [Complex.sub_re, Complex.sub_im, guideCoordinate_re, guideCoordinate_im, + add_sub_add_left_eq_sub] at hh + rw [dist_eq_norm] + nlinarith + +def guideRadius (t : ℝ) : ℝ := min t 1 / 4 + +theorem guideRadius_pos {t : ℝ} (ht : 0 < t) : 0 < guideRadius t := + div_pos (lt_min ht zero_lt_one) (by norm_num) + +/-- The guide is at most a quarter of the ribbon half-width from the +centerline; its coordinate disks have radius proportional to `min(t,1)`. -/ +theorem guideCoordinate_disk_margins + {k x t : ℝ} (hx : |x| ≤ 1 / 4) (ht : 0 < t) : + k + guideRadius t < (guideCoordinate k x t).re ∧ + |(guideCoordinate k x t).im| + guideRadius t < 1 := by + have hrt : guideRadius t ≤ t / 4 := div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num) + have hr1 : guideRadius t ≤ 1 / 4 := div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num) + have hi := (guideCoordinate_im_bound (k := k) (x := x) ht.le).trans hx + rw [guideCoordinate_re] + constructor <;> linarith + +/-- Applying the coordinate disk lemma proves the paper's guide clearance +for every positive guide time, with the actual surviving domain. -/ +theorem physical_guide_clearance + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) + (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) + (hx : |x| ≤ 1 / 4) (hK : 0 < K) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) + (hopen : ∀ t ∈ Ioc (0 : ℝ) 2, + IsOpen (P '' ball (guideCoordinate k x t) (guideRadius t))) + (hlower : ∀ t ∈ Ioc (0 : ℝ) 2, + ∀ y ∈ closedBall (guideCoordinate k x t) (guideRadius t), + dist y (guideCoordinate k x t) ≤ K * dist (P y) (P (guideCoordinate k x t))) : + ∀ t ∈ Ioc (0 : ℝ) 2, + ball (P (guideCoordinate k x t)) (min t 1 / (4 * K)) ⊆ survivingDomain γ T := by + intro t ht + obtain ⟨hfront, hedge⟩ := guideCoordinate_disk_margins (k := k) hx ht.1 + have hh := coordinate_ball_clearance P hP hi hk hH γ T hpast hunbounded + (guideCoordinate k x t) (guideRadius_pos ht.1) hK hfront hedge (hopen t ht) (hlower t ht) + simpa only [guideRadius, div_div] using hh + +/-- The fixed length-four patch appearing in the admissible ribbon class. -/ +def guidePatch (k : ℝ) : Set ℂ := {z | k ≤ z.re ∧ z.re ≤ k + 3 ∧ |z.im| ≤ 1} + +theorem guideCoordinate_disk_subset_patch + {k x t : ℝ} (hx : |x| ≤ 1 / 4) (ht : t ∈ Ioc (0 : ℝ) 2) : + closedBall (guideCoordinate k x t) (guideRadius t) ⊆ guidePatch k := by + intro y hy + have hd : ‖y - guideCoordinate k x t‖ ≤ guideRadius t := by + simpa only [mem_closedBall, dist_eq_norm] using hy + have hre := (Complex.abs_re_le_norm (y - guideCoordinate k x t)).trans hd + have him := (Complex.abs_im_le_norm (y - guideCoordinate k x t)).trans hd + simp only [Complex.sub_re, Complex.sub_im, guideCoordinate_re, guideCoordinate_im] at hre him + obtain ⟨hfront, hedge⟩ := guideCoordinate_disk_margins (k := k) hx ht.1 + have hr1 : guideRadius t ≤ 1 / 4 := + div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num) + have hi := (guideCoordinate_im_bound (k := k) (x := x) ht.1.le).trans hx + have htri : |y.im| ≤ |y.im - (guideCoordinate k x t).im| + + |(guideCoordinate k x t).im| := by + simpa only [sub_add_cancel] using + abs_add_le (y.im - (guideCoordinate k x t).im) (guideCoordinate k x t).im + change k ≤ y.re ∧ y.re ≤ k + 3 ∧ |y.im| ≤ 1 + simp only [guideCoordinate_re] at hfront + simp only [guideCoordinate_im] at htri hi + exact ⟨by linarith [(abs_le.mp hre).1], by linarith [(abs_le.mp hre).2, ht.2], + by linarith⟩ + +theorem guidePatch_subset_closedRibbon {k : ℝ} (hk : 0 ≤ k) : + guidePatch k ⊆ closedRibbon := by + intro z hz + exact ⟨hk.trans hz.1, hz.2.2⟩ + +/-- The actual fixed patch inverse-Lipschitz bound suffices for all guide +disks. The only topological parameter is the cited planar disk theorem. -/ +theorem physical_guide_clearance_of_patch + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) + (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) + (hx : |x| ≤ 1 / 4) (hK : 0 < K) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) + (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, + dist u v ≤ K * dist (P u) (P v)) : + ∀ t ∈ Ioc (0 : ℝ) 2, + ball (P (guideCoordinate k x t)) (min t 1 / (4 * K)) ⊆ survivingDomain γ T := by + apply physical_guide_clearance P hP hi hk hx hK hH γ T hpast hunbounded + · intro t ht + have hsub := (ball_subset_closedBall.trans (guideCoordinate_disk_subset_patch hx ht)).trans + (guidePatch_subset_closedRibbon hk) + exact hpublished P _ _ (guideRadius_pos ht.1) (hP.mono hsub) (hi.mono hsub) + · intro t ht y hy + have hsub := guideCoordinate_disk_subset_patch (k := k) hx ht + exact hlower y (hsub hy) _ (hsub (mem_closedBall_self (guideRadius_pos ht.1).le)) + +/-- Uniform clearance on every closed interval away from the tip is the +form used by joint physical/conformal guide compactness. -/ +theorem physical_guide_clearance_away + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) + (hi : InjOn P closedRibbon) {k x K : ℝ} (hk : 0 ≤ k) + (hx : |x| ≤ 1 / 4) (hK : 0 < K) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) + (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, + dist u v ≤ K * dist (P u) (P v)) + {δ : ℝ} (hδ : 0 < δ) (_hδ2 : δ ≤ 2) : + ∃ r : ℝ, 0 < r ∧ ∀ t ∈ Icc δ 2, + ball (P (guideCoordinate k x t)) r ⊆ survivingDomain γ T := by + refine ⟨min δ 1 / (4 * K), div_pos (lt_min hδ zero_lt_one) (by positivity), ?_⟩ + intro t ht + apply (ball_subset_ball ?_).trans (physical_guide_clearance_of_patch hpublished P hP hi + hk hx hK hH γ T hpast hunbounded hlower t ⟨hδ.trans_le ht.1, ht.2⟩) + exact div_le_div_of_nonneg_right (min_le_min_right 1 ht.1) (by positivity) + +theorem guideCoordinate_mem_patch {k x t : ℝ} + (hx : |x| ≤ 1 / 4) (ht : t ∈ Icc (0 : ℝ) 2) : + guideCoordinate k x t ∈ guidePatch k := by + change k ≤ (guideCoordinate k x t).re ∧ + (guideCoordinate k x t).re ≤ k + 3 ∧ |(guideCoordinate k x t).im| ≤ 1 + rw [guideCoordinate_re] + exact ⟨by linarith [ht.1], by linarith [ht.2], + (guideCoordinate_im_bound ht.1).trans (hx.trans (by norm_num))⟩ + +/-- The fixed coordinate patch speed bound gives the physical guide speed +bound required by the compactness and access-localization arguments. -/ +theorem physical_guide_lipschitz + (P : ℂ → ℂ) (C : ℝ≥0) {k x : ℝ} (hx : |x| ≤ 1 / 4) + (hL : LipschitzOnWith C P (guidePatch k)) : + LipschitzOnWith (2 * C) (fun t => P (guideCoordinate k x t)) (Icc (0 : ℝ) 2) := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro u hu v hv + have hp := hL.dist_le_mul _ (guideCoordinate_mem_patch hx hu) _ + (guideCoordinate_mem_patch hx hv) + have hq := guideCoordinate_upper_chord k x u v + have hcq : dist (guideCoordinate k x u) (guideCoordinate k x v) ≤ 2 * |u - v| := by + exact hq.trans (mul_le_mul_of_nonneg_right (by linarith) (abs_nonneg _)) + have hh := mul_le_mul_of_nonneg_left hcq C.coe_nonneg + simp only [NNReal.coe_mul, NNReal.coe_ofNat, Real.dist_eq] + nlinarith + +/-- The lower patch Lipschitz bound makes the physical guide uniformly +simple, including its initial boundary point. -/ +theorem physical_guide_lower_chord + (P : ℂ → ℂ) {k x K : ℝ} (hx : |x| ≤ 1 / 4) (hK : 0 < K) + (hlower : ∀ u ∈ guidePatch k, ∀ v ∈ guidePatch k, + dist u v ≤ K * dist (P u) (P v)) : + ∀ u ∈ Icc (0 : ℝ) 2, ∀ v ∈ Icc (0 : ℝ) 2, + (1 / K) * |u - v| ≤ dist (P (guideCoordinate k x u)) (P (guideCoordinate k x v)) := by + intro u hu v hv + have hh := (guideCoordinate_lower_chord k x u v).trans + (hlower _ (guideCoordinate_mem_patch hx hu) _ (guideCoordinate_mem_patch hx hv)) + rw [one_div_mul_eq_div, div_le_iff₀ hK] + simpa only [mul_comm] using hh + +@[simp] theorem guideCoordinate_zero (k x : ℝ) : + guideCoordinate k x 0 = (k : ℂ) + (x : ℂ) * Complex.I := by simp [guideCoordinate] + +@[simp] theorem guideCoordinate_waypoint (k x : ℝ) : + guideCoordinate k x (1 / 2) = ((k + 1 / 2 : ℝ) : ℂ) := by + norm_num [guideCoordinate] + +/-- The paper's escape condition on the centerline supplies the +unboundedness used to select the actual surviving component. -/ +theorem futureRibbon_unbounded_of_centerline_escape + (P : ℂ → ℂ) (k : ℝ) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : + ¬ Bornology.IsBounded (P '' futureRibbon k) := by + intro hb + obtain ⟨R, _, hR⟩ := hb.exists_pos_norm_lt + obtain ⟨s, hs, hsR⟩ := ((eventually_gt_atTop k).and + (hescape.eventually (eventually_gt_atTop R))).exists + have hz : (s : ℂ) ∈ futureRibbon k := by + change k < (s : ℂ).re ∧ |(s : ℂ).im| < 1 + simpa using And.intro hs (by norm_num : |(0 : ℝ)| < 1) + exact (not_lt_of_ge hsR.le) (hR _ (mem_image_of_mem P hz)) + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateLimit.lean b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateLimit.lean index d595d7daf11ca083ffc65d7b6d7d72d230598015..972beab4d8c6b3543ef273bc28c36123d22f620f 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateLimit.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonCoordinateLimit.lean @@ -1,138 +1,138 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateCompactness - -namespace OAI - -/-! Strict half-plane location and actual moving ribbon regions for the -compact coordinate limits. The middle region has the manuscript's fixed -longitudinal interval `(1/3,2/3)` and transverse width. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -/-- An open planar set in the closed upper half-plane cannot meet its -boundary line. -/ -theorem open_nonnegative_height_subset_upperHalfPlane - (U : Set ℂ) (hU : IsOpen U) (hheight : ∀ z ∈ U, 0 ≤ z.im) : - U ⊆ upperHalfPlane := by - intro z hz - by_contra hn - have hz0 : z.im = 0 := le_antisymm (le_of_not_gt hn) (hheight z hz) - obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp hU z hz - let w : ℂ := z - ((ε / 2 : ℝ) : ℂ) * Complex.I - have hwsub : w - z = -(((ε / 2 : ℝ) : ℂ) * Complex.I) := by dsimp [w]; ring - have hdist : dist w z = ε / 2 := by - rw [dist_eq_norm, hwsub, norm_neg, norm_mul, Complex.norm_I, mul_one] - simp only [Complex.norm_real, Real.norm_eq_abs, abs_of_pos (half_pos hε)] - have hw : w ∈ U := hball (by rw [mem_ball, hdist]; linarith) - have him : w.im = z.im - ε / 2 := by simp [w] - have hh := hheight w hw - rw [him, hz0] at hh - linarith - -/-- Uniform limits of admissible embedded ribbon patches still have -strictly positive height at every interior coordinate point. -/ -theorem uniform_limit_interior_upperHalfPlane - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) - (hP : ContinuousOn P K) (hi : InjOn P K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hheight : ∀ n z, z ∈ K → 0 ≤ (Pₙ n z).im) : - MapsTo P (interior K) upperHalfPlane := by - have hopen := isOpen_image_interior hpublished P K hP hi - have hnonneg : ∀ z ∈ P '' interior K, 0 ≤ z.im := by - rintro _ ⟨x, hx, rfl⟩ - exact ge_of_tendsto ((Complex.continuous_im.tendsto _).comp - (hconv.tendsto_at (interior_subset hx))) - (Eventually.of_forall (fun n => hheight n x (interior_subset hx))) - intro x hx - exact open_nonnegative_height_subset_upperHalfPlane _ hopen hnonneg - (mem_image_of_mem P hx) - -/-- The compactness output needed for a local ribbon patch includes -strict limiting height, without a separate positivity assumption. -/ -theorem exists_bilip_halfPlane_subsequence - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) - (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C L : ℝ≥0) (R : ℝ) - (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (htip : ∀ n, ‖Pₙ n x₀‖ ≤ R) - (hheight : ∀ n z, z ∈ K → 0 ≤ (Pₙ n z).im) : - ∃ P : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧ - TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K ∧ - LipschitzOnWith C P K ∧ - (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) ∧ - InjOn P K ∧ MapsTo P (interior K) upperHalfPlane := by - obtain ⟨P, φ, hφ, hconv, hPu, hPl, hi⟩ := - exists_bilip_uniform_subsequence Pₙ K hK x₀ hx₀ C L R hupper hlower htip - exact ⟨P, φ, hφ, hconv, hPu, hPl, hi, - uniform_limit_interior_upperHalfPlane hpublished (fun n => Pₙ (φ n)) P K - hPu.continuousOn hi hconv (fun n => hheight (φ n))⟩ - -/-- The actual physical middle strip in the source's local coordinates. -/ -def physicalMiddleRegion (P : ℂ → ℂ) (K : Set ℂ) (k ε : ℝ) : Set ℂ := - P '' {w ∈ K | (1 / 3 : ℝ) < w.re - k ∧ w.re - k < 2 / 3 ∧ |w.im| < ε} - -/-- Strict transverse margin on a compact common physical patch implies -that its moving longitudinal band is in the actual moving middle strip. -/ -theorem eventual_middle_region_inclusion - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K U : Set ℂ) (L : ℝ≥0) (k ε : ℝ) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hU : IsCompact (closure U)) (hUP : closure U ⊆ P '' K) - (hcover : ∀ᶠ n in atTop, closure U ⊆ Pₙ n '' K) - (hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε) : - ∀ᶠ n in atTop, - U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K k z ∧ - longitudinalCoordinate (Pₙ n) K k z < 2 / 3} ⊆ - physicalMiddleRegion (Pₙ n) K k ε := by - have hopen : IsOpen {w : ℂ | |w.im| < ε} := isOpen_lt Complex.continuous_im.abs continuous_const - have hcoord := eventual_compact_coordinate_region Pₙ P K (closure U) - {w : ℂ | |w.im| < ε} L hconv hlower hU hUP hcover hopen - (by rintro _ ⟨z, hz, rfl⟩; exact hwidth z hz) - filter_upwards [hcoord] with n hn - rintro z ⟨hz, hband⟩ - obtain ⟨x, hx, he⟩ := hn (subset_closure hz) - have hi := lower_lipschitz_injOn (Pₙ n) K L (hlower n) - have heq : coordinateInverse (Pₙ n) K z = x := by - rw [← he] - exact coordinateInverse_left hi hx.1 - refine ⟨x, ⟨hx.1, ?_, ?_, hx.2⟩, he⟩ - · simpa only [longitudinalCoordinate, heq] using hband.1 - · simpa only [longitudinalCoordinate, heq] using hband.2 - -/-- Physical terminal control with the inverse patch constant gives the -source's transverse coordinate bound at the attained front. -/ -theorem source_terminal_offset - (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k ε : ℝ) (z : ℂ) - (hL : 0 < L) - (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) - (hbase : ((k + 1 : ℝ) : ℂ) ∈ K) (hz : z ∈ P '' K) - (hp : longitudinalCoordinate P K k z = 1) - (hd : dist z (P ((k + 1 : ℝ) : ℂ)) < ε / (4 * (L : ℝ))) : - (coordinateInverse P K z).re = k + 1 ∧ - |(coordinateInverse P K z).im| < ε / 4 := by - constructor - · dsimp [longitudinalCoordinate] at hp - linarith - · have hh := hlower _ (coordinateInverse_mem hz) _ hbase - rw [coordinateInverse_right hz] at hh - have him : |(coordinateInverse P K z).im| ≤ - dist (coordinateInverse P K z) ((k + 1 : ℝ) : ℂ) := by - simpa only [dist_eq_norm, Complex.sub_im, Complex.ofReal_im, sub_zero] - using Complex.abs_im_le_norm (coordinateInverse P K z - ((k + 1 : ℝ) : ℂ)) - have hLr : 0 < (L : ℝ) := hL - have hden : 0 < 4 * (L : ℝ) := by positivity - have hd' := (lt_div_iff₀ hden).mp hd - have he : (L : ℝ) * dist z (P ((k + 1 : ℝ) : ℂ)) < ε / 4 := by - linarith - exact (him.trans hh).trans_lt he - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateCompactness + +namespace OAI + +/-! Strict half-plane location and actual moving ribbon regions for the +compact coordinate limits. The middle region has the manuscript's fixed +longitudinal interval `(1/3,2/3)` and transverse width. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +/-- An open planar set in the closed upper half-plane cannot meet its +boundary line. -/ +theorem open_nonnegative_height_subset_upperHalfPlane + (U : Set ℂ) (hU : IsOpen U) (hheight : ∀ z ∈ U, 0 ≤ z.im) : + U ⊆ upperHalfPlane := by + intro z hz + by_contra hn + have hz0 : z.im = 0 := le_antisymm (le_of_not_gt hn) (hheight z hz) + obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp hU z hz + let w : ℂ := z - ((ε / 2 : ℝ) : ℂ) * Complex.I + have hwsub : w - z = -(((ε / 2 : ℝ) : ℂ) * Complex.I) := by dsimp [w]; ring + have hdist : dist w z = ε / 2 := by + rw [dist_eq_norm, hwsub, norm_neg, norm_mul, Complex.norm_I, mul_one] + simp only [Complex.norm_real, Real.norm_eq_abs, abs_of_pos (half_pos hε)] + have hw : w ∈ U := hball (by rw [mem_ball, hdist]; linarith) + have him : w.im = z.im - ε / 2 := by simp [w] + have hh := hheight w hw + rw [him, hz0] at hh + linarith + +/-- Uniform limits of admissible embedded ribbon patches still have +strictly positive height at every interior coordinate point. -/ +theorem uniform_limit_interior_upperHalfPlane + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) + (hP : ContinuousOn P K) (hi : InjOn P K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hheight : ∀ n z, z ∈ K → 0 ≤ (Pₙ n z).im) : + MapsTo P (interior K) upperHalfPlane := by + have hopen := isOpen_image_interior hpublished P K hP hi + have hnonneg : ∀ z ∈ P '' interior K, 0 ≤ z.im := by + rintro _ ⟨x, hx, rfl⟩ + exact ge_of_tendsto ((Complex.continuous_im.tendsto _).comp + (hconv.tendsto_at (interior_subset hx))) + (Eventually.of_forall (fun n => hheight n x (interior_subset hx))) + intro x hx + exact open_nonnegative_height_subset_upperHalfPlane _ hopen hnonneg + (mem_image_of_mem P hx) + +/-- The compactness output needed for a local ribbon patch includes +strict limiting height, without a separate positivity assumption. -/ +theorem exists_bilip_halfPlane_subsequence + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) + (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C L : ℝ≥0) (R : ℝ) + (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (htip : ∀ n, ‖Pₙ n x₀‖ ≤ R) + (hheight : ∀ n z, z ∈ K → 0 ≤ (Pₙ n z).im) : + ∃ P : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧ + TendstoUniformlyOn (fun n => Pₙ (φ n)) P atTop K ∧ + LipschitzOnWith C P K ∧ + (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) ∧ + InjOn P K ∧ MapsTo P (interior K) upperHalfPlane := by + obtain ⟨P, φ, hφ, hconv, hPu, hPl, hi⟩ := + exists_bilip_uniform_subsequence Pₙ K hK x₀ hx₀ C L R hupper hlower htip + exact ⟨P, φ, hφ, hconv, hPu, hPl, hi, + uniform_limit_interior_upperHalfPlane hpublished (fun n => Pₙ (φ n)) P K + hPu.continuousOn hi hconv (fun n => hheight (φ n))⟩ + +/-- The actual physical middle strip in the source's local coordinates. -/ +def physicalMiddleRegion (P : ℂ → ℂ) (K : Set ℂ) (k ε : ℝ) : Set ℂ := + P '' {w ∈ K | (1 / 3 : ℝ) < w.re - k ∧ w.re - k < 2 / 3 ∧ |w.im| < ε} + +/-- Strict transverse margin on a compact common physical patch implies +that its moving longitudinal band is in the actual moving middle strip. -/ +theorem eventual_middle_region_inclusion + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K U : Set ℂ) (L : ℝ≥0) (k ε : ℝ) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hU : IsCompact (closure U)) (hUP : closure U ⊆ P '' K) + (hcover : ∀ᶠ n in atTop, closure U ⊆ Pₙ n '' K) + (hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε) : + ∀ᶠ n in atTop, + U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K k z ∧ + longitudinalCoordinate (Pₙ n) K k z < 2 / 3} ⊆ + physicalMiddleRegion (Pₙ n) K k ε := by + have hopen : IsOpen {w : ℂ | |w.im| < ε} := isOpen_lt Complex.continuous_im.abs continuous_const + have hcoord := eventual_compact_coordinate_region Pₙ P K (closure U) + {w : ℂ | |w.im| < ε} L hconv hlower hU hUP hcover hopen + (by rintro _ ⟨z, hz, rfl⟩; exact hwidth z hz) + filter_upwards [hcoord] with n hn + rintro z ⟨hz, hband⟩ + obtain ⟨x, hx, he⟩ := hn (subset_closure hz) + have hi := lower_lipschitz_injOn (Pₙ n) K L (hlower n) + have heq : coordinateInverse (Pₙ n) K z = x := by + rw [← he] + exact coordinateInverse_left hi hx.1 + refine ⟨x, ⟨hx.1, ?_, ?_, hx.2⟩, he⟩ + · simpa only [longitudinalCoordinate, heq] using hband.1 + · simpa only [longitudinalCoordinate, heq] using hband.2 + +/-- Physical terminal control with the inverse patch constant gives the +source's transverse coordinate bound at the attained front. -/ +theorem source_terminal_offset + (P : ℂ → ℂ) (K : Set ℂ) (L : ℝ≥0) (k ε : ℝ) (z : ℂ) + (hL : 0 < L) + (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) + (hbase : ((k + 1 : ℝ) : ℂ) ∈ K) (hz : z ∈ P '' K) + (hp : longitudinalCoordinate P K k z = 1) + (hd : dist z (P ((k + 1 : ℝ) : ℂ)) < ε / (4 * (L : ℝ))) : + (coordinateInverse P K z).re = k + 1 ∧ + |(coordinateInverse P K z).im| < ε / 4 := by + constructor + · dsimp [longitudinalCoordinate] at hp + linarith + · have hh := hlower _ (coordinateInverse_mem hz) _ hbase + rw [coordinateInverse_right hz] at hh + have him : |(coordinateInverse P K z).im| ≤ + dist (coordinateInverse P K z) ((k + 1 : ℝ) : ℂ) := by + simpa only [dist_eq_norm, Complex.sub_im, Complex.ofReal_im, sub_zero] + using Complex.abs_im_le_norm (coordinateInverse P K z - ((k + 1 : ℝ) : ℂ)) + have hLr : 0 < (L : ℝ) := hL + have hden : 0 < 4 * (L : ℝ) := by positivity + have hd' := (lt_div_iff₀ hden).mp hd + have he : (L : ℝ) * dist z (P ((k + 1 : ℝ) : ℂ)) < ε / 4 := by + linarith + exact (him.trans hh).trans_lt he + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonExperimentAssembly.lean b/lean/OAI/Probability/SLE/Gauge/RibbonExperimentAssembly.lean index 3ede690c854005764ba2c7432dba46f7a681ddc8..d73637a112479991a7c41e8f9d43bef859974d53 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonExperimentAssembly.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonExperimentAssembly.lean @@ -1,100 +1,100 @@ -import OAI.Probability.SLE.Gauge.RibbonMiddleMass -import OAI.Probability.SLE.Gauge.RandomPassageBounds - -namespace OAI - -/-! The dense experiment concatenates full passage-and-mass events only -on histories where every previous passage has succeeded. Their spatial -middle strips are then charged to one actual sausage. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates - -lemma adapted_successes_lower_bound_on_previous - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) - (ℱ : Filtration ℕ mΩ) (init : Set Ω) (hinit : MeasurableSet[ℱ 0] init) - (success : ℕ → Set Ω) (hmeas : ∀ n, MeasurableSet[ℱ (n + 1)] (success n)) - (q : ℝ≥0∞) - (hstep : ∀ n (A : Set Ω), MeasurableSet[ℱ n] A → - A ⊆ init ∩ ⋂ j ∈ Finset.range n, success j → q * P A ≤ P (A ∩ success n)) : - ∀ N : ℕ, q ^ N * P init ≤ P (init ∩ ⋂ n ∈ Finset.range N, success n) := by - intro N - induction N with - | zero => simp - | succ N ih => - let A := init ∩ ⋂ n ∈ Finset.range N, success n - have hA : MeasurableSet[ℱ N] A := by - apply ((ℱ.mono (Nat.zero_le N)) _ hinit).inter - apply MeasurableSet.iInter - intro n - apply MeasurableSet.iInter - intro hn - exact (ℱ.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n) - have he : (init ∩ ⋂ n ∈ Finset.range (N + 1), success n) = A ∩ success N := by - ext ω - simp only [A, mem_inter_iff, mem_iInter, Finset.mem_range] - constructor - · rintro ⟨hi, h⟩ - exact ⟨⟨hi, fun n hn => h n (Nat.lt_succ_of_lt hn)⟩, h N (Nat.lt_succ_self N)⟩ - · rintro ⟨⟨hi, h⟩, hN⟩ - refine ⟨hi, fun n hn => ?_⟩ - rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl - · exact h n hn - · exact hN - rw [he, pow_succ] - calc - q ^ N * q * P init = q * (q ^ N * P init) := by ac_rfl - _ ≤ q * P A := mul_le_mul' le_rfl ih - _ ≤ P (A ∩ success N) := hstep N A hA Subset.rfl - -lemma ribbon_probability_of_quadratic_completion - {q D : ℝ} (hq : 0 < q) (hq1 : q ≤ 1) (hD : 0 ≤ D) : - ∃ C : ℝ, 0 < C ∧ ∀ (m N : ℕ), (N : ℝ) ≤ D * (m : ℝ) ^ 2 → - Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ N := by - have hlog : Real.log q ≤ 0 := Real.log_nonpos hq.le hq1 - refine ⟨1 - D * Real.log q, by nlinarith, ?_⟩ - intro m N hN - calc - _ ≤ Real.exp ((N : ℝ) * Real.log q) := by - apply Real.exp_le_exp.mpr - have hh := mul_le_mul_of_nonpos_right hN hlog - nlinarith [sq_nonneg (m : ℝ)] - _ = q ^ N := by rw [Real.exp_nat_mul, Real.exp_log hq] - -/-- The genuine adapted successes force the desired whole-trace mass at -one prescribed mesh, using only retained disjoint middle strips. -/ -theorem ribbon_experiment_mass_event_probability - {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) - (ℱ : Filtration ℕ mΩ) (init : Set Ω) (hinit : MeasurableSet[ℱ 0] init) - (success : ℕ → Set Ω) (hmeas : ∀ n, MeasurableSet[ℱ (n + 1)] (success n)) - (q : ℝ≥0∞) - (hstep : ∀ n (A : Set Ω), MeasurableSet[ℱ n] A → - A ⊆ init ∩ ⋂ j ∈ Finset.range n, success j → q * μ A ≤ μ (A ∩ success n)) - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) (N : ℕ) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (v c ell : ℝ) - (hell : 0 < ell) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) (D : Set ℂ) - (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D) - (hmass : ∀ ω ∈ init ∩ ⋂ n ∈ Finset.range N, success n, ∀ i ∈ I, - ENNReal.ofReal (c * (ell / (m : ℝ)) ^ dimension κ) ≤ - segmentSausageCharge κ (γ ω) (s i ω) (t i ω) v - (physicalMiddleRegion P closedRibbon i ε)) : - q ^ N * μ init ≤ μ {ω | - ENNReal.ofReal (c * ell ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ - wholeSausageCharge κ (γ ω) v D} := by - refine (adapted_successes_lower_bound_on_previous μ ℱ init hinit success hmeas q hstep N).trans - (measure_mono ?_) - intro ω hω - exact dense_ribbon_mass_from_steps hpublished P hP hiP hε I κ (γ ω) - (fun i => s i ω) (fun i => t i ω) v c ell hell m hm hcard D hregions (hmass ω hω) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonMiddleMass +import OAI.Probability.SLE.Gauge.RandomPassageBounds + +namespace OAI + +/-! The dense experiment concatenates full passage-and-mass events only +on histories where every previous passage has succeeded. Their spatial +middle strips are then charged to one actual sausage. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates + +lemma adapted_successes_lower_bound_on_previous + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) + (ℱ : Filtration ℕ mΩ) (init : Set Ω) (hinit : MeasurableSet[ℱ 0] init) + (success : ℕ → Set Ω) (hmeas : ∀ n, MeasurableSet[ℱ (n + 1)] (success n)) + (q : ℝ≥0∞) + (hstep : ∀ n (A : Set Ω), MeasurableSet[ℱ n] A → + A ⊆ init ∩ ⋂ j ∈ Finset.range n, success j → q * P A ≤ P (A ∩ success n)) : + ∀ N : ℕ, q ^ N * P init ≤ P (init ∩ ⋂ n ∈ Finset.range N, success n) := by + intro N + induction N with + | zero => simp + | succ N ih => + let A := init ∩ ⋂ n ∈ Finset.range N, success n + have hA : MeasurableSet[ℱ N] A := by + apply ((ℱ.mono (Nat.zero_le N)) _ hinit).inter + apply MeasurableSet.iInter + intro n + apply MeasurableSet.iInter + intro hn + exact (ℱ.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n) + have he : (init ∩ ⋂ n ∈ Finset.range (N + 1), success n) = A ∩ success N := by + ext ω + simp only [A, mem_inter_iff, mem_iInter, Finset.mem_range] + constructor + · rintro ⟨hi, h⟩ + exact ⟨⟨hi, fun n hn => h n (Nat.lt_succ_of_lt hn)⟩, h N (Nat.lt_succ_self N)⟩ + · rintro ⟨⟨hi, h⟩, hN⟩ + refine ⟨hi, fun n hn => ?_⟩ + rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl + · exact h n hn + · exact hN + rw [he, pow_succ] + calc + q ^ N * q * P init = q * (q ^ N * P init) := by ac_rfl + _ ≤ q * P A := mul_le_mul' le_rfl ih + _ ≤ P (A ∩ success N) := hstep N A hA Subset.rfl + +lemma ribbon_probability_of_quadratic_completion + {q D : ℝ} (hq : 0 < q) (hq1 : q ≤ 1) (hD : 0 ≤ D) : + ∃ C : ℝ, 0 < C ∧ ∀ (m N : ℕ), (N : ℝ) ≤ D * (m : ℝ) ^ 2 → + Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ N := by + have hlog : Real.log q ≤ 0 := Real.log_nonpos hq.le hq1 + refine ⟨1 - D * Real.log q, by nlinarith, ?_⟩ + intro m N hN + calc + _ ≤ Real.exp ((N : ℝ) * Real.log q) := by + apply Real.exp_le_exp.mpr + have hh := mul_le_mul_of_nonpos_right hN hlog + nlinarith [sq_nonneg (m : ℝ)] + _ = q ^ N := by rw [Real.exp_nat_mul, Real.exp_log hq] + +/-- The genuine adapted successes force the desired whole-trace mass at +one prescribed mesh, using only retained disjoint middle strips. -/ +theorem ribbon_experiment_mass_event_probability + {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) + (ℱ : Filtration ℕ mΩ) (init : Set Ω) (hinit : MeasurableSet[ℱ 0] init) + (success : ℕ → Set Ω) (hmeas : ∀ n, MeasurableSet[ℱ (n + 1)] (success n)) + (q : ℝ≥0∞) + (hstep : ∀ n (A : Set Ω), MeasurableSet[ℱ n] A → + A ⊆ init ∩ ⋂ j ∈ Finset.range n, success j → q * μ A ≤ μ (A ∩ success n)) + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) (N : ℕ) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (v c ell : ℝ) + (hell : 0 < ell) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) (D : Set ℂ) + (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D) + (hmass : ∀ ω ∈ init ∩ ⋂ n ∈ Finset.range N, success n, ∀ i ∈ I, + ENNReal.ofReal (c * (ell / (m : ℝ)) ^ dimension κ) ≤ + segmentSausageCharge κ (γ ω) (s i ω) (t i ω) v + (physicalMiddleRegion P closedRibbon i ε)) : + q ^ N * μ init ≤ μ {ω | + ENNReal.ofReal (c * ell ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ + wholeSausageCharge κ (γ ω) v D} := by + refine (adapted_successes_lower_bound_on_previous μ ℱ init hinit success hmeas q hstep N).trans + (measure_mono ?_) + intro ω hω + exact dense_ribbon_mass_from_steps hpublished P hP hiP hε I κ (γ ω) + (fun i => s i ω) (fun i => t i ω) v c ell hell m hm hcard D hregions (hmass ω hω) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonFrontFreshness.lean b/lean/OAI/Probability/SLE/Gauge/RibbonFrontFreshness.lean index a07e973690229dddf2776d9108f86870510a75bf..fd81a5131df12696a32ee05ac9ca54455618a731 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonFrontFreshness.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonFrontFreshness.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry -import OAI.Probability.SLE.Loewner.FreshTipMapOut - -namespace OAI - -/-! First arrival at a ribbon front is fresh in the actual Loewner sense: -the current tip survives every strictly earlier hull. This identifies the -access prime end with the current driving point through the proved fresh-tip -chart limit. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -theorem ribbon_front_tip_survives_prior - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) - (γ x : ℝ≥0 → ℂ) (T : ℝ≥0) (hx : ContinuousOn x (Icc 0 T)) - (hx0 : (x 0).re = 0) (hcoords : MapsTo x (Icc 0 T) closedRibbon) - (hγ : EqOn γ (P ∘ x) (Icc 0 T)) - (hfront : ∀ t : ℝ≥0, t < T → (x t).re < (x T).re) - (hinterior : |(x T).im| < 1) : - ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s := by - intro s hs - have hxs := hx.mono (Icc_subset_Icc_right hs.le) - have hrex : ContinuousOn (fun t => (x t).re) (Icc 0 s) := - Complex.continuous_re.comp_continuousOn hxs - obtain ⟨t, ht, hmax⟩ := isCompact_Icc.exists_isMaxOn - (nonempty_Icc.mpr (show (0 : ℝ≥0) ≤ s from bot_le)) hrex - let k := (x t).re - have hk : 0 ≤ k := by - have hh := hmax (show (0 : ℝ≥0) ∈ Icc 0 s from ⟨le_rfl, bot_le⟩) - change (x 0).re ≤ (x t).re at hh - simpa only [hx0] using hh - have hpast : γ '' Icc 0 s ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by - rintro z ⟨u, hu, rfl⟩ - refine ⟨x u, ⟨hcoords ⟨hu.1, hu.2.trans hs.le⟩, hmax hu⟩, ?_⟩ - exact (hγ ⟨hu.1, hu.2.trans hs.le⟩).symm - have htip : x T ∈ futureRibbon k := - ⟨hfront t (ht.2.trans_lt hs), hinterior⟩ - have hsurv := futureRibbon_survives P hP hi hk hH γ s hpast - (futureRibbon_unbounded_of_centerline_escape P k hescape) - (mem_image_of_mem P htip) - rw [hγ ⟨bot_le, le_rfl⟩] - exact hsurv - -/-- The actual capacity chart tends to the current driver along the -surviving domain at a first ribbon front. -/ -theorem IsCapacityTwoWitness.ribbon_front_chart_tendsto - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) - (x : ℝ≥0 → ℂ) (T : ℝ≥0) (hT : 0 < T) (hx : ContinuousOn x (Icc 0 T)) - (hx0 : (x 0).re = 0) (hcoords : MapsTo x (Icc 0 T) closedRibbon) - (hγ : EqOn γ (P ∘ x) (Icc 0 T)) - (hfront : ∀ t : ℝ≥0, t < T → (x t).re < (x T).re) - (hinterior : |(x T).im| < 1) : - Tendsto (G T) (𝓝[survivingDomain γ T] (γ T)) (𝓝 (U T : ℂ)) := by - exact hw.fresh_tip_chart_tendsto hU hT - (ribbon_front_tip_survives_prior P hP hi hH hescape γ x T hx hx0 hcoords hγ hfront hinterior) - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry +import OAI.Probability.SLE.Loewner.FreshTipMapOut + +namespace OAI + +/-! First arrival at a ribbon front is fresh in the actual Loewner sense: +the current tip survives every strictly earlier hull. This identifies the +access prime end with the current driving point through the proved fresh-tip +chart limit. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +theorem ribbon_front_tip_survives_prior + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) + (γ x : ℝ≥0 → ℂ) (T : ℝ≥0) (hx : ContinuousOn x (Icc 0 T)) + (hx0 : (x 0).re = 0) (hcoords : MapsTo x (Icc 0 T) closedRibbon) + (hγ : EqOn γ (P ∘ x) (Icc 0 T)) + (hfront : ∀ t : ℝ≥0, t < T → (x t).re < (x T).re) + (hinterior : |(x T).im| < 1) : + ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s := by + intro s hs + have hxs := hx.mono (Icc_subset_Icc_right hs.le) + have hrex : ContinuousOn (fun t => (x t).re) (Icc 0 s) := + Complex.continuous_re.comp_continuousOn hxs + obtain ⟨t, ht, hmax⟩ := isCompact_Icc.exists_isMaxOn + (nonempty_Icc.mpr (show (0 : ℝ≥0) ≤ s from bot_le)) hrex + let k := (x t).re + have hk : 0 ≤ k := by + have hh := hmax (show (0 : ℝ≥0) ∈ Icc 0 s from ⟨le_rfl, bot_le⟩) + change (x 0).re ≤ (x t).re at hh + simpa only [hx0] using hh + have hpast : γ '' Icc 0 s ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by + rintro z ⟨u, hu, rfl⟩ + refine ⟨x u, ⟨hcoords ⟨hu.1, hu.2.trans hs.le⟩, hmax hu⟩, ?_⟩ + exact (hγ ⟨hu.1, hu.2.trans hs.le⟩).symm + have htip : x T ∈ futureRibbon k := + ⟨hfront t (ht.2.trans_lt hs), hinterior⟩ + have hsurv := futureRibbon_survives P hP hi hk hH γ s hpast + (futureRibbon_unbounded_of_centerline_escape P k hescape) + (mem_image_of_mem P htip) + rw [hγ ⟨bot_le, le_rfl⟩] + exact hsurv + +/-- The actual capacity chart tends to the current driver along the +surviving domain at a first ribbon front. -/ +theorem IsCapacityTwoWitness.ribbon_front_chart_tendsto + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) + (x : ℝ≥0 → ℂ) (T : ℝ≥0) (hT : 0 < T) (hx : ContinuousOn x (Icc 0 T)) + (hx0 : (x 0).re = 0) (hcoords : MapsTo x (Icc 0 T) closedRibbon) + (hγ : EqOn γ (P ∘ x) (Icc 0 T)) + (hfront : ∀ t : ℝ≥0, t < T → (x t).re < (x T).re) + (hinterior : |(x T).im| < 1) : + Tendsto (G T) (𝓝[survivingDomain γ T] (γ T)) (𝓝 (U T : ℂ)) := by + exact hw.fresh_tip_chart_tendsto hU hT + (ribbon_front_tip_survives_prior P hP hi hH hescape γ x T hx hx0 hcoords hγ hfront hinterior) + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonFrontStops.lean b/lean/OAI/Probability/SLE/Gauge/RibbonFrontStops.lean index fec7e3499cd3b0973321d2e7c8fb9c4987b80170..1b87d6697a0ffe4454d24ddbf73d6fadf0e5f763 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonFrontStops.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonFrontStops.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Gauge.ProperRibbonCoordinates - -namespace OAI - -/-! One globally continuous ribbon progress coordinate supplies all -integer first-front stopping times. Their order is pathwise, including -infinite failure times, and successful histories identify them exactly. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates FirstFrontPassage - -lemma firstFrontTime_mono_of_initial - (x : ℝ≥0 → ℝ) (hx : Continuous x) {a b : ℝ} (h0 : x 0 ≤ a) (hab : a ≤ b) : - firstFrontTime x a ≤ firstFrontTime x b := by - by_cases hb : ∃ t : ℝ≥0, 0 ≤ t ∧ x t - b ∈ ({0} : Set ℝ) - · have ha : ∃ t : ℝ≥0, 0 ≤ t ∧ x t - a ∈ ({0} : Set ℝ) := by - obtain ⟨t, ht, htb⟩ := hb - have he : x t = b := sub_eq_zero.mp htb - obtain ⟨u, _, hue⟩ := intermediate_value_Icc ht hx.continuousOn - ⟨h0, by rw [he]; exact hab⟩ - exact ⟨u, bot_le, by simp only [hue, sub_self, mem_singleton_iff]⟩ - unfold firstFrontTime hittingAfter - rw [ite_eq_left ha, ite_eq_left hb, WithTop.coe_le_coe] - apply le_csInf hb - intro t ht - have he : x t = b := sub_eq_zero.mp ht.2 - obtain ⟨u, hu, hue⟩ := intermediate_value_Icc ht.1 hx.continuousOn - ⟨h0, by rw [he]; exact hab⟩ - have hle : sInf {t : ℝ≥0 | 0 ≤ t ∧ x t - a ∈ ({0} : Set ℝ)} ≤ u := - csInf_le (OrderBot.bddBelow _) ⟨bot_le, by simp only [hue, sub_self, mem_singleton_iff]⟩ - exact hle.trans hu.2 - · have he : firstFrontTime x b = ⊤ := by - unfold firstFrontTime hittingAfter - rw [ite_eq_right hb] - rw [he] - exact le_top - -def ribbonFrontStop {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) - (k : ℝ) (ω : Ω) : WithTop ℝ≥0 := firstFrontTime (fun t => g (γ ω t)) k - -lemma ribbonFrontStop_stopping - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) - (ha : Adapted ℱ (fun t ω => γ ω t)) (hc : ∀ ω, Continuous (γ ω)) (k : ℝ) : - IsStoppingTime ℱ (ribbonFrontStop g γ k) := - firstFrontTime_isStoppingTime ℱ (fun t ω => g (γ ω t)) - (fun t => g.continuous.measurable.comp (ha t)) (fun ω => g.continuous.comp (hc ω)) k - -lemma ribbonFrontStop_monotone - {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) - (hc : ∀ ω, Continuous (γ ω)) (h0 : ∀ ω, g (γ ω 0) = 0) : - ∀ ω, Monotone (fun n : ℕ => ribbonFrontStop g γ (n : ℝ) ω) := by - intro ω n k hnk - exact firstFrontTime_mono_of_initial _ (g.continuous.comp (hc ω)) - (by change g (γ ω 0) ≤ (n : ℝ); rw [h0 ω]; positivity) (by exact_mod_cast hnk) - -/-- The global first front agrees with the actual time in a successful -history, irrespective of the extension away from the physical ribbon. -/ -lemma RibbonHistory.global_frontStop_eq - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (g : C(ℂ, ℝ)) - (hg : ∀ z ∈ closedRibbon, g (d.chart z) = z.re) : - firstFrontTime (fun t => g (d.trace t)) d.front = (d.time : WithTop ℝ≥0) := by - have heq (t : ℝ≥0) (ht : t ∈ Icc 0 d.time) : - g (d.trace t) = (d.coordinates t).re := by - rw [d.trace_coordinates ht] - exact hg _ (d.coordinates_ribbon ht) - have hattain : g (d.trace d.time) = d.front := by - rw [heq _ ⟨bot_le, le_rfl⟩, d.attained] - simp - have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ g (d.trace t) - d.front ∈ ({0} : Set ℝ)} - d.time := by - refine ⟨⟨bot_le, by simp only [hattain, sub_self, mem_singleton_iff]⟩, ?_⟩ - intro t ht - by_contra hn - have hlt : t < d.time := lt_of_not_ge hn - have hh := d.strict_past t hlt - rw [← heq t ⟨bot_le, hlt.le⟩, sub_eq_zero.mp ht.2] at hh - exact (lt_irrefl _ hh) - unfold firstFrontTime hittingAfter - rw [ite_eq_left ⟨d.time, hleast.1⟩, hleast.csInf_eq] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ProperRibbonCoordinates + +namespace OAI + +/-! One globally continuous ribbon progress coordinate supplies all +integer first-front stopping times. Their order is pathwise, including +infinite failure times, and successful histories identify them exactly. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates FirstFrontPassage + +lemma firstFrontTime_mono_of_initial + (x : ℝ≥0 → ℝ) (hx : Continuous x) {a b : ℝ} (h0 : x 0 ≤ a) (hab : a ≤ b) : + firstFrontTime x a ≤ firstFrontTime x b := by + by_cases hb : ∃ t : ℝ≥0, 0 ≤ t ∧ x t - b ∈ ({0} : Set ℝ) + · have ha : ∃ t : ℝ≥0, 0 ≤ t ∧ x t - a ∈ ({0} : Set ℝ) := by + obtain ⟨t, ht, htb⟩ := hb + have he : x t = b := sub_eq_zero.mp htb + obtain ⟨u, _, hue⟩ := intermediate_value_Icc ht hx.continuousOn + ⟨h0, by rw [he]; exact hab⟩ + exact ⟨u, bot_le, by simp only [hue, sub_self, mem_singleton_iff]⟩ + unfold firstFrontTime hittingAfter + rw [ite_eq_left ha, ite_eq_left hb, WithTop.coe_le_coe] + apply le_csInf hb + intro t ht + have he : x t = b := sub_eq_zero.mp ht.2 + obtain ⟨u, hu, hue⟩ := intermediate_value_Icc ht.1 hx.continuousOn + ⟨h0, by rw [he]; exact hab⟩ + have hle : sInf {t : ℝ≥0 | 0 ≤ t ∧ x t - a ∈ ({0} : Set ℝ)} ≤ u := + csInf_le (OrderBot.bddBelow _) ⟨bot_le, by simp only [hue, sub_self, mem_singleton_iff]⟩ + exact hle.trans hu.2 + · have he : firstFrontTime x b = ⊤ := by + unfold firstFrontTime hittingAfter + rw [ite_eq_right hb] + rw [he] + exact le_top + +def ribbonFrontStop {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) + (k : ℝ) (ω : Ω) : WithTop ℝ≥0 := firstFrontTime (fun t => g (γ ω t)) k + +lemma ribbonFrontStop_stopping + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) + (ha : Adapted ℱ (fun t ω => γ ω t)) (hc : ∀ ω, Continuous (γ ω)) (k : ℝ) : + IsStoppingTime ℱ (ribbonFrontStop g γ k) := + firstFrontTime_isStoppingTime ℱ (fun t ω => g (γ ω t)) + (fun t => g.continuous.measurable.comp (ha t)) (fun ω => g.continuous.comp (hc ω)) k + +lemma ribbonFrontStop_monotone + {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) + (hc : ∀ ω, Continuous (γ ω)) (h0 : ∀ ω, g (γ ω 0) = 0) : + ∀ ω, Monotone (fun n : ℕ => ribbonFrontStop g γ (n : ℝ) ω) := by + intro ω n k hnk + exact firstFrontTime_mono_of_initial _ (g.continuous.comp (hc ω)) + (by change g (γ ω 0) ≤ (n : ℝ); rw [h0 ω]; positivity) (by exact_mod_cast hnk) + +/-- The global first front agrees with the actual time in a successful +history, irrespective of the extension away from the physical ribbon. -/ +lemma RibbonHistory.global_frontStop_eq + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (g : C(ℂ, ℝ)) + (hg : ∀ z ∈ closedRibbon, g (d.chart z) = z.re) : + firstFrontTime (fun t => g (d.trace t)) d.front = (d.time : WithTop ℝ≥0) := by + have heq (t : ℝ≥0) (ht : t ∈ Icc 0 d.time) : + g (d.trace t) = (d.coordinates t).re := by + rw [d.trace_coordinates ht] + exact hg _ (d.coordinates_ribbon ht) + have hattain : g (d.trace d.time) = d.front := by + rw [heq _ ⟨bot_le, le_rfl⟩, d.attained] + simp + have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ g (d.trace t) - d.front ∈ ({0} : Set ℝ)} + d.time := by + refine ⟨⟨bot_le, by simp only [hattain, sub_self, mem_singleton_iff]⟩, ?_⟩ + intro t ht + by_contra hn + have hlt : t < d.time := lt_of_not_ge hn + have hh := d.strict_past t hlt + rw [← heq t ⟨bot_le, hlt.le⟩, sub_eq_zero.mp ht.2] at hh + exact (lt_irrefl _ hh) + unfold firstFrontTime hittingAfter + rw [ite_eq_left ⟨d.time, hleast.1⟩, hleast.csInf_eq] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonGoodPrefix.lean b/lean/OAI/Probability/SLE/Gauge/RibbonGoodPrefix.lean index 32287622db7654dadd8152b4d0af9f78d12a913d..245572969a18e07d1abcab2cb3d292efede5e05d 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonGoodPrefix.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonGoodPrefix.lean @@ -1,129 +1,129 @@ -import OAI.Probability.SLE.Gauge.ActualRibbonHistory - -namespace OAI - -/-! Narrow actual prefixes are measurable in the possibly infinite front -past. They are exactly the physical information used to reconstruct the -history at the next integer front. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability - -local instance goodPrefixPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance goodPrefixPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -def ribbonCore (P : ℂ → ℂ) (ε : ℝ) : Set ℂ := - P '' {z ∈ closedRibbon | |z.im| ≤ ε} - -lemma ribbonCore_isClosed (P : ℂ → ℂ) - (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) (ε : ℝ) : - IsClosed (ribbonCore P ε) := by - have hs : IsClosed {z : closedRibbon | |(z : ℂ).im| ≤ ε} := - isClosed_le (Complex.continuous_im.comp continuous_subtype_val).abs continuous_const - have hh := he.isClosedMap _ hs - have hset : (fun z : closedRibbon => P z) '' {z : closedRibbon | |(z : ℂ).im| ≤ ε} = - ribbonCore P ε := by - ext w - constructor - · rintro ⟨z, hz, rfl⟩ - exact ⟨z, ⟨z.2, hz⟩, rfl⟩ - · rintro ⟨z, ⟨hz, hi⟩, rfl⟩ - exact ⟨⟨z, hz⟩, hi, rfl⟩ - exact hset ▸ hh - -def GoodRibbonPrefix {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) - (τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε : ℝ) : Set Ω := - {ω | τ ω ≠ ⊤ ∧ MapsTo (γ ω) (Icc 0 ((τ ω).untopD 0)) (ribbonCore P ε) ∧ - γ ω ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)} - -lemma stoppedAtPath_eval_time (f : C(ℝ≥0, ℂ)) (τ : WithTop ℝ≥0) : - stoppedAtPath f τ (τ.untopD 0) = f (τ.untopD 0) := by - cases τ with - | top => simp only [stoppedAtPath, ite_true] - | coe t => simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe, - clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, min_self, zero_le, - max_eq_right] - -lemma stoppedAtPath_range_iff (f : C(ℝ≥0, ℂ)) (t : ℝ≥0) (K : Set ℂ) : - MapsTo (stoppedAtPath f (t : WithTop ℝ≥0)) univ K ↔ MapsTo f (Icc 0 t) K := by - simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe] - constructor - · intro h u hu - have hh := h (mem_univ u) - simpa only [clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, - min_eq_left hu.2, max_eq_right hu.1] using hh - · intro h u _ - exact h ⟨le_max_left _ _, max_le bot_le (min_le_right _ _)⟩ - -lemma measurableSet_path_range_closed - {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) - (K : Set ℂ) (hK : IsClosed K) : - MeasurableSet {ω | MapsTo (f ω) univ K} := by - have hclosed : IsClosed {p : C(ℝ≥0, ℂ) | MapsTo p univ K} := by - have he : {p : C(ℝ≥0, ℂ) | MapsTo p univ K} = - ⋂ t : ℝ≥0, (fun p : C(ℝ≥0, ℂ) => p t) ⁻¹' K := by - ext p - simp only [mem_ofPred_eq, MapsTo, mem_univ, forall_const, mem_iInter, mem_preimage] - rw [he] - exact isClosed_iInter fun t => hK.preimage (continuous_eval_const t) - exact hclosed.measurableSet.preimage hf - -/-- The successful physical prefix belongs to the actual front past; -failed samples need not have a finite front time. -/ -theorem goodRibbonPrefix_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) - (hc : ∀ ω, Continuous (γ ω)) - (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) - (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) (ε : ℝ) : - MeasurableSet[hτ.measurableSpace] (GoodRibbonPrefix γ τ P ε) := by - let f := fun ω => stoppedAtPath (sourceTracePath γ hc ω) (τ ω) - have hfm : Measurable[hτ.measurableSpace] f := stoppedAtPath_measurable ℱ γ ha hc τ hτ - have htm : Measurable[hτ.measurableSpace] (fun ω => (τ ω).untopD 0) := hτ.measurable.untopD 0 - have hem : Measurable[hτ.measurableSpace] (fun ω => f ω ((τ ω).untopD 0)) := - continuous_eval.measurable.comp (hfm.prodMk htm) - let : MeasurableSpace Ω := hτ.measurableSpace - have hm := (measurableSet_eq_fun hτ.measurable - (measurable_const : Measurable (fun _ : Ω => (⊤ : WithTop ℝ≥0)))).compl.inter - ((measurableSet_path_range_closed f hfm _ (ribbonCore_isClosed P he ε)).inter - ((ribbonCore_isClosed P he (ε / 4)).measurableSet.preimage hem)) - convert hm using 1 - ext ω - change (τ ω ≠ ⊤ ∧ MapsTo (γ ω) (Icc 0 ((τ ω).untopD 0)) (ribbonCore P ε) ∧ - γ ω ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)) ↔ - (τ ω ≠ ⊤ ∧ MapsTo (stoppedAtPath (sourceTracePath γ hc ω) (τ ω)) univ (ribbonCore P ε) ∧ - stoppedAtPath (sourceTracePath γ hc ω) (τ ω) ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)) - cases ht : τ ω with - | top => simp only [ne_eq, not_true_eq_false, false_and] - | coe t => - simp only [ne_eq, WithTop.coe_ne_top, not_false_eq_true, WithTop.untopD_coe, true_and] - rw [stoppedAtPath_range_iff] - have hv := stoppedAtPath_eval_time (sourceTracePath γ hc ω) (t : WithTop ℝ≥0) - have hv' : stoppedAtPath (sourceTracePath γ hc ω) (t : WithTop ℝ≥0) t = γ ω t := by - simpa only [WithTop.untopD_coe, sourceTracePath, ContinuousMap.coe_mk] using hv - rw [hv'] - rfl - -lemma RibbonHistory.goodRibbonPrefix - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) : - () ∈ GoodRibbonPrefix (fun _ : Unit => d.trace) (fun _ => (d.time : WithTop ℝ≥0)) - d.chart ε := by - refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ - · intro t ht - refine ⟨d.coordinates t, ⟨d.coordinates_ribbon ht, (d.past_core t ht).2⟩, ?_⟩ - exact (d.trace_coordinates ht).symm - · refine ⟨d.coordinates d.time, ⟨d.coordinates_ribbon ⟨bot_le, le_rfl⟩, ?_⟩, ?_⟩ - · rw [d.attained] - simpa only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_im, mul_one, Complex.I_re, mul_zero, add_zero, zero_add] using d.offset_bound - · exact (d.trace_coordinates ⟨bot_le, le_rfl⟩).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ActualRibbonHistory + +namespace OAI + +/-! Narrow actual prefixes are measurable in the possibly infinite front +past. They are exactly the physical information used to reconstruct the +history at the next integer front. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability + +local instance goodPrefixPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance goodPrefixPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +def ribbonCore (P : ℂ → ℂ) (ε : ℝ) : Set ℂ := + P '' {z ∈ closedRibbon | |z.im| ≤ ε} + +lemma ribbonCore_isClosed (P : ℂ → ℂ) + (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) (ε : ℝ) : + IsClosed (ribbonCore P ε) := by + have hs : IsClosed {z : closedRibbon | |(z : ℂ).im| ≤ ε} := + isClosed_le (Complex.continuous_im.comp continuous_subtype_val).abs continuous_const + have hh := he.isClosedMap _ hs + have hset : (fun z : closedRibbon => P z) '' {z : closedRibbon | |(z : ℂ).im| ≤ ε} = + ribbonCore P ε := by + ext w + constructor + · rintro ⟨z, hz, rfl⟩ + exact ⟨z, ⟨z.2, hz⟩, rfl⟩ + · rintro ⟨z, ⟨hz, hi⟩, rfl⟩ + exact ⟨⟨z, hz⟩, hi, rfl⟩ + exact hset ▸ hh + +def GoodRibbonPrefix {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) + (τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε : ℝ) : Set Ω := + {ω | τ ω ≠ ⊤ ∧ MapsTo (γ ω) (Icc 0 ((τ ω).untopD 0)) (ribbonCore P ε) ∧ + γ ω ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)} + +lemma stoppedAtPath_eval_time (f : C(ℝ≥0, ℂ)) (τ : WithTop ℝ≥0) : + stoppedAtPath f τ (τ.untopD 0) = f (τ.untopD 0) := by + cases τ with + | top => simp only [stoppedAtPath, ite_true] + | coe t => simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe, + clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, min_self, zero_le, + max_eq_right] + +lemma stoppedAtPath_range_iff (f : C(ℝ≥0, ℂ)) (t : ℝ≥0) (K : Set ℂ) : + MapsTo (stoppedAtPath f (t : WithTop ℝ≥0)) univ K ↔ MapsTo f (Icc 0 t) K := by + simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe] + constructor + · intro h u hu + have hh := h (mem_univ u) + simpa only [clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, + min_eq_left hu.2, max_eq_right hu.1] using hh + · intro h u _ + exact h ⟨le_max_left _ _, max_le bot_le (min_le_right _ _)⟩ + +lemma measurableSet_path_range_closed + {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) + (K : Set ℂ) (hK : IsClosed K) : + MeasurableSet {ω | MapsTo (f ω) univ K} := by + have hclosed : IsClosed {p : C(ℝ≥0, ℂ) | MapsTo p univ K} := by + have he : {p : C(ℝ≥0, ℂ) | MapsTo p univ K} = + ⋂ t : ℝ≥0, (fun p : C(ℝ≥0, ℂ) => p t) ⁻¹' K := by + ext p + simp only [mem_ofPred_eq, MapsTo, mem_univ, forall_const, mem_iInter, mem_preimage] + rw [he] + exact isClosed_iInter fun t => hK.preimage (continuous_eval_const t) + exact hclosed.measurableSet.preimage hf + +/-- The successful physical prefix belongs to the actual front past; +failed samples need not have a finite front time. -/ +theorem goodRibbonPrefix_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) + (hc : ∀ ω, Continuous (γ ω)) + (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) + (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) (ε : ℝ) : + MeasurableSet[hτ.measurableSpace] (GoodRibbonPrefix γ τ P ε) := by + let f := fun ω => stoppedAtPath (sourceTracePath γ hc ω) (τ ω) + have hfm : Measurable[hτ.measurableSpace] f := stoppedAtPath_measurable ℱ γ ha hc τ hτ + have htm : Measurable[hτ.measurableSpace] (fun ω => (τ ω).untopD 0) := hτ.measurable.untopD 0 + have hem : Measurable[hτ.measurableSpace] (fun ω => f ω ((τ ω).untopD 0)) := + continuous_eval.measurable.comp (hfm.prodMk htm) + let : MeasurableSpace Ω := hτ.measurableSpace + have hm := (measurableSet_eq_fun hτ.measurable + (measurable_const : Measurable (fun _ : Ω => (⊤ : WithTop ℝ≥0)))).compl.inter + ((measurableSet_path_range_closed f hfm _ (ribbonCore_isClosed P he ε)).inter + ((ribbonCore_isClosed P he (ε / 4)).measurableSet.preimage hem)) + convert hm using 1 + ext ω + change (τ ω ≠ ⊤ ∧ MapsTo (γ ω) (Icc 0 ((τ ω).untopD 0)) (ribbonCore P ε) ∧ + γ ω ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)) ↔ + (τ ω ≠ ⊤ ∧ MapsTo (stoppedAtPath (sourceTracePath γ hc ω) (τ ω)) univ (ribbonCore P ε) ∧ + stoppedAtPath (sourceTracePath γ hc ω) (τ ω) ((τ ω).untopD 0) ∈ ribbonCore P (ε / 4)) + cases ht : τ ω with + | top => simp only [ne_eq, not_true_eq_false, false_and] + | coe t => + simp only [ne_eq, WithTop.coe_ne_top, not_false_eq_true, WithTop.untopD_coe, true_and] + rw [stoppedAtPath_range_iff] + have hv := stoppedAtPath_eval_time (sourceTracePath γ hc ω) (t : WithTop ℝ≥0) + have hv' : stoppedAtPath (sourceTracePath γ hc ω) (t : WithTop ℝ≥0) t = γ ω t := by + simpa only [WithTop.untopD_coe, sourceTracePath, ContinuousMap.coe_mk] using hv + rw [hv'] + rfl + +lemma RibbonHistory.goodRibbonPrefix + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) : + () ∈ GoodRibbonPrefix (fun _ : Unit => d.trace) (fun _ => (d.time : WithTop ℝ≥0)) + d.chart ε := by + refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ + · intro t ht + refine ⟨d.coordinates t, ⟨d.coordinates_ribbon ht, (d.past_core t ht).2⟩, ?_⟩ + exact (d.trace_coordinates ht).symm + · refine ⟨d.coordinates d.time, ⟨d.coordinates_ribbon ⟨bot_le, le_rfl⟩, ?_⟩, ?_⟩ + · rw [d.attained] + simpa only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_im, mul_one, Complex.I_re, mul_zero, add_zero, zero_add] using d.offset_bound + · exact (d.trace_coordinates ⟨bot_le, le_rfl⟩).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonHistoryAdvance.lean b/lean/OAI/Probability/SLE/Gauge/RibbonHistoryAdvance.lean index 2a5048e857bdd23ee1fa942026a55b21b9fffce6..016df2c3d808d8fd951abec4d4e0321a5ee22f2a 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonHistoryAdvance.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonHistoryAdvance.lean @@ -1,263 +1,263 @@ -import OAI.Probability.SLE.Gauge.UniformRibbonStep - -namespace OAI - -/-! An actual successful passage gives another admissible stopped ribbon -history. Inverse coordinates are used only on the compact physical patch -where the successful path is contained. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates BoundaryScaleLaw FirstFrontPassage PassageSuccess - -lemma strict_before_attained_front - (x : ℝ≥0 → ℝ) (q : ℝ) (S : ℝ≥0) - (hc : ContinuousOn x (Icc 0 S)) (h0 : x 0 < q) - (hfirst : firstFrontTime x q = (S : WithTop ℝ≥0)) : - ∀ t : ℝ≥0, t < S → x t < q := by - intro t ht - by_contra hn - obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le) - (hc.mono (Icc_subset_Icc_right ht.le)) ⟨h0.le, le_of_not_gt hn⟩ - have hh := hittingAfter_le_of_mem (u := fun t (_ : Unit) => x t - q) - (ω := ()) (show (0 : ℝ≥0) ≤ u from bot_le) - (show x u - q ∈ ({0} : Set ℝ) by simp only [hue, sub_self, mem_singleton_iff]) - change firstFrontTime x q ≤ (u : WithTop ℝ≥0) at hh - rw [hfirst, WithTop.coe_le_coe] at hh - exact (not_le_of_gt ht) (hh.trans hu.2) - -def RibbonHistory.futureCoordinates {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (t : ℝ≥0) : ℂ := - (d.front : ℂ) + coordinateInverse d.stepChart ribbonStepCompact (d.trace (d.time + t)) - -def RibbonHistory.advanceCoordinates {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (t : ℝ≥0) : ℂ := - if t ≤ d.time then d.coordinates (min t d.time) else d.futureCoordinates (t - d.time) - -def RibbonHistory.futureTrace {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : C(ℝ≥0, ℂ) := - ⟨fun t => d.trace (d.time + t), - d.witness.1.comp (continuous_const.add continuous_id)⟩ - -lemma RibbonHistory.tip_in_stepCompact {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : - (d.offset : ℂ) * Complex.I ∈ ribbonStepCompact := by - change -(1 / 2 : ℝ) ≤ ((d.offset : ℂ) * Complex.I).re ∧ - ((d.offset : ℂ) * Complex.I).re ≤ 5 / 2 ∧ - |((d.offset : ℂ) * Complex.I).im| ≤ 1 / 2 - simp only [Complex.mul_re, Complex.ofReal_re, Complex.I_re, mul_zero, - Complex.ofReal_im, Complex.I_im, sub_self, Complex.mul_im, mul_one, add_zero] - exact ⟨by norm_num, by norm_num, by linarith [d.offset_bound]⟩ - -lemma RibbonHistory.futureCoordinates_zero {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : - d.futureCoordinates 0 = d.coordinates d.time := by - have htip : d.trace d.time = d.stepChart ((d.offset : ℂ) * Complex.I) := by - rw [RibbonHistory.stepChart, ← d.attained] - exact d.trace_coordinates ⟨bot_le, le_rfl⟩ - simp only [RibbonHistory.futureCoordinates, add_zero, htip, - coordinateInverse_left (lower_lipschitz_injOn _ _ L d.step_lower) - (d.tip_in_stepCompact hε), d.attained] - -/-- The completed actual trace retains continuous ribbon coordinates, -the narrow core, and the strict first-front property required for the -next application of the uniform step theorem. -/ -theorem RibbonHistory.successful_continuation - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) - (hε : ε ≤ 1 / 8) (hL : 0 < L) - (S : ℝ≥0) - (hfirst : firstFrontTime - (fun t => longitudinalCoordinate d.chart closedRibbon d.front - (d.trace (d.time + t))) 1 = (S : WithTop ℝ≥0)) - (hhit : longitudinalCoordinate d.chart closedRibbon d.front - (d.trace (d.time + S)) = 1) - (hS : 0 < S) - (hpath : MapsTo (fun t => d.trace (d.time + t)) (Icc 0 S) - (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L))) - (hoffset : ‖d.trace (d.time + S) - d.chart ((d.front : ℂ) + 1)‖ < (ε / L) / 4) : - ContinuousOn d.advanceCoordinates (Icc 0 (d.time + S)) ∧ - EqOn d.trace (d.chart ∘ d.advanceCoordinates) (Icc 0 (d.time + S)) ∧ - (d.advanceCoordinates 0).re = 0 ∧ - d.advanceCoordinates (d.time + S) = - ((d.front + 1 : ℝ) : ℂ) + ((d.futureCoordinates S).im : ℂ) * Complex.I ∧ - |(d.futureCoordinates S).im| ≤ ε / 4 ∧ - MapsTo d.advanceCoordinates (Icc 0 (d.time + S)) closedRibbon ∧ - (∀ t ∈ Icc 0 (d.time + S), - (d.advanceCoordinates t).re ≤ d.front + 1 ∧ |(d.advanceCoordinates t).im| ≤ ε) ∧ - ∀ t : ℝ≥0, t < d.time + S → (d.advanceCoordinates t).re < d.front + 1 := by - let p := longitudinalCoordinate d.chart closedRibbon d.front - let η := fun t => d.trace (d.time + t) - have hi := lower_lipschitz_injOn d.stepChart ribbonStepCompact L d.step_lower - have hmap : MapsTo η (Icc 0 S) (d.stepChart '' ribbonStepCompact) := by - intro t ht - exact image_mono inter_subset_left (hpath ht) - have hη : Continuous η := d.witness.1.comp (continuous_const.add continuous_id) - have hq : ContinuousOn d.futureCoordinates (Icc 0 S) := - continuousOn_const.add ((coordinateInverse_lipschitz d.stepChart ribbonStepCompact L - d.step_lower).continuousOn.comp hη.continuousOn hmap) - have hqzero := d.futureCoordinates_zero hε - have hpeq (t : ℝ≥0) (ht : t ∈ Icc 0 S) : - p (η t) = (d.futureCoordinates t).re - d.front := by - dsimp only [p] - rw [d.global_progress_eq_step (hmap ht)] - simp only [η, longitudinalCoordinate, RibbonHistory.futureCoordinates, Complex.add_re, - Complex.ofReal_re, add_sub_cancel_left, sub_zero] - have hp : ContinuousOn (fun t => p (η t)) (Icc 0 S) := - ((Complex.continuous_re.comp_continuousOn hq).sub continuousOn_const).congr - (fun t ht => hpeq t ht) - have hp0 : p (η 0) < 1 := by - rw [hpeq 0 ⟨bot_le, bot_le⟩, hqzero, d.attained] - simp - have hprior := strict_before_attained_front _ 1 S hp hp0 hfirst - have hqre : (d.futureCoordinates S).re = d.front + 1 := by - have hh := (hpeq S ⟨bot_le, le_rfl⟩).symm.trans hhit - linarith - have hqcore (t : ℝ≥0) (ht : t ∈ Icc 0 S) : - d.futureCoordinates t ∈ closedRibbon ∧ |(d.futureCoordinates t).im| ≤ ε := by - obtain ⟨w, ⟨hwK, hwV⟩, he⟩ := hpath ht - have hqw : d.futureCoordinates t = (d.front : ℂ) + w := by - simp only [RibbonHistory.futureCoordinates, ← he, coordinateInverse_left hi hwK] - rw [hqw] - exact ⟨d.stepCompact_maps hwK, by - simpa only [Complex.add_im, Complex.ofReal_im, zero_add] using hwV.2.2.1.le⟩ - have hqoffset : |(d.futureCoordinates S).im| ≤ ε / 4 := by - have h1 : (1 : ℂ) ∈ ribbonStepCompact := by - change -(1 / 2 : ℝ) ≤ 1 ∧ 1 ≤ 5 / 2 ∧ |(0 : ℝ)| ≤ 1 / 2 - norm_num - have hd := d.step_lower _ (coordinateInverse_mem (hmap ⟨bot_le, le_rfl⟩)) 1 h1 - rw [coordinateInverse_right (hmap ⟨bot_le, le_rfl⟩)] at hd - have habs : |(d.futureCoordinates S).im| ≤ - dist (coordinateInverse d.stepChart ribbonStepCompact (η S)) 1 := by - simpa only [RibbonHistory.futureCoordinates, Complex.add_im, Complex.ofReal_im, - zero_add, dist_eq_norm, Complex.sub_im, Complex.one_im, sub_zero] using - Complex.abs_im_le_norm (coordinateInverse d.stepChart ribbonStepCompact (η S) - 1) - have hnorm : dist (η S) (d.stepChart 1) < (ε / L) / 4 := by - simpa only [dist_eq_norm, η, RibbonHistory.stepChart] using hoffset - have hmul := mul_lt_mul_of_pos_left hnorm (show 0 < (L : ℝ) from hL) - have he : (L : ℝ) * ((ε / L) / 4) = ε / 4 := by field_simp - exact (habs.trans hd).trans (he ▸ hmul.le) - have hsub (t : ℝ≥0) (ht : t ∈ Icc 0 (d.time + S)) : t - d.time ∈ Icc 0 S := - ⟨bot_le, (tsub_le_iff_right).mpr (by simpa only [add_comm] using ht.2)⟩ - have hleft : ContinuousOn (fun t => d.coordinates (min t d.time)) - (Icc 0 (d.time + S)) := d.coordinates_continuous.comp - (continuous_id.min continuous_const).continuousOn (fun _ _ => ⟨bot_le, min_le_right _ _⟩) - have hright : ContinuousOn (fun t => d.futureCoordinates (t - d.time)) - (Icc 0 (d.time + S)) := hq.comp - (continuous_id.sub continuous_const).continuousOn hsub - have hcont : ContinuousOn d.advanceCoordinates (Icc 0 (d.time + S)) := by - unfold RibbonHistory.advanceCoordinates - apply ContinuousOn.if ?_ (hleft.mono inter_subset_left) (hright.mono inter_subset_left) - intro t ht - have htT : t = d.time := by - have hf : t ∈ frontier (Iic d.time) := ht.2 - simpa only [frontier_Iic, mem_singleton_iff] using hf - simp only [htT, min_self, tsub_self, hqzero] - have hold (t : ℝ≥0) (ht : t ≤ d.time) : d.advanceCoordinates t = d.coordinates t := by - simp only [RibbonHistory.advanceCoordinates, ite_eq_left ht, min_eq_left ht] - have hnew (t : ℝ≥0) (ht : d.time < t) : - d.advanceCoordinates t = d.futureCoordinates (t - d.time) := by - simp only [RibbonHistory.advanceCoordinates, ite_eq_right (not_le_of_gt ht)] - have hTS : d.time < d.time + S := lt_add_of_pos_right _ hS - refine ⟨hcont, ?_, ?_, ?_, hqoffset, ?_, ?_, ?_⟩ - · intro t ht - by_cases htT : t ≤ d.time - · rw [Function.comp_apply, hold t htT] - exact d.trace_coordinates ⟨bot_le, htT⟩ - · rw [Function.comp_apply, hnew t (lt_of_not_ge htT), RibbonHistory.futureCoordinates] - change d.trace t = d.stepChart (coordinateInverse d.stepChart ribbonStepCompact - (η (t - d.time))) - rw [coordinateInverse_right (hmap (hsub t ht))] - simp only [η, add_tsub_cancel_of_le (le_of_not_ge htT)] - · rw [hold 0 bot_le] - exact d.initial_front - · rw [hnew _ hTS, add_tsub_cancel_left] - apply Complex.ext <;> simp [hqre] - · intro t ht - by_cases htT : t ≤ d.time - · rw [hold t htT] - exact d.coordinates_ribbon ⟨bot_le, htT⟩ - · rw [hnew t (lt_of_not_ge htT)] - exact (hqcore _ (hsub t ht)).1 - · intro t ht - by_cases htT : t ≤ d.time - · rw [hold t htT] - obtain ⟨hr, hi⟩ := d.past_core t ⟨bot_le, htT⟩ - exact ⟨by linarith, hi⟩ - · rw [hnew t (lt_of_not_ge htT)] - refine ⟨?_, (hqcore _ (hsub t ht)).2⟩ - rcases (hsub t ht).2.eq_or_lt with he | hlt - · rw [he, hqre] - · have hh := hprior _ hlt - rw [hpeq _ (hsub t ht)] at hh - linarith - · intro t ht - by_cases htT : t ≤ d.time - · rw [hold t htT] - linarith [(d.past_core t ⟨bot_le, htT⟩).1] - · rw [hnew t (lt_of_not_ge htT)] - have hlt : t - d.time < S := by - exact (tsub_lt_iff_left (le_of_not_ge htT)).mpr (by simpa only [add_comm] using ht) - have hh := hprior _ hlt - rw [hpeq _ ⟨bot_le, hlt.le⟩] at hh - linarith - -/-- A successful full step produces the actual history at the next -integer front, with the same driver, trace, and Loewner maps. The only -new geometric data are the deterministic bounds on the next patch. -/ -theorem RibbonHistory.next_history_of_stepEvent - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) - (hε : ε ≤ 1 / 8) (hL : 0 < L) (κ v c : ℝ) - (hstep : d.futureTrace ∈ d.stepEvent κ v c) - (hupper : LipschitzOnWith C d.chart (ribbonStepPatch (d.front + 1))) - (hlower : ∀ u ∈ ribbonStepPatch (d.front + 1), - ∀ v ∈ ribbonStepPatch (d.front + 1), dist u v ≤ L * dist (d.chart u) (d.chart v)) : - ∃ d' : RibbonHistory C L ε, - d'.front = d.front + 1 ∧ - d'.time = d.time + frontTime - (fun t => longitudinalCoordinate d.chart closedRibbon d.front (d.futureTrace t)) 1 ∧ - d'.chart = d.chart ∧ d'.U = d.U ∧ d'.trace = d.trace ∧ - d'.forward = d.forward ∧ d'.inverse = d.inverse ∧ - d'.coordinates = d.advanceCoordinates := by - let S := frontTime - (fun t => longitudinalCoordinate d.chart closedRibbon d.front (d.futureTrace t)) 1 - obtain ⟨hfirst, hhit, hS, hpath, hoffset⟩ := hstep.1 - obtain ⟨hcont, htrace, hzero, hattain, hoff, hribbon, hcore, hstrict⟩ := - d.successful_continuation hε hL S hfirst hhit hS hpath hoffset - let d' : RibbonHistory C L ε := { - U := d.U - trace := d.trace - forward := d.forward - inverse := d.inverse - driver_continuous := d.driver_continuous - witness := d.witness - chart := d.chart - chart_continuous := d.chart_continuous - chart_injective := d.chart_injective - chart_upper := d.chart_upper - chart_escape := d.chart_escape - front := d.front + 1 - offset := (d.futureCoordinates S).im - front_ge_one := by linarith [d.front_ge_one] - offset_bound := hoff - patch_upper := hupper - patch_lower := hlower - time := d.time + S - time_pos := d.time_pos.trans_le (le_add_of_nonneg_right bot_le) - coordinates := d.advanceCoordinates - coordinates_continuous := hcont - trace_coordinates := htrace - initial_front := hzero - attained := hattain - coordinates_ribbon := hribbon - past_core := hcore - strict_past := hstrict } - exact ⟨d', rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.UniformRibbonStep + +namespace OAI + +/-! An actual successful passage gives another admissible stopped ribbon +history. Inverse coordinates are used only on the compact physical patch +where the successful path is contained. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates BoundaryScaleLaw FirstFrontPassage PassageSuccess + +lemma strict_before_attained_front + (x : ℝ≥0 → ℝ) (q : ℝ) (S : ℝ≥0) + (hc : ContinuousOn x (Icc 0 S)) (h0 : x 0 < q) + (hfirst : firstFrontTime x q = (S : WithTop ℝ≥0)) : + ∀ t : ℝ≥0, t < S → x t < q := by + intro t ht + by_contra hn + obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le) + (hc.mono (Icc_subset_Icc_right ht.le)) ⟨h0.le, le_of_not_gt hn⟩ + have hh := hittingAfter_le_of_mem (u := fun t (_ : Unit) => x t - q) + (ω := ()) (show (0 : ℝ≥0) ≤ u from bot_le) + (show x u - q ∈ ({0} : Set ℝ) by simp only [hue, sub_self, mem_singleton_iff]) + change firstFrontTime x q ≤ (u : WithTop ℝ≥0) at hh + rw [hfirst, WithTop.coe_le_coe] at hh + exact (not_le_of_gt ht) (hh.trans hu.2) + +def RibbonHistory.futureCoordinates {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (t : ℝ≥0) : ℂ := + (d.front : ℂ) + coordinateInverse d.stepChart ribbonStepCompact (d.trace (d.time + t)) + +def RibbonHistory.advanceCoordinates {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (t : ℝ≥0) : ℂ := + if t ≤ d.time then d.coordinates (min t d.time) else d.futureCoordinates (t - d.time) + +def RibbonHistory.futureTrace {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : C(ℝ≥0, ℂ) := + ⟨fun t => d.trace (d.time + t), + d.witness.1.comp (continuous_const.add continuous_id)⟩ + +lemma RibbonHistory.tip_in_stepCompact {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : + (d.offset : ℂ) * Complex.I ∈ ribbonStepCompact := by + change -(1 / 2 : ℝ) ≤ ((d.offset : ℂ) * Complex.I).re ∧ + ((d.offset : ℂ) * Complex.I).re ≤ 5 / 2 ∧ + |((d.offset : ℂ) * Complex.I).im| ≤ 1 / 2 + simp only [Complex.mul_re, Complex.ofReal_re, Complex.I_re, mul_zero, + Complex.ofReal_im, Complex.I_im, sub_self, Complex.mul_im, mul_one, add_zero] + exact ⟨by norm_num, by norm_num, by linarith [d.offset_bound]⟩ + +lemma RibbonHistory.futureCoordinates_zero {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : + d.futureCoordinates 0 = d.coordinates d.time := by + have htip : d.trace d.time = d.stepChart ((d.offset : ℂ) * Complex.I) := by + rw [RibbonHistory.stepChart, ← d.attained] + exact d.trace_coordinates ⟨bot_le, le_rfl⟩ + simp only [RibbonHistory.futureCoordinates, add_zero, htip, + coordinateInverse_left (lower_lipschitz_injOn _ _ L d.step_lower) + (d.tip_in_stepCompact hε), d.attained] + +/-- The completed actual trace retains continuous ribbon coordinates, +the narrow core, and the strict first-front property required for the +next application of the uniform step theorem. -/ +theorem RibbonHistory.successful_continuation + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) + (hε : ε ≤ 1 / 8) (hL : 0 < L) + (S : ℝ≥0) + (hfirst : firstFrontTime + (fun t => longitudinalCoordinate d.chart closedRibbon d.front + (d.trace (d.time + t))) 1 = (S : WithTop ℝ≥0)) + (hhit : longitudinalCoordinate d.chart closedRibbon d.front + (d.trace (d.time + S)) = 1) + (hS : 0 < S) + (hpath : MapsTo (fun t => d.trace (d.time + t)) (Icc 0 S) + (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L))) + (hoffset : ‖d.trace (d.time + S) - d.chart ((d.front : ℂ) + 1)‖ < (ε / L) / 4) : + ContinuousOn d.advanceCoordinates (Icc 0 (d.time + S)) ∧ + EqOn d.trace (d.chart ∘ d.advanceCoordinates) (Icc 0 (d.time + S)) ∧ + (d.advanceCoordinates 0).re = 0 ∧ + d.advanceCoordinates (d.time + S) = + ((d.front + 1 : ℝ) : ℂ) + ((d.futureCoordinates S).im : ℂ) * Complex.I ∧ + |(d.futureCoordinates S).im| ≤ ε / 4 ∧ + MapsTo d.advanceCoordinates (Icc 0 (d.time + S)) closedRibbon ∧ + (∀ t ∈ Icc 0 (d.time + S), + (d.advanceCoordinates t).re ≤ d.front + 1 ∧ |(d.advanceCoordinates t).im| ≤ ε) ∧ + ∀ t : ℝ≥0, t < d.time + S → (d.advanceCoordinates t).re < d.front + 1 := by + let p := longitudinalCoordinate d.chart closedRibbon d.front + let η := fun t => d.trace (d.time + t) + have hi := lower_lipschitz_injOn d.stepChart ribbonStepCompact L d.step_lower + have hmap : MapsTo η (Icc 0 S) (d.stepChart '' ribbonStepCompact) := by + intro t ht + exact image_mono inter_subset_left (hpath ht) + have hη : Continuous η := d.witness.1.comp (continuous_const.add continuous_id) + have hq : ContinuousOn d.futureCoordinates (Icc 0 S) := + continuousOn_const.add ((coordinateInverse_lipschitz d.stepChart ribbonStepCompact L + d.step_lower).continuousOn.comp hη.continuousOn hmap) + have hqzero := d.futureCoordinates_zero hε + have hpeq (t : ℝ≥0) (ht : t ∈ Icc 0 S) : + p (η t) = (d.futureCoordinates t).re - d.front := by + dsimp only [p] + rw [d.global_progress_eq_step (hmap ht)] + simp only [η, longitudinalCoordinate, RibbonHistory.futureCoordinates, Complex.add_re, + Complex.ofReal_re, add_sub_cancel_left, sub_zero] + have hp : ContinuousOn (fun t => p (η t)) (Icc 0 S) := + ((Complex.continuous_re.comp_continuousOn hq).sub continuousOn_const).congr + (fun t ht => hpeq t ht) + have hp0 : p (η 0) < 1 := by + rw [hpeq 0 ⟨bot_le, bot_le⟩, hqzero, d.attained] + simp + have hprior := strict_before_attained_front _ 1 S hp hp0 hfirst + have hqre : (d.futureCoordinates S).re = d.front + 1 := by + have hh := (hpeq S ⟨bot_le, le_rfl⟩).symm.trans hhit + linarith + have hqcore (t : ℝ≥0) (ht : t ∈ Icc 0 S) : + d.futureCoordinates t ∈ closedRibbon ∧ |(d.futureCoordinates t).im| ≤ ε := by + obtain ⟨w, ⟨hwK, hwV⟩, he⟩ := hpath ht + have hqw : d.futureCoordinates t = (d.front : ℂ) + w := by + simp only [RibbonHistory.futureCoordinates, ← he, coordinateInverse_left hi hwK] + rw [hqw] + exact ⟨d.stepCompact_maps hwK, by + simpa only [Complex.add_im, Complex.ofReal_im, zero_add] using hwV.2.2.1.le⟩ + have hqoffset : |(d.futureCoordinates S).im| ≤ ε / 4 := by + have h1 : (1 : ℂ) ∈ ribbonStepCompact := by + change -(1 / 2 : ℝ) ≤ 1 ∧ 1 ≤ 5 / 2 ∧ |(0 : ℝ)| ≤ 1 / 2 + norm_num + have hd := d.step_lower _ (coordinateInverse_mem (hmap ⟨bot_le, le_rfl⟩)) 1 h1 + rw [coordinateInverse_right (hmap ⟨bot_le, le_rfl⟩)] at hd + have habs : |(d.futureCoordinates S).im| ≤ + dist (coordinateInverse d.stepChart ribbonStepCompact (η S)) 1 := by + simpa only [RibbonHistory.futureCoordinates, Complex.add_im, Complex.ofReal_im, + zero_add, dist_eq_norm, Complex.sub_im, Complex.one_im, sub_zero] using + Complex.abs_im_le_norm (coordinateInverse d.stepChart ribbonStepCompact (η S) - 1) + have hnorm : dist (η S) (d.stepChart 1) < (ε / L) / 4 := by + simpa only [dist_eq_norm, η, RibbonHistory.stepChart] using hoffset + have hmul := mul_lt_mul_of_pos_left hnorm (show 0 < (L : ℝ) from hL) + have he : (L : ℝ) * ((ε / L) / 4) = ε / 4 := by field_simp + exact (habs.trans hd).trans (he ▸ hmul.le) + have hsub (t : ℝ≥0) (ht : t ∈ Icc 0 (d.time + S)) : t - d.time ∈ Icc 0 S := + ⟨bot_le, (tsub_le_iff_right).mpr (by simpa only [add_comm] using ht.2)⟩ + have hleft : ContinuousOn (fun t => d.coordinates (min t d.time)) + (Icc 0 (d.time + S)) := d.coordinates_continuous.comp + (continuous_id.min continuous_const).continuousOn (fun _ _ => ⟨bot_le, min_le_right _ _⟩) + have hright : ContinuousOn (fun t => d.futureCoordinates (t - d.time)) + (Icc 0 (d.time + S)) := hq.comp + (continuous_id.sub continuous_const).continuousOn hsub + have hcont : ContinuousOn d.advanceCoordinates (Icc 0 (d.time + S)) := by + unfold RibbonHistory.advanceCoordinates + apply ContinuousOn.if ?_ (hleft.mono inter_subset_left) (hright.mono inter_subset_left) + intro t ht + have htT : t = d.time := by + have hf : t ∈ frontier (Iic d.time) := ht.2 + simpa only [frontier_Iic, mem_singleton_iff] using hf + simp only [htT, min_self, tsub_self, hqzero] + have hold (t : ℝ≥0) (ht : t ≤ d.time) : d.advanceCoordinates t = d.coordinates t := by + simp only [RibbonHistory.advanceCoordinates, ite_eq_left ht, min_eq_left ht] + have hnew (t : ℝ≥0) (ht : d.time < t) : + d.advanceCoordinates t = d.futureCoordinates (t - d.time) := by + simp only [RibbonHistory.advanceCoordinates, ite_eq_right (not_le_of_gt ht)] + have hTS : d.time < d.time + S := lt_add_of_pos_right _ hS + refine ⟨hcont, ?_, ?_, ?_, hqoffset, ?_, ?_, ?_⟩ + · intro t ht + by_cases htT : t ≤ d.time + · rw [Function.comp_apply, hold t htT] + exact d.trace_coordinates ⟨bot_le, htT⟩ + · rw [Function.comp_apply, hnew t (lt_of_not_ge htT), RibbonHistory.futureCoordinates] + change d.trace t = d.stepChart (coordinateInverse d.stepChart ribbonStepCompact + (η (t - d.time))) + rw [coordinateInverse_right (hmap (hsub t ht))] + simp only [η, add_tsub_cancel_of_le (le_of_not_ge htT)] + · rw [hold 0 bot_le] + exact d.initial_front + · rw [hnew _ hTS, add_tsub_cancel_left] + apply Complex.ext <;> simp [hqre] + · intro t ht + by_cases htT : t ≤ d.time + · rw [hold t htT] + exact d.coordinates_ribbon ⟨bot_le, htT⟩ + · rw [hnew t (lt_of_not_ge htT)] + exact (hqcore _ (hsub t ht)).1 + · intro t ht + by_cases htT : t ≤ d.time + · rw [hold t htT] + obtain ⟨hr, hi⟩ := d.past_core t ⟨bot_le, htT⟩ + exact ⟨by linarith, hi⟩ + · rw [hnew t (lt_of_not_ge htT)] + refine ⟨?_, (hqcore _ (hsub t ht)).2⟩ + rcases (hsub t ht).2.eq_or_lt with he | hlt + · rw [he, hqre] + · have hh := hprior _ hlt + rw [hpeq _ (hsub t ht)] at hh + linarith + · intro t ht + by_cases htT : t ≤ d.time + · rw [hold t htT] + linarith [(d.past_core t ⟨bot_le, htT⟩).1] + · rw [hnew t (lt_of_not_ge htT)] + have hlt : t - d.time < S := by + exact (tsub_lt_iff_left (le_of_not_ge htT)).mpr (by simpa only [add_comm] using ht) + have hh := hprior _ hlt + rw [hpeq _ ⟨bot_le, hlt.le⟩] at hh + linarith + +/-- A successful full step produces the actual history at the next +integer front, with the same driver, trace, and Loewner maps. The only +new geometric data are the deterministic bounds on the next patch. -/ +theorem RibbonHistory.next_history_of_stepEvent + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) + (hε : ε ≤ 1 / 8) (hL : 0 < L) (κ v c : ℝ) + (hstep : d.futureTrace ∈ d.stepEvent κ v c) + (hupper : LipschitzOnWith C d.chart (ribbonStepPatch (d.front + 1))) + (hlower : ∀ u ∈ ribbonStepPatch (d.front + 1), + ∀ v ∈ ribbonStepPatch (d.front + 1), dist u v ≤ L * dist (d.chart u) (d.chart v)) : + ∃ d' : RibbonHistory C L ε, + d'.front = d.front + 1 ∧ + d'.time = d.time + frontTime + (fun t => longitudinalCoordinate d.chart closedRibbon d.front (d.futureTrace t)) 1 ∧ + d'.chart = d.chart ∧ d'.U = d.U ∧ d'.trace = d.trace ∧ + d'.forward = d.forward ∧ d'.inverse = d.inverse ∧ + d'.coordinates = d.advanceCoordinates := by + let S := frontTime + (fun t => longitudinalCoordinate d.chart closedRibbon d.front (d.futureTrace t)) 1 + obtain ⟨hfirst, hhit, hS, hpath, hoffset⟩ := hstep.1 + obtain ⟨hcont, htrace, hzero, hattain, hoff, hribbon, hcore, hstrict⟩ := + d.successful_continuation hε hL S hfirst hhit hS hpath hoffset + let d' : RibbonHistory C L ε := { + U := d.U + trace := d.trace + forward := d.forward + inverse := d.inverse + driver_continuous := d.driver_continuous + witness := d.witness + chart := d.chart + chart_continuous := d.chart_continuous + chart_injective := d.chart_injective + chart_upper := d.chart_upper + chart_escape := d.chart_escape + front := d.front + 1 + offset := (d.futureCoordinates S).im + front_ge_one := by linarith [d.front_ge_one] + offset_bound := hoff + patch_upper := hupper + patch_lower := hlower + time := d.time + S + time_pos := d.time_pos.trans_le (le_add_of_nonneg_right bot_le) + coordinates := d.advanceCoordinates + coordinates_continuous := hcont + trace_coordinates := htrace + initial_front := hzero + attained := hattain + coordinates_ribbon := hribbon + past_core := hcore + strict_past := hstrict } + exact ⟨d', rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonLimitPassageGeometry.lean b/lean/OAI/Probability/SLE/Gauge/RibbonLimitPassageGeometry.lean index 57f5695de5e26a4d4bcde33fb733786e292ce7dc..84e1e5d0d2fc2dc13a6a224f54b33012cf072491 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonLimitPassageGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonLimitPassageGeometry.lean @@ -1,199 +1,199 @@ -import OAI.Probability.SLE.Gauge.RibbonPassageNeighborhood - -namespace OAI - -/-! Deterministic ribbon regions used by the limiting splice. The envelope -extends beyond the attained front, while its front slice is narrowed to -supply the prescribed physical terminal tolerance. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -def passageCoordinateRegion (C : ℝ≥0) (ε η : ℝ) : Set ℂ := - {w | -(1 / 4 : ℝ) < w.re ∧ w.re < 2 ∧ |w.im| < ε ∧ - (w.re < 1 / 2 ∨ |w.im| < η / (4 * (C : ℝ)))} - -def physicalPassageRegion (P : ℂ → ℂ) (K : Set ℂ) (C : ℝ≥0) (ε η : ℝ) : Set ℂ := - P '' (K ∩ passageCoordinateRegion C ε η) - -theorem passageCoordinateRegion_isOpen (C : ℝ≥0) (ε η : ℝ) : - IsOpen (passageCoordinateRegion C ε η) := by - apply IsOpen.inter (isOpen_lt continuous_const Complex.continuous_re) - apply IsOpen.inter (isOpen_lt Complex.continuous_re continuous_const) - apply IsOpen.inter (isOpen_lt Complex.continuous_im.abs continuous_const) - exact (isOpen_lt Complex.continuous_re continuous_const).union - (isOpen_lt Complex.continuous_im.abs continuous_const) - -/-- The prescribed access guide stays in the envelope, with its entire -late part on the centerline, independently of the terminal tolerance. -/ -theorem guideCoordinate_mem_passageRegion - (C : ℝ≥0) (hC : 0 < C) {ε η x t : ℝ} - (hx : |x| < ε) (hη : 0 < η) (ht : t ∈ Icc (0 : ℝ) (3 / 2)) : - guideCoordinate 0 x t ∈ passageCoordinateRegion C ε η := by - have him := (guideCoordinate_im_bound (k := 0) (x := x) ht.1).trans_lt hx - refine ⟨by rw [guideCoordinate_re]; linarith [ht.1], - by rw [guideCoordinate_re]; linarith [ht.2], him, ?_⟩ - by_cases hsmall : t < 1 / 2 - · left - simpa only [guideCoordinate_re, zero_add] using hsmall - · right - have htlarge : 1 / 2 ≤ t := le_of_not_gt hsmall - have him0 : (guideCoordinate 0 x t).im = 0 := by - rw [guideCoordinate_im, max_eq_right (by linarith), mul_zero] - rw [him0, abs_zero] - exact div_pos hη (mul_pos (by norm_num) hC) - -/-- The thin front slice gives the terminal physical tolerance required by -`FrontSuccess`, using the actual upper patch Lipschitz constant. -/ -theorem physicalPassageRegion_terminal - (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0) (ε η : ℝ) - (hC : 0 < C) (hupper : LipschitzOnWith C P K) - (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) - (hbase : (1 : ℂ) ∈ K) {z : ℂ} - (hz : z ∈ physicalPassageRegion P K C ε η) - (hp : longitudinalCoordinate P K 0 z = 1) : - ‖z - P 1‖ < η / 4 := by - obtain ⟨w, ⟨hwK, hwV⟩, rfl⟩ := hz - have hi := lower_lipschitz_injOn P K L hlower - have hwre : w.re = 1 := by - simpa only [longitudinalCoordinate, coordinateInverse_left hi hwK, sub_zero] using hp - have him : |w.im| < η / (4 * (C : ℝ)) := by - rcases hwV.2.2.2 with hsmall | hthin - · rw [hwre] at hsmall - norm_num at hsmall - · exact hthin - have hd : dist w (1 : ℂ) = |w.im| := by - have he : w - 1 = (w.im : ℂ) * Complex.I := by - apply Complex.ext <;> simp [hwre] - rw [dist_eq_norm, he, norm_mul, Complex.norm_I, mul_one, Complex.norm_real, Real.norm_eq_abs] - have hbound := hupper.dist_le_mul w hwK 1 hbase - rw [hd] at hbound - have hCpos : 0 < (C : ℝ) := hC - have hthin : (C : ℝ) * |w.im| < η / 4 := by - have hh := (lt_div_iff₀ (mul_pos (by norm_num : (0 : ℝ) < 4) hCpos)).mp him - linarith - simpa only [dist_eq_norm] using hbound.trans_lt hthin - -/-- The actual limiting guide supplies a common physical neighborhood and -all varying-region conditions used by the retained-mass passage theorem. -/ -theorem limiting_guide_passage_neighborhood - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0) - (hC : 0 < C) (hL : 0 < L) - (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hP : ContinuousOn P K) (hPH : MapsTo P (interior K) upperHalfPlane) - (α : ℝ → ℂ) (hα : ContinuousOn α (Icc 0 2)) (x ε η : ℝ) - (hx : |x| < ε) (hη : 0 < η) - (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2)) - (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K) - (hbase : (1 : ℂ) ∈ K) : - ∃ U : Set ℂ, IsOpen U ∧ α '' Icc (0 : ℝ) (3 / 2) ⊆ U ∧ - IsCompact (closure U) ∧ closure U ⊆ P '' interior K ∧ - U ⊆ upperHalfPlane ∧ - ContinuousOn (longitudinalCoordinate P K 0) U ∧ - (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K 0) U) ∧ - TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K 0) - (longitudinalCoordinate P K 0) atTop U ∧ - (∀ᶠ n in atTop, U ⊆ physicalPassageRegion (Pₙ n) K C ε η) ∧ - (∀ n z, z ∈ physicalPassageRegion (Pₙ n) K C ε η → - longitudinalCoordinate (Pₙ n) K 0 z = 1 → ‖z - Pₙ n 1‖ < η / 4) ∧ - (∀ᶠ n in atTop, - U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K 0 z ∧ - longitudinalCoordinate (Pₙ n) K 0 z < 2 / 3} ⊆ physicalMiddleRegion (Pₙ n) K 0 ε) := by - let V := passageCoordinateRegion C ε η - let A := α '' Icc (0 : ℝ) (3 / 2) - have hA : IsCompact A := isCompact_Icc.image_of_continuousOn - (hα.mono (Icc_subset_Icc_right (by norm_num))) - have hAV : A ⊆ P '' (interior K ∩ V) := by - rintro _ ⟨t, ht, rfl⟩ - exact ⟨guideCoordinate 0 x t, ⟨hmem t ht, - guideCoordinate_mem_passageRegion C hC hx hη ht⟩, - (hshape ⟨ht.1, ht.2.trans (by norm_num)⟩).symm⟩ - obtain ⟨U, hU, hAU, hUc, hUPV, hVlate, hKlate, _⟩ := - common_coordinate_region_neighborhood hpublished Pₙ P K A V L hL hP - (fun n => (hupper n).continuousOn) hconv hlower hA - (passageCoordinateRegion_isOpen C ε η) hAV - have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left) - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hi := lower_lipschitz_injOn P K L hlim - have hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε := by - intro z hz - obtain ⟨w, hw, he⟩ := hUPV hz - rw [← he, coordinateInverse_left hi (interior_subset hw.1)] - exact hw.2.2.2.1 - refine ⟨U, hU, hAU, hUc, hUP, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ - · exact (subset_closure.trans hUP).trans (image_subset_iff.mpr hPH) - · exact (longitudinalCoordinate_continuousOn P K L 0 hlim).mono - (subset_closure.trans (hUP.trans (image_mono interior_subset))) - · filter_upwards [hKlate] with n hn - exact (longitudinalCoordinate_continuousOn (Pₙ n) K L 0 (hlower n)).mono - (subset_closure.trans (hn.trans (image_mono interior_subset))) - · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L 0 hL hP - (fun n => (hupper n).continuousOn) hconv hlower).mono (subset_closure.trans hUP) - · exact hVlate.mono (fun _ hn => subset_closure.trans hn) - · exact fun n z hz hp => physicalPassageRegion_terminal (Pₙ n) K C L ε η hC - (hupper n) (hlower n) hbase hz hp - · exact eventual_middle_region_inclusion Pₙ P K U L 0 ε hconv hlower hUc - (hUP.trans (image_mono interior_subset)) - (hKlate.mono (fun _ hn => hn.trans (image_mono interior_subset))) hwidth - -/-- Local coordinate convergence and convergence of the starting tips -supply the fixed early ball used by localization. -/ -theorem eventual_early_coordinate_ball - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (aₙ : ℕ → ℂ) (a : ℂ) (ha : a ∈ U) (hpa : p a = 0) - (halim : Tendsto aₙ atTop (𝓝 a)) : - ∃ r : ℝ, 0 < r ∧ ball a (r / 2) ⊆ U ∧ - (∀ z ∈ ball a (r / 4), p z < 1 / 3) ∧ - ∀ᶠ n in atTop, ball (aₙ n) r ⊆ U ∩ {z | pₙ n z < 1} := by - let W := U ∩ p ⁻¹' Iio (1 / 6 : ℝ) - have hW : IsOpen W := hp.isOpen_inter_preimage hU isOpen_Iio - have haW : a ∈ W := ⟨ha, by simp only [mem_preimage, mem_Iio, hpa]; norm_num⟩ - obtain ⟨d, hd, hball⟩ := Metric.isOpen_iff.mp hW a haW - let r := d / 4 - have hr : 0 < r := by dsimp [r]; positivity - let C := closedBall a (d / 2) - have hCW : C ⊆ W := by - intro z hz - have hz' : dist z a ≤ d / 2 := hz - exact hball (hz'.trans_lt (by linarith)) - have hCU : C ⊆ U := hCW.trans inter_subset_left - have hu : TendstoUniformlyOn pₙ p atTop C := - (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact - (isCompact_closedBall a (d / 2))).mp (hconv.mono hCU) - refine ⟨r, hr, ?_, ?_, ?_⟩ - · intro z hz - have hrd : r / 2 < d := by dsimp only [r]; linarith - exact (hball (lt_trans hz hrd)).1 - · intro z hz - have hrd : r / 4 < d := by dsimp only [r]; linarith - have hh := (hball (lt_trans hz hrd)).2 - exact hh.trans (by norm_num) - · filter_upwards [halim.eventually (Metric.ball_mem_nhds a hr), - Metric.tendstoUniformlyOn_iff.mp hu (1 / 6) (by norm_num)] with n hn hclose - intro z hz - have hza : dist z a < d / 2 := by - have hh := (dist_triangle z (aₙ n) a).trans_lt (add_lt_add hz hn) - dsimp [r] at hh - linarith - have hzC : z ∈ C := hza.le - have hc := hclose z hzC - rw [Real.dist_eq] at hc - have hpz := (hCW hzC).2 - refine ⟨hCU hzC, ?_⟩ - have habs := (abs_lt.mp hc).1 - change p z < 1 / 6 at hpz - change pₙ n z < 1 - linarith - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonPassageNeighborhood + +namespace OAI + +/-! Deterministic ribbon regions used by the limiting splice. The envelope +extends beyond the attained front, while its front slice is narrowed to +supply the prescribed physical terminal tolerance. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +def passageCoordinateRegion (C : ℝ≥0) (ε η : ℝ) : Set ℂ := + {w | -(1 / 4 : ℝ) < w.re ∧ w.re < 2 ∧ |w.im| < ε ∧ + (w.re < 1 / 2 ∨ |w.im| < η / (4 * (C : ℝ)))} + +def physicalPassageRegion (P : ℂ → ℂ) (K : Set ℂ) (C : ℝ≥0) (ε η : ℝ) : Set ℂ := + P '' (K ∩ passageCoordinateRegion C ε η) + +theorem passageCoordinateRegion_isOpen (C : ℝ≥0) (ε η : ℝ) : + IsOpen (passageCoordinateRegion C ε η) := by + apply IsOpen.inter (isOpen_lt continuous_const Complex.continuous_re) + apply IsOpen.inter (isOpen_lt Complex.continuous_re continuous_const) + apply IsOpen.inter (isOpen_lt Complex.continuous_im.abs continuous_const) + exact (isOpen_lt Complex.continuous_re continuous_const).union + (isOpen_lt Complex.continuous_im.abs continuous_const) + +/-- The prescribed access guide stays in the envelope, with its entire +late part on the centerline, independently of the terminal tolerance. -/ +theorem guideCoordinate_mem_passageRegion + (C : ℝ≥0) (hC : 0 < C) {ε η x t : ℝ} + (hx : |x| < ε) (hη : 0 < η) (ht : t ∈ Icc (0 : ℝ) (3 / 2)) : + guideCoordinate 0 x t ∈ passageCoordinateRegion C ε η := by + have him := (guideCoordinate_im_bound (k := 0) (x := x) ht.1).trans_lt hx + refine ⟨by rw [guideCoordinate_re]; linarith [ht.1], + by rw [guideCoordinate_re]; linarith [ht.2], him, ?_⟩ + by_cases hsmall : t < 1 / 2 + · left + simpa only [guideCoordinate_re, zero_add] using hsmall + · right + have htlarge : 1 / 2 ≤ t := le_of_not_gt hsmall + have him0 : (guideCoordinate 0 x t).im = 0 := by + rw [guideCoordinate_im, max_eq_right (by linarith), mul_zero] + rw [him0, abs_zero] + exact div_pos hη (mul_pos (by norm_num) hC) + +/-- The thin front slice gives the terminal physical tolerance required by +`FrontSuccess`, using the actual upper patch Lipschitz constant. -/ +theorem physicalPassageRegion_terminal + (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0) (ε η : ℝ) + (hC : 0 < C) (hupper : LipschitzOnWith C P K) + (hlower : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ L * dist (P x) (P y)) + (hbase : (1 : ℂ) ∈ K) {z : ℂ} + (hz : z ∈ physicalPassageRegion P K C ε η) + (hp : longitudinalCoordinate P K 0 z = 1) : + ‖z - P 1‖ < η / 4 := by + obtain ⟨w, ⟨hwK, hwV⟩, rfl⟩ := hz + have hi := lower_lipschitz_injOn P K L hlower + have hwre : w.re = 1 := by + simpa only [longitudinalCoordinate, coordinateInverse_left hi hwK, sub_zero] using hp + have him : |w.im| < η / (4 * (C : ℝ)) := by + rcases hwV.2.2.2 with hsmall | hthin + · rw [hwre] at hsmall + norm_num at hsmall + · exact hthin + have hd : dist w (1 : ℂ) = |w.im| := by + have he : w - 1 = (w.im : ℂ) * Complex.I := by + apply Complex.ext <;> simp [hwre] + rw [dist_eq_norm, he, norm_mul, Complex.norm_I, mul_one, Complex.norm_real, Real.norm_eq_abs] + have hbound := hupper.dist_le_mul w hwK 1 hbase + rw [hd] at hbound + have hCpos : 0 < (C : ℝ) := hC + have hthin : (C : ℝ) * |w.im| < η / 4 := by + have hh := (lt_div_iff₀ (mul_pos (by norm_num : (0 : ℝ) < 4) hCpos)).mp him + linarith + simpa only [dist_eq_norm] using hbound.trans_lt hthin + +/-- The actual limiting guide supplies a common physical neighborhood and +all varying-region conditions used by the retained-mass passage theorem. -/ +theorem limiting_guide_passage_neighborhood + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0) + (hC : 0 < C) (hL : 0 < L) + (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hP : ContinuousOn P K) (hPH : MapsTo P (interior K) upperHalfPlane) + (α : ℝ → ℂ) (hα : ContinuousOn α (Icc 0 2)) (x ε η : ℝ) + (hx : |x| < ε) (hη : 0 < η) + (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2)) + (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K) + (hbase : (1 : ℂ) ∈ K) : + ∃ U : Set ℂ, IsOpen U ∧ α '' Icc (0 : ℝ) (3 / 2) ⊆ U ∧ + IsCompact (closure U) ∧ closure U ⊆ P '' interior K ∧ + U ⊆ upperHalfPlane ∧ + ContinuousOn (longitudinalCoordinate P K 0) U ∧ + (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K 0) U) ∧ + TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K 0) + (longitudinalCoordinate P K 0) atTop U ∧ + (∀ᶠ n in atTop, U ⊆ physicalPassageRegion (Pₙ n) K C ε η) ∧ + (∀ n z, z ∈ physicalPassageRegion (Pₙ n) K C ε η → + longitudinalCoordinate (Pₙ n) K 0 z = 1 → ‖z - Pₙ n 1‖ < η / 4) ∧ + (∀ᶠ n in atTop, + U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K 0 z ∧ + longitudinalCoordinate (Pₙ n) K 0 z < 2 / 3} ⊆ physicalMiddleRegion (Pₙ n) K 0 ε) := by + let V := passageCoordinateRegion C ε η + let A := α '' Icc (0 : ℝ) (3 / 2) + have hA : IsCompact A := isCompact_Icc.image_of_continuousOn + (hα.mono (Icc_subset_Icc_right (by norm_num))) + have hAV : A ⊆ P '' (interior K ∩ V) := by + rintro _ ⟨t, ht, rfl⟩ + exact ⟨guideCoordinate 0 x t, ⟨hmem t ht, + guideCoordinate_mem_passageRegion C hC hx hη ht⟩, + (hshape ⟨ht.1, ht.2.trans (by norm_num)⟩).symm⟩ + obtain ⟨U, hU, hAU, hUc, hUPV, hVlate, hKlate, _⟩ := + common_coordinate_region_neighborhood hpublished Pₙ P K A V L hL hP + (fun n => (hupper n).continuousOn) hconv hlower hA + (passageCoordinateRegion_isOpen C ε η) hAV + have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left) + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hi := lower_lipschitz_injOn P K L hlim + have hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε := by + intro z hz + obtain ⟨w, hw, he⟩ := hUPV hz + rw [← he, coordinateInverse_left hi (interior_subset hw.1)] + exact hw.2.2.2.1 + refine ⟨U, hU, hAU, hUc, hUP, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ + · exact (subset_closure.trans hUP).trans (image_subset_iff.mpr hPH) + · exact (longitudinalCoordinate_continuousOn P K L 0 hlim).mono + (subset_closure.trans (hUP.trans (image_mono interior_subset))) + · filter_upwards [hKlate] with n hn + exact (longitudinalCoordinate_continuousOn (Pₙ n) K L 0 (hlower n)).mono + (subset_closure.trans (hn.trans (image_mono interior_subset))) + · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L 0 hL hP + (fun n => (hupper n).continuousOn) hconv hlower).mono (subset_closure.trans hUP) + · exact hVlate.mono (fun _ hn => subset_closure.trans hn) + · exact fun n z hz hp => physicalPassageRegion_terminal (Pₙ n) K C L ε η hC + (hupper n) (hlower n) hbase hz hp + · exact eventual_middle_region_inclusion Pₙ P K U L 0 ε hconv hlower hUc + (hUP.trans (image_mono interior_subset)) + (hKlate.mono (fun _ hn => hn.trans (image_mono interior_subset))) hwidth + +/-- Local coordinate convergence and convergence of the starting tips +supply the fixed early ball used by localization. -/ +theorem eventual_early_coordinate_ball + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (aₙ : ℕ → ℂ) (a : ℂ) (ha : a ∈ U) (hpa : p a = 0) + (halim : Tendsto aₙ atTop (𝓝 a)) : + ∃ r : ℝ, 0 < r ∧ ball a (r / 2) ⊆ U ∧ + (∀ z ∈ ball a (r / 4), p z < 1 / 3) ∧ + ∀ᶠ n in atTop, ball (aₙ n) r ⊆ U ∩ {z | pₙ n z < 1} := by + let W := U ∩ p ⁻¹' Iio (1 / 6 : ℝ) + have hW : IsOpen W := hp.isOpen_inter_preimage hU isOpen_Iio + have haW : a ∈ W := ⟨ha, by simp only [mem_preimage, mem_Iio, hpa]; norm_num⟩ + obtain ⟨d, hd, hball⟩ := Metric.isOpen_iff.mp hW a haW + let r := d / 4 + have hr : 0 < r := by dsimp [r]; positivity + let C := closedBall a (d / 2) + have hCW : C ⊆ W := by + intro z hz + have hz' : dist z a ≤ d / 2 := hz + exact hball (hz'.trans_lt (by linarith)) + have hCU : C ⊆ U := hCW.trans inter_subset_left + have hu : TendstoUniformlyOn pₙ p atTop C := + (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact + (isCompact_closedBall a (d / 2))).mp (hconv.mono hCU) + refine ⟨r, hr, ?_, ?_, ?_⟩ + · intro z hz + have hrd : r / 2 < d := by dsimp only [r]; linarith + exact (hball (lt_trans hz hrd)).1 + · intro z hz + have hrd : r / 4 < d := by dsimp only [r]; linarith + have hh := (hball (lt_trans hz hrd)).2 + exact hh.trans (by norm_num) + · filter_upwards [halim.eventually (Metric.ball_mem_nhds a hr), + Metric.tendstoUniformlyOn_iff.mp hu (1 / 6) (by norm_num)] with n hn hclose + intro z hz + have hza : dist z a < d / 2 := by + have hh := (dist_triangle z (aₙ n) a).trans_lt (add_lt_add hz hn) + dsimp [r] at hh + linarith + have hzC : z ∈ C := hza.le + have hc := hclose z hzC + rw [Real.dist_eq] at hc + have hpz := (hCW hzC).2 + refine ⟨hCU hzC, ?_⟩ + have habs := (abs_lt.mp hc).1 + change p z < 1 / 6 at hpz + change pₙ n z < 1 + linarith + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonLimitSplice.lean b/lean/OAI/Probability/SLE/Gauge/RibbonLimitSplice.lean index b3652ce665935a1b1378da0047c39271b06019dd..f37822938f2d172196c5efdfd22765d1d622f766 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonLimitSplice.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonLimitSplice.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry -import OAI.Probability.SLE.Geometry.GuideSplice - -namespace OAI - -/-! The limiting coordinate guide supplies the actual small-circle splice, -its open continuation region, and a target strictly beyond the next front. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -def limitingFrontTarget (F : ℂ → ℂ) (p : ℂ → ℝ) (U : Set ℂ) : Set ℂ := - upperHalfPlane ∩ F ⁻¹' (U ∩ p ⁻¹' Ioi 1) - -theorem limitingFrontTarget_isOpen - (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) - (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) (hp : ContinuousOn p U) : - IsOpen (limitingFrontTarget F p U) := - hF.isOpen_inter_preimage isOpen_upperHalfPlane (hp.isOpen_inter_preimage hU isOpen_Ioi) - -/-- No small-image or last-crossing hypothesis is needed for the limiting -splice: both follow from the actual continuous inverse guide and its -boundary value. -/ -theorem limiting_guide_splice_geometry - (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) - (hi : InjOn F upperHalfPlane) (α β : ℝ → ℂ) - (hα : ContinuousOn α (Icc 0 (3 / 2 : ℝ))) - (hβ : ContinuousOn β (Icc 0 (3 / 2 : ℝ))) (hβ0 : β 0 = 0) - (hβH : MapsTo β (Ioc 0 (3 / 2 : ℝ)) upperHalfPlane) - (hrel : EqOn (F ∘ β) α (Ioc 0 (3 / 2 : ℝ))) - (homit : α 0 ∉ F '' upperHalfPlane) - (U : Set ℂ) (hU : IsOpen U) (p : ℂ → ℝ) (hp : ContinuousOn p U) - (hαU : α '' Icc (0 : ℝ) (3 / 2) ⊆ U) - (hprogress : ∀ t ∈ Icc (0 : ℝ) (3 / 2), p (α t) = t) - (r m : ℝ) (hr : 0 < r) (hm : 0 < m) : - ∃ B tB : ℝ, 0 < B ∧ B / 2 ≤ m ∧ 0 < tB ∧ tB < 3 / 2 ∧ - ‖β tB‖ = B ∧ (∀ t ∈ Ioc tB (3 / 2 : ℝ), B < ‖β t‖) ∧ - (∀ θ ∈ Ioo 0 Real.pi, - ‖F (GuideSplice.circlePoint B θ) - α 0‖ < r / 4) ∧ - (∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - α 0‖ < r / 4) ∧ - F '' (β '' Icc tB (3 / 2 : ℝ)) ⊆ U ∧ - IsOpen (limitingFrontTarget F p U) ∧ - β (3 / 2) ∈ limitingFrontTarget F p U ∧ - (∀ z ∈ limitingFrontTarget F p U, 1 < p (F z)) := by - have hT : (0 : ℝ) < 3 / 2 := by norm_num - have ha0 := hα 0 ⟨le_rfl, hT.le⟩ - have haIoc : Tendsto α (𝓝[Ioc (0 : ℝ) (3 / 2)] 0) (𝓝 (α 0)) := - ha0.mono_left (nhdsWithin_mono _ Ioc_subset_Icc_self) - have heq : (F ∘ β) =ᶠ[𝓝[Ioc (0 : ℝ) (3 / 2)] 0] α := - eventually_mem_nhdsWithin.mono (fun _ ht => hrel ht) - have hlimit : Tendsto (F ∘ β) (𝓝[>] 0) (𝓝 (α 0)) := by - rw [← nhdsWithin_Ioc_eq_nhdsGT hT] - exact haIoc.congr' heq.symm - have homit' : ∀ z, 0 < z.im → F z ≠ α 0 := by - intro z hz he - exact homit ⟨z, hz, he⟩ - have hendH : 0 < (β (3 / 2)).im := hβH ⟨hT, le_rfl⟩ - have hnorm : 0 < ‖β (3 / 2)‖ := - (lt_of_lt_of_le hendH (Complex.im_le_norm _)) - have hbound : 0 < min (2 * m) ‖β (3 / 2)‖ := lt_min (by positivity) hnorm - obtain ⟨B, hB, hcircle, hsector⟩ := - SmallImageSemicircle.access_guide_has_arbitrarily_small_quarter_image_sector hF hi homit' - hr hT hβ hβ0 (fun t ht => hβH ht) hlimit _ hbound - have hBend : B < ‖β (3 / 2)‖ := hB.2.trans_le (min_le_right _ _) - obtain ⟨tB, htB, hcross, hafter⟩ := GuideSplice.exists_last_circle_crossing hB.1 hT le_rfl - hβ hβ0 (fun t ht => by have he : t = 3 / 2 := le_antisymm ht.2 ht.1; simpa only [he] using hBend) - refine ⟨B, tB, hB.1, ?_, htB.1, htB.2, hcross, hafter, hcircle, hsector, ?_, - limitingFrontTarget_isOpen F hF.continuousOn p U hU hp, ?_, ?_⟩ - · have hh := hB.2.trans_le (min_le_left _ _) - linarith - · rintro _ ⟨z, ⟨t, ht, rfl⟩, rfl⟩ - rw [show F (β t) = α t from hrel ⟨htB.1.trans_le ht.1, ht.2⟩] - exact hαU (mem_image_of_mem α ⟨htB.1.le.trans ht.1, ht.2⟩) - · refine ⟨hendH, ?_, ?_⟩ - · rw [show F (β (3 / 2)) = α (3 / 2) from hrel ⟨hT, le_rfl⟩] - exact hαU (mem_image_of_mem α ⟨hT.le, le_rfl⟩) - · change 1 < p (F (β (3 / 2))) - rw [show F (β (3 / 2)) = α (3 / 2) from hrel ⟨hT, le_rfl⟩, - hprogress _ ⟨hT.le, le_rfl⟩] - norm_num - · intro z hz - exact hz.2.2 - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry +import OAI.Probability.SLE.Geometry.GuideSplice + +namespace OAI + +/-! The limiting coordinate guide supplies the actual small-circle splice, +its open continuation region, and a target strictly beyond the next front. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +def limitingFrontTarget (F : ℂ → ℂ) (p : ℂ → ℝ) (U : Set ℂ) : Set ℂ := + upperHalfPlane ∩ F ⁻¹' (U ∩ p ⁻¹' Ioi 1) + +theorem limitingFrontTarget_isOpen + (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) + (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) (hp : ContinuousOn p U) : + IsOpen (limitingFrontTarget F p U) := + hF.isOpen_inter_preimage isOpen_upperHalfPlane (hp.isOpen_inter_preimage hU isOpen_Ioi) + +/-- No small-image or last-crossing hypothesis is needed for the limiting +splice: both follow from the actual continuous inverse guide and its +boundary value. -/ +theorem limiting_guide_splice_geometry + (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) + (hi : InjOn F upperHalfPlane) (α β : ℝ → ℂ) + (hα : ContinuousOn α (Icc 0 (3 / 2 : ℝ))) + (hβ : ContinuousOn β (Icc 0 (3 / 2 : ℝ))) (hβ0 : β 0 = 0) + (hβH : MapsTo β (Ioc 0 (3 / 2 : ℝ)) upperHalfPlane) + (hrel : EqOn (F ∘ β) α (Ioc 0 (3 / 2 : ℝ))) + (homit : α 0 ∉ F '' upperHalfPlane) + (U : Set ℂ) (hU : IsOpen U) (p : ℂ → ℝ) (hp : ContinuousOn p U) + (hαU : α '' Icc (0 : ℝ) (3 / 2) ⊆ U) + (hprogress : ∀ t ∈ Icc (0 : ℝ) (3 / 2), p (α t) = t) + (r m : ℝ) (hr : 0 < r) (hm : 0 < m) : + ∃ B tB : ℝ, 0 < B ∧ B / 2 ≤ m ∧ 0 < tB ∧ tB < 3 / 2 ∧ + ‖β tB‖ = B ∧ (∀ t ∈ Ioc tB (3 / 2 : ℝ), B < ‖β t‖) ∧ + (∀ θ ∈ Ioo 0 Real.pi, + ‖F (GuideSplice.circlePoint B θ) - α 0‖ < r / 4) ∧ + (∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - α 0‖ < r / 4) ∧ + F '' (β '' Icc tB (3 / 2 : ℝ)) ⊆ U ∧ + IsOpen (limitingFrontTarget F p U) ∧ + β (3 / 2) ∈ limitingFrontTarget F p U ∧ + (∀ z ∈ limitingFrontTarget F p U, 1 < p (F z)) := by + have hT : (0 : ℝ) < 3 / 2 := by norm_num + have ha0 := hα 0 ⟨le_rfl, hT.le⟩ + have haIoc : Tendsto α (𝓝[Ioc (0 : ℝ) (3 / 2)] 0) (𝓝 (α 0)) := + ha0.mono_left (nhdsWithin_mono _ Ioc_subset_Icc_self) + have heq : (F ∘ β) =ᶠ[𝓝[Ioc (0 : ℝ) (3 / 2)] 0] α := + eventually_mem_nhdsWithin.mono (fun _ ht => hrel ht) + have hlimit : Tendsto (F ∘ β) (𝓝[>] 0) (𝓝 (α 0)) := by + rw [← nhdsWithin_Ioc_eq_nhdsGT hT] + exact haIoc.congr' heq.symm + have homit' : ∀ z, 0 < z.im → F z ≠ α 0 := by + intro z hz he + exact homit ⟨z, hz, he⟩ + have hendH : 0 < (β (3 / 2)).im := hβH ⟨hT, le_rfl⟩ + have hnorm : 0 < ‖β (3 / 2)‖ := + (lt_of_lt_of_le hendH (Complex.im_le_norm _)) + have hbound : 0 < min (2 * m) ‖β (3 / 2)‖ := lt_min (by positivity) hnorm + obtain ⟨B, hB, hcircle, hsector⟩ := + SmallImageSemicircle.access_guide_has_arbitrarily_small_quarter_image_sector hF hi homit' + hr hT hβ hβ0 (fun t ht => hβH ht) hlimit _ hbound + have hBend : B < ‖β (3 / 2)‖ := hB.2.trans_le (min_le_right _ _) + obtain ⟨tB, htB, hcross, hafter⟩ := GuideSplice.exists_last_circle_crossing hB.1 hT le_rfl + hβ hβ0 (fun t ht => by have he : t = 3 / 2 := le_antisymm ht.2 ht.1; simpa only [he] using hBend) + refine ⟨B, tB, hB.1, ?_, htB.1, htB.2, hcross, hafter, hcircle, hsector, ?_, + limitingFrontTarget_isOpen F hF.continuousOn p U hU hp, ?_, ?_⟩ + · have hh := hB.2.trans_le (min_le_left _ _) + linarith + · rintro _ ⟨z, ⟨t, ht, rfl⟩, rfl⟩ + rw [show F (β t) = α t from hrel ⟨htB.1.trans_le ht.1, ht.2⟩] + exact hαU (mem_image_of_mem α ⟨htB.1.le.trans ht.1, ht.2⟩) + · refine ⟨hendH, ?_, ?_⟩ + · rw [show F (β (3 / 2)) = α (3 / 2) from hrel ⟨hT, le_rfl⟩] + exact hαU (mem_image_of_mem α ⟨hT.le, le_rfl⟩) + · change 1 < p (F (β (3 / 2))) + rw [show F (β (3 / 2)) = α (3 / 2) from hrel ⟨hT, le_rfl⟩, + hprogress _ ⟨hT.le, le_rfl⟩] + norm_num + · intro z hz + exact hz.2.2 + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonMiddleMass.lean b/lean/OAI/Probability/SLE/Gauge/RibbonMiddleMass.lean index 64fd4f98b8d649daa55f7e2dc7a7dd162ea48e8b..0c89be7ca9d8b9c0d66b1ca5135b7138d802876f 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonMiddleMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonMiddleMass.lean @@ -1,151 +1,151 @@ -import OAI.Probability.SLE.Gauge.RibbonConcatenation -import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit -import OAI.Probability.SLE.Gauge.FiniteCoverAssembly -import OAI.Probability.SLE.Mass.StoppedConformalPassageMass - -namespace OAI - -/-! The actual source middle strips of an embedded ribbon are disjoint, -measurable spatial regions. Their stopped passage masses can therefore be -charged to one whole-trace sausage at the common prescribed mesh. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RibbonCoordinates - -def middleStrip (k ε : ℝ) : Set ℂ := - {z | (1 / 3 : ℝ) < z.re - k ∧ z.re - k < 2 / 3 ∧ |z.im| < ε} - -theorem middleStrip_open (k ε : ℝ) : IsOpen (middleStrip k ε) := - (isOpen_lt continuous_const (Complex.continuous_re.sub continuous_const)).inter - ((isOpen_lt (Complex.continuous_re.sub continuous_const) continuous_const).inter - (isOpen_lt Complex.continuous_im.abs continuous_const)) - -theorem middleStrip_subset_closedRibbon {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : - middleStrip k ε ⊆ closedRibbon := by - intro z hz - exact ⟨by linarith [hz.1], hz.2.2.le.trans hε⟩ - -theorem physicalMiddleRegion_eq_image_middleStrip - (P : ℂ → ℂ) {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : - physicalMiddleRegion P closedRibbon k ε = P '' middleStrip k ε := by - unfold physicalMiddleRegion - apply congrArg (fun R : Set ℂ => P '' R) - ext z - exact ⟨fun hz => hz.2, - fun hz => ⟨middleStrip_subset_closedRibbon hk hε hz, hz⟩⟩ - -theorem middleStrip_pairwise_disjoint (ε : ℝ) : - Pairwise (fun i j : ℕ => Disjoint (middleStrip i ε) (middleStrip j ε)) := by - intro i j hij - rw [disjoint_left] - intro z hi hj - have hh : (i : ℝ) < (j : ℝ) ∨ (j : ℝ) < (i : ℝ) := by - exact_mod_cast lt_or_gt_of_ne hij - rcases hh with hh | hh - · have hn : i + 1 ≤ j := by exact_mod_cast hh - have hnR : (i : ℝ) + 1 ≤ j := by exact_mod_cast hn - linarith [hi.2.1, hj.1] - · have hn : j + 1 ≤ i := by exact_mod_cast hh - have hnR : (j : ℝ) + 1 ≤ i := by exact_mod_cast hn - linarith [hi.1, hj.2.1] - -theorem physical_middle_regions_pairwise_disjoint - (P : ℂ → ℂ) (hiP : InjOn P closedRibbon) {ε : ℝ} (hε : ε ≤ 1) : - Pairwise (fun i j : ℕ => Disjoint - (physicalMiddleRegion P closedRibbon i ε) (physicalMiddleRegion P closedRibbon j ε)) := by - intro i j hij - rw [physicalMiddleRegion_eq_image_middleStrip P (Nat.cast_nonneg _) hε, - physicalMiddleRegion_eq_image_middleStrip P (Nat.cast_nonneg _) hε, disjoint_left] - rintro z ⟨u, hu, he⟩ ⟨v, hv, hve⟩ - have huv : u = v := hiP - (middleStrip_subset_closedRibbon (Nat.cast_nonneg _) hε hu) - (middleStrip_subset_closedRibbon (Nat.cast_nonneg _) hε hv) (he.trans hve.symm) - exact Set.disjoint_left.mp (middleStrip_pairwise_disjoint ε hij) hu (huv ▸ hv) - -theorem isOpen_embedded_middle_region - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : - IsOpen (physicalMiddleRegion P closedRibbon k ε) := by - rw [physicalMiddleRegion_eq_image_middleStrip P hk hε] - apply isOpen_iff_mem_nhds.mpr - rintro z ⟨u, hu, rfl⟩ - obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp (middleStrip_open k ε) u hu - have hclosed := hball.trans (middleStrip_subset_closedRibbon hk hε) - exact mem_of_superset - ((hpublished P u r hr (hP.mono hclosed) (hiP.mono hclosed)).mem_nhds - (mem_image_of_mem P (mem_ball_self hr))) (image_mono hball) - -theorem segmentSausageCharge_le_whole - (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (v : ℝ) (R : Set ℂ) : - segmentSausageCharge κ γ s t v R ≤ wholeSausageCharge κ γ v R := by - apply mul_le_mul' le_rfl - apply measure_mono - rintro z ⟨hz, u, _, hd⟩ - exact ⟨hz, u, by simpa only [dist_comm] using hd⟩ - -/-- Exactly the source middle strips are counted, even if their sausages -also meet other parts of the curve. Spatial disjointness alone suffices. -/ -theorem whole_sausage_charge_from_ribbon_steps - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) - (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℕ → ℝ≥0) (v b : ℝ) (A : Set ℂ) - (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ A) - (hmass : ∀ i ∈ I, ENNReal.ofReal b ≤ - segmentSausageCharge κ γ (s i) (t i) v (physicalMiddleRegion P closedRibbon i ε)) : - (I.card : ℝ≥0∞) * ENNReal.ofReal b ≤ wholeSausageCharge κ γ v A := by - let ν := wholeSausageMeasure κ γ v - have hmeas i : MeasurableSet (physicalMiddleRegion P closedRibbon (i : ℕ) ε) := - (isOpen_embedded_middle_region hpublished P hP hiP (Nat.cast_nonneg _) hε).measurableSet - calc - _ = ∑ _i ∈ I, ENNReal.ofReal b := by rw [Finset.sum_const, nsmul_eq_mul] - _ ≤ ∑ i ∈ I, ν (physicalMiddleRegion P closedRibbon i ε) := by - apply Finset.sum_le_sum - intro i hi - rw [wholeSausageMeasure_apply κ γ v (hmeas i)] - exact (hmass i hi).trans (segmentSausageCharge_le_whole κ γ (s i) (t i) v _) - _ = ν (⋃ i ∈ I, physicalMiddleRegion P closedRibbon i ε) := - (measure_biUnion_finset - (fun i _ j _ hij => physical_middle_regions_pairwise_disjoint P hiP hε hij) - (fun i _ => hmeas i)).symm - _ ≤ wholeSausageCharge κ γ v A := by - rw [wholeSausageMeasure_apply κ γ v - (I.measurableSet_biUnion (fun i _ => hmeas i))] - apply mul_le_mul' le_rfl - exact measure_mono (inter_subset_inter (iUnion₂_subset hregions) Subset.rfl) - -theorem dense_ribbon_mass_from_steps - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) - (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℕ → ℝ≥0) (v c ell : ℝ) - (hell : 0 < ell) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) - (A : Set ℂ) - (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ A) - (hmass : ∀ i ∈ I, ENNReal.ofReal (c * (ell / (m : ℝ)) ^ dimension κ) ≤ - segmentSausageCharge κ γ (s i) (t i) v (physicalMiddleRegion P closedRibbon i ε)) : - ENNReal.ofReal (c * ell ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ - wholeSausageCharge κ γ v A := by - have hmR : (0 : ℝ) < m := by exact_mod_cast hm - have he : (m : ℝ) ^ 2 * (c * (ell / (m : ℝ)) ^ dimension κ) = - c * ell ^ dimension κ * (m : ℝ) ^ codimension κ := by - rw [div_eq_mul_one_div, Real.mul_rpow hell.le (one_div_nonneg.mpr hmR.le)] - calc - _ = c * ell ^ dimension κ * - ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by ring - _ = _ := by rw [ribbon_mass_power_cancellation κ hmR] - have hh := whole_sausage_charge_from_ribbon_steps hpublished P hP hiP hε I κ γ s t v - (c * (ell / (m : ℝ)) ^ dimension κ) A hregions hmass - rw [hcard, ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _), - Nat.cast_pow, he] at hh - exact hh - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonConcatenation +import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit +import OAI.Probability.SLE.Gauge.FiniteCoverAssembly +import OAI.Probability.SLE.Mass.StoppedConformalPassageMass + +namespace OAI + +/-! The actual source middle strips of an embedded ribbon are disjoint, +measurable spatial regions. Their stopped passage masses can therefore be +charged to one whole-trace sausage at the common prescribed mesh. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RibbonCoordinates + +def middleStrip (k ε : ℝ) : Set ℂ := + {z | (1 / 3 : ℝ) < z.re - k ∧ z.re - k < 2 / 3 ∧ |z.im| < ε} + +theorem middleStrip_open (k ε : ℝ) : IsOpen (middleStrip k ε) := + (isOpen_lt continuous_const (Complex.continuous_re.sub continuous_const)).inter + ((isOpen_lt (Complex.continuous_re.sub continuous_const) continuous_const).inter + (isOpen_lt Complex.continuous_im.abs continuous_const)) + +theorem middleStrip_subset_closedRibbon {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : + middleStrip k ε ⊆ closedRibbon := by + intro z hz + exact ⟨by linarith [hz.1], hz.2.2.le.trans hε⟩ + +theorem physicalMiddleRegion_eq_image_middleStrip + (P : ℂ → ℂ) {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : + physicalMiddleRegion P closedRibbon k ε = P '' middleStrip k ε := by + unfold physicalMiddleRegion + apply congrArg (fun R : Set ℂ => P '' R) + ext z + exact ⟨fun hz => hz.2, + fun hz => ⟨middleStrip_subset_closedRibbon hk hε hz, hz⟩⟩ + +theorem middleStrip_pairwise_disjoint (ε : ℝ) : + Pairwise (fun i j : ℕ => Disjoint (middleStrip i ε) (middleStrip j ε)) := by + intro i j hij + rw [disjoint_left] + intro z hi hj + have hh : (i : ℝ) < (j : ℝ) ∨ (j : ℝ) < (i : ℝ) := by + exact_mod_cast lt_or_gt_of_ne hij + rcases hh with hh | hh + · have hn : i + 1 ≤ j := by exact_mod_cast hh + have hnR : (i : ℝ) + 1 ≤ j := by exact_mod_cast hn + linarith [hi.2.1, hj.1] + · have hn : j + 1 ≤ i := by exact_mod_cast hh + have hnR : (j : ℝ) + 1 ≤ i := by exact_mod_cast hn + linarith [hi.1, hj.2.1] + +theorem physical_middle_regions_pairwise_disjoint + (P : ℂ → ℂ) (hiP : InjOn P closedRibbon) {ε : ℝ} (hε : ε ≤ 1) : + Pairwise (fun i j : ℕ => Disjoint + (physicalMiddleRegion P closedRibbon i ε) (physicalMiddleRegion P closedRibbon j ε)) := by + intro i j hij + rw [physicalMiddleRegion_eq_image_middleStrip P (Nat.cast_nonneg _) hε, + physicalMiddleRegion_eq_image_middleStrip P (Nat.cast_nonneg _) hε, disjoint_left] + rintro z ⟨u, hu, he⟩ ⟨v, hv, hve⟩ + have huv : u = v := hiP + (middleStrip_subset_closedRibbon (Nat.cast_nonneg _) hε hu) + (middleStrip_subset_closedRibbon (Nat.cast_nonneg _) hε hv) (he.trans hve.symm) + exact Set.disjoint_left.mp (middleStrip_pairwise_disjoint ε hij) hu (huv ▸ hv) + +theorem isOpen_embedded_middle_region + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {k ε : ℝ} (hk : 0 ≤ k) (hε : ε ≤ 1) : + IsOpen (physicalMiddleRegion P closedRibbon k ε) := by + rw [physicalMiddleRegion_eq_image_middleStrip P hk hε] + apply isOpen_iff_mem_nhds.mpr + rintro z ⟨u, hu, rfl⟩ + obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp (middleStrip_open k ε) u hu + have hclosed := hball.trans (middleStrip_subset_closedRibbon hk hε) + exact mem_of_superset + ((hpublished P u r hr (hP.mono hclosed) (hiP.mono hclosed)).mem_nhds + (mem_image_of_mem P (mem_ball_self hr))) (image_mono hball) + +theorem segmentSausageCharge_le_whole + (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (v : ℝ) (R : Set ℂ) : + segmentSausageCharge κ γ s t v R ≤ wholeSausageCharge κ γ v R := by + apply mul_le_mul' le_rfl + apply measure_mono + rintro z ⟨hz, u, _, hd⟩ + exact ⟨hz, u, by simpa only [dist_comm] using hd⟩ + +/-- Exactly the source middle strips are counted, even if their sausages +also meet other parts of the curve. Spatial disjointness alone suffices. -/ +theorem whole_sausage_charge_from_ribbon_steps + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) + (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℕ → ℝ≥0) (v b : ℝ) (A : Set ℂ) + (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ A) + (hmass : ∀ i ∈ I, ENNReal.ofReal b ≤ + segmentSausageCharge κ γ (s i) (t i) v (physicalMiddleRegion P closedRibbon i ε)) : + (I.card : ℝ≥0∞) * ENNReal.ofReal b ≤ wholeSausageCharge κ γ v A := by + let ν := wholeSausageMeasure κ γ v + have hmeas i : MeasurableSet (physicalMiddleRegion P closedRibbon (i : ℕ) ε) := + (isOpen_embedded_middle_region hpublished P hP hiP (Nat.cast_nonneg _) hε).measurableSet + calc + _ = ∑ _i ∈ I, ENNReal.ofReal b := by rw [Finset.sum_const, nsmul_eq_mul] + _ ≤ ∑ i ∈ I, ν (physicalMiddleRegion P closedRibbon i ε) := by + apply Finset.sum_le_sum + intro i hi + rw [wholeSausageMeasure_apply κ γ v (hmeas i)] + exact (hmass i hi).trans (segmentSausageCharge_le_whole κ γ (s i) (t i) v _) + _ = ν (⋃ i ∈ I, physicalMiddleRegion P closedRibbon i ε) := + (measure_biUnion_finset + (fun i _ j _ hij => physical_middle_regions_pairwise_disjoint P hiP hε hij) + (fun i _ => hmeas i)).symm + _ ≤ wholeSausageCharge κ γ v A := by + rw [wholeSausageMeasure_apply κ γ v + (I.measurableSet_biUnion (fun i _ => hmeas i))] + apply mul_le_mul' le_rfl + exact measure_mono (inter_subset_inter (iUnion₂_subset hregions) Subset.rfl) + +theorem dense_ribbon_mass_from_steps + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {ε : ℝ} (hε : ε ≤ 1) (I : Finset ℕ) + (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℕ → ℝ≥0) (v c ell : ℝ) + (hell : 0 < ell) (m : ℕ) (hm : 0 < m) (hcard : I.card = m ^ 2) + (A : Set ℂ) + (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ A) + (hmass : ∀ i ∈ I, ENNReal.ofReal (c * (ell / (m : ℝ)) ^ dimension κ) ≤ + segmentSausageCharge κ γ (s i) (t i) v (physicalMiddleRegion P closedRibbon i ε)) : + ENNReal.ofReal (c * ell ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ + wholeSausageCharge κ γ v A := by + have hmR : (0 : ℝ) < m := by exact_mod_cast hm + have he : (m : ℝ) ^ 2 * (c * (ell / (m : ℝ)) ^ dimension κ) = + c * ell ^ dimension κ * (m : ℝ) ^ codimension κ := by + rw [div_eq_mul_one_div, Real.mul_rpow hell.le (one_div_nonneg.mpr hmR.le)] + calc + _ = c * ell ^ dimension κ * + ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by ring + _ = _ := by rw [ribbon_mass_power_cancellation κ hmR] + have hh := whole_sausage_charge_from_ribbon_steps hpublished P hP hiP hε I κ γ s t v + (c * (ell / (m : ℝ)) ^ dimension κ) A hregions hmass + rw [hcard, ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _), + Nat.cast_pow, he] at hh + exact hh + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonPassageMass.lean b/lean/OAI/Probability/SLE/Gauge/RibbonPassageMass.lean index 3dc73395405df036fe7ee7ae36346531ef7cd48d..f7338f0d739a317b0c5a4d45ab38246bcaa6b359 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonPassageMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonPassageMass.lean @@ -1,119 +1,119 @@ -import OAI.Probability.SLE.SausageSeedAll - -namespace OAI - -/-! -The last probabilistic transfer in the ribbon step: a finite deterministic -list of retained time intervals permits subtracting one unconditional -small-sausage exception from the varying passage events. The sausage -estimate is never conditioned on passage. --/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -/-- The finite-list subtraction step of `rc:step`. Passage may select a -different deterministic interval on each sample and at each mesh. -/ -theorem ribbon_passage_mass_of_eventually_finite_intervals - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n)) - (physicalMass : ℕ → Ω → ℝ≥0∞) (passage : ℕ → Ω → Prop) - (c : ℝ) (hc : 0 < c) - (htransfer : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → passage n ω ∧ - ∃ J ∈ I, ENNReal.ofReal c * - sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 2) ≤ - P {ω | ω ∈ G n ∧ passage n ω ∧ ENNReal.ofReal a ≤ physicalMass n ω} := by - obtain ⟨b, e₀, hb, he₀, hsmall⟩ := sourceSmallSausageMain_proved - Ω inferInstance P inferInstance κ hκ hκ8 γ hSLE I hI (q / 2) (by positivity) - refine ⟨c * b, mul_pos hc hb, ?_⟩ - have hevent : ∀ᶠ n in atTop, v n < e₀ := - (tendsto_order.mp hlim).2 e₀ he₀ - filter_upwards [hevent, hG, htransfer] with n hn hGn htransfern - let bad : Set Ω := {ω | ∃ J ∈ I, - sourceSausageMass κ (γ ω) J.1 J.2 (v n) < ENNReal.ofReal b} - let good : Set Ω := {ω | ω ∈ G n ∧ passage n ω ∧ - ENNReal.ofReal (c * b) ≤ physicalMass n ω} - have hbad : P bad < ENNReal.ofReal (q / 2) := hsmall (v n) (hv n) hn - have hcover : G n ⊆ good ∪ bad := by - intro ω hω - by_cases hbadω : ω ∈ bad - · exact Or.inr hbadω - · obtain ⟨hpassage, J, hJI, hmass⟩ := htransfern ω hω - left - refine ⟨hω, hpassage, ?_⟩ - have hsausage : ENNReal.ofReal b ≤ sourceSausageMass κ (γ ω) J.1 J.2 (v n) := by - by_contra h - exact hbadω ⟨J, hJI, lt_of_not_ge h⟩ - rw [ENNReal.ofReal_mul hc.le] - exact (mul_le_mul' le_rfl hsausage).trans hmass - change ENNReal.ofReal (q / 2) ≤ P good - by_contra hgood - have hhalf : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = - ENNReal.ofReal q := by - rw [← ENNReal.ofReal_add (by positivity) (by positivity)] - congr 1 - ring - have hlt : P good + P bad < ENNReal.ofReal q := by - rw [← hhalf] - exact ENNReal.add_lt_add (lt_of_not_ge hgood) hbad - exact (not_lt_of_ge hGn) - (((measure_mono hcover).trans (measure_union_le good bad)).trans_lt hlt) - -/-- A version with transfer and probability bounds at every index. -/ -theorem ribbon_passage_mass_of_finite_intervals - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n)) - (physicalMass : ℕ → Ω → ℝ≥0∞) (passage : ℕ → Ω → Prop) - (c : ℝ) (hc : 0 < c) - (htransfer : ∀ n ω, ω ∈ G n → passage n ω ∧ - ∃ J ∈ I, ENNReal.ofReal c * - sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 2) ≤ - P {ω | ω ∈ G n ∧ passage n ω ∧ ENNReal.ofReal a ≤ physicalMass n ω} := - ribbon_passage_mass_of_eventually_finite_intervals P κ hκ hκ8 γ hSLE I hI - v hv hlim G q hq (Eventually.of_forall hG) physicalMass passage c hc - (Eventually.of_forall htransfer) - -/-- The same finite-list transfer with passage already included in `G n`. -/ -theorem ribbon_mass_of_finite_intervals - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n)) - (physicalMass : ℕ → Ω → ℝ≥0∞) (c : ℝ) (hc : 0 < c) - (htransfer : ∀ n ω, ω ∈ G n → - ∃ J ∈ I, ENNReal.ofReal c * - sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 2) ≤ - P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤ physicalMass n ω} := by - obtain ⟨a, ha, hevent⟩ := ribbon_passage_mass_of_finite_intervals - P κ hκ hκ8 γ hSLE I hI v hv hlim G q hq hG physicalMass - (fun _ _ => True) c hc (fun n ω hω => ⟨trivial, htransfer n ω hω⟩) - refine ⟨a, ha, ?_⟩ - simpa only [true_and] using hevent - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.SausageSeedAll + +namespace OAI + +/-! +The last probabilistic transfer in the ribbon step: a finite deterministic +list of retained time intervals permits subtracting one unconditional +small-sausage exception from the varying passage events. The sausage +estimate is never conditioned on passage. +-/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +/-- The finite-list subtraction step of `rc:step`. Passage may select a +different deterministic interval on each sample and at each mesh. -/ +theorem ribbon_passage_mass_of_eventually_finite_intervals + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n)) + (physicalMass : ℕ → Ω → ℝ≥0∞) (passage : ℕ → Ω → Prop) + (c : ℝ) (hc : 0 < c) + (htransfer : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → passage n ω ∧ + ∃ J ∈ I, ENNReal.ofReal c * + sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 2) ≤ + P {ω | ω ∈ G n ∧ passage n ω ∧ ENNReal.ofReal a ≤ physicalMass n ω} := by + obtain ⟨b, e₀, hb, he₀, hsmall⟩ := sourceSmallSausageMain_proved + Ω inferInstance P inferInstance κ hκ hκ8 γ hSLE I hI (q / 2) (by positivity) + refine ⟨c * b, mul_pos hc hb, ?_⟩ + have hevent : ∀ᶠ n in atTop, v n < e₀ := + (tendsto_order.mp hlim).2 e₀ he₀ + filter_upwards [hevent, hG, htransfer] with n hn hGn htransfern + let bad : Set Ω := {ω | ∃ J ∈ I, + sourceSausageMass κ (γ ω) J.1 J.2 (v n) < ENNReal.ofReal b} + let good : Set Ω := {ω | ω ∈ G n ∧ passage n ω ∧ + ENNReal.ofReal (c * b) ≤ physicalMass n ω} + have hbad : P bad < ENNReal.ofReal (q / 2) := hsmall (v n) (hv n) hn + have hcover : G n ⊆ good ∪ bad := by + intro ω hω + by_cases hbadω : ω ∈ bad + · exact Or.inr hbadω + · obtain ⟨hpassage, J, hJI, hmass⟩ := htransfern ω hω + left + refine ⟨hω, hpassage, ?_⟩ + have hsausage : ENNReal.ofReal b ≤ sourceSausageMass κ (γ ω) J.1 J.2 (v n) := by + by_contra h + exact hbadω ⟨J, hJI, lt_of_not_ge h⟩ + rw [ENNReal.ofReal_mul hc.le] + exact (mul_le_mul' le_rfl hsausage).trans hmass + change ENNReal.ofReal (q / 2) ≤ P good + by_contra hgood + have hhalf : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = + ENNReal.ofReal q := by + rw [← ENNReal.ofReal_add (by positivity) (by positivity)] + congr 1 + ring + have hlt : P good + P bad < ENNReal.ofReal q := by + rw [← hhalf] + exact ENNReal.add_lt_add (lt_of_not_ge hgood) hbad + exact (not_lt_of_ge hGn) + (((measure_mono hcover).trans (measure_union_le good bad)).trans_lt hlt) + +/-- A version with transfer and probability bounds at every index. -/ +theorem ribbon_passage_mass_of_finite_intervals + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n)) + (physicalMass : ℕ → Ω → ℝ≥0∞) (passage : ℕ → Ω → Prop) + (c : ℝ) (hc : 0 < c) + (htransfer : ∀ n ω, ω ∈ G n → passage n ω ∧ + ∃ J ∈ I, ENNReal.ofReal c * + sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 2) ≤ + P {ω | ω ∈ G n ∧ passage n ω ∧ ENNReal.ofReal a ≤ physicalMass n ω} := + ribbon_passage_mass_of_eventually_finite_intervals P κ hκ hκ8 γ hSLE I hI + v hv hlim G q hq (Eventually.of_forall hG) physicalMass passage c hc + (Eventually.of_forall htransfer) + +/-- The same finite-list transfer with passage already included in `G n`. -/ +theorem ribbon_mass_of_finite_intervals + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n)) + (physicalMass : ℕ → Ω → ℝ≥0∞) (c : ℝ) (hc : 0 < c) + (htransfer : ∀ n ω, ω ∈ G n → + ∃ J ∈ I, ENNReal.ofReal c * + sourceSausageMass κ (γ ω) J.1 J.2 (v n) ≤ physicalMass n ω) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 2) ≤ + P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤ physicalMass n ω} := by + obtain ⟨a, ha, hevent⟩ := ribbon_passage_mass_of_finite_intervals + P κ hκ hκ8 γ hSLE I hI v hv hlim G q hq hG physicalMass + (fun _ _ => True) c hc (fun n ω hω => ⟨trivial, htransfer n ω hω⟩) + refine ⟨a, ha, ?_⟩ + simpa only [true_and] using hevent + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonPassageNeighborhood.lean b/lean/OAI/Probability/SLE/Gauge/RibbonPassageNeighborhood.lean index a4f91a6af700a236f269175447e78fdebeb27343..560c9fa81ecb0d06520c410c420ae466fe218372 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonPassageNeighborhood.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonPassageNeighborhood.lean @@ -1,96 +1,96 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit - -namespace OAI - -/-! Common physical neighborhoods of compact passage routes. Strict -coordinate margins are retained on the closure before passing to moving -ribbon charts. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -/-- An embedded coordinate chart maps every open coordinate subset of its -controlled patch to an open physical subset. -/ -theorem isOpen_image_coordinate_subset - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (K V : Set ℂ) (hP : ContinuousOn P K) (hi : InjOn P K) - (hV : IsOpen V) (hVK : V ⊆ K) : IsOpen (P '' V) := by - apply isOpen_iff_mem_nhds.mpr - rintro z ⟨x, hx, rfl⟩ - obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp hV x hx - have hsub : ball x r ⊆ K := hball.trans hVK - have hopen := hpublished P x r hr (hP.mono hsub) (hi.mono hsub) - exact mem_of_superset - (hopen.mem_nhds (mem_image_of_mem P (mem_ball_self hr))) - (image_mono hball) - -/-- A compact route strictly inside a coordinate region admits one open -physical neighborhood whose closure lies in the same region for every -sufficiently late ribbon chart. -/ -theorem common_coordinate_region_neighborhood - (hpublished : PublishedPlanarDiskOpenness) - (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A V : Set ℂ) (L : ℝ≥0) - (hL : 0 < L) (hP : ContinuousOn P K) - (hPn : ∀ n, ContinuousOn (Pₙ n) K) - (hconv : TendstoUniformlyOn Pₙ P atTop K) - (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) - (hA : IsCompact A) (hV : IsOpen V) - (hAV : A ⊆ P '' (interior K ∩ V)) : - ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ - closure U ⊆ P '' (interior K ∩ V) ∧ - (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' (K ∩ V)) ∧ - (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' interior K) ∧ - TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) - (coordinateInverse P K) atTop (closure U) := by - have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower - have hi := lower_lipschitz_injOn P K L hlim - have hopen := isOpen_image_coordinate_subset hpublished P K (interior K ∩ V) - hP hi (isOpen_interior.inter hV) (inter_subset_left.trans interior_subset) - obtain ⟨ρ, hρ, hthick⟩ := hA.exists_cthickening_subset_open hopen hAV - let U := thickening ρ A - have hcl : closure U ⊆ cthickening ρ A := closure_thickening_subset_cthickening ρ A - have hUc : IsCompact (closure U) := hA.cthickening.of_isClosed_subset isClosed_closure hcl - have hUPV : closure U ⊆ P '' (interior K ∩ V) := hcl.trans hthick - have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left) - have hlate := eventual_compact_image_interior hpublished Pₙ P K (closure U) L hL hPn - hconv hlower hUc hUP - have hUV : coordinateInverse P K '' closure U ⊆ V := by - rintro _ ⟨z, hz, rfl⟩ - obtain ⟨x, hx, he⟩ := hUPV hz - have heq : coordinateInverse P K z = x := by - rw [← he] - exact coordinateInverse_left hi (interior_subset hx.1) - simpa only [heq] using hx.2 - have hcover : ∀ᶠ n in atTop, closure U ⊆ Pₙ n '' K := - hlate.mono (fun _ h => h.trans (image_mono interior_subset)) - have hregions := eventual_compact_coordinate_region Pₙ P K (closure U) V L hconv hlower - hUc (hUP.trans (image_mono interior_subset)) hcover hV hUV - exact ⟨U, isOpen_thickening, self_subset_thickening hρ A, hUc, hUPV, hregions, hlate, - coordinateInverse_uniform_on Pₙ P K (closure U) L hconv hlower - (hUP.trans (image_mono interior_subset)) hcover⟩ - -/-- Limiting coordinate guides have exact longitudinal progress and retain -the source transverse margin. -/ -theorem limit_guide_coordinates - (P : ℂ → ℂ) (K : Set ℂ) (hi : InjOn P K) - (α : ℝ → ℂ) (x : ℝ) - (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2)) - (hmem : ∀ t ∈ Icc (0 : ℝ) 2, guideCoordinate 0 x t ∈ K) : - ∀ t ∈ Icc (0 : ℝ) 2, - longitudinalCoordinate P K 0 (α t) = t ∧ - |(coordinateInverse P K (α t)).im| ≤ |x| := by - intro t ht - rw [hshape ht, coordinateInverse_left hi (hmem t ht)] - constructor - · simp only [longitudinalCoordinate, coordinateInverse_left hi (hmem t ht), - guideCoordinate_re, zero_add, sub_zero] - · exact guideCoordinate_im_bound ht.1 - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit + +namespace OAI + +/-! Common physical neighborhoods of compact passage routes. Strict +coordinate margins are retained on the closure before passing to moving +ribbon charts. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +/-- An embedded coordinate chart maps every open coordinate subset of its +controlled patch to an open physical subset. -/ +theorem isOpen_image_coordinate_subset + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (K V : Set ℂ) (hP : ContinuousOn P K) (hi : InjOn P K) + (hV : IsOpen V) (hVK : V ⊆ K) : IsOpen (P '' V) := by + apply isOpen_iff_mem_nhds.mpr + rintro z ⟨x, hx, rfl⟩ + obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp hV x hx + have hsub : ball x r ⊆ K := hball.trans hVK + have hopen := hpublished P x r hr (hP.mono hsub) (hi.mono hsub) + exact mem_of_superset + (hopen.mem_nhds (mem_image_of_mem P (mem_ball_self hr))) + (image_mono hball) + +/-- A compact route strictly inside a coordinate region admits one open +physical neighborhood whose closure lies in the same region for every +sufficiently late ribbon chart. -/ +theorem common_coordinate_region_neighborhood + (hpublished : PublishedPlanarDiskOpenness) + (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K A V : Set ℂ) (L : ℝ≥0) + (hL : 0 < L) (hP : ContinuousOn P K) + (hPn : ∀ n, ContinuousOn (Pₙ n) K) + (hconv : TendstoUniformlyOn Pₙ P atTop K) + (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y)) + (hA : IsCompact A) (hV : IsOpen V) + (hAV : A ⊆ P '' (interior K ∩ V)) : + ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ IsCompact (closure U) ∧ + closure U ⊆ P '' (interior K ∩ V) ∧ + (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' (K ∩ V)) ∧ + (∀ᶠ n in atTop, closure U ⊆ Pₙ n '' interior K) ∧ + TendstoUniformlyOn (fun n => coordinateInverse (Pₙ n) K) + (coordinateInverse P K) atTop (closure U) := by + have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower + have hi := lower_lipschitz_injOn P K L hlim + have hopen := isOpen_image_coordinate_subset hpublished P K (interior K ∩ V) + hP hi (isOpen_interior.inter hV) (inter_subset_left.trans interior_subset) + obtain ⟨ρ, hρ, hthick⟩ := hA.exists_cthickening_subset_open hopen hAV + let U := thickening ρ A + have hcl : closure U ⊆ cthickening ρ A := closure_thickening_subset_cthickening ρ A + have hUc : IsCompact (closure U) := hA.cthickening.of_isClosed_subset isClosed_closure hcl + have hUPV : closure U ⊆ P '' (interior K ∩ V) := hcl.trans hthick + have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left) + have hlate := eventual_compact_image_interior hpublished Pₙ P K (closure U) L hL hPn + hconv hlower hUc hUP + have hUV : coordinateInverse P K '' closure U ⊆ V := by + rintro _ ⟨z, hz, rfl⟩ + obtain ⟨x, hx, he⟩ := hUPV hz + have heq : coordinateInverse P K z = x := by + rw [← he] + exact coordinateInverse_left hi (interior_subset hx.1) + simpa only [heq] using hx.2 + have hcover : ∀ᶠ n in atTop, closure U ⊆ Pₙ n '' K := + hlate.mono (fun _ h => h.trans (image_mono interior_subset)) + have hregions := eventual_compact_coordinate_region Pₙ P K (closure U) V L hconv hlower + hUc (hUP.trans (image_mono interior_subset)) hcover hV hUV + exact ⟨U, isOpen_thickening, self_subset_thickening hρ A, hUc, hUPV, hregions, hlate, + coordinateInverse_uniform_on Pₙ P K (closure U) L hconv hlower + (hUP.trans (image_mono interior_subset)) hcover⟩ + +/-- Limiting coordinate guides have exact longitudinal progress and retain +the source transverse margin. -/ +theorem limit_guide_coordinates + (P : ℂ → ℂ) (K : Set ℂ) (hi : InjOn P K) + (α : ℝ → ℂ) (x : ℝ) + (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2)) + (hmem : ∀ t ∈ Icc (0 : ℝ) 2, guideCoordinate 0 x t ∈ K) : + ∀ t ∈ Icc (0 : ℝ) 2, + longitudinalCoordinate P K 0 (α t) = t ∧ + |(coordinateInverse P K (α t)).im| ≤ |x| := by + intro t ht + rw [hshape ht, coordinateInverse_left hi (hmem t ht)] + constructor + · simp only [longitudinalCoordinate, coordinateInverse_left hi (hmem t ht), + guideCoordinate_re, zero_add, sub_zero] + · exact guideCoordinate_im_bound ht.1 + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonPassageScaling.lean b/lean/OAI/Probability/SLE/Gauge/RibbonPassageScaling.lean index 848962004186f5e7d52668152b8d0b404ff738ba..44d94c1e83c529dbf48c7e5bc2ed6d17d2e24edf 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonPassageScaling.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonPassageScaling.lean @@ -1,173 +1,173 @@ -import OAI.Probability.SLE.Gauge.UniformFrontPassage -import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar - -namespace OAI - -/-! Positive affine rescaling of the physical one-step passage. A mesh -`v` becomes `ell * v`; its normalized sausage mass is multiplied by -`ell ^ dimension κ`, exactly as in the width-rescaled ribbon statement. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal Pointwise - -namespace SLEExactGauge.RibbonPassageScaling -open FirstFrontPassage PassageSuccess - -/-- Translate and dilate the physical coordinates without changing the -curve parameter. -/ -def affine (a : ℂ) (ell : ℝ) (z : ℂ) : ℂ := a + (ell : ℂ) * z - -theorem affine_dist (a : ℂ) {ell : ℝ} (hell : 0 ≤ ell) (z w : ℂ) : - dist (affine a ell z) (affine a ell w) = ell * dist z w := by - simp only [affine, dist_eq_norm, add_sub_add_left_eq_sub, ← mul_sub, norm_mul, - Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hell] - -theorem affine_bijective (a : ℂ) {ell : ℝ} (hell : 0 < ell) : - Function.Bijective (affine a ell) := by - have he : (ell : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hell.ne' - constructor - · intro z w h - exact mul_left_cancel₀ he (add_left_cancel h) - · intro z - refine ⟨(z - a) / (ell : ℂ), ?_⟩ - dsimp [affine] - field_simp - ring - -/-- Planar area scales by the square of the physical length scale, also -for sets for which no measurability has been assumed. -/ -theorem affine_volume (a : ℂ) {ell : ℝ} (hell : 0 ≤ ell) (S : Set ℂ) : - volume (affine a ell '' S) = ENNReal.ofReal (ell ^ 2) * volume S := by - have he : affine a ell '' S = (fun z : ℂ => a + z) '' (ell • S) := by - rw [← image_smul, image_image] - rfl - rw [he, image_add_left, measure_preimage_add, Measure.addHaar_smul_of_nonneg volume hell] - rw [Complex.finrank_real_complex] - -/-- The full sausage, including its intersection with a physical middle -strip, scales exactly under a positive affine map. -/ -theorem sourceSausageSet_affine - (a : ℂ) {ell : ℝ} (hell : 0 < ell) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (v : ℝ) : - sourceSausageSet (fun u => affine a ell (γ u)) s t (ell * v) = - affine a ell '' sourceSausageSet γ s t v := by - ext z - obtain ⟨w, rfl⟩ := (affine_bijective a hell).2 z - constructor - · rintro ⟨u, hu, hd⟩ - rw [affine_dist a hell.le] at hd - exact ⟨w, ⟨u, hu, (mul_lt_mul_iff_right₀ hell).mp hd⟩, rfl⟩ - · rintro ⟨w', ⟨u, hu, hd⟩, he⟩ - have hw' : w' = w := (affine_bijective a hell).1 he - subst w' - exact ⟨u, hu, by rw [affine_dist a hell.le]; exact mul_lt_mul_of_pos_left hd hell⟩ - -theorem segmentSausageCharge_affine - (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) - (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : - segmentSausageCharge κ (fun u => affine a ell (γ u)) s t (ell * v) - (affine a ell '' R) = - ENNReal.ofReal (ell ^ dimension κ) * segmentSausageCharge κ γ s t v R := by - have he : (ell * v) ^ (-codimension κ) * ell ^ 2 = - ell ^ dimension κ * v ^ (-codimension κ) := by - rw [Real.mul_rpow hell.le hv.le] - calc - ell ^ (-codimension κ) * v ^ (-codimension κ) * ell ^ 2 = - (ell ^ (-codimension κ) * ell ^ (2 : ℝ)) * v ^ (-codimension κ) := by - rw [Real.rpow_two] - ring - _ = ell ^ (-codimension κ + 2) * v ^ (-codimension κ) := by - rw [Real.rpow_add hell] - _ = _ := by congr 2; unfold codimension; ring - simp only [segmentSausageCharge, sourceSausageSet_affine a hell] - rw [← image_inter (affine_bijective a hell).1, affine_volume a hell.le] - rw [← mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg (mul_pos hell hv).le _), he, - ENNReal.ofReal_mul (Real.rpow_nonneg hell.le _), mul_assoc] - -/-- The prescribed mass threshold scales by `ell^d`. -/ -theorem affine_mass_threshold_iff - (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) - (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {v : ℝ} (hv : 0 < v) (R : Set ℂ) (c : ℝ) : - ENNReal.ofReal (c * ell ^ dimension κ) ≤ - segmentSausageCharge κ (fun u => affine a ell (γ u)) s t (ell * v) - (affine a ell '' R) ↔ - ENNReal.ofReal c ≤ segmentSausageCharge κ γ s t v R := by - rw [segmentSausageCharge_affine κ a hell γ s t hv R, mul_comm c, - ENNReal.ofReal_mul (Real.rpow_pos_of_pos hell _).le] - exact ENNReal.mul_le_mul_iff_right ((ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hell _)).ne') - ENNReal.ofReal_ne_top - -/-- Pulling back the longitudinal coordinate leaves the actual first-front -clock unchanged. Spatial rescaling does not change curve order. -/ -theorem firstFrontTime_affine - (a : ℂ) (ell : ℝ) (γ : ℝ≥0 → ℂ) (p p' : ℂ → ℝ) - (hp : ∀ z, p' (affine a ell z) = p z) : - firstFrontTime (fun t => p' (affine a ell (γ t))) 1 = - firstFrontTime (fun t => p (γ t)) 1 := by - congr 1 - funext t - exact hp (γ t) - -/-- Affine scaling preserves all actual front-success conditions, including -whole-prefix containment and the terminal offset. -/ -theorem frontSuccess_affine_iff - {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ) - (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ) - (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ) - (hp : ∀ z, p' n (affine a ell z) = p n z) : - FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ - (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n = - FrontSuccess η p σ V b ε n := by - ext ω - have hmaps (T : ℝ≥0) : - MapsTo (fun t => affine a ell (η n ω t)) (Icc 0 T) (affine a ell '' V n) ↔ - MapsTo (η n ω) (Icc 0 T) (V n) := by - constructor - · intro h t ht - obtain ⟨z, hz, he⟩ := h ht - exact (affine_bijective a hell).1 he ▸ hz - · intro h t ht - exact mem_image_of_mem _ (h ht) - have hterminal (T : ℝ≥0) : - ‖affine a ell (η n ω T) - affine a ell (b n)‖ < ell * ε / 4 ↔ - ‖η n ω T - b n‖ < ε / 4 := by - have hd := affine_dist a hell.le (η n ω T) (b n) - rw [dist_eq_norm, dist_eq_norm] at hd - rw [hd, mul_div_assoc, mul_lt_mul_iff_right₀ hell] - simp only [FrontSuccess, mem_ofPred_eq, hp, hmaps, hterminal] - -/-- The complete one-step event has the same probability after simultaneous -spatial, mesh and mass rescaling. This is a deterministic event identity; -the trace law need not itself have been scaled. -/ -theorem frontSuccess_mass_probability_affine - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ) - (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ) - (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ) - (hp : ∀ z, p' n (affine a ell z) = p n z) - (κ : ℝ) (R : Set ℂ) {v : ℝ} (hv : 0 < v) (c : ℝ) : - P {ω | ω ∈ FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ - (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n ∧ - ENNReal.ofReal (c * ell ^ dimension κ) ≤ - segmentSausageCharge κ (fun t => affine a ell (η n ω t)) (σ ω) - (frontTime (fun t => p' n (affine a ell (η n ω t))) 1) - (ell * v) (affine a ell '' R)} = - P {ω | ω ∈ FrontSuccess η p σ V b ε n ∧ ENNReal.ofReal c ≤ - segmentSausageCharge κ (η n ω) (σ ω) - (frontTime (fun t => p n (η n ω t)) 1) v R} := by - congr 1 - ext ω - have hevent := frontSuccess_affine_iff η p p' σ V b ε a hell n hp - have ht : frontTime (fun t => p' n (affine a ell (η n ω t))) 1 = - frontTime (fun t => p n (η n ω t)) 1 := by - congr 1 - funext t - exact hp (η n ω t) - simp only [mem_ofPred_eq, hevent, ht, - affine_mass_threshold_iff κ a hell (η n ω) (σ ω) _ hv R c] - -end SLEExactGauge.RibbonPassageScaling - -end - -end OAI +import OAI.Probability.SLE.Gauge.UniformFrontPassage +import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar + +namespace OAI + +/-! Positive affine rescaling of the physical one-step passage. A mesh +`v` becomes `ell * v`; its normalized sausage mass is multiplied by +`ell ^ dimension κ`, exactly as in the width-rescaled ribbon statement. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal Pointwise + +namespace SLEExactGauge.RibbonPassageScaling +open FirstFrontPassage PassageSuccess + +/-- Translate and dilate the physical coordinates without changing the +curve parameter. -/ +def affine (a : ℂ) (ell : ℝ) (z : ℂ) : ℂ := a + (ell : ℂ) * z + +theorem affine_dist (a : ℂ) {ell : ℝ} (hell : 0 ≤ ell) (z w : ℂ) : + dist (affine a ell z) (affine a ell w) = ell * dist z w := by + simp only [affine, dist_eq_norm, add_sub_add_left_eq_sub, ← mul_sub, norm_mul, + Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hell] + +theorem affine_bijective (a : ℂ) {ell : ℝ} (hell : 0 < ell) : + Function.Bijective (affine a ell) := by + have he : (ell : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hell.ne' + constructor + · intro z w h + exact mul_left_cancel₀ he (add_left_cancel h) + · intro z + refine ⟨(z - a) / (ell : ℂ), ?_⟩ + dsimp [affine] + field_simp + ring + +/-- Planar area scales by the square of the physical length scale, also +for sets for which no measurability has been assumed. -/ +theorem affine_volume (a : ℂ) {ell : ℝ} (hell : 0 ≤ ell) (S : Set ℂ) : + volume (affine a ell '' S) = ENNReal.ofReal (ell ^ 2) * volume S := by + have he : affine a ell '' S = (fun z : ℂ => a + z) '' (ell • S) := by + rw [← image_smul, image_image] + rfl + rw [he, image_add_left, measure_preimage_add, Measure.addHaar_smul_of_nonneg volume hell] + rw [Complex.finrank_real_complex] + +/-- The full sausage, including its intersection with a physical middle +strip, scales exactly under a positive affine map. -/ +theorem sourceSausageSet_affine + (a : ℂ) {ell : ℝ} (hell : 0 < ell) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (v : ℝ) : + sourceSausageSet (fun u => affine a ell (γ u)) s t (ell * v) = + affine a ell '' sourceSausageSet γ s t v := by + ext z + obtain ⟨w, rfl⟩ := (affine_bijective a hell).2 z + constructor + · rintro ⟨u, hu, hd⟩ + rw [affine_dist a hell.le] at hd + exact ⟨w, ⟨u, hu, (mul_lt_mul_iff_right₀ hell).mp hd⟩, rfl⟩ + · rintro ⟨w', ⟨u, hu, hd⟩, he⟩ + have hw' : w' = w := (affine_bijective a hell).1 he + subst w' + exact ⟨u, hu, by rw [affine_dist a hell.le]; exact mul_lt_mul_of_pos_left hd hell⟩ + +theorem segmentSausageCharge_affine + (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) + (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : + segmentSausageCharge κ (fun u => affine a ell (γ u)) s t (ell * v) + (affine a ell '' R) = + ENNReal.ofReal (ell ^ dimension κ) * segmentSausageCharge κ γ s t v R := by + have he : (ell * v) ^ (-codimension κ) * ell ^ 2 = + ell ^ dimension κ * v ^ (-codimension κ) := by + rw [Real.mul_rpow hell.le hv.le] + calc + ell ^ (-codimension κ) * v ^ (-codimension κ) * ell ^ 2 = + (ell ^ (-codimension κ) * ell ^ (2 : ℝ)) * v ^ (-codimension κ) := by + rw [Real.rpow_two] + ring + _ = ell ^ (-codimension κ + 2) * v ^ (-codimension κ) := by + rw [Real.rpow_add hell] + _ = _ := by congr 2; unfold codimension; ring + simp only [segmentSausageCharge, sourceSausageSet_affine a hell] + rw [← image_inter (affine_bijective a hell).1, affine_volume a hell.le] + rw [← mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg (mul_pos hell hv).le _), he, + ENNReal.ofReal_mul (Real.rpow_nonneg hell.le _), mul_assoc] + +/-- The prescribed mass threshold scales by `ell^d`. -/ +theorem affine_mass_threshold_iff + (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) + (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {v : ℝ} (hv : 0 < v) (R : Set ℂ) (c : ℝ) : + ENNReal.ofReal (c * ell ^ dimension κ) ≤ + segmentSausageCharge κ (fun u => affine a ell (γ u)) s t (ell * v) + (affine a ell '' R) ↔ + ENNReal.ofReal c ≤ segmentSausageCharge κ γ s t v R := by + rw [segmentSausageCharge_affine κ a hell γ s t hv R, mul_comm c, + ENNReal.ofReal_mul (Real.rpow_pos_of_pos hell _).le] + exact ENNReal.mul_le_mul_iff_right ((ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hell _)).ne') + ENNReal.ofReal_ne_top + +/-- Pulling back the longitudinal coordinate leaves the actual first-front +clock unchanged. Spatial rescaling does not change curve order. -/ +theorem firstFrontTime_affine + (a : ℂ) (ell : ℝ) (γ : ℝ≥0 → ℂ) (p p' : ℂ → ℝ) + (hp : ∀ z, p' (affine a ell z) = p z) : + firstFrontTime (fun t => p' (affine a ell (γ t))) 1 = + firstFrontTime (fun t => p (γ t)) 1 := by + congr 1 + funext t + exact hp (γ t) + +/-- Affine scaling preserves all actual front-success conditions, including +whole-prefix containment and the terminal offset. -/ +theorem frontSuccess_affine_iff + {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ) + (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ) + (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ) + (hp : ∀ z, p' n (affine a ell z) = p n z) : + FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ + (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n = + FrontSuccess η p σ V b ε n := by + ext ω + have hmaps (T : ℝ≥0) : + MapsTo (fun t => affine a ell (η n ω t)) (Icc 0 T) (affine a ell '' V n) ↔ + MapsTo (η n ω) (Icc 0 T) (V n) := by + constructor + · intro h t ht + obtain ⟨z, hz, he⟩ := h ht + exact (affine_bijective a hell).1 he ▸ hz + · intro h t ht + exact mem_image_of_mem _ (h ht) + have hterminal (T : ℝ≥0) : + ‖affine a ell (η n ω T) - affine a ell (b n)‖ < ell * ε / 4 ↔ + ‖η n ω T - b n‖ < ε / 4 := by + have hd := affine_dist a hell.le (η n ω T) (b n) + rw [dist_eq_norm, dist_eq_norm] at hd + rw [hd, mul_div_assoc, mul_lt_mul_iff_right₀ hell] + simp only [FrontSuccess, mem_ofPred_eq, hp, hmaps, hterminal] + +/-- The complete one-step event has the same probability after simultaneous +spatial, mesh and mass rescaling. This is a deterministic event identity; +the trace law need not itself have been scaled. -/ +theorem frontSuccess_mass_probability_affine + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ) + (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ) + (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ) + (hp : ∀ z, p' n (affine a ell z) = p n z) + (κ : ℝ) (R : Set ℂ) {v : ℝ} (hv : 0 < v) (c : ℝ) : + P {ω | ω ∈ FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ + (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n ∧ + ENNReal.ofReal (c * ell ^ dimension κ) ≤ + segmentSausageCharge κ (fun t => affine a ell (η n ω t)) (σ ω) + (frontTime (fun t => p' n (affine a ell (η n ω t))) 1) + (ell * v) (affine a ell '' R)} = + P {ω | ω ∈ FrontSuccess η p σ V b ε n ∧ ENNReal.ofReal c ≤ + segmentSausageCharge κ (η n ω) (σ ω) + (frontTime (fun t => p n (η n ω t)) 1) v R} := by + congr 1 + ext ω + have hevent := frontSuccess_affine_iff η p p' σ V b ε a hell n hp + have ht : frontTime (fun t => p' n (affine a ell (η n ω t))) 1 = + frontTime (fun t => p n (η n ω t)) 1 := by + congr 1 + funext t + exact hp (η n ω t) + simp only [mem_ofPred_eq, hevent, ht, + affine_mass_threshold_iff κ a hell (η n ω) (σ ω) _ hv R c] + +end SLEExactGauge.RibbonPassageScaling + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonPrefixTransport.lean b/lean/OAI/Probability/SLE/Gauge/RibbonPrefixTransport.lean index 15f33b3e7d2c9b943b36e45ba955e5a87d4b6ba9..7fd91866ae878e5c3c66f91ca8a50c1e302a9e59 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonPrefixTransport.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonPrefixTransport.lean @@ -1,67 +1,67 @@ -import OAI.Probability.SLE.Gauge.WholeSausageScaling -import OAI.Probability.SLE.Gauge.RibbonGoodPrefix - -namespace OAI - -/-! A successful narrow physical prefix is preserved by the exact SLE -spatial/capacity scaling. The front is transported as its actual first hit, -not chosen as a separate completion certificate. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates FirstFrontPassage FrontClockChange ScaleTraceLaw FrontSuccessMeasurability - -lemma scalePath_at_scaled_time (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (t : ℝ≥0) : - scalePath c p (c ^ 2 * t) = (c : ℂ) * p t := by - change (c : ℂ) * p ((c ^ 2 * t) / c ^ 2) = _ - rw [mul_div_cancel_left₀ t (pow_ne_zero 2 hc.ne')] - -theorem goodRibbonPrefix_scalePath - (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (P : ℂ → ℂ) - (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) - (k ε : ℝ) - (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => (p : ℝ≥0 → ℂ)) - (fun _ => firstFrontTime (fun t => g (p t)) k) P ε) : - () ∈ GoodRibbonPrefix (fun _ : Unit => (scalePath c p : ℝ≥0 → ℂ)) - (fun _ => firstFrontTime (fun t => g' (scalePath c p t)) k) - (fun z => (c : ℂ) * P z) ε := by - cases hτ : firstFrontTime (fun t => g (p t)) k with - | top => exact False.elim (hpref.1 hτ) - | coe T => - let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) - have heval (t : ℝ≥0) : scalePath c p (e t) = (c : ℂ) * p t := - scalePath_at_scaled_time c hc p t - have hclock : (fun t => g' (scalePath c p (e t))) = (fun t => g (p t)) := by - funext t - rw [heval, hgg'] - have hhit := firstFrontTime_attained _ (g.continuous.comp p.continuous) k T hτ - have hnew := firstFrontTime_of_clock_hit e (fun t => g' (scalePath c p t)) k T - (by rw [hclock]; exact hτ) (by rw [heval, hgg']; exact hhit) - have hprefix : MapsTo p (Icc 0 T) (ribbonCore P ε) := by - simpa only [hτ, WithTop.untopD_coe] using hpref.2.1 - have hend : p T ∈ ribbonCore P (ε / 4) := by - simpa only [hτ, WithTop.untopD_coe] using hpref.2.2 - change firstFrontTime (fun t => g' (scalePath c p t)) k ≠ ⊤ ∧ _ - rw [hnew] - refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ - · intro t ht - have htt : t ≤ e T := by simpa only [WithTop.untopD_coe] using ht.2 - have hu : e.symm t ∈ Icc 0 T := ⟨bot_le, by - simpa only [e.symm_apply_apply] using e.symm.monotone htt⟩ - obtain ⟨z, hz, hez⟩ := hprefix hu - refine ⟨z, hz, ?_⟩ - change (c : ℂ) * P z = scalePath c p t - rw [hez, ← heval, e.apply_symm_apply] - · obtain ⟨z, hz, hez⟩ := hend - refine ⟨z, hz, ?_⟩ - change (c : ℂ) * P z = scalePath c p (e T) - rw [heval, hez] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.WholeSausageScaling +import OAI.Probability.SLE.Gauge.RibbonGoodPrefix + +namespace OAI + +/-! A successful narrow physical prefix is preserved by the exact SLE +spatial/capacity scaling. The front is transported as its actual first hit, +not chosen as a separate completion certificate. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates FirstFrontPassage FrontClockChange ScaleTraceLaw FrontSuccessMeasurability + +lemma scalePath_at_scaled_time (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (t : ℝ≥0) : + scalePath c p (c ^ 2 * t) = (c : ℂ) * p t := by + change (c : ℂ) * p ((c ^ 2 * t) / c ^ 2) = _ + rw [mul_div_cancel_left₀ t (pow_ne_zero 2 hc.ne')] + +theorem goodRibbonPrefix_scalePath + (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (P : ℂ → ℂ) + (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) + (k ε : ℝ) + (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => (p : ℝ≥0 → ℂ)) + (fun _ => firstFrontTime (fun t => g (p t)) k) P ε) : + () ∈ GoodRibbonPrefix (fun _ : Unit => (scalePath c p : ℝ≥0 → ℂ)) + (fun _ => firstFrontTime (fun t => g' (scalePath c p t)) k) + (fun z => (c : ℂ) * P z) ε := by + cases hτ : firstFrontTime (fun t => g (p t)) k with + | top => exact False.elim (hpref.1 hτ) + | coe T => + let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) + have heval (t : ℝ≥0) : scalePath c p (e t) = (c : ℂ) * p t := + scalePath_at_scaled_time c hc p t + have hclock : (fun t => g' (scalePath c p (e t))) = (fun t => g (p t)) := by + funext t + rw [heval, hgg'] + have hhit := firstFrontTime_attained _ (g.continuous.comp p.continuous) k T hτ + have hnew := firstFrontTime_of_clock_hit e (fun t => g' (scalePath c p t)) k T + (by rw [hclock]; exact hτ) (by rw [heval, hgg']; exact hhit) + have hprefix : MapsTo p (Icc 0 T) (ribbonCore P ε) := by + simpa only [hτ, WithTop.untopD_coe] using hpref.2.1 + have hend : p T ∈ ribbonCore P (ε / 4) := by + simpa only [hτ, WithTop.untopD_coe] using hpref.2.2 + change firstFrontTime (fun t => g' (scalePath c p t)) k ≠ ⊤ ∧ _ + rw [hnew] + refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ + · intro t ht + have htt : t ≤ e T := by simpa only [WithTop.untopD_coe] using ht.2 + have hu : e.symm t ∈ Icc 0 T := ⟨bot_le, by + simpa only [e.symm_apply_apply] using e.symm.monotone htt⟩ + obtain ⟨z, hz, hez⟩ := hprefix hu + refine ⟨z, hz, ?_⟩ + change (c : ℂ) * P z = scalePath c p t + rw [hez, ← heval, e.apply_symm_apply] + · obtain ⟨z, hz, hez⟩ := hend + refine ⟨z, hz, ?_⟩ + change (c : ℂ) * P z = scalePath c p (e T) + rw [heval, hez] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonRouteLifting.lean b/lean/OAI/Probability/SLE/Gauge/RibbonRouteLifting.lean index bcf2276d70f00fc5542a783bb7eef277f067225a..afb7d7a6cfb7225b160e2dff7824392d4fb2e595 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonRouteLifting.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonRouteLifting.lean @@ -1,116 +1,116 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry -import OAI.Probability.SLE.Geometry.RobustTransverseCrossing - -namespace OAI - -/-! Physical perturbations of a transverse ribbon route admit continuous -coordinate lifts. The fixed inverse Lipschitz bound then turns a physical -tube event into a forced hit of the actual stopped past. -/ - -noncomputable section -open Set Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates -open RectangleCrossing - -theorem inverse_coordinate_lipschitz - (P : ℂ → ℂ) (S : Set ℂ) (K : ℝ≥0) - (hlower : ∀ u ∈ S, ∀ v ∈ S, dist u v ≤ K * dist (P u) (P v)) : - LipschitzOnWith K (invFunOn P S) (P '' S) := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro u hu v hv - have hh := hlower _ (invFunOn_mem hu) _ (invFunOn_mem hv) - simpa only [invFunOn_eq hu, invFunOn_eq hv] using hh - -/-- Invariance of domain supplies a physical tube; the patch inverse is -continuous with exactly the given inverse Lipschitz bound. -/ -theorem physical_tube_has_coordinate_lift - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (S : Set ℂ) (hP : ContinuousOn P S) (hi : InjOn P S) - (K : ℝ≥0) (hK : 0 < (K : ℝ)) - (hlower : ∀ u ∈ S, ∀ v ∈ S, dist u v ≤ K * dist (P u) (P v)) - (ξ ζ : ℝ → ℂ) (I : Set ℝ) {r : ℝ} (hr : 0 < r) - (hξ : ∀ t ∈ I, closedBall (ξ t) r ⊆ S) - (hζ : ContinuousOn ζ I) - (hnear : ∀ t ∈ I, dist (ζ t) (P (ξ t)) < r / K) : - let q := invFunOn P S ∘ ζ - ContinuousOn q I ∧ MapsTo q I S ∧ EqOn (P ∘ q) ζ I ∧ - ∀ t ∈ I, dist (q t) (ξ t) < r := by - have hlift t (ht : t ∈ I) : - ∃ z ∈ ball (ξ t) r, P z = ζ t := by - apply ball_subset_image_ball P (ξ t) hr hK (hP.mono (hξ t ht)) - (hpublished P _ _ hr (hP.mono (ball_subset_closedBall.trans (hξ t ht))) - (hi.mono (ball_subset_closedBall.trans (hξ t ht)))) - (fun z hz => hlower z (hξ t ht hz) _ (hξ t ht (mem_closedBall_self hr.le))) - exact hnear t ht - have hζS : MapsTo ζ I (P '' S) := by - intro t ht - obtain ⟨z, hz, he⟩ := hlift t ht - exact ⟨z, hξ t ht (ball_subset_closedBall hz), he⟩ - dsimp only - refine ⟨(inverse_coordinate_lipschitz P S K hlower).continuousOn.comp hζ hζS, - (fun t ht => invFunOn_mem (hζS ht)), (fun t ht => invFunOn_eq (hζS ht)), ?_⟩ - intro t ht - obtain ⟨z, hz, he⟩ := hlift t ht - have hzS := hξ t ht (ball_subset_closedBall hz) - have heq : invFunOn P S (ζ t) = z := - hi (invFunOn_mem (hζS ht)) hzS ((invFunOn_eq (hζS ht)).trans he.symm) - simpa only [Function.comp_apply, heq, mem_ball] using hz - -def crossingPatch (k : ℝ) : Set ℂ := - {z | k - 1 ≤ z.re ∧ z.re ≤ k + 1 ∧ |z.im| ≤ 1} - -theorem crossingPatch_subset_closedRibbon {k : ℝ} (hk : 1 ≤ k) : - crossingPatch k ⊆ closedRibbon := by - intro z hz - exact ⟨by linarith [hz.1], hz.2.2⟩ - -theorem transverse_template_ball_subset_patch (k : ℝ) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) : - closedBall (transverseTemplate k t) (1 / 8) ⊆ crossingPatch k := by - intro z hz - have hn : ‖z - transverseTemplate k t‖ ≤ 1 / 8 := by - simpa only [mem_closedBall, dist_eq_norm] using hz - have hre := (Complex.abs_re_le_norm (z - transverseTemplate k t)).trans hn - have him := (Complex.abs_im_le_norm (z - transverseTemplate k t)).trans hn - simp only [Complex.sub_re, Complex.sub_im, transverseTemplate, Complex.add_re, - Complex.add_im, Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, - Complex.mul_im, Complex.I_re, Complex.I_im, mul_zero, mul_one, - add_zero, zero_add, sub_zero] at hre him - have hr := abs_le.mp hre - have hi := abs_le.mp him - refine ⟨by linarith, by linarith, abs_le.mpr ⟨by linarith [ht.1], by linarith [ht.2]⟩⟩ - -/-- A single physical tube radius, independent of the past, forces a hit. -Every coordinate and continuity condition for the perturbed route is -derived from the actual patch embedding rather than assumed. -/ -theorem physical_transverse_tube_hits_past - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (K : ℝ≥0) (hK : 0 < (K : ℝ)) {k ε T : ℝ} - (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) (hT : 0 < T) - (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, - dist u v ≤ K * dist (P u) (P v)) - (γ xp ζ : ℝ → ℂ) (hxp : ContinuousOn xp (Icc 0 T)) - (hγ : EqOn γ (P ∘ xp) (Icc 0 T)) - (hxp0 : (xp 0).re = 0) (hxpT : (xp T).re = k) - (hcore : ∀ t ∈ Icc (0 : ℝ) T, |(xp t).im| ≤ ε) - (hζ : ContinuousOn ζ (Icc 0 1)) - (hnear : ∀ t ∈ Icc (0 : ℝ) 1, - dist (ζ t) (P (transverseTemplate k t)) < (1 / 8 : ℝ) / K) : - ∃ s ∈ Icc (0 : ℝ) T, ∃ t ∈ Icc (0 : ℝ) 1, γ s = ζ t := by - have hsub := crossingPatch_subset_closedRibbon hk - obtain ⟨hq, _, hmap, hnearq⟩ := physical_tube_has_coordinate_lift hpublished - P (crossingPatch k) (hP.mono hsub) (hi.mono hsub) K hK hlower - (transverseTemplate k) ζ (Icc 0 1) (by norm_num : (0 : ℝ) < 1 / 8) - (fun _ ht => transverse_template_ball_subset_patch k ht) hζ hnear - obtain ⟨s, hs, t, ht, he⟩ := near_transverse_template_hits_past xp - (invFunOn P (crossingPatch k) ∘ ζ) hT hk hε hεsmall le_rfl hxp hq - hxp0 hxpT hcore hnearq - exact ⟨s, hs, t, ht, (hγ hs).trans ((congrArg P he).trans (hmap ht))⟩ - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry +import OAI.Probability.SLE.Geometry.RobustTransverseCrossing + +namespace OAI + +/-! Physical perturbations of a transverse ribbon route admit continuous +coordinate lifts. The fixed inverse Lipschitz bound then turns a physical +tube event into a forced hit of the actual stopped past. -/ + +noncomputable section +open Set Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates +open RectangleCrossing + +theorem inverse_coordinate_lipschitz + (P : ℂ → ℂ) (S : Set ℂ) (K : ℝ≥0) + (hlower : ∀ u ∈ S, ∀ v ∈ S, dist u v ≤ K * dist (P u) (P v)) : + LipschitzOnWith K (invFunOn P S) (P '' S) := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro u hu v hv + have hh := hlower _ (invFunOn_mem hu) _ (invFunOn_mem hv) + simpa only [invFunOn_eq hu, invFunOn_eq hv] using hh + +/-- Invariance of domain supplies a physical tube; the patch inverse is +continuous with exactly the given inverse Lipschitz bound. -/ +theorem physical_tube_has_coordinate_lift + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (S : Set ℂ) (hP : ContinuousOn P S) (hi : InjOn P S) + (K : ℝ≥0) (hK : 0 < (K : ℝ)) + (hlower : ∀ u ∈ S, ∀ v ∈ S, dist u v ≤ K * dist (P u) (P v)) + (ξ ζ : ℝ → ℂ) (I : Set ℝ) {r : ℝ} (hr : 0 < r) + (hξ : ∀ t ∈ I, closedBall (ξ t) r ⊆ S) + (hζ : ContinuousOn ζ I) + (hnear : ∀ t ∈ I, dist (ζ t) (P (ξ t)) < r / K) : + let q := invFunOn P S ∘ ζ + ContinuousOn q I ∧ MapsTo q I S ∧ EqOn (P ∘ q) ζ I ∧ + ∀ t ∈ I, dist (q t) (ξ t) < r := by + have hlift t (ht : t ∈ I) : + ∃ z ∈ ball (ξ t) r, P z = ζ t := by + apply ball_subset_image_ball P (ξ t) hr hK (hP.mono (hξ t ht)) + (hpublished P _ _ hr (hP.mono (ball_subset_closedBall.trans (hξ t ht))) + (hi.mono (ball_subset_closedBall.trans (hξ t ht)))) + (fun z hz => hlower z (hξ t ht hz) _ (hξ t ht (mem_closedBall_self hr.le))) + exact hnear t ht + have hζS : MapsTo ζ I (P '' S) := by + intro t ht + obtain ⟨z, hz, he⟩ := hlift t ht + exact ⟨z, hξ t ht (ball_subset_closedBall hz), he⟩ + dsimp only + refine ⟨(inverse_coordinate_lipschitz P S K hlower).continuousOn.comp hζ hζS, + (fun t ht => invFunOn_mem (hζS ht)), (fun t ht => invFunOn_eq (hζS ht)), ?_⟩ + intro t ht + obtain ⟨z, hz, he⟩ := hlift t ht + have hzS := hξ t ht (ball_subset_closedBall hz) + have heq : invFunOn P S (ζ t) = z := + hi (invFunOn_mem (hζS ht)) hzS ((invFunOn_eq (hζS ht)).trans he.symm) + simpa only [Function.comp_apply, heq, mem_ball] using hz + +def crossingPatch (k : ℝ) : Set ℂ := + {z | k - 1 ≤ z.re ∧ z.re ≤ k + 1 ∧ |z.im| ≤ 1} + +theorem crossingPatch_subset_closedRibbon {k : ℝ} (hk : 1 ≤ k) : + crossingPatch k ⊆ closedRibbon := by + intro z hz + exact ⟨by linarith [hz.1], hz.2.2⟩ + +theorem transverse_template_ball_subset_patch (k : ℝ) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) : + closedBall (transverseTemplate k t) (1 / 8) ⊆ crossingPatch k := by + intro z hz + have hn : ‖z - transverseTemplate k t‖ ≤ 1 / 8 := by + simpa only [mem_closedBall, dist_eq_norm] using hz + have hre := (Complex.abs_re_le_norm (z - transverseTemplate k t)).trans hn + have him := (Complex.abs_im_le_norm (z - transverseTemplate k t)).trans hn + simp only [Complex.sub_re, Complex.sub_im, transverseTemplate, Complex.add_re, + Complex.add_im, Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, + Complex.mul_im, Complex.I_re, Complex.I_im, mul_zero, mul_one, + add_zero, zero_add, sub_zero] at hre him + have hr := abs_le.mp hre + have hi := abs_le.mp him + refine ⟨by linarith, by linarith, abs_le.mpr ⟨by linarith [ht.1], by linarith [ht.2]⟩⟩ + +/-- A single physical tube radius, independent of the past, forces a hit. +Every coordinate and continuity condition for the perturbed route is +derived from the actual patch embedding rather than assumed. -/ +theorem physical_transverse_tube_hits_past + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (K : ℝ≥0) (hK : 0 < (K : ℝ)) {k ε T : ℝ} + (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) (hT : 0 < T) + (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, + dist u v ≤ K * dist (P u) (P v)) + (γ xp ζ : ℝ → ℂ) (hxp : ContinuousOn xp (Icc 0 T)) + (hγ : EqOn γ (P ∘ xp) (Icc 0 T)) + (hxp0 : (xp 0).re = 0) (hxpT : (xp T).re = k) + (hcore : ∀ t ∈ Icc (0 : ℝ) T, |(xp t).im| ≤ ε) + (hζ : ContinuousOn ζ (Icc 0 1)) + (hnear : ∀ t ∈ Icc (0 : ℝ) 1, + dist (ζ t) (P (transverseTemplate k t)) < (1 / 8 : ℝ) / K) : + ∃ s ∈ Icc (0 : ℝ) T, ∃ t ∈ Icc (0 : ℝ) 1, γ s = ζ t := by + have hsub := crossingPatch_subset_closedRibbon hk + obtain ⟨hq, _, hmap, hnearq⟩ := physical_tube_has_coordinate_lift hpublished + P (crossingPatch k) (hP.mono hsub) (hi.mono hsub) K hK hlower + (transverseTemplate k) ζ (Icc 0 1) (by norm_num : (0 : ℝ) < 1 / 8) + (fun _ ht => transverse_template_ball_subset_patch k ht) hζ hnear + obtain ⟨s, hs, t, ht, he⟩ := near_transverse_template_hits_past xp + (invFunOn P (crossingPatch k) ∘ ζ) hT hk hε hεsmall le_rfl hxp hq + hxp0 hxpT hcore hnearq + exact ⟨s, hs, t, ht, (hγ hs).trans ((congrArg P he).trans (hmap ht))⟩ + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonSequenceCompactness.lean b/lean/OAI/Probability/SLE/Gauge/RibbonSequenceCompactness.lean index 4079c57668c0929763cd8649d372621c839c4977..f6ad0f96ecd5445a7fcbe82560b497f5a36e4dd7 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonSequenceCompactness.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonSequenceCompactness.lean @@ -1,142 +1,142 @@ -import OAI.Probability.SLE.Geometry.ActualGuideCompactness -import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit - -namespace OAI - -/-! Joint extraction of normalized Loewner charts, physical and parameter -guides, and local ribbon coordinate maps under a raw waypoint-angle bound. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -private theorem uniform_subsequence {X Y : Type*} [UniformSpace Y] - {f : ℕ → X → Y} {g : X → Y} {S : Set X} - (h : TendstoUniformlyOn f g atTop S) {ψ : ℕ → ℕ} (hψ : StrictMono ψ) : - TendstoUniformlyOn (fun n => f (ψ n)) g atTop S := by - intro u hu - exact hψ.tendsto_atTop.eventually (h u hu) - -private theorem locally_uniform_subsequence {X Y : Type*} [TopologicalSpace X] [UniformSpace Y] - {f : ℕ → X → Y} {g : X → Y} {S : Set X} - (h : TendstoLocallyUniformlyOn f g atTop S) {ψ : ℕ → ℕ} (hψ : StrictMono ψ) : - TendstoLocallyUniformlyOn (fun n => f (ψ n)) g atTop S := by - intro u hu x hx - obtain ⟨V, hV, he⟩ := h u hu x hx - exact ⟨V, hV, hψ.tendsto_atTop.eventually he⟩ - -/-- The physical guide in the joint limit is the guide drawn in the -limiting coordinate chart. This uses uniform chart convergence, rather -than assuming the guide identity survives moving evaluation points. -/ -theorem coordinate_guide_limit_identity - (Qₙ : ℕ → ℂ → ℂ) (Q : ℂ → ℂ) (K : Set ℂ) (hK : IsClosed K) - (hQ : ContinuousOn Q K) (hQconv : TendstoUniformlyOn Qₙ Q atTop K) - (xₙ : ℕ → ℝ) (x : ℝ) (hx : Tendsto xₙ atTop (𝓝 x)) - (αₙ : ℕ → ℝ → ℂ) (α : ℝ → ℂ) - (hαconv : TendstoUniformlyOn αₙ α atTop (Icc 0 2)) - (hmem : ∀ n t, t ∈ Icc (0 : ℝ) 2 → - RibbonCoordinates.guideCoordinate 0 (xₙ n) t ∈ K) - (hshape : ∀ n t, t ∈ Icc (0 : ℝ) 2 → - αₙ n t = Qₙ n (RibbonCoordinates.guideCoordinate 0 (xₙ n) t)) : - EqOn α (fun t => Q (RibbonCoordinates.guideCoordinate 0 x t)) (Icc 0 2) := by - intro t ht - have hqc : Continuous (fun y : ℝ => RibbonCoordinates.guideCoordinate 0 y t) := by - unfold RibbonCoordinates.guideCoordinate - fun_prop - have hqt := (hqc.tendsto x).comp hx - have hqK : RibbonCoordinates.guideCoordinate 0 x t ∈ K := - hK.mem_of_tendsto hqt (Eventually.of_forall (fun n => hmem n t ht)) - have hqwithin : Tendsto (fun n => RibbonCoordinates.guideCoordinate 0 (xₙ n) t) - atTop (𝓝[K] (RibbonCoordinates.guideCoordinate 0 x t)) := - tendsto_nhdsWithin_iff.mpr ⟨hqt, Eventually.of_forall (fun n => hmem n t ht)⟩ - have hlim := hQconv.tendsto_comp (hQ _ hqK) hqwithin - have halim : Tendsto (fun n => αₙ n t) atTop - (𝓝 (Q (RibbonCoordinates.guideCoordinate 0 x t))) := by - convert hlim using 1 - funext n - exact hshape n t ht - exact tendsto_nhds_unique (hαconv.tendsto_at ht) halim - -/-- The normalized Loewner charts and coordinate charts admit one common -subsequence under the raw waypoint-angle normalization hypothesis. -/ -theorem actual_ribbon_joint_compactness - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedLoewnerBoundaryExtension) - (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) - (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) - (hU : ∀ n, Continuous (U n)) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) - (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) - (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) - (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) - (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) - (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) - (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) - (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → - c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) - (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) - (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) - (Qₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) - (x₀ : ℂ) (hx₀ : x₀ ∈ K) (Cq Lq : ℝ≥0) (Rq : ℝ) - (hQupper : ∀ n, LipschitzOnWith Cq (Qₙ n) K) - (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) - (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) - (hQheight : ∀ n z, z ∈ K → 0 ≤ (Qₙ n z).im) - (hplanar : PublishedPlanarDiskOpenness) - (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / - ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : - ∃ (Q : ℂ → ℂ) (b : ℕ → ℝ) (Fb : ℕ → ℂ → OnePoint ℂ) (βₙ : ℕ → ℝ → ℂ) - (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), - (∀ n, 0 < b n) ∧ - (∀ n, ContinuousOn (Fb n) {z : ℂ | 0 ≤ z.im}) ∧ - (∀ n z, z ∈ upperHalfPlane → Fb n z = - (normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) z : OnePoint ℂ)) ∧ - (∀ n, ContinuousOn (βₙ n) (Icc 0 2)) ∧ - (∀ n, βₙ n 0 = 0) ∧ - (∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im) ∧ - (∀ n t, t ∈ Ioc (0 : ℝ) 2 → βₙ n t = - normalizedForward (G n (T n)) (U n (T n)) (b n) (αₙ n t)) ∧ - (∀ n t, t ∈ Icc (0 : ℝ) 2 → Fb n (βₙ n t) = (αₙ n t : OnePoint ℂ)) ∧ - StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ - (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ - TendstoLocallyUniformlyOn - (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) - F atTop upperHalfPlane ∧ - TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ - TendstoUniformlyOn (fun n => βₙ (φ n)) β atTop (Icc 0 2) ∧ - Tendsto (fun n => γₙ (φ n) (T (φ n))) atTop (𝓝 (α 0)) ∧ - LipschitzOnWith C α (Icc 0 2) ∧ ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ - β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ - MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) ∧ - TendstoUniformlyOn (fun n => Qₙ (φ n)) Q atTop K ∧ - LipschitzOnWith Cq Q K ∧ - (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ Lq * dist (Q x) (Q y)) ∧ - InjOn Q K ∧ MapsTo Q (interior K) upperHalfPlane := by - obtain ⟨b, Fb, βₙ, φ, F, α, β, hb, hbar, hext, hβc, hβ0, hβH, hβeq, - hbarα, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, - hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := - actual_guide_joint_compactness_with_family P κ hκ hκ8 γ hSLE hcompact hboundary - U γₙ G F₀ hU hw T hT hfresh αₙ C s c A hC hs hc hαL hα0 haBound hchord hαD hclear hangle - obtain ⟨Q, ψ, hψ, hQconv, hQu, hQl, hQi, hQH⟩ := - RibbonCoordinates.exists_bilip_halfPlane_subsequence hplanar (fun n => Qₙ (φ n)) - K hK x₀ hx₀ Cq Lq Rq (fun n => hQupper (φ n)) (fun n => hQlower (φ n)) - (fun n => hQtip (φ n)) (fun n => hQheight (φ n)) - have hFsub := locally_uniform_subsequence hFconv hψ - have hαsub := uniform_subsequence hαconv hψ - have hβsub := uniform_subsequence hβconv hψ - exact ⟨Q, b, Fb, βₙ, φ ∘ ψ, F, α, β, hb, hbar, hext, hβc, hβ0, hβH, - hβeq, hbarα, hφ.comp hψ, hF, hiF, hFne, homitlim, hFsub, hαsub, hβsub, - hatip.comp hψ.tendsto_atTop, hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, - hβlimH, hlimrel, hQconv, hQu, hQl, hQi, hQH⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ActualGuideCompactness +import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit + +namespace OAI + +/-! Joint extraction of normalized Loewner charts, physical and parameter +guides, and local ribbon coordinate maps under a raw waypoint-angle bound. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +private theorem uniform_subsequence {X Y : Type*} [UniformSpace Y] + {f : ℕ → X → Y} {g : X → Y} {S : Set X} + (h : TendstoUniformlyOn f g atTop S) {ψ : ℕ → ℕ} (hψ : StrictMono ψ) : + TendstoUniformlyOn (fun n => f (ψ n)) g atTop S := by + intro u hu + exact hψ.tendsto_atTop.eventually (h u hu) + +private theorem locally_uniform_subsequence {X Y : Type*} [TopologicalSpace X] [UniformSpace Y] + {f : ℕ → X → Y} {g : X → Y} {S : Set X} + (h : TendstoLocallyUniformlyOn f g atTop S) {ψ : ℕ → ℕ} (hψ : StrictMono ψ) : + TendstoLocallyUniformlyOn (fun n => f (ψ n)) g atTop S := by + intro u hu x hx + obtain ⟨V, hV, he⟩ := h u hu x hx + exact ⟨V, hV, hψ.tendsto_atTop.eventually he⟩ + +/-- The physical guide in the joint limit is the guide drawn in the +limiting coordinate chart. This uses uniform chart convergence, rather +than assuming the guide identity survives moving evaluation points. -/ +theorem coordinate_guide_limit_identity + (Qₙ : ℕ → ℂ → ℂ) (Q : ℂ → ℂ) (K : Set ℂ) (hK : IsClosed K) + (hQ : ContinuousOn Q K) (hQconv : TendstoUniformlyOn Qₙ Q atTop K) + (xₙ : ℕ → ℝ) (x : ℝ) (hx : Tendsto xₙ atTop (𝓝 x)) + (αₙ : ℕ → ℝ → ℂ) (α : ℝ → ℂ) + (hαconv : TendstoUniformlyOn αₙ α atTop (Icc 0 2)) + (hmem : ∀ n t, t ∈ Icc (0 : ℝ) 2 → + RibbonCoordinates.guideCoordinate 0 (xₙ n) t ∈ K) + (hshape : ∀ n t, t ∈ Icc (0 : ℝ) 2 → + αₙ n t = Qₙ n (RibbonCoordinates.guideCoordinate 0 (xₙ n) t)) : + EqOn α (fun t => Q (RibbonCoordinates.guideCoordinate 0 x t)) (Icc 0 2) := by + intro t ht + have hqc : Continuous (fun y : ℝ => RibbonCoordinates.guideCoordinate 0 y t) := by + unfold RibbonCoordinates.guideCoordinate + fun_prop + have hqt := (hqc.tendsto x).comp hx + have hqK : RibbonCoordinates.guideCoordinate 0 x t ∈ K := + hK.mem_of_tendsto hqt (Eventually.of_forall (fun n => hmem n t ht)) + have hqwithin : Tendsto (fun n => RibbonCoordinates.guideCoordinate 0 (xₙ n) t) + atTop (𝓝[K] (RibbonCoordinates.guideCoordinate 0 x t)) := + tendsto_nhdsWithin_iff.mpr ⟨hqt, Eventually.of_forall (fun n => hmem n t ht)⟩ + have hlim := hQconv.tendsto_comp (hQ _ hqK) hqwithin + have halim : Tendsto (fun n => αₙ n t) atTop + (𝓝 (Q (RibbonCoordinates.guideCoordinate 0 x t))) := by + convert hlim using 1 + funext n + exact hshape n t ht + exact tendsto_nhds_unique (hαconv.tendsto_at ht) halim + +/-- The normalized Loewner charts and coordinate charts admit one common +subsequence under the raw waypoint-angle normalization hypothesis. -/ +theorem actual_ribbon_joint_compactness + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedLoewnerBoundaryExtension) + (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) + (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) + (hU : ∀ n, Continuous (U n)) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) + (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) + (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) + (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) + (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) + (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) + (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) + (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → + c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) + (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) + (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) + (Qₙ : ℕ → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) + (x₀ : ℂ) (hx₀ : x₀ ∈ K) (Cq Lq : ℝ≥0) (Rq : ℝ) + (hQupper : ∀ n, LipschitzOnWith Cq (Qₙ n) K) + (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) + (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) + (hQheight : ∀ n z, z ∈ K → 0 ≤ (Qₙ n z).im) + (hplanar : PublishedPlanarDiskOpenness) + (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / + ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : + ∃ (Q : ℂ → ℂ) (b : ℕ → ℝ) (Fb : ℕ → ℂ → OnePoint ℂ) (βₙ : ℕ → ℝ → ℂ) + (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), + (∀ n, 0 < b n) ∧ + (∀ n, ContinuousOn (Fb n) {z : ℂ | 0 ≤ z.im}) ∧ + (∀ n z, z ∈ upperHalfPlane → Fb n z = + (normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) z : OnePoint ℂ)) ∧ + (∀ n, ContinuousOn (βₙ n) (Icc 0 2)) ∧ + (∀ n, βₙ n 0 = 0) ∧ + (∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im) ∧ + (∀ n t, t ∈ Ioc (0 : ℝ) 2 → βₙ n t = + normalizedForward (G n (T n)) (U n (T n)) (b n) (αₙ n t)) ∧ + (∀ n t, t ∈ Icc (0 : ℝ) 2 → Fb n (βₙ n t) = (αₙ n t : OnePoint ℂ)) ∧ + StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ + (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ + TendstoLocallyUniformlyOn + (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) + F atTop upperHalfPlane ∧ + TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ + TendstoUniformlyOn (fun n => βₙ (φ n)) β atTop (Icc 0 2) ∧ + Tendsto (fun n => γₙ (φ n) (T (φ n))) atTop (𝓝 (α 0)) ∧ + LipschitzOnWith C α (Icc 0 2) ∧ ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ + β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ + MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) ∧ + TendstoUniformlyOn (fun n => Qₙ (φ n)) Q atTop K ∧ + LipschitzOnWith Cq Q K ∧ + (∀ x ∈ K, ∀ y ∈ K, dist x y ≤ Lq * dist (Q x) (Q y)) ∧ + InjOn Q K ∧ MapsTo Q (interior K) upperHalfPlane := by + obtain ⟨b, Fb, βₙ, φ, F, α, β, hb, hbar, hext, hβc, hβ0, hβH, hβeq, + hbarα, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, + hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := + actual_guide_joint_compactness_with_family P κ hκ hκ8 γ hSLE hcompact hboundary + U γₙ G F₀ hU hw T hT hfresh αₙ C s c A hC hs hc hαL hα0 haBound hchord hαD hclear hangle + obtain ⟨Q, ψ, hψ, hQconv, hQu, hQl, hQi, hQH⟩ := + RibbonCoordinates.exists_bilip_halfPlane_subsequence hplanar (fun n => Qₙ (φ n)) + K hK x₀ hx₀ Cq Lq Rq (fun n => hQupper (φ n)) (fun n => hQlower (φ n)) + (fun n => hQtip (φ n)) (fun n => hQheight (φ n)) + have hFsub := locally_uniform_subsequence hFconv hψ + have hαsub := uniform_subsequence hαconv hψ + have hβsub := uniform_subsequence hβconv hψ + exact ⟨Q, b, Fb, βₙ, φ ∘ ψ, F, α, β, hb, hbar, hext, hβc, hβ0, hβH, + hβeq, hbarα, hφ.comp hψ, hF, hiF, hFne, homitlim, hFsub, hαsub, hβsub, + hatip.comp hψ.tendsto_atTop, hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, + hβlimH, hlimrel, hQconv, hQu, hQl, hQi, hQH⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonSideCorridors.lean b/lean/OAI/Probability/SLE/Gauge/RibbonSideCorridors.lean index ae6be295051de9efecdbbda2faac26f08f017524..f6d28df519f6b5697dbad909f8fe3de240824150 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonSideCorridors.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonSideCorridors.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry - -namespace OAI - -/-! The two strips outside the narrow core are in the actual surviving -component. Their connection to the unbounded future excludes filled hulls, -even when the stopped past is not simple. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -def sideCorridor (a b : ℝ) : Set ℂ := - {z | 0 < z.re ∧ z.im ∈ Ioo a b} - -theorem sideCorridor_convex (a b : ℝ) : Convex ℝ (sideCorridor a b) := by - exact (convex_halfSpace_gt Complex.reLm.isLinear 0).inter - ((convex_Ioo a b).is_linear_preimage Complex.imLm.isLinear) - -/-- A strip strictly outside the past's core connects to the free future -and hence is in the unbounded surviving domain. This covers either side. -/ -theorem sideCorridor_survives - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) {k ε a b : ℝ} (hk : 0 ≤ k) - (hab : a < b) (ha : -1 ≤ a) (hb : b ≤ 1) - (hgap : ∀ x ∈ Ioo a b, ε < |x|) - (hpast : γ '' Icc 0 T ⊆ - P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) : - P '' sideCorridor a b ⊆ survivingDomain γ T := by - have hsideSub : sideCorridor a b ⊆ closedRibbon := by - intro z hz - exact ⟨hz.1.le, (abs_lt.mpr ⟨ha.trans_lt hz.2.1, hz.2.2.trans_le hb⟩).le⟩ - have hfutureSub := futureRibbon_subset_closedRibbon hk - let S := sideCorridor a b ∪ futureRibbon k - have hSsub : S ⊆ closedRibbon := union_subset hsideSub hfutureSub - let z : ℂ := ((k + 1 : ℝ) : ℂ) + (((a + b) / 2 : ℝ) : ℂ) * Complex.I - have hzre : z.re = k + 1 := by simp [z] - have hzim : z.im = (a + b) / 2 := by simp [z] - have hzm : z.im ∈ Ioo a b := by rw [hzim]; constructor <;> linarith - have hzside : z ∈ sideCorridor a b := ⟨by rw [hzre]; linarith, hzm⟩ - have hzfuture : z ∈ futureRibbon k := - ⟨by rw [hzre]; linarith, - abs_lt.mpr ⟨ha.trans_lt hzm.1, hzm.2.trans_le hb⟩⟩ - have hSconn : IsPreconnected S := - (sideCorridor_convex a b).isPreconnected.union z hzside hzfuture - (futureRibbon_convex k).isPreconnected - have hfree : P '' S ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by - rintro y ⟨u, hu, rfl⟩ - have huH : 0 < u.re := by - rcases hu with hu | hu - · exact hu.1 - · exact hk.trans_lt hu.1 - refine ⟨hH u (hSsub hu) huH, ?_⟩ - intro hp - obtain ⟨v, hv, he⟩ := hpast hp - have hvu := hi hv.1 (hSsub hu) he - rw [hvu] at hv - rcases hu with hu | hu - · exact (not_le_of_gt (hgap u.im hu.2)) hv.2.2 - · exact (not_le_of_gt hu.1) hv.2.1 - have hSunbounded : ¬ Bornology.IsBounded (P '' S) := by - intro hbound - exact hunbounded (hbound.subset (image_mono subset_union_right)) - exact (image_mono subset_union_left).trans - (free_preconnected_subset_surviving γ T (P '' S) - (hSconn.image P (hP.mono hSsub)) hfree hSunbounded) - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry + +namespace OAI + +/-! The two strips outside the narrow core are in the actual surviving +component. Their connection to the unbounded future excludes filled hulls, +even when the stopped past is not simple. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +def sideCorridor (a b : ℝ) : Set ℂ := + {z | 0 < z.re ∧ z.im ∈ Ioo a b} + +theorem sideCorridor_convex (a b : ℝ) : Convex ℝ (sideCorridor a b) := by + exact (convex_halfSpace_gt Complex.reLm.isLinear 0).inter + ((convex_Ioo a b).is_linear_preimage Complex.imLm.isLinear) + +/-- A strip strictly outside the past's core connects to the free future +and hence is in the unbounded surviving domain. This covers either side. -/ +theorem sideCorridor_survives + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) {k ε a b : ℝ} (hk : 0 ≤ k) + (hab : a < b) (ha : -1 ≤ a) (hb : b ≤ 1) + (hgap : ∀ x ∈ Ioo a b, ε < |x|) + (hpast : γ '' Icc 0 T ⊆ + P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) : + P '' sideCorridor a b ⊆ survivingDomain γ T := by + have hsideSub : sideCorridor a b ⊆ closedRibbon := by + intro z hz + exact ⟨hz.1.le, (abs_lt.mpr ⟨ha.trans_lt hz.2.1, hz.2.2.trans_le hb⟩).le⟩ + have hfutureSub := futureRibbon_subset_closedRibbon hk + let S := sideCorridor a b ∪ futureRibbon k + have hSsub : S ⊆ closedRibbon := union_subset hsideSub hfutureSub + let z : ℂ := ((k + 1 : ℝ) : ℂ) + (((a + b) / 2 : ℝ) : ℂ) * Complex.I + have hzre : z.re = k + 1 := by simp [z] + have hzim : z.im = (a + b) / 2 := by simp [z] + have hzm : z.im ∈ Ioo a b := by rw [hzim]; constructor <;> linarith + have hzside : z ∈ sideCorridor a b := ⟨by rw [hzre]; linarith, hzm⟩ + have hzfuture : z ∈ futureRibbon k := + ⟨by rw [hzre]; linarith, + abs_lt.mpr ⟨ha.trans_lt hzm.1, hzm.2.trans_le hb⟩⟩ + have hSconn : IsPreconnected S := + (sideCorridor_convex a b).isPreconnected.union z hzside hzfuture + (futureRibbon_convex k).isPreconnected + have hfree : P '' S ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by + rintro y ⟨u, hu, rfl⟩ + have huH : 0 < u.re := by + rcases hu with hu | hu + · exact hu.1 + · exact hk.trans_lt hu.1 + refine ⟨hH u (hSsub hu) huH, ?_⟩ + intro hp + obtain ⟨v, hv, he⟩ := hpast hp + have hvu := hi hv.1 (hSsub hu) he + rw [hvu] at hv + rcases hu with hu | hu + · exact (not_le_of_gt (hgap u.im hu.2)) hv.2.2 + · exact (not_le_of_gt hu.1) hv.2.1 + have hSunbounded : ¬ Bornology.IsBounded (P '' S) := by + intro hbound + exact hunbounded (hbound.subset (image_mono subset_union_right)) + exact (image_mono subset_union_left).trans + (free_preconnected_subset_surviving γ T (P '' S) + (hSconn.image P (hP.mono hSsub)) hfree hSunbounded) + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonStepPatch.lean b/lean/OAI/Probability/SLE/Gauge/RibbonStepPatch.lean index 93d17f1a9b41a767fa783b2f1277cea052537249..baf5e55843dd5617e8e46eb4d79a23064d9591ba 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonStepPatch.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonStepPatch.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Gauge.RibbonAccessGuide -import OAI.Probability.SLE.Gauge.RibbonRouteLifting -import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry - -namespace OAI - -/-! A fixed compact coordinate patch for the actual one-step ribbon. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -/-- The length-four patch used for one step and its two-sided waypoint routes. -/ -def ribbonStepPatch (k : ℝ) : Set ℂ := - {z | k - 1 ≤ z.re ∧ z.re ≤ k + 3 ∧ |z.im| ≤ 1} - -/-- A padded compact patch in coordinates relative to the current front. -/ -def ribbonStepCompact : Set ℂ := - {z | -(1 / 2 : ℝ) ≤ z.re ∧ z.re ≤ 5 / 2 ∧ |z.im| ≤ 1 / 2} - -theorem ribbonStepCompact_isCompact : IsCompact ribbonStepCompact := by - have hc : IsClosed ribbonStepCompact := - (isClosed_le continuous_const Complex.continuous_re).inter - ((isClosed_le Complex.continuous_re continuous_const).inter - (isClosed_le Complex.continuous_im.abs continuous_const)) - apply (isCompact_closedBall (0 : ℂ) 4).of_isClosed_subset hc - intro z hz - have hre : |z.re| ≤ 5 / 2 := abs_le.mpr ⟨by linarith [hz.1], hz.2.1⟩ - have hn := Complex.norm_le_abs_re_add_abs_im z - simpa only [mem_closedBall, dist_zero_right] using - (show ‖z‖ ≤ 4 by linarith [hz.2.2]) - -theorem ribbonStepCompact_interior - {z : ℂ} (hre : -(1 / 2 : ℝ) < z.re ∧ z.re < 5 / 2) - (him : |z.im| < 1 / 2) : z ∈ interior ribbonStepCompact := by - let V : Set ℂ := {z | -(1 / 2 : ℝ) < z.re ∧ z.re < 5 / 2 ∧ |z.im| < 1 / 2} - have hV : IsOpen V := (isOpen_lt continuous_const Complex.continuous_re).inter - ((isOpen_lt Complex.continuous_re continuous_const).inter - (isOpen_lt Complex.continuous_im.abs continuous_const)) - have hVK : V ⊆ ribbonStepCompact := by - intro w hw - exact ⟨hw.1.le, hw.2.1.le, hw.2.2.le⟩ - exact interior_maximal hVK hV ⟨hre.1, hre.2, him⟩ - -theorem ribbonStepCompact_zero : (0 : ℂ) ∈ ribbonStepCompact := by - change -(1 / 2 : ℝ) ≤ 0 ∧ (0 : ℝ) ≤ 5 / 2 ∧ |(0 : ℝ)| ≤ 1 / 2 - norm_num - -theorem ribbonStepCompact_core {ε : ℝ} (hε : ε ≤ 1 / 8) : - {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ - interior ribbonStepCompact := by - intro z hz - apply ribbonStepCompact_interior - · constructor <;> linarith [hz.1, hz.2.1] - · linarith [hz.2.2] - -theorem ribbonStepCompact_passage {ε : ℝ} (hε : ε ≤ 1 / 8) - (C : ℝ≥0) (η : ℝ) : - ribbonStepCompact ∩ passageCoordinateRegion C ε η ⊆ interior ribbonStepCompact := by - intro z hz - apply ribbonStepCompact_interior - · constructor <;> linarith [hz.2.1, hz.2.2.1] - · linarith [hz.2.2.2.1] - -theorem guidePatch_subset_ribbonStepPatch (k : ℝ) : - guidePatch k ⊆ ribbonStepPatch k := by - intro z hz - exact ⟨by linarith [hz.1], hz.2.1, hz.2.2⟩ - -theorem crossingPatch_subset_ribbonStepPatch (k : ℝ) : - crossingPatch k ⊆ ribbonStepPatch k := by - intro z hz - exact ⟨hz.1, by linarith [hz.2.1], hz.2.2⟩ - -theorem translate_ribbonStepCompact {k : ℝ} {z : ℂ} (hz : z ∈ ribbonStepCompact) : - (k : ℂ) + z ∈ ribbonStepPatch k := by - change k - 1 ≤ k + z.re ∧ k + z.re ≤ k + 3 ∧ |0 + z.im| ≤ 1 - exact ⟨by linarith [hz.1], by linarith [hz.2.1], by simpa using hz.2.2.trans (by norm_num)⟩ - -theorem guideCoordinate_translate (k x t : ℝ) : - (k : ℂ) + guideCoordinate 0 x t = guideCoordinate k x t := by - apply Complex.ext <;> simp [guideCoordinate] - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonAccessGuide +import OAI.Probability.SLE.Gauge.RibbonRouteLifting +import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry + +namespace OAI + +/-! A fixed compact coordinate patch for the actual one-step ribbon. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +/-- The length-four patch used for one step and its two-sided waypoint routes. -/ +def ribbonStepPatch (k : ℝ) : Set ℂ := + {z | k - 1 ≤ z.re ∧ z.re ≤ k + 3 ∧ |z.im| ≤ 1} + +/-- A padded compact patch in coordinates relative to the current front. -/ +def ribbonStepCompact : Set ℂ := + {z | -(1 / 2 : ℝ) ≤ z.re ∧ z.re ≤ 5 / 2 ∧ |z.im| ≤ 1 / 2} + +theorem ribbonStepCompact_isCompact : IsCompact ribbonStepCompact := by + have hc : IsClosed ribbonStepCompact := + (isClosed_le continuous_const Complex.continuous_re).inter + ((isClosed_le Complex.continuous_re continuous_const).inter + (isClosed_le Complex.continuous_im.abs continuous_const)) + apply (isCompact_closedBall (0 : ℂ) 4).of_isClosed_subset hc + intro z hz + have hre : |z.re| ≤ 5 / 2 := abs_le.mpr ⟨by linarith [hz.1], hz.2.1⟩ + have hn := Complex.norm_le_abs_re_add_abs_im z + simpa only [mem_closedBall, dist_zero_right] using + (show ‖z‖ ≤ 4 by linarith [hz.2.2]) + +theorem ribbonStepCompact_interior + {z : ℂ} (hre : -(1 / 2 : ℝ) < z.re ∧ z.re < 5 / 2) + (him : |z.im| < 1 / 2) : z ∈ interior ribbonStepCompact := by + let V : Set ℂ := {z | -(1 / 2 : ℝ) < z.re ∧ z.re < 5 / 2 ∧ |z.im| < 1 / 2} + have hV : IsOpen V := (isOpen_lt continuous_const Complex.continuous_re).inter + ((isOpen_lt Complex.continuous_re continuous_const).inter + (isOpen_lt Complex.continuous_im.abs continuous_const)) + have hVK : V ⊆ ribbonStepCompact := by + intro w hw + exact ⟨hw.1.le, hw.2.1.le, hw.2.2.le⟩ + exact interior_maximal hVK hV ⟨hre.1, hre.2, him⟩ + +theorem ribbonStepCompact_zero : (0 : ℂ) ∈ ribbonStepCompact := by + change -(1 / 2 : ℝ) ≤ 0 ∧ (0 : ℝ) ≤ 5 / 2 ∧ |(0 : ℝ)| ≤ 1 / 2 + norm_num + +theorem ribbonStepCompact_core {ε : ℝ} (hε : ε ≤ 1 / 8) : + {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ + interior ribbonStepCompact := by + intro z hz + apply ribbonStepCompact_interior + · constructor <;> linarith [hz.1, hz.2.1] + · linarith [hz.2.2] + +theorem ribbonStepCompact_passage {ε : ℝ} (hε : ε ≤ 1 / 8) + (C : ℝ≥0) (η : ℝ) : + ribbonStepCompact ∩ passageCoordinateRegion C ε η ⊆ interior ribbonStepCompact := by + intro z hz + apply ribbonStepCompact_interior + · constructor <;> linarith [hz.2.1, hz.2.2.1] + · linarith [hz.2.2.2.1] + +theorem guidePatch_subset_ribbonStepPatch (k : ℝ) : + guidePatch k ⊆ ribbonStepPatch k := by + intro z hz + exact ⟨by linarith [hz.1], hz.2.1, hz.2.2⟩ + +theorem crossingPatch_subset_ribbonStepPatch (k : ℝ) : + crossingPatch k ⊆ ribbonStepPatch k := by + intro z hz + exact ⟨hz.1, by linarith [hz.2.1], hz.2.2⟩ + +theorem translate_ribbonStepCompact {k : ℝ} {z : ℂ} (hz : z ∈ ribbonStepCompact) : + (k : ℂ) + z ∈ ribbonStepPatch k := by + change k - 1 ≤ k + z.re ∧ k + z.re ≤ k + 3 ∧ |0 + z.im| ≤ 1 + exact ⟨by linarith [hz.1], by linarith [hz.2.1], by simpa using hz.2.2.trans (by norm_num)⟩ + +theorem guideCoordinate_translate (k x t : ℝ) : + (k : ℂ) + guideCoordinate 0 x t = guideCoordinate k x t := by + apply Complex.ext <;> simp [guideCoordinate] + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonStepSuccess.lean b/lean/OAI/Probability/SLE/Gauge/RibbonStepSuccess.lean index 8a8d97bce164cb684cba53ccaa655161bbe721c6..f2fa0f2ff4f71a69d65155d9cfd6597ab0c42af2 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonStepSuccess.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonStepSuccess.lean @@ -1,85 +1,85 @@ -import OAI.Probability.SLE.Gauge.RibbonGoodPrefix - -namespace OAI - -/-! Actual passage-and-mass successes are measured in the next global -front past, with the earlier global front as the mass window's start. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability - -local instance stepSuccessPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance stepSuccessPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -def RibbonStepSuccess {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) - (σ τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε κ v c : ℝ) (R : Set ℂ) : Set Ω := - GoodRibbonPrefix γ τ P ε ∩ {ω | ENNReal.ofReal c ≤ - segmentSausageCharge κ (γ ω) ((σ ω).untopD 0) ((τ ω).untopD 0) v R} - -lemma stoppedAtPath_eq_on_prefix (f : C(ℝ≥0, ℂ)) (t : ℝ≥0) : - EqOn (stoppedAtPath f (t : WithTop ℝ≥0)) f (Icc 0 t) := by - intro u hu - simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe, - clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, min_eq_left hu.2, - max_eq_right hu.1] - -lemma segmentSausageCharge_shift - (κ : ℝ) (γ : ℝ≥0 → ℂ) (T S : ℝ≥0) (v : ℝ) (R : Set ℂ) : - segmentSausageCharge κ (fun t => γ (T + t)) 0 S v R = - segmentSausageCharge κ γ T (T + S) v R := by - have hs : sourceSausageSet (fun t => γ (T + t)) 0 S v = - sourceSausageSet γ T (T + S) v := by - ext z - constructor - · rintro ⟨t, ht, hd⟩ - exact ⟨T + t, ⟨le_add_of_nonneg_right bot_le, add_le_add le_rfl ht.2⟩, hd⟩ - · rintro ⟨t, ht, hd⟩ - refine ⟨t - T, ⟨bot_le, ?_⟩, ?_⟩ - · exact (tsub_le_iff_right).mpr (by simpa only [add_comm] using ht.2) - · simpa only [add_tsub_cancel_of_le ht.1] using hd - simp only [segmentSausageCharge, hs] - -/-- The whole actual step and its sausage lower bound are adapted to the -next front. Both global front times may be infinite off success. -/ -theorem ribbonStepSuccess_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) - (hc : ∀ ω, Continuous (γ ω)) - (σ τ : Ω → WithTop ℝ≥0) (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ) - (hστ : σ ≤ τ) (P : ℂ → ℂ) - (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) - (ε κ v c : ℝ) (R : Set ℂ) : - MeasurableSet[hτ.measurableSpace] (RibbonStepSuccess γ σ τ P ε κ v c R) := by - let f := fun ω => stoppedAtPath (sourceTracePath γ hc ω) (τ ω) - have hfm : Measurable[hτ.measurableSpace] f := stoppedAtPath_measurable ℱ γ ha hc τ hτ - have htm : Measurable[hτ.measurableSpace] (fun ω => (τ ω).untopD 0) := hτ.measurable.untopD 0 - have hsm : Measurable[hτ.measurableSpace] (fun ω => (σ ω).untopD 0) := - (hσ.measurable.mono (IsStoppingTime.measurableSpace_mono hσ hτ hστ) le_rfl).untopD 0 - have hgood := goodRibbonPrefix_measurable ℱ γ ha hc τ hτ P he ε - let : MeasurableSpace Ω := hτ.measurableSpace - have hm := hgood.inter - (measurableSet_le (measurable_const : Measurable (fun _ : Ω => ENNReal.ofReal c)) - (segmentSausageCharge_measurable f hfm _ _ hsm htm κ v R)) - convert hm using 1 - ext ω - by_cases hω : ω ∈ GoodRibbonPrefix γ τ P ε - · have ht := hω.1 - lift τ ω to ℝ≥0 using ht with t hτt - have hs := sourceSausageSet_congr_interval - (s := (σ ω).untopD 0) (t := t) - (fun u hu => stoppedAtPath_eq_on_prefix (sourceTracePath γ hc ω) t - ⟨bot_le, hu.2⟩) v - simp only [RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq, hω, true_and, - f, ← hτt, WithTop.untopD_coe, segmentSausageCharge, hs] - rfl - · simp only [RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq, hω, false_and] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonGoodPrefix + +namespace OAI + +/-! Actual passage-and-mass successes are measured in the next global +front past, with the earlier global front as the mass window's start. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability + +local instance stepSuccessPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance stepSuccessPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +def RibbonStepSuccess {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) + (σ τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε κ v c : ℝ) (R : Set ℂ) : Set Ω := + GoodRibbonPrefix γ τ P ε ∩ {ω | ENNReal.ofReal c ≤ + segmentSausageCharge κ (γ ω) ((σ ω).untopD 0) ((τ ω).untopD 0) v R} + +lemma stoppedAtPath_eq_on_prefix (f : C(ℝ≥0, ℂ)) (t : ℝ≥0) : + EqOn (stoppedAtPath f (t : WithTop ℝ≥0)) f (Icc 0 t) := by + intro u hu + simp only [stoppedAtPath, WithTop.coe_ne_top, ite_false, WithTop.untopD_coe, + clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk, min_eq_left hu.2, + max_eq_right hu.1] + +lemma segmentSausageCharge_shift + (κ : ℝ) (γ : ℝ≥0 → ℂ) (T S : ℝ≥0) (v : ℝ) (R : Set ℂ) : + segmentSausageCharge κ (fun t => γ (T + t)) 0 S v R = + segmentSausageCharge κ γ T (T + S) v R := by + have hs : sourceSausageSet (fun t => γ (T + t)) 0 S v = + sourceSausageSet γ T (T + S) v := by + ext z + constructor + · rintro ⟨t, ht, hd⟩ + exact ⟨T + t, ⟨le_add_of_nonneg_right bot_le, add_le_add le_rfl ht.2⟩, hd⟩ + · rintro ⟨t, ht, hd⟩ + refine ⟨t - T, ⟨bot_le, ?_⟩, ?_⟩ + · exact (tsub_le_iff_right).mpr (by simpa only [add_comm] using ht.2) + · simpa only [add_tsub_cancel_of_le ht.1] using hd + simp only [segmentSausageCharge, hs] + +/-- The whole actual step and its sausage lower bound are adapted to the +next front. Both global front times may be infinite off success. -/ +theorem ribbonStepSuccess_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) + (hc : ∀ ω, Continuous (γ ω)) + (σ τ : Ω → WithTop ℝ≥0) (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ) + (hστ : σ ≤ τ) (P : ℂ → ℂ) + (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) + (ε κ v c : ℝ) (R : Set ℂ) : + MeasurableSet[hτ.measurableSpace] (RibbonStepSuccess γ σ τ P ε κ v c R) := by + let f := fun ω => stoppedAtPath (sourceTracePath γ hc ω) (τ ω) + have hfm : Measurable[hτ.measurableSpace] f := stoppedAtPath_measurable ℱ γ ha hc τ hτ + have htm : Measurable[hτ.measurableSpace] (fun ω => (τ ω).untopD 0) := hτ.measurable.untopD 0 + have hsm : Measurable[hτ.measurableSpace] (fun ω => (σ ω).untopD 0) := + (hσ.measurable.mono (IsStoppingTime.measurableSpace_mono hσ hτ hστ) le_rfl).untopD 0 + have hgood := goodRibbonPrefix_measurable ℱ γ ha hc τ hτ P he ε + let : MeasurableSpace Ω := hτ.measurableSpace + have hm := hgood.inter + (measurableSet_le (measurable_const : Measurable (fun _ : Ω => ENNReal.ofReal c)) + (segmentSausageCharge_measurable f hfm _ _ hsm htm κ v R)) + convert hm using 1 + ext ω + by_cases hω : ω ∈ GoodRibbonPrefix γ τ P ε + · have ht := hω.1 + lift τ ω to ℝ≥0 using ht with t hτt + have hs := sourceSausageSet_congr_interval + (s := (σ ω).untopD 0) (t := t) + (fun u hu => stoppedAtPath_eq_on_prefix (sourceTracePath γ hc ω) t + ⟨bot_le, hu.2⟩) v + simp only [RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq, hω, true_and, + f, ← hτt, WithTop.untopD_coe, segmentSausageCharge, hs] + rfl + · simp only [RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq, hω, false_and] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonStepTranslation.lean b/lean/OAI/Probability/SLE/Gauge/RibbonStepTranslation.lean index 98670a00241cb4a2c3d3fd0b1bae99613a1a07ed..8ac8b2b7f98a2373cc95103b74617d5e5028bbb7 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonStepTranslation.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonStepTranslation.lean @@ -1,134 +1,134 @@ -import OAI.Probability.SLE.Gauge.UniformActualRibbonPassage -import OAI.Probability.SLE.Gauge.LocalAffinePassage -import OAI.Probability.SLE.Gauge.RibbonMiddleMass - -namespace OAI - -/-! The recentered passage uses precisely the original ribbon coordinate -and original middle strip after translating back by the actual tip. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates RibbonPassageScaling BoundaryScaleLaw - -def RibbonHistory.stepChart {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (z : ℂ) : ℂ := d.chart ((d.front : ℂ) + z) - -lemma RibbonHistory.stepCompact_maps {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : - MapsTo (fun z : ℂ => (d.front : ℂ) + z) ribbonStepCompact closedRibbon := by - intro z hz - change 0 ≤ d.front + z.re ∧ |0 + z.im| ≤ 1 - exact ⟨by linarith [d.front_ge_one, hz.1], by simpa using hz.2.2.trans (by norm_num)⟩ - -lemma RibbonHistory.recenteredChart_injOn {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : InjOn d.recenteredChart ribbonStepCompact := by - intro u hu v hv huv - have he := d.chart_injective (d.stepCompact_maps hu) (d.stepCompact_maps hv) - (sub_left_inj.mp huv) - exact add_left_cancel he - -lemma RibbonHistory.step_progress_agrees {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : - ∀ z ∈ d.recenteredChart '' ribbonStepCompact, - longitudinalCoordinate d.chart closedRibbon d.front (affine (d.trace d.time) 1 z) = - longitudinalCoordinate d.recenteredChart ribbonStepCompact 0 z := by - rintro z ⟨w, hw, rfl⟩ - have he : affine (d.trace d.time) 1 (d.recenteredChart w) = - d.chart ((d.front : ℂ) + w) := by - simp only [affine, Complex.ofReal_one, one_mul, RibbonHistory.recenteredChart] - abel - rw [he] - simp only [longitudinalCoordinate, - coordinateInverse_left d.chart_injective (d.stepCompact_maps hw), - coordinateInverse_left d.recenteredChart_injOn hw, Complex.add_re, - Complex.ofReal_re, add_sub_cancel_left, sub_zero] - -lemma RibbonHistory.affine_recentered {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : - (fun z => affine (d.trace d.time) 1 (d.recenteredChart z)) = d.stepChart := by - funext z - simp only [affine, Complex.ofReal_one, one_mul, RibbonHistory.recenteredChart, - RibbonHistory.stepChart] - abel - -lemma RibbonHistory.passage_translate {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : - affine (d.trace d.time) 1 '' - physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L) = - physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L) := by - unfold physicalPassageRegion - rw [← image_comp] - exact congrArg (fun f : ℂ → ℂ => f '' _) d.affine_recentered - -lemma RibbonHistory.middle_translate {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : - affine (d.trace d.time) 1 '' - physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε = - physicalMiddleRegion d.chart closedRibbon d.front ε := by - unfold physicalMiddleRegion - rw [← image_comp] - rw [show affine (d.trace d.time) 1 ∘ d.recenteredChart = d.stepChart from - d.affine_recentered] - ext z - constructor - · rintro ⟨w, ⟨hw, hwl, hwu, hwi⟩, rfl⟩ - refine ⟨(d.front : ℂ) + w, ⟨d.stepCompact_maps hw, ?_, ?_, ?_⟩, rfl⟩ - · simpa only [Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, sub_zero] using hwl - · simpa only [Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, sub_zero] using hwu - · simpa only [Complex.add_im, Complex.ofReal_im, zero_add] using hwi - · rintro ⟨w, ⟨_, hwl, hwu, hwi⟩, rfl⟩ - refine ⟨w - (d.front : ℂ), ⟨?_, ?_, ?_, ?_⟩, ?_⟩ - · change -(1 / 2 : ℝ) ≤ w.re - d.front ∧ w.re - d.front ≤ 5 / 2 ∧ - |w.im - 0| ≤ 1 / 2 - exact ⟨by linarith, by linarith, by simpa using hwi.le.trans (by linarith)⟩ - · simpa only [Complex.sub_re, Complex.ofReal_re, sub_zero] using hwl - · simpa only [Complex.sub_re, Complex.ofReal_re, sub_zero] using hwu - · simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hwi - · simp only [RibbonHistory.stepChart, add_sub_cancel] - -/-- Translating the actual finite boundary image back to its original tip -preserves the full probability, including the original middle-strip mass. -/ -theorem RibbonHistory.frontMass_probability_translate - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) - (η : Ω → ℝ≥0 → ℂ) (κ v c : ℝ) (hv : 0 < v) : - P {ω | (fun t => affine (d.trace d.time) 1 (η ω t)) ∈ - {f : ℝ≥0 → ℂ | f ∈ PassageSuccess.FrontSuccess - (fun _ (f : ℝ≥0 → ℂ) => f) - (fun _ => longitudinalCoordinate d.chart closedRibbon d.front) (fun _ => 0) - (fun _ => physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) - (fun _ => d.chart ((d.front : ℂ) + 1)) (ε / L) 0 ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ f 0 - (FirstFrontPassage.frontTime - (fun t => longitudinalCoordinate d.chart closedRibbon d.front (f t)) 1) - v (physicalMiddleRegion d.chart closedRibbon d.front ε)}} = - P {ω | η ω ∈ PassageSuccess.FrontSuccess - (fun _ (f : ℝ≥0 → ℂ) => f) - (fun _ => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) (fun _ => 0) - (fun _ => physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) - (fun _ => d.recenteredChart 1) (ε / L) 0 ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ (η ω) 0 - (FirstFrontPassage.frontTime - (fun t => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0 (η ω t)) 1) - v (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by - have hh := frontSuccess_mass_probability_affine_local P - (fun _ => η) (fun _ => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) - (fun _ => longitudinalCoordinate d.chart closedRibbon d.front) (fun _ => 0) - (fun _ => physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) - (fun _ => d.recenteredChart 1) (ε / L) (d.trace d.time) (by norm_num : (0 : ℝ) < 1) 0 - (fun z hz => d.step_progress_agrees z (image_mono inter_subset_left hz)) - κ (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε) hv c - have htarget : affine (d.trace d.time) 1 (d.recenteredChart 1) = - d.chart ((d.front : ℂ) + 1) := congrFun d.affine_recentered 1 - simpa only [d.passage_translate, d.middle_translate hε, htarget, one_mul, - Real.one_rpow, mul_one, mem_ofPred_eq, PassageSuccess.FrontSuccess] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.UniformActualRibbonPassage +import OAI.Probability.SLE.Gauge.LocalAffinePassage +import OAI.Probability.SLE.Gauge.RibbonMiddleMass + +namespace OAI + +/-! The recentered passage uses precisely the original ribbon coordinate +and original middle strip after translating back by the actual tip. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates RibbonPassageScaling BoundaryScaleLaw + +def RibbonHistory.stepChart {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (z : ℂ) : ℂ := d.chart ((d.front : ℂ) + z) + +lemma RibbonHistory.stepCompact_maps {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : + MapsTo (fun z : ℂ => (d.front : ℂ) + z) ribbonStepCompact closedRibbon := by + intro z hz + change 0 ≤ d.front + z.re ∧ |0 + z.im| ≤ 1 + exact ⟨by linarith [d.front_ge_one, hz.1], by simpa using hz.2.2.trans (by norm_num)⟩ + +lemma RibbonHistory.recenteredChart_injOn {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : InjOn d.recenteredChart ribbonStepCompact := by + intro u hu v hv huv + have he := d.chart_injective (d.stepCompact_maps hu) (d.stepCompact_maps hv) + (sub_left_inj.mp huv) + exact add_left_cancel he + +lemma RibbonHistory.step_progress_agrees {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : + ∀ z ∈ d.recenteredChart '' ribbonStepCompact, + longitudinalCoordinate d.chart closedRibbon d.front (affine (d.trace d.time) 1 z) = + longitudinalCoordinate d.recenteredChart ribbonStepCompact 0 z := by + rintro z ⟨w, hw, rfl⟩ + have he : affine (d.trace d.time) 1 (d.recenteredChart w) = + d.chart ((d.front : ℂ) + w) := by + simp only [affine, Complex.ofReal_one, one_mul, RibbonHistory.recenteredChart] + abel + rw [he] + simp only [longitudinalCoordinate, + coordinateInverse_left d.chart_injective (d.stepCompact_maps hw), + coordinateInverse_left d.recenteredChart_injOn hw, Complex.add_re, + Complex.ofReal_re, add_sub_cancel_left, sub_zero] + +lemma RibbonHistory.affine_recentered {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : + (fun z => affine (d.trace d.time) 1 (d.recenteredChart z)) = d.stepChart := by + funext z + simp only [affine, Complex.ofReal_one, one_mul, RibbonHistory.recenteredChart, + RibbonHistory.stepChart] + abel + +lemma RibbonHistory.passage_translate {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : + affine (d.trace d.time) 1 '' + physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L) = + physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L) := by + unfold physicalPassageRegion + rw [← image_comp] + exact congrArg (fun f : ℂ → ℂ => f '' _) d.affine_recentered + +lemma RibbonHistory.middle_translate {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) : + affine (d.trace d.time) 1 '' + physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε = + physicalMiddleRegion d.chart closedRibbon d.front ε := by + unfold physicalMiddleRegion + rw [← image_comp] + rw [show affine (d.trace d.time) 1 ∘ d.recenteredChart = d.stepChart from + d.affine_recentered] + ext z + constructor + · rintro ⟨w, ⟨hw, hwl, hwu, hwi⟩, rfl⟩ + refine ⟨(d.front : ℂ) + w, ⟨d.stepCompact_maps hw, ?_, ?_, ?_⟩, rfl⟩ + · simpa only [Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, sub_zero] using hwl + · simpa only [Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, sub_zero] using hwu + · simpa only [Complex.add_im, Complex.ofReal_im, zero_add] using hwi + · rintro ⟨w, ⟨_, hwl, hwu, hwi⟩, rfl⟩ + refine ⟨w - (d.front : ℂ), ⟨?_, ?_, ?_, ?_⟩, ?_⟩ + · change -(1 / 2 : ℝ) ≤ w.re - d.front ∧ w.re - d.front ≤ 5 / 2 ∧ + |w.im - 0| ≤ 1 / 2 + exact ⟨by linarith, by linarith, by simpa using hwi.le.trans (by linarith)⟩ + · simpa only [Complex.sub_re, Complex.ofReal_re, sub_zero] using hwl + · simpa only [Complex.sub_re, Complex.ofReal_re, sub_zero] using hwu + · simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hwi + · simp only [RibbonHistory.stepChart, add_sub_cancel] + +/-- Translating the actual finite boundary image back to its original tip +preserves the full probability, including the original middle-strip mass. -/ +theorem RibbonHistory.frontMass_probability_translate + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) + (η : Ω → ℝ≥0 → ℂ) (κ v c : ℝ) (hv : 0 < v) : + P {ω | (fun t => affine (d.trace d.time) 1 (η ω t)) ∈ + {f : ℝ≥0 → ℂ | f ∈ PassageSuccess.FrontSuccess + (fun _ (f : ℝ≥0 → ℂ) => f) + (fun _ => longitudinalCoordinate d.chart closedRibbon d.front) (fun _ => 0) + (fun _ => physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) + (fun _ => d.chart ((d.front : ℂ) + 1)) (ε / L) 0 ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ f 0 + (FirstFrontPassage.frontTime + (fun t => longitudinalCoordinate d.chart closedRibbon d.front (f t)) 1) + v (physicalMiddleRegion d.chart closedRibbon d.front ε)}} = + P {ω | η ω ∈ PassageSuccess.FrontSuccess + (fun _ (f : ℝ≥0 → ℂ) => f) + (fun _ => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) (fun _ => 0) + (fun _ => physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) + (fun _ => d.recenteredChart 1) (ε / L) 0 ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ (η ω) 0 + (FirstFrontPassage.frontTime + (fun t => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0 (η ω t)) 1) + v (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by + have hh := frontSuccess_mass_probability_affine_local P + (fun _ => η) (fun _ => longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) + (fun _ => longitudinalCoordinate d.chart closedRibbon d.front) (fun _ => 0) + (fun _ => physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) + (fun _ => d.recenteredChart 1) (ε / L) (d.trace d.time) (by norm_num : (0 : ℝ) < 1) 0 + (fun z hz => d.step_progress_agrees z (image_mono inter_subset_left hz)) + κ (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε) hv c + have htarget : affine (d.trace d.time) 1 (d.recenteredChart 1) = + d.chart ((d.front : ℂ) + 1) := congrFun d.affine_recentered 1 + simpa only [d.passage_translate, d.middle_translate hε, htarget, one_mul, + Real.one_rpow, mul_one, mem_ofPred_eq, PassageSuccess.FrontSuccess] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassLaw.lean b/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassLaw.lean index 6f20843ca613208a6189c5e19a227993112eb629..eff852a861ac3b3f2d0602a7781354165ac531d8 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassLaw.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassLaw.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Gauge.RibbonStoppedMassTransport - -namespace OAI - -/-! Transfer a completed-ribbon experiment through the actual SLE scaling -law on continuous traces. No uniform step estimate is reapplied to a chart -whose spatial scale varies with the number of strips. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates ScaleTraceLaw RibbonPassageScaling - -local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -lemma ribbonStoppedMass_scale_probability - {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) - (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) - (c : ℝ≥0) (hc : 0 < c) - (hlaw : μ.map (fun ω => scalePath c (f ω)) = μ.map f) - (P : ℂ → ℂ) - (he : Topology.IsClosedEmbedding (fun z : closedRibbon => (c : ℂ) * P z)) - (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) - (k ε κ : ℝ) {v : ℝ} (hv : 0 < v) (D : Set ℂ) (b : ℝ≥0∞) : - μ {ω | f ω ∈ ribbonStoppedMassEvent P g k ε κ v D b} ≤ - μ {ω | f ω ∈ ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ - ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b)} := by - let E := ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ - ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b) - have hE : MeasurableSet E := measurableSet_ribbonStoppedMassEvent _ he g' k ε κ _ _ _ - calc - _ ≤ μ {ω | scalePath c (f ω) ∈ E} := measure_mono (fun ω hω => - mem_ribbonStoppedMassEvent_scalePath c hc (f ω) P g g' hgg' k ε κ hv D b hω) - _ = (μ.map (fun ω => scalePath c (f ω))) E := - (Measure.map_apply ((measurable_scalePath c).comp hf) hE).symm - _ = (μ.map f) E := by rw [hlaw] - _ = _ := Measure.map_apply hf hE - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonStoppedMassTransport + +namespace OAI + +/-! Transfer a completed-ribbon experiment through the actual SLE scaling +law on continuous traces. No uniform step estimate is reapplied to a chart +whose spatial scale varies with the number of strips. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates ScaleTraceLaw RibbonPassageScaling + +local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +lemma ribbonStoppedMass_scale_probability + {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) + (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) + (c : ℝ≥0) (hc : 0 < c) + (hlaw : μ.map (fun ω => scalePath c (f ω)) = μ.map f) + (P : ℂ → ℂ) + (he : Topology.IsClosedEmbedding (fun z : closedRibbon => (c : ℂ) * P z)) + (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) + (k ε κ : ℝ) {v : ℝ} (hv : 0 < v) (D : Set ℂ) (b : ℝ≥0∞) : + μ {ω | f ω ∈ ribbonStoppedMassEvent P g k ε κ v D b} ≤ + μ {ω | f ω ∈ ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ + ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b)} := by + let E := ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ + ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b) + have hE : MeasurableSet E := measurableSet_ribbonStoppedMassEvent _ he g' k ε κ _ _ _ + calc + _ ≤ μ {ω | scalePath c (f ω) ∈ E} := measure_mono (fun ω hω => + mem_ribbonStoppedMassEvent_scalePath c hc (f ω) P g g' hgg' k ε κ hv D b hω) + _ = (μ.map (fun ω => scalePath c (f ω))) E := + (Measure.map_apply ((measurable_scalePath c).comp hf) hE).symm + _ = (μ.map f) E := by rw [hlaw] + _ = _ := Measure.map_apply hf hE + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassTransport.lean b/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassTransport.lean index 0495884b77b7b43d619b175ca1bd33ae1e690693..fd4c067fe95d6219ea8742353f15b39cd6449d5a 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassTransport.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonStoppedMassTransport.lean @@ -1,105 +1,105 @@ -import OAI.Probability.SLE.Gauge.RibbonPrefixTransport -import OAI.Probability.SLE.Gauge.RibbonStepSuccess - -namespace OAI - -/-! Exact spatial and capacity transport of a completed ribbon's stopped -mass. The completion is the actual first front of the transported trace. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates FirstFrontPassage FrontClockChange ScaleTraceLaw -open FrontSuccessMeasurability RibbonPassageScaling - -local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -lemma firstFrontTime_scalePath_of_finite - (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (g g' : C(ℂ, ℝ)) - (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) (k : ℝ) (T : ℝ≥0) - (hT : firstFrontTime (fun t => g (p t)) k = (T : WithTop ℝ≥0)) : - firstFrontTime (fun t => g' (scalePath c p t)) k = ((c ^ 2 * T : ℝ≥0) : WithTop ℝ≥0) := by - let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) - have heval (t : ℝ≥0) : scalePath c p (e t) = (c : ℂ) * p t := - scalePath_at_scaled_time c hc p t - have hclock : (fun t => g' (scalePath c p (e t))) = (fun t => g (p t)) := by - funext t - rw [heval, hgg'] - have hhit := firstFrontTime_attained _ (g.continuous.comp p.continuous) k T hT - exact firstFrontTime_of_clock_hit e (fun t => g' (scalePath c p t)) k T - (by rw [hclock]; exact hT) (by rw [heval, hgg']; exact hhit) - -lemma segmentSausageCharge_scalePath - (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (T : ℝ≥0) - {v : ℝ} (hv : 0 < v) (D : Set ℂ) : - segmentSausageCharge κ (scalePath c p) 0 (c ^ 2 * T) ((c : ℝ) * v) (affine 0 c '' D) = - ENNReal.ofReal ((c : ℝ) ^ dimension κ) * segmentSausageCharge κ p 0 T v D := by - let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) - have he0 : e 0 = 0 := by change c ^ 2 * 0 = 0; exact mul_zero _ - have hpath : (fun t => scalePath c p (e t)) = (fun t => affine 0 c (p t)) := by - funext t - change scalePath c p (c ^ 2 * t) = affine 0 c (p t) - simpa only [affine, zero_add] using scalePath_at_scaled_time c hc p t - calc - _ = segmentSausageCharge κ (fun t => scalePath c p (e t)) 0 T ((c : ℝ) * v) - (affine 0 c '' D) := by rw [segmentSausageCharge_clock, he0]; rfl - _ = segmentSausageCharge κ (fun t => affine 0 c (p t)) 0 T ((c : ℝ) * v) - (affine 0 c '' D) := by rw [hpath] - _ = _ := segmentSausageCharge_affine κ 0 (show 0 < (c : ℝ) from hc) p 0 T hv D - -def ribbonStoppedMassEvent (P : ℂ → ℂ) (g : C(ℂ, ℝ)) (k ε κ v : ℝ) - (D : Set ℂ) (b : ℝ≥0∞) : Set C(ℝ≥0, ℂ) := - {p | p ∈ GoodRibbonPrefix (fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ)) - (ribbonFrontStop g (fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ)) k) P ε ∧ - b ≤ segmentSausageCharge κ p 0 ((firstFrontTime (fun t => g (p t)) k).untopD 0) v D} - -lemma measurableSet_ribbonStoppedMassEvent - (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) - (g : C(ℂ, ℝ)) (k ε κ v : ℝ) (D : Set ℂ) (b : ℝ≥0∞) : - MeasurableSet (ribbonStoppedMassEvent P g k ε κ v D b) := by - let ℱ : Filtration ℝ≥0 (borel C(ℝ≥0, ℂ)) := - Filtration.const ℝ≥0 (borel C(ℝ≥0, ℂ)) le_rfl - let γ := fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ) - have ha : Adapted ℱ (fun t p => γ p t) := by - intro t - change @Measurable C(ℝ≥0, ℂ) ℂ (borel C(ℝ≥0, ℂ)) _ (fun p => p t) - exact (continuous_eval_const t).measurable - have hτ := ribbonFrontStop_stopping ℱ g γ ha (fun p => p.continuous) k - have hg : MeasurableSet (GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) := - hτ.measurableSpace_le _ (goodRibbonPrefix_measurable ℱ γ ha (fun p => p.continuous) _ hτ P he ε) - exact hg.inter (measurableSet_le measurable_const - (segmentSausageCharge_measurable id measurable_id (fun _ => 0) - (fun p => (firstFrontTime (fun t => g (p t)) k).untopD 0) - measurable_const (hτ.measurable'.untopD 0) κ v D)) - -/-- Transport uses the same actual completed prefix and a deterministically -scaled mesh, region and mass threshold. -/ -theorem mem_ribbonStoppedMassEvent_scalePath - (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (P : ℂ → ℂ) - (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) - (k ε κ : ℝ) {v : ℝ} (hv : 0 < v) (D : Set ℂ) (b : ℝ≥0∞) - (hp : p ∈ ribbonStoppedMassEvent P g k ε κ v D b) : - scalePath c p ∈ ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ - ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b) := by - have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => (p : ℝ≥0 → ℂ)) - (fun _ => firstFrontTime (fun t => g (p t)) k) P ε := hp.1 - have hnew := goodRibbonPrefix_scalePath c hc p P g g' hgg' k ε hpref - refine ⟨hnew, ?_⟩ - cases hT : firstFrontTime (fun t => g (p t)) k with - | top => exact False.elim (hpref.1 hT) - | coe T => - have ht := firstFrontTime_scalePath_of_finite c hc p g g' hgg' k T hT - change _ ≤ segmentSausageCharge κ (scalePath c p) 0 - ((firstFrontTime (fun t => g' (scalePath c p t)) k).untopD 0) _ _ - rw [ht, WithTop.untopD_coe, segmentSausageCharge_scalePath κ c hc p T hv] - apply mul_le_mul' le_rfl - simpa only [hT, WithTop.untopD_coe] using hp.2 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonPrefixTransport +import OAI.Probability.SLE.Gauge.RibbonStepSuccess + +namespace OAI + +/-! Exact spatial and capacity transport of a completed ribbon's stopped +mass. The completion is the actual first front of the transported trace. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates FirstFrontPassage FrontClockChange ScaleTraceLaw +open FrontSuccessMeasurability RibbonPassageScaling + +local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +lemma firstFrontTime_scalePath_of_finite + (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (g g' : C(ℂ, ℝ)) + (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) (k : ℝ) (T : ℝ≥0) + (hT : firstFrontTime (fun t => g (p t)) k = (T : WithTop ℝ≥0)) : + firstFrontTime (fun t => g' (scalePath c p t)) k = ((c ^ 2 * T : ℝ≥0) : WithTop ℝ≥0) := by + let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) + have heval (t : ℝ≥0) : scalePath c p (e t) = (c : ℂ) * p t := + scalePath_at_scaled_time c hc p t + have hclock : (fun t => g' (scalePath c p (e t))) = (fun t => g (p t)) := by + funext t + rw [heval, hgg'] + have hhit := firstFrontTime_attained _ (g.continuous.comp p.continuous) k T hT + exact firstFrontTime_of_clock_hit e (fun t => g' (scalePath c p t)) k T + (by rw [hclock]; exact hT) (by rw [heval, hgg']; exact hhit) + +lemma segmentSausageCharge_scalePath + (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (T : ℝ≥0) + {v : ℝ} (hv : 0 < v) (D : Set ℂ) : + segmentSausageCharge κ (scalePath c p) 0 (c ^ 2 * T) ((c : ℝ) * v) (affine 0 c '' D) = + ENNReal.ofReal ((c : ℝ) ^ dimension κ) * segmentSausageCharge κ p 0 T v D := by + let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) + have he0 : e 0 = 0 := by change c ^ 2 * 0 = 0; exact mul_zero _ + have hpath : (fun t => scalePath c p (e t)) = (fun t => affine 0 c (p t)) := by + funext t + change scalePath c p (c ^ 2 * t) = affine 0 c (p t) + simpa only [affine, zero_add] using scalePath_at_scaled_time c hc p t + calc + _ = segmentSausageCharge κ (fun t => scalePath c p (e t)) 0 T ((c : ℝ) * v) + (affine 0 c '' D) := by rw [segmentSausageCharge_clock, he0]; rfl + _ = segmentSausageCharge κ (fun t => affine 0 c (p t)) 0 T ((c : ℝ) * v) + (affine 0 c '' D) := by rw [hpath] + _ = _ := segmentSausageCharge_affine κ 0 (show 0 < (c : ℝ) from hc) p 0 T hv D + +def ribbonStoppedMassEvent (P : ℂ → ℂ) (g : C(ℂ, ℝ)) (k ε κ v : ℝ) + (D : Set ℂ) (b : ℝ≥0∞) : Set C(ℝ≥0, ℂ) := + {p | p ∈ GoodRibbonPrefix (fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ)) + (ribbonFrontStop g (fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ)) k) P ε ∧ + b ≤ segmentSausageCharge κ p 0 ((firstFrontTime (fun t => g (p t)) k).untopD 0) v D} + +lemma measurableSet_ribbonStoppedMassEvent + (P : ℂ → ℂ) (he : Topology.IsClosedEmbedding (fun z : closedRibbon => P z)) + (g : C(ℂ, ℝ)) (k ε κ v : ℝ) (D : Set ℂ) (b : ℝ≥0∞) : + MeasurableSet (ribbonStoppedMassEvent P g k ε κ v D b) := by + let ℱ : Filtration ℝ≥0 (borel C(ℝ≥0, ℂ)) := + Filtration.const ℝ≥0 (borel C(ℝ≥0, ℂ)) le_rfl + let γ := fun p : C(ℝ≥0, ℂ) => (p : ℝ≥0 → ℂ) + have ha : Adapted ℱ (fun t p => γ p t) := by + intro t + change @Measurable C(ℝ≥0, ℂ) ℂ (borel C(ℝ≥0, ℂ)) _ (fun p => p t) + exact (continuous_eval_const t).measurable + have hτ := ribbonFrontStop_stopping ℱ g γ ha (fun p => p.continuous) k + have hg : MeasurableSet (GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) := + hτ.measurableSpace_le _ (goodRibbonPrefix_measurable ℱ γ ha (fun p => p.continuous) _ hτ P he ε) + exact hg.inter (measurableSet_le measurable_const + (segmentSausageCharge_measurable id measurable_id (fun _ => 0) + (fun p => (firstFrontTime (fun t => g (p t)) k).untopD 0) + measurable_const (hτ.measurable'.untopD 0) κ v D)) + +/-- Transport uses the same actual completed prefix and a deterministically +scaled mesh, region and mass threshold. -/ +theorem mem_ribbonStoppedMassEvent_scalePath + (c : ℝ≥0) (hc : 0 < c) (p : C(ℝ≥0, ℂ)) (P : ℂ → ℂ) + (g g' : C(ℂ, ℝ)) (hgg' : ∀ z : ℂ, g' ((c : ℂ) * z) = g z) + (k ε κ : ℝ) {v : ℝ} (hv : 0 < v) (D : Set ℂ) (b : ℝ≥0∞) + (hp : p ∈ ribbonStoppedMassEvent P g k ε κ v D b) : + scalePath c p ∈ ribbonStoppedMassEvent (fun z => (c : ℂ) * P z) g' k ε κ + ((c : ℝ) * v) (affine 0 c '' D) (ENNReal.ofReal ((c : ℝ) ^ dimension κ) * b) := by + have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => (p : ℝ≥0 → ℂ)) + (fun _ => firstFrontTime (fun t => g (p t)) k) P ε := hp.1 + have hnew := goodRibbonPrefix_scalePath c hc p P g g' hgg' k ε hpref + refine ⟨hnew, ?_⟩ + cases hT : firstFrontTime (fun t => g (p t)) k with + | top => exact False.elim (hpref.1 hT) + | coe T => + have ht := firstFrontTime_scalePath_of_finite c hc p g g' hgg' k T hT + change _ ≤ segmentSausageCharge κ (scalePath c p) 0 + ((firstFrontTime (fun t => g' (scalePath c p t)) k).untopD 0) _ _ + rw [ht, WithTop.untopD_coe, segmentSausageCharge_scalePath κ c hc p T hv] + apply mul_le_mul' le_rfl + simpa only [hT, WithTop.untopD_coe] using hp.2 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/RibbonSurvival.lean b/lean/OAI/Probability/SLE/Gauge/RibbonSurvival.lean index 0ca5f21750929b469c6edada7be40dc5ba51dbfb..c96d98143b652367abdbda24504142702026e1c0 100644 --- a/lean/OAI/Probability/SLE/Gauge/RibbonSurvival.lean +++ b/lean/OAI/Probability/SLE/Gauge/RibbonSurvival.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Geometry.GuideTube -import OAI.Probability.SLE.NearRealContraction - -namespace OAI - -/-! Actual surviving-component assertions for access rays and the splice. -Containment of the past in the initial rectangle implies that every point -above or beside it belongs to the unbounded surviving component. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem survives_of_unbounded_free_curve - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (α : ℝ≥0 → ℂ) (hα : Continuous α) - (hfree : range α ⊆ upperHalfPlane \ (γ '' Icc 0 T)) - (hunbounded : ¬ Bornology.IsBounded (range α)) : - range α ⊆ survivingDomain γ T := by - rintro z ⟨u, rfl⟩ - refine ⟨hfree (mem_range_self u), ?_⟩ - intro hb - apply hunbounded - exact hb.subset ((isPreconnected_range hα).subset_connectedComponentIn - (mem_range_self u) hfree) - -def upwardRay (z : ℂ) (u : ℝ≥0) : ℂ := z + (u : ℝ) * Complex.I - -theorem upwardRay_continuous (z : ℂ) : Continuous (upwardRay z) := - continuous_const.add ((Complex.continuous_ofReal.comp NNReal.continuous_coe).mul continuous_const) - -theorem upwardRay_unbounded (z : ℂ) : ¬ Bornology.IsBounded (range (upwardRay z)) := by - intro hb - obtain ⟨R, hR, hbound⟩ := hb.exists_pos_norm_lt - let u : ℝ≥0 := ⟨R + |z.im| + 1, by positivity⟩ - have hu := hbound (upwardRay z u) (mem_range_self u) - have hi := Complex.im_le_norm (upwardRay z u) - simp only [upwardRay, Complex.add_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_im, mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] at hi - change z.im + (R + |z.im| + 1) ≤ ‖upwardRay z u‖ at hi - linarith [neg_le_abs z.im] - -theorem outside_initial_rectangle_survives - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (B : ℝ) - (hpast : γ '' Icc 0 T ⊆ GuideSplice.closedInitialRectangle B) : - upperHalfPlane \ GuideSplice.closedInitialRectangle B ⊆ survivingDomain γ T := by - intro z hz - have hfree : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by - rintro w ⟨u, rfl⟩ - have him : (upwardRay z u).im = z.im + (u : ℝ) := by simp [upwardRay] - have hre : (upwardRay z u).re = z.re := by simp [upwardRay] - refine ⟨by change 0 < (upwardRay z u).im; rw [him]; exact add_pos_of_pos_of_nonneg hz.1 u.coe_nonneg, ?_⟩ - intro hp - have hrect := hpast hp - change |(upwardRay z u).re| ≤ B / 16 ∧ - 0 ≤ (upwardRay z u).im ∧ (upwardRay z u).im ≤ B / 4 at hrect - apply hz.2 - change |z.re| ≤ B / 16 ∧ 0 ≤ z.im ∧ z.im ≤ B / 4 - refine ⟨by simpa only [hre] using hrect.1, hz.1.le, ?_⟩ - rw [him] at hrect - linarith [u.coe_nonneg, hrect.2.2] - have hs := survives_of_unbounded_free_curve γ T (upwardRay z) - (upwardRay_continuous z) hfree (upwardRay_unbounded z) (mem_range_self 0) - simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hs - -/-- The open continuation tube avoids the actual filled hull whenever the -past lies in the containing rectangle; no filled-hull containment is assumed. -/ -theorem continuationTube_subset_survivingDomain - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (B : ℝ) - (hpast : γ '' Icc 0 T ⊆ GuideSplice.closedInitialRectangle B) - (F : ℂ → ℂ) (a : ℂ) (r : ℝ) (O : Set ℂ) : - GuideSplice.continuationTube F a r B O ⊆ survivingDomain γ T := by - intro z hz - exact outside_initial_rectangle_survives γ T B hpast ⟨hz.1.1, hz.2⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.GuideTube +import OAI.Probability.SLE.NearRealContraction + +namespace OAI + +/-! Actual surviving-component assertions for access rays and the splice. +Containment of the past in the initial rectangle implies that every point +above or beside it belongs to the unbounded surviving component. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem survives_of_unbounded_free_curve + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (α : ℝ≥0 → ℂ) (hα : Continuous α) + (hfree : range α ⊆ upperHalfPlane \ (γ '' Icc 0 T)) + (hunbounded : ¬ Bornology.IsBounded (range α)) : + range α ⊆ survivingDomain γ T := by + rintro z ⟨u, rfl⟩ + refine ⟨hfree (mem_range_self u), ?_⟩ + intro hb + apply hunbounded + exact hb.subset ((isPreconnected_range hα).subset_connectedComponentIn + (mem_range_self u) hfree) + +def upwardRay (z : ℂ) (u : ℝ≥0) : ℂ := z + (u : ℝ) * Complex.I + +theorem upwardRay_continuous (z : ℂ) : Continuous (upwardRay z) := + continuous_const.add ((Complex.continuous_ofReal.comp NNReal.continuous_coe).mul continuous_const) + +theorem upwardRay_unbounded (z : ℂ) : ¬ Bornology.IsBounded (range (upwardRay z)) := by + intro hb + obtain ⟨R, hR, hbound⟩ := hb.exists_pos_norm_lt + let u : ℝ≥0 := ⟨R + |z.im| + 1, by positivity⟩ + have hu := hbound (upwardRay z u) (mem_range_self u) + have hi := Complex.im_le_norm (upwardRay z u) + simp only [upwardRay, Complex.add_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_im, mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] at hi + change z.im + (R + |z.im| + 1) ≤ ‖upwardRay z u‖ at hi + linarith [neg_le_abs z.im] + +theorem outside_initial_rectangle_survives + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (B : ℝ) + (hpast : γ '' Icc 0 T ⊆ GuideSplice.closedInitialRectangle B) : + upperHalfPlane \ GuideSplice.closedInitialRectangle B ⊆ survivingDomain γ T := by + intro z hz + have hfree : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by + rintro w ⟨u, rfl⟩ + have him : (upwardRay z u).im = z.im + (u : ℝ) := by simp [upwardRay] + have hre : (upwardRay z u).re = z.re := by simp [upwardRay] + refine ⟨by change 0 < (upwardRay z u).im; rw [him]; exact add_pos_of_pos_of_nonneg hz.1 u.coe_nonneg, ?_⟩ + intro hp + have hrect := hpast hp + change |(upwardRay z u).re| ≤ B / 16 ∧ + 0 ≤ (upwardRay z u).im ∧ (upwardRay z u).im ≤ B / 4 at hrect + apply hz.2 + change |z.re| ≤ B / 16 ∧ 0 ≤ z.im ∧ z.im ≤ B / 4 + refine ⟨by simpa only [hre] using hrect.1, hz.1.le, ?_⟩ + rw [him] at hrect + linarith [u.coe_nonneg, hrect.2.2] + have hs := survives_of_unbounded_free_curve γ T (upwardRay z) + (upwardRay_continuous z) hfree (upwardRay_unbounded z) (mem_range_self 0) + simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hs + +/-- The open continuation tube avoids the actual filled hull whenever the +past lies in the containing rectangle; no filled-hull containment is assumed. -/ +theorem continuationTube_subset_survivingDomain + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (B : ℝ) + (hpast : γ '' Icc 0 T ⊆ GuideSplice.closedInitialRectangle B) + (F : ℂ → ℂ) (a : ℂ) (r : ℝ) (O : Set ℂ) : + GuideSplice.continuationTube F a r B O ⊆ survivingDomain γ T := by + intro z hz + exact outside_initial_rectangle_survives γ T B hpast ⟨hz.1.1, hz.2⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SequencePassage.lean b/lean/OAI/Probability/SLE/Gauge/SequencePassage.lean index ca80016d0989dc6fac483aa6281db2fc949d724d..1a6b259c4e5257515feaa59320ef2e74675a31fc 100644 --- a/lean/OAI/Probability/SLE/Gauge/SequencePassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/SequencePassage.lean @@ -1,124 +1,124 @@ -import OAI.Probability.SLE.Geometry.BoundedSpliceProbability -import OAI.Probability.SLE.Gauge.FirstFrontPassage -import OAI.Probability.SLE.Geometry.UniformAccessLocalization - -namespace OAI - -/-! Sequencewise ribbon passage from the actual bounded first-height event. -The early event is stopped-image containment, and the retained mass window is -the actual first front crossing constructed from the continuation. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.SequencePassage -open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage - -/-- The actual stopped-image events give one positive splice probability. -The finite initial part of the sequence does not need an early-loss estimate. -/ -theorem eventual_bounded_splice_probability - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) - (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) - (Fbar : ℕ → ℂ → OnePoint ℂ) - (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) - (aₙ : ℕ → ℂ) (ρ : ℝ) - (hpositive : 0 < P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) - (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ - stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) ρ) ≤ - P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : - ∃ q : ℝ, 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P - (stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) ρ ∩ - (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc ∩ - stoppedPhysicalTube γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (parameterContinuationRegion F a r O) W)) := by - classical - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc - let E := stoppedNarrowCertificate γ σ B Tc - let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ - let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W - have hH : ∀ n, MeasurableSet[hσ.measurableSpace] (H n) := fun n => - stoppedImageEvent_measurable ℱ γ ha (fun ω => (hw ω).1) - (fun ω t => (hw ω).2.2.1 t) σ hσ Tc - (fun ω => boundedFirstHeightTime_le (γ ω) B Tc) (Fbar n) (hbar n) (aₙ n) ρ - let H' : ℕ → Set Ω := fun n => if P (E \ H n) ≤ P E / 2 then H n else univ - have hH' : ∀ n, MeasurableSet[hσ.measurableSpace] (H' n) := by - intro n - dsimp only [H'] - split_ifs - · exact hH n - · exact MeasurableSet.univ - have hmiss : ∀ n, P (E \ H' n) ≤ P E / 2 := by - intro n - dsimp only [H'] - split_ifs with hn - · exact hn - · simp only [sdiff_univ, measure_empty, zero_le] - obtain ⟨q, hq, hqH⟩ := bounded_firstHeight_splice_varying_probability - κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport B r hB hr Tc - F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend - H' hH' hpositive hmiss - have hqtop : q ≠ ⊤ := ne_top_of_le_ne_top (measure_ne_top P _) (hqH 0) - refine ⟨q.toReal, ENNReal.toReal_pos hq.ne' hqtop, ?_⟩ - filter_upwards [hmissing] with n hn - have heq : H' n = H n := ite_eq_left hn - rw [ENNReal.ofReal_toReal hqtop] - have hh := hqH n - rw [heq] at hh - exact hh.trans (measure_mono (fun _ hω => ⟨hω.1.2, hω.1.1, hω.2⟩)) - -/-- The published support input and proved scaling law give a real positive -lower probability, uniformly in the first-height scale. -/ -theorem uniform_bounded_firstHeight_real_probability - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hsupport : PublishedGeometricTubeSupport P γ) : - ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q : ℝ, 0 < q ∧ ∀ B : ℝ, 0 < B → - ENNReal.ofReal q ≤ P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) - B (T₀ * (Real.toNNReal B) ^ 2)) := by - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - obtain ⟨T₀, hT₀, q, hq, hprob⟩ := uniform_boundedFirstHeight_probability - κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport - have hqtop : q ≠ ⊤ := ne_top_of_le_ne_top (measure_ne_top P _) (hprob 1 zero_lt_one) - refine ⟨T₀, hT₀, q.toReal, ENNReal.toReal_pos hq.ne' hqtop, ?_⟩ - intro B hB - have hh := hprob (Real.toNNReal B) (Real.toNNReal_pos.mpr hB) - simpa only [ENNReal.ofReal_toReal hqtop, Real.coe_toNNReal B hB.le] using hh - -end SLEExactGauge.SequencePassage - -end - -end OAI +import OAI.Probability.SLE.Geometry.BoundedSpliceProbability +import OAI.Probability.SLE.Gauge.FirstFrontPassage +import OAI.Probability.SLE.Geometry.UniformAccessLocalization + +namespace OAI + +/-! Sequencewise ribbon passage from the actual bounded first-height event. +The early event is stopped-image containment, and the retained mass window is +the actual first front crossing constructed from the continuation. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.SequencePassage +open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage + +/-- The actual stopped-image events give one positive splice probability. +The finite initial part of the sequence does not need an early-loss estimate. -/ +theorem eventual_bounded_splice_probability + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) + (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) + (Fbar : ℕ → ℂ → OnePoint ℂ) + (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) + (aₙ : ℕ → ℂ) (ρ : ℝ) + (hpositive : 0 < P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) + (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ + stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) ρ) ≤ + P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : + ∃ q : ℝ, 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P + (stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) ρ ∩ + (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc ∩ + stoppedPhysicalTube γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (parameterContinuationRegion F a r O) W)) := by + classical + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc + let E := stoppedNarrowCertificate γ σ B Tc + let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ + let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W + have hH : ∀ n, MeasurableSet[hσ.measurableSpace] (H n) := fun n => + stoppedImageEvent_measurable ℱ γ ha (fun ω => (hw ω).1) + (fun ω t => (hw ω).2.2.1 t) σ hσ Tc + (fun ω => boundedFirstHeightTime_le (γ ω) B Tc) (Fbar n) (hbar n) (aₙ n) ρ + let H' : ℕ → Set Ω := fun n => if P (E \ H n) ≤ P E / 2 then H n else univ + have hH' : ∀ n, MeasurableSet[hσ.measurableSpace] (H' n) := by + intro n + dsimp only [H'] + split_ifs + · exact hH n + · exact MeasurableSet.univ + have hmiss : ∀ n, P (E \ H' n) ≤ P E / 2 := by + intro n + dsimp only [H'] + split_ifs with hn + · exact hn + · simp only [sdiff_univ, measure_empty, zero_le] + obtain ⟨q, hq, hqH⟩ := bounded_firstHeight_splice_varying_probability + κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport B r hB hr Tc + F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend + H' hH' hpositive hmiss + have hqtop : q ≠ ⊤ := ne_top_of_le_ne_top (measure_ne_top P _) (hqH 0) + refine ⟨q.toReal, ENNReal.toReal_pos hq.ne' hqtop, ?_⟩ + filter_upwards [hmissing] with n hn + have heq : H' n = H n := ite_eq_left hn + rw [ENNReal.ofReal_toReal hqtop] + have hh := hqH n + rw [heq] at hh + exact hh.trans (measure_mono (fun _ hω => ⟨hω.1.2, hω.1.1, hω.2⟩)) + +/-- The published support input and proved scaling law give a real positive +lower probability, uniformly in the first-height scale. -/ +theorem uniform_bounded_firstHeight_real_probability + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hsupport : PublishedGeometricTubeSupport P γ) : + ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q : ℝ, 0 < q ∧ ∀ B : ℝ, 0 < B → + ENNReal.ofReal q ≤ P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) + B (T₀ * (Real.toNNReal B) ^ 2)) := by + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + obtain ⟨T₀, hT₀, q, hq, hprob⟩ := uniform_boundedFirstHeight_probability + κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport + have hqtop : q ≠ ⊤ := ne_top_of_le_ne_top (measure_ne_top P _) (hprob 1 zero_lt_one) + refine ⟨T₀, hT₀, q.toReal, ENNReal.toReal_pos hq.ne' hqtop, ?_⟩ + intro B hB + have hh := hprob (Real.toNNReal B) (Real.toNNReal_pos.mpr hB) + simpa only [ENNReal.ofReal_toReal hqtop, Real.coe_toNNReal B hB.le] using hh + +end SLEExactGauge.SequencePassage + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SequencePassageRegions.lean b/lean/OAI/Probability/SLE/Gauge/SequencePassageRegions.lean index 5c9f370ac5cb087e1461465c3d3ae4b55cdae606..f57758a5db180b126b7941c54fb593ed3e928e89 100644 --- a/lean/OAI/Probability/SLE/Gauge/SequencePassageRegions.lean +++ b/lean/OAI/Probability/SLE/Gauge/SequencePassageRegions.lean @@ -1,253 +1,253 @@ -import OAI.Probability.SLE.Gauge.SequencePassage -import OAI.Probability.SLE.Gauge.PassageSuccessRegions - -namespace OAI - -/-! Sequence passage with compact-dependent moving middle-strip margins. -No inclusion of the entire limiting open middle strip is required. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.SequencePassage -open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess - -local instance regionSequencePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance regionSequencePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem bounded_splice_sequence_success_middle_regions - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) - (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) - (Fbar : ℕ → ℂ → OnePoint ℂ) - (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) - (aₙ : ℕ → ℂ) (ρ : ℝ) - (Fₙ : ℕ → ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpn : ∀ᶠ n in atTop, ContinuousOn (pₙ n) U) - (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (hOU : ball a (r / 2) ∪ O ⊆ U) - (hWfront : ∀ z ∈ W, 1 < p (F z)) - (hstartBall : ∀ z ∈ ball a (r / 4), p z < 1 / 3) - (hearlyBall : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ U ∩ {z | pₙ n z < 1}) - (V : Set ℂ) (hV : IsOpen V) (hOV : ball a (r / 2) ∪ O ⊆ V) - (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n) - (bₙ : ℕ → ℂ) (ε : ℝ) - (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - bₙ n‖ < ε / 4) - (hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ Vₙ n) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) - (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) - (Rₙ : ℕ → Set ℂ) (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n) - (hpositive : 0 < P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) - (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ - stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) ρ) ≤ - P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : - ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | - ω ∈ FrontSuccess η pₙ (fun ω => boundedFirstHeightTime (γ ω) B Tc) Vₙ bₙ ε n ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) - (boundedFirstHeightTime (γ ω) B Tc) - (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ - have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc - have hσm : Measurable σ := stopping_time_measurable_nnreal hσ - let f := sourceTracePath γ (fun ω => (hw ω).1) - have hfm : Measurable f := source_path_measurable f hγm - let Uparam := parameterContinuationRegion F a r O - have hUparam : IsOpen Uparam := parameterContinuationRegion_isOpen hF.continuousOn a r hO - let C := stoppedPhysicalTube γ σ Uparam W - let E := stoppedNarrowCertificate γ σ B Tc ∩ C - let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ - have hEm : MeasurableSet E := by - apply MeasurableSet.inter - · exact hσ.measurableSpace_le _ - (boundedFirstHeight_certificate_measurable ℱ γ ha (fun ω => (hw ω).1) B Tc) - · exact measurableSet_stoppedPhysicalTube γ (fun ω => (hw ω).1) hγm σ hσm Uparam W hUparam hW - have hstart : ∀ ω ∈ E, p (F (γ ω (σ ω))) < 1 / 3 := by - intro ω hω - have hnorm := bounded_firstHeight_norm (f ω) (hw ω).2.1 (hw ω).2.2.1 - hB Tc hω.1 (σ ω) ⟨bot_le, le_rfl⟩ - change ‖γ ω (σ ω)‖ < B / 2 at hnorm - have hheight : (γ ω (σ ω)).im = B / 4 := hω.1.2.1 - have him := Complex.im_le_norm (γ ω (σ ω)) - have hsmall := hsector (γ ω (σ ω)) (by linarith) (by linarith) - (by rw [hheight]; linarith) - exact hstartBall _ (by simpa only [mem_ball, dist_eq_norm] using hsmall) - obtain ⟨q, hq, hprob⟩ := eventual_bounded_splice_probability κ hκ hκ8 B₀ hB₀ hm hc - γ G F₀ hw hboundary hsupport B r hB hr Tc F hF.continuousOn a O W hO hW - β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend Fbar hbar aₙ ρ - hpositive hmissing - have hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → γ ω t ∈ upperHalfPlane → - η n ω t = Fₙ n (γ ω t) := by - intro n - filter_upwards [hηbar n] with ω hω - intro t _ ht - exact OnePoint.coe_injective ((hω t).symm.trans (hext n _ ht)) - have hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → - MapsTo (η n ω) (Icc 0 (σ ω)) U ∧ - ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1 := by - filter_upwards [hearlyBall] with n hn - filter_upwards [hηbar n] with ω hω - intro hmem - have himage : MapsTo (η n ω) (Icc 0 (σ ω)) (ball (aₙ n) ρ) := by - intro t ht - obtain ⟨z, hz, he⟩ := hmem.1 t ht - have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) - exact heq ▸ hz - exact ⟨fun _ ht => (hn (himage ht)).1, fun _ ht => (hn (himage ht)).2⟩ - have hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → - MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n) := by - filter_upwards [hearlyAllowed] with n hn - filter_upwards [hηbar n] with ω hω - intro hmem t ht - obtain ⟨z, hz, he⟩ := hmem.1 t ht - have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) - exact hn (heq ▸ hz) - obtain ⟨c, hcpos, hmass⟩ := retained_tube_firstFront_success_mass_middle_regions P κ hκ hκ8 - f hfm hSLE σ hσm Fₙ F hFn hinj hF hne hconv Uparam W hUparam inter_subset_left - pₙ p U hU hp hpn hpconv (fun _ hz => hOU hz.2) hWfront E hEm inter_subset_right - hstart V hV (fun _ hz => hOV hz.2) Vₙ hVVₙ bₙ ε hterminal - v hv hlim η Rₙ hregion H q hq hprob hη hηc hearly hearlyV - exact ⟨c, q / 8, hcpos, by positivity, hmass⟩ - - -theorem localized_sequence_passage_success_middle_regions - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) - (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) - (δ : ℝ) (hδ : 0 < δ) (βₙ : ℕ → ℝ → ℂ) - (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 δ)) (hβ0 : ∀ n, βₙ n 0 = 0) - (hβHn : ∀ n t, t ∈ Icc 0 δ → 0 ≤ (βₙ n t).im) - (z : ℂ) (hz : z ≠ 0) (hβconv : Tendsto (fun n => βₙ n δ) atTop (𝓝 z)) - (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0) - (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 δ)) (hα0 : ∀ n, αₙ n 0 = aₙ n) - (hmap : ∀ n t, t ∈ Icc 0 δ → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ)) - (r₀ : ℝ) (hr₀ : 0 < r₀) (hsmallGuide : (Cg : ℝ) * δ ≤ r₀) : - ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ, - 0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧ - ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 → - ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b → - ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane → - ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W → - ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T → - ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) → - (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B → - (∀ t ∈ Ioc tB T, B < ‖β t‖) → - (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) → - (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) → - F '' (β '' Icc tB T) ⊆ O → β T ∈ W → - (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) → - TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane → - ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U → - ContinuousOn p U → (∀ᶠ n in atTop, ContinuousOn (pₙ n) U) → - TendstoLocallyUniformlyOn pₙ p atTop U → - ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) → - (∀ w ∈ ball a (r / 4), p w < 1 / 3) → - (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) → - ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V → - ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) → - ∀ (bₙ : ℕ → ℂ) (ε : ℝ), - (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) → - (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) → - ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) → - ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) → - (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) → - ∀ (Rₙ : ℕ → Set ℂ), - (∀ᶠ n in atTop, U ∩ {w | (1 / 3 : ℝ) < pₙ n w ∧ pₙ n w < 2 / 3} ⊆ Rₙ n) → - ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | - ω ∈ FrontSuccess η pₙ - (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) - (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) - (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ - obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability - κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport - obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ := - common_access_gives_firstHeight_localization κ hκ hκ8 - obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1) - (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t) - Fₙ Fbar hFn hinj hbar hext δ hδ βₙ hβc hβ0 hβHn z hz hβconv - αₙ aₙ Cg hα hα0 hmap r₀ hr₀ hsmallGuide - refine ⟨T₀, hT₀, q₀, b₀, b, A, d, L₀, hq₀, hb₀, hb, hA, hd, hL₀, ?_⟩ - intro L hL hsmall B r hB hr hBL F hF a O W hO hW β tB T htB htBT hβ hi hβH - hcross hafter hcircle hsector htail hend hne hconv pₙ p U hU hp hpn hpconv hOU - hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed - v hv hlim η hηc hηbar Rₙ hregion - let Tc := T₀ * (Real.toNNReal B) ^ 2 - have hheightB := hheight B hB - have hpositive : 0 < P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) := - (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB - have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ - stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) (L * r₀)) ≤ - P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by - filter_upwards [hlocseq] with n hn - calc - _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hn L hL B hB hBL Tc - _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall - _ = ENNReal.ofReal q₀ / 2 := by - rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] - norm_num - _ ≤ _ := ENNReal.div_le_div_right hheightB 2 - exact bounded_splice_sequence_success_middle_regions κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport - B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector - htail hend Fbar hbar aₙ (L * r₀) Fₙ hFn hinj hne hconv hext pₙ p U hU hp hpn hpconv - hOU hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed - v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing - -end SLEExactGauge.SequencePassage - -end - -end OAI +import OAI.Probability.SLE.Gauge.SequencePassage +import OAI.Probability.SLE.Gauge.PassageSuccessRegions + +namespace OAI + +/-! Sequence passage with compact-dependent moving middle-strip margins. +No inclusion of the entire limiting open middle strip is required. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.SequencePassage +open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess + +local instance regionSequencePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance regionSequencePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem bounded_splice_sequence_success_middle_regions + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) + (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) + (Fbar : ℕ → ℂ → OnePoint ℂ) + (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) + (aₙ : ℕ → ℂ) (ρ : ℝ) + (Fₙ : ℕ → ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpn : ∀ᶠ n in atTop, ContinuousOn (pₙ n) U) + (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (hOU : ball a (r / 2) ∪ O ⊆ U) + (hWfront : ∀ z ∈ W, 1 < p (F z)) + (hstartBall : ∀ z ∈ ball a (r / 4), p z < 1 / 3) + (hearlyBall : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ U ∩ {z | pₙ n z < 1}) + (V : Set ℂ) (hV : IsOpen V) (hOV : ball a (r / 2) ∪ O ⊆ V) + (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n) + (bₙ : ℕ → ℂ) (ε : ℝ) + (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - bₙ n‖ < ε / 4) + (hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ Vₙ n) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) + (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) + (Rₙ : ℕ → Set ℂ) (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n) + (hpositive : 0 < P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) + (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ + stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) ρ) ≤ + P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : + ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | + ω ∈ FrontSuccess η pₙ (fun ω => boundedFirstHeightTime (γ ω) B Tc) Vₙ bₙ ε n ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) + (boundedFirstHeightTime (γ ω) B Tc) + (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ + have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc + have hσm : Measurable σ := stopping_time_measurable_nnreal hσ + let f := sourceTracePath γ (fun ω => (hw ω).1) + have hfm : Measurable f := source_path_measurable f hγm + let Uparam := parameterContinuationRegion F a r O + have hUparam : IsOpen Uparam := parameterContinuationRegion_isOpen hF.continuousOn a r hO + let C := stoppedPhysicalTube γ σ Uparam W + let E := stoppedNarrowCertificate γ σ B Tc ∩ C + let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ + have hEm : MeasurableSet E := by + apply MeasurableSet.inter + · exact hσ.measurableSpace_le _ + (boundedFirstHeight_certificate_measurable ℱ γ ha (fun ω => (hw ω).1) B Tc) + · exact measurableSet_stoppedPhysicalTube γ (fun ω => (hw ω).1) hγm σ hσm Uparam W hUparam hW + have hstart : ∀ ω ∈ E, p (F (γ ω (σ ω))) < 1 / 3 := by + intro ω hω + have hnorm := bounded_firstHeight_norm (f ω) (hw ω).2.1 (hw ω).2.2.1 + hB Tc hω.1 (σ ω) ⟨bot_le, le_rfl⟩ + change ‖γ ω (σ ω)‖ < B / 2 at hnorm + have hheight : (γ ω (σ ω)).im = B / 4 := hω.1.2.1 + have him := Complex.im_le_norm (γ ω (σ ω)) + have hsmall := hsector (γ ω (σ ω)) (by linarith) (by linarith) + (by rw [hheight]; linarith) + exact hstartBall _ (by simpa only [mem_ball, dist_eq_norm] using hsmall) + obtain ⟨q, hq, hprob⟩ := eventual_bounded_splice_probability κ hκ hκ8 B₀ hB₀ hm hc + γ G F₀ hw hboundary hsupport B r hB hr Tc F hF.continuousOn a O W hO hW + β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend Fbar hbar aₙ ρ + hpositive hmissing + have hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → γ ω t ∈ upperHalfPlane → + η n ω t = Fₙ n (γ ω t) := by + intro n + filter_upwards [hηbar n] with ω hω + intro t _ ht + exact OnePoint.coe_injective ((hω t).symm.trans (hext n _ ht)) + have hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → + MapsTo (η n ω) (Icc 0 (σ ω)) U ∧ + ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1 := by + filter_upwards [hearlyBall] with n hn + filter_upwards [hηbar n] with ω hω + intro hmem + have himage : MapsTo (η n ω) (Icc 0 (σ ω)) (ball (aₙ n) ρ) := by + intro t ht + obtain ⟨z, hz, he⟩ := hmem.1 t ht + have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) + exact heq ▸ hz + exact ⟨fun _ ht => (hn (himage ht)).1, fun _ ht => (hn (himage ht)).2⟩ + have hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → + MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n) := by + filter_upwards [hearlyAllowed] with n hn + filter_upwards [hηbar n] with ω hω + intro hmem t ht + obtain ⟨z, hz, he⟩ := hmem.1 t ht + have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) + exact hn (heq ▸ hz) + obtain ⟨c, hcpos, hmass⟩ := retained_tube_firstFront_success_mass_middle_regions P κ hκ hκ8 + f hfm hSLE σ hσm Fₙ F hFn hinj hF hne hconv Uparam W hUparam inter_subset_left + pₙ p U hU hp hpn hpconv (fun _ hz => hOU hz.2) hWfront E hEm inter_subset_right + hstart V hV (fun _ hz => hOV hz.2) Vₙ hVVₙ bₙ ε hterminal + v hv hlim η Rₙ hregion H q hq hprob hη hηc hearly hearlyV + exact ⟨c, q / 8, hcpos, by positivity, hmass⟩ + + +theorem localized_sequence_passage_success_middle_regions + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) + (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) + (δ : ℝ) (hδ : 0 < δ) (βₙ : ℕ → ℝ → ℂ) + (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 δ)) (hβ0 : ∀ n, βₙ n 0 = 0) + (hβHn : ∀ n t, t ∈ Icc 0 δ → 0 ≤ (βₙ n t).im) + (z : ℂ) (hz : z ≠ 0) (hβconv : Tendsto (fun n => βₙ n δ) atTop (𝓝 z)) + (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0) + (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 δ)) (hα0 : ∀ n, αₙ n 0 = aₙ n) + (hmap : ∀ n t, t ∈ Icc 0 δ → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ)) + (r₀ : ℝ) (hr₀ : 0 < r₀) (hsmallGuide : (Cg : ℝ) * δ ≤ r₀) : + ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ, + 0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧ + ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 → + ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b → + ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane → + ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W → + ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T → + ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) → + (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B → + (∀ t ∈ Ioc tB T, B < ‖β t‖) → + (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) → + (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) → + F '' (β '' Icc tB T) ⊆ O → β T ∈ W → + (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) → + TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane → + ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U → + ContinuousOn p U → (∀ᶠ n in atTop, ContinuousOn (pₙ n) U) → + TendstoLocallyUniformlyOn pₙ p atTop U → + ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) → + (∀ w ∈ ball a (r / 4), p w < 1 / 3) → + (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) → + ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V → + ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) → + ∀ (bₙ : ℕ → ℂ) (ε : ℝ), + (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) → + (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) → + ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) → + ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) → + (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) → + ∀ (Rₙ : ℕ → Set ℂ), + (∀ᶠ n in atTop, U ∩ {w | (1 / 3 : ℝ) < pₙ n w ∧ pₙ n w < 2 / 3} ⊆ Rₙ n) → + ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | + ω ∈ FrontSuccess η pₙ + (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) + (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) + (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ + obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability + κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport + obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ := + common_access_gives_firstHeight_localization κ hκ hκ8 + obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1) + (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t) + Fₙ Fbar hFn hinj hbar hext δ hδ βₙ hβc hβ0 hβHn z hz hβconv + αₙ aₙ Cg hα hα0 hmap r₀ hr₀ hsmallGuide + refine ⟨T₀, hT₀, q₀, b₀, b, A, d, L₀, hq₀, hb₀, hb, hA, hd, hL₀, ?_⟩ + intro L hL hsmall B r hB hr hBL F hF a O W hO hW β tB T htB htBT hβ hi hβH + hcross hafter hcircle hsector htail hend hne hconv pₙ p U hU hp hpn hpconv hOU + hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed + v hv hlim η hηc hηbar Rₙ hregion + let Tc := T₀ * (Real.toNNReal B) ^ 2 + have hheightB := hheight B hB + have hpositive : 0 < P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) := + (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB + have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ + stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) (L * r₀)) ≤ + P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by + filter_upwards [hlocseq] with n hn + calc + _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hn L hL B hB hBL Tc + _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall + _ = ENNReal.ofReal q₀ / 2 := by + rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] + norm_num + _ ≤ _ := ENNReal.div_le_div_right hheightB 2 + exact bounded_splice_sequence_success_middle_regions κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport + B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector + htail hend Fbar hbar aₙ (L * r₀) Fₙ hFn hinj hne hconv hext pₙ p U hU hp hpn hpconv + hOU hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed + v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing + +end SLEExactGauge.SequencePassage + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SequencePassageSuccess.lean b/lean/OAI/Probability/SLE/Gauge/SequencePassageSuccess.lean index c588c6c3950b7ce09cc0614ae5a4ff8f268dbefd..beb1cbd2ae53dce7acdebbc2f7814e1b8172aa00 100644 --- a/lean/OAI/Probability/SLE/Gauge/SequencePassageSuccess.lean +++ b/lean/OAI/Probability/SLE/Gauge/SequencePassageSuccess.lean @@ -1,256 +1,256 @@ -import OAI.Probability.SLE.Gauge.SequencePassage -import OAI.Probability.SLE.Gauge.PassageSuccess - -namespace OAI - -/-! Successful sequence passage: full allowed-ribbon containment and the -terminal tip offset accompany the actual stopped sausage lower bound. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.SequencePassage -open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess - -local instance successSequencePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance successSequencePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem bounded_splice_sequence_success - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) - (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) - (Fbar : ℕ → ℂ → OnePoint ℂ) - (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) - (aₙ : ℕ → ℂ) (ρ : ℝ) - (Fₙ : ℕ → ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) - (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) - (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U) - (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) - (hOU : ball a (r / 2) ∪ O ⊆ U) - (hWfront : ∀ z ∈ W, 1 < p (F z)) - (hstartBall : ∀ z ∈ ball a (r / 4), p z < 1 / 3) - (hearlyBall : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ U ∩ {z | pₙ n z < 1}) - (V : Set ℂ) (hV : IsOpen V) (hOV : ball a (r / 2) ∪ O ⊆ V) - (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n) - (bₙ : ℕ → ℂ) (ε : ℝ) - (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - bₙ n‖ < ε / 4) - (hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ Vₙ n) - (R : Set ℂ) (hR : IsOpen R) - (hmiddle : ∀ z ∈ ball a (r / 2) ∪ O, 1 / 3 < p z → p z < 2 / 3 → z ∈ R) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) - (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) - (Rₙ : ℕ → Set ℂ) (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) - (hpositive : 0 < P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) - (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ - stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) ρ) ≤ - P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : - ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | - ω ∈ FrontSuccess η pₙ (fun ω => boundedFirstHeightTime (γ ω) B Tc) Vₙ bₙ ε n ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) - (boundedFirstHeightTime (γ ω) B Tc) - (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ - have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc - have hσm : Measurable σ := stopping_time_measurable_nnreal hσ - let f := sourceTracePath γ (fun ω => (hw ω).1) - have hfm : Measurable f := source_path_measurable f hγm - let Uparam := parameterContinuationRegion F a r O - have hUparam : IsOpen Uparam := parameterContinuationRegion_isOpen hF.continuousOn a r hO - let C := stoppedPhysicalTube γ σ Uparam W - let E := stoppedNarrowCertificate γ σ B Tc ∩ C - let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ - have hEm : MeasurableSet E := by - apply MeasurableSet.inter - · exact hσ.measurableSpace_le _ - (boundedFirstHeight_certificate_measurable ℱ γ ha (fun ω => (hw ω).1) B Tc) - · exact measurableSet_stoppedPhysicalTube γ (fun ω => (hw ω).1) hγm σ hσm Uparam W hUparam hW - have hstart : ∀ ω ∈ E, p (F (γ ω (σ ω))) < 1 / 3 := by - intro ω hω - have hnorm := bounded_firstHeight_norm (f ω) (hw ω).2.1 (hw ω).2.2.1 - hB Tc hω.1 (σ ω) ⟨bot_le, le_rfl⟩ - change ‖γ ω (σ ω)‖ < B / 2 at hnorm - have hheight : (γ ω (σ ω)).im = B / 4 := hω.1.2.1 - have him := Complex.im_le_norm (γ ω (σ ω)) - have hsmall := hsector (γ ω (σ ω)) (by linarith) (by linarith) - (by rw [hheight]; linarith) - exact hstartBall _ (by simpa only [mem_ball, dist_eq_norm] using hsmall) - obtain ⟨q, hq, hprob⟩ := eventual_bounded_splice_probability κ hκ hκ8 B₀ hB₀ hm hc - γ G F₀ hw hboundary hsupport B r hB hr Tc F hF.continuousOn a O W hO hW - β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend Fbar hbar aₙ ρ - hpositive hmissing - have hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → γ ω t ∈ upperHalfPlane → - η n ω t = Fₙ n (γ ω t) := by - intro n - filter_upwards [hηbar n] with ω hω - intro t _ ht - exact OnePoint.coe_injective ((hω t).symm.trans (hext n _ ht)) - have hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → - MapsTo (η n ω) (Icc 0 (σ ω)) U ∧ - ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1 := by - filter_upwards [hearlyBall] with n hn - filter_upwards [hηbar n] with ω hω - intro hmem - have himage : MapsTo (η n ω) (Icc 0 (σ ω)) (ball (aₙ n) ρ) := by - intro t ht - obtain ⟨z, hz, he⟩ := hmem.1 t ht - have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) - exact heq ▸ hz - exact ⟨fun _ ht => (hn (himage ht)).1, fun _ ht => (hn (himage ht)).2⟩ - have hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → - MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n) := by - filter_upwards [hearlyAllowed] with n hn - filter_upwards [hηbar n] with ω hω - intro hmem t ht - obtain ⟨z, hz, he⟩ := hmem.1 t ht - have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) - exact hn (heq ▸ hz) - obtain ⟨c, hcpos, hmass⟩ := retained_tube_firstFront_success_mass P κ hκ hκ8 - f hfm hSLE σ hσm Fₙ F hFn hinj hF hne hconv Uparam W hUparam inter_subset_left - pₙ p U hU hp hpn hpconv (fun _ hz => hOU hz.2) hWfront E hEm inter_subset_right - hstart V hV (fun _ hz => hOV hz.2) Vₙ hVVₙ bₙ ε hterminal - R hR (fun _ hz => hmiddle _ hz.2) v hv hlim η Rₙ hregion H q hq hprob hη hηc hearly hearlyV - exact ⟨c, q / 8, hcpos, by positivity, hmass⟩ - - -theorem localized_sequence_passage_success - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) - (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) - (δ : ℝ) (hδ : 0 < δ) (βₙ : ℕ → ℝ → ℂ) - (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 δ)) (hβ0 : ∀ n, βₙ n 0 = 0) - (hβHn : ∀ n t, t ∈ Icc 0 δ → 0 ≤ (βₙ n t).im) - (z : ℂ) (hz : z ≠ 0) (hβconv : Tendsto (fun n => βₙ n δ) atTop (𝓝 z)) - (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0) - (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 δ)) (hα0 : ∀ n, αₙ n 0 = aₙ n) - (hmap : ∀ n t, t ∈ Icc 0 δ → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ)) - (r₀ : ℝ) (hr₀ : 0 < r₀) (hsmallGuide : (Cg : ℝ) * δ ≤ r₀) : - ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ, - 0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧ - ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 → - ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b → - ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane → - ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W → - ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T → - ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) → - (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B → - (∀ t ∈ Ioc tB T, B < ‖β t‖) → - (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) → - (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) → - F '' (β '' Icc tB T) ⊆ O → β T ∈ W → - (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) → - TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane → - ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U → - ContinuousOn p U → (∀ n, ContinuousOn (pₙ n) U) → - TendstoLocallyUniformlyOn pₙ p atTop U → - ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) → - (∀ w ∈ ball a (r / 4), p w < 1 / 3) → - (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) → - ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V → - ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) → - ∀ (bₙ : ℕ → ℂ) (ε : ℝ), - (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) → - (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) → - ∀ (R : Set ℂ), IsOpen R → - (∀ w ∈ ball a (r / 2) ∪ O, 1 / 3 < p w → p w < 2 / 3 → w ∈ R) → - ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) → - ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) → - (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) → - ∀ (Rₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, R ⊆ Rₙ n) → - ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | - ω ∈ FrontSuccess η pₙ - (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) - (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) - (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ - obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability - κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport - obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ := - common_access_gives_firstHeight_localization κ hκ hκ8 - obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1) - (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t) - Fₙ Fbar hFn hinj hbar hext δ hδ βₙ hβc hβ0 hβHn z hz hβconv - αₙ aₙ Cg hα hα0 hmap r₀ hr₀ hsmallGuide - refine ⟨T₀, hT₀, q₀, b₀, b, A, d, L₀, hq₀, hb₀, hb, hA, hd, hL₀, ?_⟩ - intro L hL hsmall B r hB hr hBL F hF a O W hO hW β tB T htB htBT hβ hi hβH - hcross hafter hcircle hsector htail hend hne hconv pₙ p U hU hp hpn hpconv hOU - hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed - R hR hmiddle v hv hlim η hηc hηbar Rₙ hregion - let Tc := T₀ * (Real.toNNReal B) ^ 2 - have hheightB := hheight B hB - have hpositive : 0 < P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) := - (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB - have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ - (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ - stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) - (Fbar n) (aₙ n) (L * r₀)) ≤ - P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by - filter_upwards [hlocseq] with n hn - calc - _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hn L hL B hB hBL Tc - _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall - _ = ENNReal.ofReal q₀ / 2 := by - rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] - norm_num - _ ≤ _ := ENNReal.div_le_div_right hheightB 2 - exact bounded_splice_sequence_success κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport - B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector - htail hend Fbar hbar aₙ (L * r₀) Fₙ hFn hinj hne hconv hext pₙ p U hU hp hpn hpconv - hOU hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed - R hR hmiddle v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing - -end SLEExactGauge.SequencePassage - -end - -end OAI +import OAI.Probability.SLE.Gauge.SequencePassage +import OAI.Probability.SLE.Gauge.PassageSuccess + +namespace OAI + +/-! Successful sequence passage: full allowed-ribbon containment and the +terminal tip offset accompany the actual stopped sausage lower bound. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.SequencePassage +open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess + +local instance successSequencePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance successSequencePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem bounded_splice_sequence_success + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) + (F : ℂ → ℂ) (hF : DifferentiableOn ℂ F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) + (Fbar : ℕ → ℂ → OnePoint ℂ) + (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) + (aₙ : ℕ → ℂ) (ρ : ℝ) + (Fₙ : ℕ → ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) + (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U) + (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U) + (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U) + (hOU : ball a (r / 2) ∪ O ⊆ U) + (hWfront : ∀ z ∈ W, 1 < p (F z)) + (hstartBall : ∀ z ∈ ball a (r / 4), p z < 1 / 3) + (hearlyBall : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ U ∩ {z | pₙ n z < 1}) + (V : Set ℂ) (hV : IsOpen V) (hOV : ball a (r / 2) ∪ O ⊆ V) + (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n) + (bₙ : ℕ → ℂ) (ε : ℝ) + (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - bₙ n‖ < ε / 4) + (hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) ρ ⊆ Vₙ n) + (R : Set ℂ) (hR : IsOpen R) + (hmiddle : ∀ z ∈ ball a (r / 2) ∪ O, 1 / 3 < p z → p z < 2 / 3 → z ∈ R) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) + (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) + (Rₙ : ℕ → Set ℂ) (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n) + (hpositive : 0 < P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc)) + (hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ + stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) ρ) ≤ + P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2) : + ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | + ω ∈ FrontSuccess η pₙ (fun ω => boundedFirstHeightTime (γ ω) B Tc) Vₙ bₙ ε n ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) + (boundedFirstHeightTime (γ ω) B Tc) + (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => boundedFirstHeightTime (γ ω) B Tc + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ + have hσ := boundedFirstHeightTime_isStoppingTime ℱ γ ha (fun ω => (hw ω).1) B Tc + have hσm : Measurable σ := stopping_time_measurable_nnreal hσ + let f := sourceTracePath γ (fun ω => (hw ω).1) + have hfm : Measurable f := source_path_measurable f hγm + let Uparam := parameterContinuationRegion F a r O + have hUparam : IsOpen Uparam := parameterContinuationRegion_isOpen hF.continuousOn a r hO + let C := stoppedPhysicalTube γ σ Uparam W + let E := stoppedNarrowCertificate γ σ B Tc ∩ C + let H : ℕ → Set Ω := fun n => stoppedImageEvent γ σ (Fbar n) (aₙ n) ρ + have hEm : MeasurableSet E := by + apply MeasurableSet.inter + · exact hσ.measurableSpace_le _ + (boundedFirstHeight_certificate_measurable ℱ γ ha (fun ω => (hw ω).1) B Tc) + · exact measurableSet_stoppedPhysicalTube γ (fun ω => (hw ω).1) hγm σ hσm Uparam W hUparam hW + have hstart : ∀ ω ∈ E, p (F (γ ω (σ ω))) < 1 / 3 := by + intro ω hω + have hnorm := bounded_firstHeight_norm (f ω) (hw ω).2.1 (hw ω).2.2.1 + hB Tc hω.1 (σ ω) ⟨bot_le, le_rfl⟩ + change ‖γ ω (σ ω)‖ < B / 2 at hnorm + have hheight : (γ ω (σ ω)).im = B / 4 := hω.1.2.1 + have him := Complex.im_le_norm (γ ω (σ ω)) + have hsmall := hsector (γ ω (σ ω)) (by linarith) (by linarith) + (by rw [hheight]; linarith) + exact hstartBall _ (by simpa only [mem_ball, dist_eq_norm] using hsmall) + obtain ⟨q, hq, hprob⟩ := eventual_bounded_splice_probability κ hκ hκ8 B₀ hB₀ hm hc + γ G F₀ hw hboundary hsupport B r hB hr Tc F hF.continuousOn a O W hO hW + β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend Fbar hbar aₙ ρ + hpositive hmissing + have hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → γ ω t ∈ upperHalfPlane → + η n ω t = Fₙ n (γ ω t) := by + intro n + filter_upwards [hηbar n] with ω hω + intro t _ ht + exact OnePoint.coe_injective ((hω t).symm.trans (hext n _ ht)) + have hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → + MapsTo (η n ω) (Icc 0 (σ ω)) U ∧ + ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1 := by + filter_upwards [hearlyBall] with n hn + filter_upwards [hηbar n] with ω hω + intro hmem + have himage : MapsTo (η n ω) (Icc 0 (σ ω)) (ball (aₙ n) ρ) := by + intro t ht + obtain ⟨z, hz, he⟩ := hmem.1 t ht + have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) + exact heq ▸ hz + exact ⟨fun _ ht => (hn (himage ht)).1, fun _ ht => (hn (himage ht)).2⟩ + have hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ H n ∩ E → + MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n) := by + filter_upwards [hearlyAllowed] with n hn + filter_upwards [hηbar n] with ω hω + intro hmem t ht + obtain ⟨z, hz, he⟩ := hmem.1 t ht + have heq : z = η n ω t := OnePoint.coe_injective (he.trans (hω t)) + exact hn (heq ▸ hz) + obtain ⟨c, hcpos, hmass⟩ := retained_tube_firstFront_success_mass P κ hκ hκ8 + f hfm hSLE σ hσm Fₙ F hFn hinj hF hne hconv Uparam W hUparam inter_subset_left + pₙ p U hU hp hpn hpconv (fun _ hz => hOU hz.2) hWfront E hEm inter_subset_right + hstart V hV (fun _ hz => hOV hz.2) Vₙ hVVₙ bₙ ε hterminal + R hR (fun _ hz => hmiddle _ hz.2) v hv hlim η Rₙ hregion H q hq hprob hη hηc hearly hearlyV + exact ⟨c, q / 8, hcpos, by positivity, hmass⟩ + + +theorem localized_sequence_passage_success + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane) + (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ)) + (δ : ℝ) (hδ : 0 < δ) (βₙ : ℕ → ℝ → ℂ) + (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 δ)) (hβ0 : ∀ n, βₙ n 0 = 0) + (hβHn : ∀ n t, t ∈ Icc 0 δ → 0 ≤ (βₙ n t).im) + (z : ℂ) (hz : z ≠ 0) (hβconv : Tendsto (fun n => βₙ n δ) atTop (𝓝 z)) + (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0) + (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 δ)) (hα0 : ∀ n, αₙ n 0 = aₙ n) + (hmap : ∀ n t, t ∈ Icc 0 δ → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ)) + (r₀ : ℝ) (hr₀ : 0 < r₀) (hsmallGuide : (Cg : ℝ) * δ ≤ r₀) : + ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ, + 0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧ + ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 → + ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b → + ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane → + ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W → + ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T → + ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) → + (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B → + (∀ t ∈ Ioc tB T, B < ‖β t‖) → + (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) → + (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) → + F '' (β '' Icc tB T) ⊆ O → β T ∈ W → + (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) → + TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane → + ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U → + ContinuousOn p U → (∀ n, ContinuousOn (pₙ n) U) → + TendstoLocallyUniformlyOn pₙ p atTop U → + ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) → + (∀ w ∈ ball a (r / 4), p w < 1 / 3) → + (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) → + ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V → + ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) → + ∀ (bₙ : ℕ → ℂ) (ε : ℝ), + (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) → + (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) → + ∀ (R : Set ℂ), IsOpen R → + (∀ w ∈ ball a (r / 2) ∪ O, 1 / 3 < p w → p w < 2 / 3 → w ∈ R) → + ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) → + ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) → + (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) → + ∀ (Rₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, R ⊆ Rₙ n) → + ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω | + ω ∈ FrontSuccess η pₙ + (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω) + (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) + (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩ + obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability + κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport + obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ := + common_access_gives_firstHeight_localization κ hκ hκ8 + obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1) + (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t) + Fₙ Fbar hFn hinj hbar hext δ hδ βₙ hβc hβ0 hβHn z hz hβconv + αₙ aₙ Cg hα hα0 hmap r₀ hr₀ hsmallGuide + refine ⟨T₀, hT₀, q₀, b₀, b, A, d, L₀, hq₀, hb₀, hb, hA, hd, hL₀, ?_⟩ + intro L hL hsmall B r hB hr hBL F hF a O W hO hW β tB T htB htBT hβ hi hβH + hcross hafter hcircle hsector htail hend hne hconv pₙ p U hU hp hpn hpconv hOU + hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed + R hR hmiddle v hv hlim η hηc hηbar Rₙ hregion + let Tc := T₀ * (Real.toNNReal B) ^ 2 + have hheightB := hheight B hB + have hpositive : 0 < P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) := + (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB + have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ + (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \ + stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) + (Fbar n) (aₙ n) (L * r₀)) ≤ + P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by + filter_upwards [hlocseq] with n hn + calc + _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hn L hL B hB hBL Tc + _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall + _ = ENNReal.ofReal q₀ / 2 := by + rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] + norm_num + _ ≤ _ := ENNReal.div_le_div_right hheightB 2 + exact bounded_splice_sequence_success κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport + B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector + htail hend Fbar hbar aₙ (L * r₀) Fₙ hFn hinj hne hconv hext pₙ p U hU hp hpn hpconv + hOU hWfront hstartBall hearlyBall V hV hOV Vₙ hVVₙ bₙ ε hterminal hearlyAllowed + R hR hmiddle v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing + +end SLEExactGauge.SequencePassage + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngle.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngle.lean index 4a08efcfa34ee684aa8d43f15c72d3128f94d9b4..eb831014350f222822b6105b592229746ceb15dd 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngle.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngle.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Gauge.SerpentineActualGuide -import OAI.Probability.SLE.Geometry.GoodPrefixCoordinates -import OAI.Probability.SLE.Geometry.MarkedAccessAngle - -namespace OAI - -/-! Successful actual serpentine completion gives the marked-point angle -uniformly in the number of retained strips and in the trace history. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates FirstFrontPassage Target - -theorem uniform_serpentine_actual_angle_nonneg - (hopen : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hsupport : PublishedPlanarWienerUnitSupport) (W : WienerLaw) (hW : IsWiener W) - {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : - ∃ s₀ : ℝ, 0 < s₀ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → - ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → 0 ≤ w.re → - ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (physicalFitChart s w m z) = z.re) → - ∀ ε : ℝ, ε ≤ 1 / 8 → - () ∈ GoodRibbonPrefix (fun _ : Unit => γ) - (fun _ => firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)) - (physicalFitChart s w m) ε → - let T := (firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)).untopD 0 - s₀ ≤ (G T w - (U T : ℂ)).im / ‖G T w - (U T : ℂ)‖ := by - obtain ⟨s₀, hs₀, hangle⟩ := uniform_marked_access_angle hopen hstraight hexit hsupport W hW - (show 0 < s / 20 by positivity) (show s / 20 ≤ 1 by linarith) - refine ⟨s₀, hs₀, ?_⟩ - intro U γ G F hw hU w hwn hwy hwr m hm g hg ε hε hpref - have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by - funext z - simp only [physicalExperimentChart, ite_eq_left hwr] - have hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => physicalFitChart s w m z) := by - simpa only [hchart] using physicalExperimentChart_closedEmbedding hs w hwy hm - obtain ⟨hP, hi, hH, hescape⟩ := physicalFitChart_geometry hs w hwy hm - have hg0 : g (γ 0) = 0 := by - rw [hw.2.1, ← physicalFitChart_zero hs w hwy hm] - exact hg 0 (by simp [closedRibbon]) - have hk : 0 < (fitCompletion s w m : ℝ) := by - have hbound := (fitCompletion_bounds hs w hwy hm).1 - have hEntry : 0 ≤ fitEntry s w m := - (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hm) - have hcol : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ - have hmR : (0 : ℝ) < m := by exact_mod_cast hm - linarith - obtain ⟨T, x, xp, hfirst, hT, hx, hxpc, hγeq, hxp0, htip, hcoords, hcore, hstrict⟩ := - goodRibbonPrefix_coordinates γ hw.1 _ hemb hi g hg hg0 hk hpref - have hxq : |x| ≤ 1 / 4 := hx.trans (by linarith) - have hx1 : |x| ≤ 1 := hxq.trans (by norm_num) - have htipre : (xp T).re = fitCompletion s w m := by simp [htip] - have htipim : (xp T).im = x := by simp [htip] - have hfresh := ribbon_front_tip_survives_prior _ hP.continuousOn hi hH hescape γ xp T - hxpc hxp0 hcoords hγeq (fun t ht => by rw [htipre]; exact hstrict t ht) - (by rw [htipim]; exact hxq.trans_lt (by norm_num)) - have htipphysical : physicalFitChart s w m ((fitCompletion s w m : ℂ) + (x : ℂ) * Complex.I) = γ T := by - push_cast at htip - rw [← htip] - exact (hγeq ⟨bot_le, le_rfl⟩).symm - let α := fun u : ℝ≥0 => completeMarkedAccess s w m x u - have hαc : Continuous α := - (completeMarkedAccess_continuous hs hs1 w hwy hwn hwr hm x).comp NNReal.continuous_coe - have hαi : Function.Injective α := by - intro t u he - apply NNReal.coe_injective - exact completeMarkedAccess_injective hs hs1 w hwy hwn hwr hm hx1 t.coe_nonneg u.coe_nonneg he - have hα0 : α 0 = γ T := by - change completeMarkedAccess s w m x (0 : ℝ) = γ T - rw [completeMarkedAccess_zero hs hs1 w hwy hwn hm x, htipphysical] - have hαD := completeMarkedAccess_survives hs hs1 w hwy hwn hwr hm hxq γ xp T hcoords hγeq - (fun t ht => (hcore t ht).1) - have hαescape : Tendsto α atTop (cocompact ℂ) := by - apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 - simp only [dist_zero_right] - exact (completeMarkedAccess_norm_tendsto hs w hwn hwr m x).comp - (NNReal.tendsto_coe_atTop.mpr tendsto_id) - have hlocus (z : ℂ) (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : - (∃ u : ℝ≥0, 0 < u ∧ α u = markedCornerChart w z) ↔ z.im = 0 := - completeMarkedAccess_corner_nnreal_iff hs hs1 w hwy hwn hwr hm hx1 hz - have hclear (right : Bool) (t : ℝ) (ht : t ∈ Icc (1 / 64 : ℝ) 3) (u : ℝ≥0) (_hu : 0 < u) : - (s / 20) / 1000 ≤ dist (markedWindowRoute right w (s / 20) t) (α u) := by - have hh := completeMarkedAccess_route_clearance hs hs1 w hwy hwn hwr hm hx1 - (show 1 / 64 ≤ 4 / 3 * t by linarith [ht.1]) u.coe_nonneg - cases right - · exact hh.1 - · exact hh.2 - have hpast := physicalFit_prefix_low_right hs w hwy hwr hm γ xp T hcoords hγeq - have hh := hangle U γ G F hw hU T hT hfresh w hwn (show 2 * (s / 20) < w.im by linarith) - hpast α hαc hαi hα0 hαD hαescape hlocus hclear - simpa only [hfirst, WithTop.untopD_coe] using hh - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineActualGuide +import OAI.Probability.SLE.Geometry.GoodPrefixCoordinates +import OAI.Probability.SLE.Geometry.MarkedAccessAngle + +namespace OAI + +/-! Successful actual serpentine completion gives the marked-point angle +uniformly in the number of retained strips and in the trace history. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates FirstFrontPassage Target + +theorem uniform_serpentine_actual_angle_nonneg + (hopen : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hsupport : PublishedPlanarWienerUnitSupport) (W : WienerLaw) (hW : IsWiener W) + {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : + ∃ s₀ : ℝ, 0 < s₀ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → + ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → 0 ≤ w.re → + ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (physicalFitChart s w m z) = z.re) → + ∀ ε : ℝ, ε ≤ 1 / 8 → + () ∈ GoodRibbonPrefix (fun _ : Unit => γ) + (fun _ => firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)) + (physicalFitChart s w m) ε → + let T := (firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)).untopD 0 + s₀ ≤ (G T w - (U T : ℂ)).im / ‖G T w - (U T : ℂ)‖ := by + obtain ⟨s₀, hs₀, hangle⟩ := uniform_marked_access_angle hopen hstraight hexit hsupport W hW + (show 0 < s / 20 by positivity) (show s / 20 ≤ 1 by linarith) + refine ⟨s₀, hs₀, ?_⟩ + intro U γ G F hw hU w hwn hwy hwr m hm g hg ε hε hpref + have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by + funext z + simp only [physicalExperimentChart, ite_eq_left hwr] + have hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => physicalFitChart s w m z) := by + simpa only [hchart] using physicalExperimentChart_closedEmbedding hs w hwy hm + obtain ⟨hP, hi, hH, hescape⟩ := physicalFitChart_geometry hs w hwy hm + have hg0 : g (γ 0) = 0 := by + rw [hw.2.1, ← physicalFitChart_zero hs w hwy hm] + exact hg 0 (by simp [closedRibbon]) + have hk : 0 < (fitCompletion s w m : ℝ) := by + have hbound := (fitCompletion_bounds hs w hwy hm).1 + have hEntry : 0 ≤ fitEntry s w m := + (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hm) + have hcol : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ + have hmR : (0 : ℝ) < m := by exact_mod_cast hm + linarith + obtain ⟨T, x, xp, hfirst, hT, hx, hxpc, hγeq, hxp0, htip, hcoords, hcore, hstrict⟩ := + goodRibbonPrefix_coordinates γ hw.1 _ hemb hi g hg hg0 hk hpref + have hxq : |x| ≤ 1 / 4 := hx.trans (by linarith) + have hx1 : |x| ≤ 1 := hxq.trans (by norm_num) + have htipre : (xp T).re = fitCompletion s w m := by simp [htip] + have htipim : (xp T).im = x := by simp [htip] + have hfresh := ribbon_front_tip_survives_prior _ hP.continuousOn hi hH hescape γ xp T + hxpc hxp0 hcoords hγeq (fun t ht => by rw [htipre]; exact hstrict t ht) + (by rw [htipim]; exact hxq.trans_lt (by norm_num)) + have htipphysical : physicalFitChart s w m ((fitCompletion s w m : ℂ) + (x : ℂ) * Complex.I) = γ T := by + push_cast at htip + rw [← htip] + exact (hγeq ⟨bot_le, le_rfl⟩).symm + let α := fun u : ℝ≥0 => completeMarkedAccess s w m x u + have hαc : Continuous α := + (completeMarkedAccess_continuous hs hs1 w hwy hwn hwr hm x).comp NNReal.continuous_coe + have hαi : Function.Injective α := by + intro t u he + apply NNReal.coe_injective + exact completeMarkedAccess_injective hs hs1 w hwy hwn hwr hm hx1 t.coe_nonneg u.coe_nonneg he + have hα0 : α 0 = γ T := by + change completeMarkedAccess s w m x (0 : ℝ) = γ T + rw [completeMarkedAccess_zero hs hs1 w hwy hwn hm x, htipphysical] + have hαD := completeMarkedAccess_survives hs hs1 w hwy hwn hwr hm hxq γ xp T hcoords hγeq + (fun t ht => (hcore t ht).1) + have hαescape : Tendsto α atTop (cocompact ℂ) := by + apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 + simp only [dist_zero_right] + exact (completeMarkedAccess_norm_tendsto hs w hwn hwr m x).comp + (NNReal.tendsto_coe_atTop.mpr tendsto_id) + have hlocus (z : ℂ) (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : + (∃ u : ℝ≥0, 0 < u ∧ α u = markedCornerChart w z) ↔ z.im = 0 := + completeMarkedAccess_corner_nnreal_iff hs hs1 w hwy hwn hwr hm hx1 hz + have hclear (right : Bool) (t : ℝ) (ht : t ∈ Icc (1 / 64 : ℝ) 3) (u : ℝ≥0) (_hu : 0 < u) : + (s / 20) / 1000 ≤ dist (markedWindowRoute right w (s / 20) t) (α u) := by + have hh := completeMarkedAccess_route_clearance hs hs1 w hwy hwn hwr hm hx1 + (show 1 / 64 ≤ 4 / 3 * t by linarith [ht.1]) u.coe_nonneg + cases right + · exact hh.1 + · exact hh.2 + have hpast := physicalFit_prefix_low_right hs w hwy hwr hm γ xp T hcoords hγeq + have hh := hangle U γ G F hw hU T hT hfresh w hwn (show 2 * (s / 20) < w.im by linarith) + hpast α hαc hαi hα0 hαD hαescape hlocus hclear + simpa only [hfirst, WithTop.untopD_coe] using hh + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngleReflection.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngleReflection.lean index 6430371b0a46a9c1f06542fa40462500c1a748ef..d808ba8101cbca20d57f2687bd067dd2b1274d94 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngleReflection.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineActualAngleReflection.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Gauge.SerpentineActualAngle -import OAI.Probability.SLE.Geometry.MarkedWitnessReflection - -namespace OAI - -/-! Reflection removes the sign restriction on the marked point, with -exactly the same uniform angle constant as the right-hand experiment. -/ - -noncomputable section -open Set Filter Metric ComplexConjugate -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates FirstFrontPassage Target - -theorem uniform_serpentine_actual_angle - (hopen : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hsupport : PublishedPlanarWienerUnitSupport) (W : WienerLaw) (hW : IsWiener W) - {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : - ∃ s₀ : ℝ, 0 < s₀ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → - ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → - ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → - ∀ ε : ℝ, ε ≤ 1 / 8 → - () ∈ GoodRibbonPrefix (fun _ : Unit => γ) - (fun _ => firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)) - (physicalExperimentChart s w m) ε → - let T := (firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)).untopD 0 - s₀ ≤ (G T w - (U T : ℂ)).im / ‖G T w - (U T : ℂ)‖ := by - obtain ⟨s₀, hs₀, hangle⟩ := - uniform_serpentine_actual_angle_nonneg hopen hstraight hexit hsupport W hW hs hs1 - refine ⟨s₀, hs₀, ?_⟩ - intro U γ G F hw hU w hwn hwy m hm g hg ε hε hpref - by_cases hwr : 0 ≤ w.re - · have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by - funext z - simp only [physicalExperimentChart, ite_eq_left hwr] - exact hangle U γ G F hw hU w hwn hwy hwr m hm g - (by simpa only [hchart] using hg) ε hε (by simpa only [hchart] using hpref) - · have hwneg : w.re < 0 := lt_of_not_ge hwr - let γ' : ℝ≥0 → ℂ := loewnerMirror ∘ γ - let U' : ℝ≥0 → ℝ := fun t => -U t - let G' : ℝ≥0 → ℂ → ℂ := fun t z => loewnerMirror (G t (loewnerMirror z)) - let F' : ℝ≥0 → ℂ → ℂ := fun t z => loewnerMirror (F t (loewnerMirror z)) - let g' := mirroredProgress g - have hg' : ∀ z ∈ closedRibbon, - g' (physicalFitChart s (loewnerMirror w) m z) = z.re := by - intro z hz - have hh := hg (conj z) (conj_mem_closedRibbon hz) - rw [physicalExperimentChart_negative hwneg m] at hh - simpa only [g', mirroredProgress, ContinuousMap.coe_mk, reflectedChart, - Complex.conj_conj, Complex.conj_re] using hh - have hclock : firstFrontTime (fun t => g' (γ' t)) - (fitCompletion s (loewnerMirror w) m) = - firstFrontTime (fun t => g (γ t)) (fitCompletion s w m) := by - rw [fitCompletion_mirror] - exact mirroredProgress_firstFrontTime g γ _ - have hpref' : () ∈ GoodRibbonPrefix (fun _ : Unit => γ') - (fun _ => firstFrontTime (fun t => g' (γ' t)) - (fitCompletion s (loewnerMirror w) m)) - (physicalFitChart s (loewnerMirror w) m) ε := by - rw [hclock] - rw [goodRibbonPrefix_physical_negative _ _ hwneg m ε] at hpref - simpa only [γ', Function.comp_def] using hpref - have hh := hangle U' γ' G' F' hw.mirror hU.neg (loewnerMirror w) - (by simpa only [loewnerMirror_norm] using hwn) - (by simpa only [loewnerMirror_im] using hwy) - (by simp only [loewnerMirror_re]; linarith) - m hm g' hg' ε hε hpref' - dsimp only at hh ⊢ - rw [hclock] at hh - simpa only [U', G', loewnerMirror_twice, mirrored_centered_sine] using hh - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineActualAngle +import OAI.Probability.SLE.Geometry.MarkedWitnessReflection + +namespace OAI + +/-! Reflection removes the sign restriction on the marked point, with +exactly the same uniform angle constant as the right-hand experiment. -/ + +noncomputable section +open Set Filter Metric ComplexConjugate +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates FirstFrontPassage Target + +theorem uniform_serpentine_actual_angle + (hopen : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hsupport : PublishedPlanarWienerUnitSupport) (W : WienerLaw) (hW : IsWiener W) + {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : + ∃ s₀ : ℝ, 0 < s₀ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → + ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → + ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → + ∀ ε : ℝ, ε ≤ 1 / 8 → + () ∈ GoodRibbonPrefix (fun _ : Unit => γ) + (fun _ => firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)) + (physicalExperimentChart s w m) ε → + let T := (firstFrontTime (fun t => g (γ t)) (fitCompletion s w m)).untopD 0 + s₀ ≤ (G T w - (U T : ℂ)).im / ‖G T w - (U T : ℂ)‖ := by + obtain ⟨s₀, hs₀, hangle⟩ := + uniform_serpentine_actual_angle_nonneg hopen hstraight hexit hsupport W hW hs hs1 + refine ⟨s₀, hs₀, ?_⟩ + intro U γ G F hw hU w hwn hwy m hm g hg ε hε hpref + by_cases hwr : 0 ≤ w.re + · have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by + funext z + simp only [physicalExperimentChart, ite_eq_left hwr] + exact hangle U γ G F hw hU w hwn hwy hwr m hm g + (by simpa only [hchart] using hg) ε hε (by simpa only [hchart] using hpref) + · have hwneg : w.re < 0 := lt_of_not_ge hwr + let γ' : ℝ≥0 → ℂ := loewnerMirror ∘ γ + let U' : ℝ≥0 → ℝ := fun t => -U t + let G' : ℝ≥0 → ℂ → ℂ := fun t z => loewnerMirror (G t (loewnerMirror z)) + let F' : ℝ≥0 → ℂ → ℂ := fun t z => loewnerMirror (F t (loewnerMirror z)) + let g' := mirroredProgress g + have hg' : ∀ z ∈ closedRibbon, + g' (physicalFitChart s (loewnerMirror w) m z) = z.re := by + intro z hz + have hh := hg (conj z) (conj_mem_closedRibbon hz) + rw [physicalExperimentChart_negative hwneg m] at hh + simpa only [g', mirroredProgress, ContinuousMap.coe_mk, reflectedChart, + Complex.conj_conj, Complex.conj_re] using hh + have hclock : firstFrontTime (fun t => g' (γ' t)) + (fitCompletion s (loewnerMirror w) m) = + firstFrontTime (fun t => g (γ t)) (fitCompletion s w m) := by + rw [fitCompletion_mirror] + exact mirroredProgress_firstFrontTime g γ _ + have hpref' : () ∈ GoodRibbonPrefix (fun _ : Unit => γ') + (fun _ => firstFrontTime (fun t => g' (γ' t)) + (fitCompletion s (loewnerMirror w) m)) + (physicalFitChart s (loewnerMirror w) m) ε := by + rw [hclock] + rw [goodRibbonPrefix_physical_negative _ _ hwneg m ε] at hpref + simpa only [γ', Function.comp_def] using hpref + have hh := hangle U' γ' G' F' hw.mirror hU.neg (loewnerMirror w) + (by simpa only [loewnerMirror_norm] using hwn) + (by simpa only [loewnerMirror_im] using hwy) + (by simp only [loewnerMirror_re]; linarith) + m hm g' hg' ε hε hpref' + dsimp only at hh ⊢ + rw [hclock] at hh + simpa only [U', G', loewnerMirror_twice, mirrored_centered_sine] using hh + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineActualGuide.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineActualGuide.lean index fb9dc02ff77f24b4a8393ca47d16898dd2f4c56d..73f6aa3a13dc9ae55b6a9e7f6c11840b3e583239 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineActualGuide.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineActualGuide.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Gauge.SerpentinePhysicalGeometry -import OAI.Probability.SLE.Gauge.SerpentineMarkedRouteClearance -import OAI.Probability.SLE.Geometry.ActualWaypointStraightening - -namespace OAI - -/-! The concrete terminal access guide lies in the actual surviving -component of a successful stopped serpentine trace. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma physicalFit_prefix_low_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) - (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) - (hcoords : MapsTo xp (Icc 0 T) closedRibbon) - (hγ : EqOn γ (physicalFitChart s w m ∘ xp) (Icc 0 T)) : - ∀ q ∈ γ '' Icc 0 T, q.im ≤ w.im - 2 * (s / 20) ∨ w.re + 2 * (s / 20) ≤ q.re := by - rintro q ⟨t, ht, rfl⟩ - rw [hγ ht] - rcases physicalFitChart_low_or_right hs w hw hm (hcoords ht).2 with hlo | hright - · left - change (physicalFitChart s w m (xp t)).im ≤ _ - linarith - · right - rw [abs_of_nonneg hwr] at hright - change _ ≤ (physicalFitChart s w m (xp t)).re - linarith - -theorem completeMarkedAccess_survives {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1 / 4) - (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) - (hcoords : MapsTo xp (Icc 0 T) closedRibbon) - (hγ : EqOn γ (physicalFitChart s w m ∘ xp) (Icc 0 T)) - (hfront : ∀ t ∈ Icc 0 T, (xp t).re ≤ fitCompletion s w m) : - ∀ u : ℝ≥0, 0 < u → completeMarkedAccess s w m x u ∈ survivingDomain γ T := by - obtain ⟨hP, hi, hH, hescape⟩ := physicalFitChart_geometry hs w hw hm - have hpast : γ '' Icc 0 T ⊆ physicalFitChart s w m '' - {z ∈ closedRibbon | z.re ≤ fitCompletion s w m} := by - rintro q ⟨t, ht, rfl⟩ - exact ⟨xp t, ⟨hcoords ht, hfront t ht⟩, (hγ ht).symm⟩ - have hfuture := futureRibbon_survives _ hP.continuousOn hi - (Nat.cast_nonneg (fitCompletion s w m)) hH γ T hpast - (futureRibbon_unbounded_of_centerline_escape _ (fitCompletion s w m) hescape) - have hwi := (Complex.im_le_norm w).trans hw1 - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - have hhook := markedHook_survives γ T w (lt_of_lt_of_le hs hw) hR - (show w.im < 2 by linarith) (show 0 < s / 20 by positivity) (by - rintro q ⟨t, ht, rfl⟩ - rw [hγ ht] - refine ⟨physicalFitChart_finite_height_lt_two hs hs1 w hw hw1 hm (hcoords ht) (hfront t ht), ?_⟩ - rcases physicalFitChart_low_or_right hs w hw hm (hcoords ht).2 with hlo | hright - · left - change (physicalFitChart s w m (xp t)).im < w.im - linarith - · right - rw [abs_of_nonneg hwr] at hright - change w.re < (physicalFitChart s w m (xp t)).re - linarith) - intro u hu - by_cases huT : (u : ℝ) ≤ fitHighTime s w m - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] - apply hfuture - exact mem_image_of_mem _ (guideCoordinate_mem_future hx (by exact_mod_cast hu)) - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] - exact hhook ⟨⟨(u : ℝ) - fitHighTime s w m, by linarith⟩, rfl⟩ - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentinePhysicalGeometry +import OAI.Probability.SLE.Gauge.SerpentineMarkedRouteClearance +import OAI.Probability.SLE.Geometry.ActualWaypointStraightening + +namespace OAI + +/-! The concrete terminal access guide lies in the actual surviving +component of a successful stopped serpentine trace. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma physicalFit_prefix_low_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) + (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) + (hcoords : MapsTo xp (Icc 0 T) closedRibbon) + (hγ : EqOn γ (physicalFitChart s w m ∘ xp) (Icc 0 T)) : + ∀ q ∈ γ '' Icc 0 T, q.im ≤ w.im - 2 * (s / 20) ∨ w.re + 2 * (s / 20) ≤ q.re := by + rintro q ⟨t, ht, rfl⟩ + rw [hγ ht] + rcases physicalFitChart_low_or_right hs w hw hm (hcoords ht).2 with hlo | hright + · left + change (physicalFitChart s w m (xp t)).im ≤ _ + linarith + · right + rw [abs_of_nonneg hwr] at hright + change _ ≤ (physicalFitChart s w m (xp t)).re + linarith + +theorem completeMarkedAccess_survives {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1 / 4) + (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0) + (hcoords : MapsTo xp (Icc 0 T) closedRibbon) + (hγ : EqOn γ (physicalFitChart s w m ∘ xp) (Icc 0 T)) + (hfront : ∀ t ∈ Icc 0 T, (xp t).re ≤ fitCompletion s w m) : + ∀ u : ℝ≥0, 0 < u → completeMarkedAccess s w m x u ∈ survivingDomain γ T := by + obtain ⟨hP, hi, hH, hescape⟩ := physicalFitChart_geometry hs w hw hm + have hpast : γ '' Icc 0 T ⊆ physicalFitChart s w m '' + {z ∈ closedRibbon | z.re ≤ fitCompletion s w m} := by + rintro q ⟨t, ht, rfl⟩ + exact ⟨xp t, ⟨hcoords ht, hfront t ht⟩, (hγ ht).symm⟩ + have hfuture := futureRibbon_survives _ hP.continuousOn hi + (Nat.cast_nonneg (fitCompletion s w m)) hH γ T hpast + (futureRibbon_unbounded_of_centerline_escape _ (fitCompletion s w m) hescape) + have hwi := (Complex.im_le_norm w).trans hw1 + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + have hhook := markedHook_survives γ T w (lt_of_lt_of_le hs hw) hR + (show w.im < 2 by linarith) (show 0 < s / 20 by positivity) (by + rintro q ⟨t, ht, rfl⟩ + rw [hγ ht] + refine ⟨physicalFitChart_finite_height_lt_two hs hs1 w hw hw1 hm (hcoords ht) (hfront t ht), ?_⟩ + rcases physicalFitChart_low_or_right hs w hw hm (hcoords ht).2 with hlo | hright + · left + change (physicalFitChart s w m (xp t)).im < w.im + linarith + · right + rw [abs_of_nonneg hwr] at hright + change w.re < (physicalFitChart s w m (xp t)).re + linarith) + intro u hu + by_cases huT : (u : ℝ) ≤ fitHighTime s w m + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] + apply hfuture + exact mem_image_of_mem _ (guideCoordinate_mem_future hx (by exact_mod_cast hu)) + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] + exact hhook ⟨⟨(u : ℝ) - fitHighTime s w m, by linarith⟩, rfl⟩ + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineActualRadius.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineActualRadius.lean index 2cb9acb89fdbaa1104b9825b5b2c4523ced27338..3dd643e5f39bdf1502ff6f6d27230eb39efeddaa 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineActualRadius.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineActualRadius.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Gauge.SerpentineActualGuide -import OAI.Probability.SLE.Gauge.SerpentineMarkedReflection -import OAI.Probability.SLE.Gauge.ExperimentClearance -import OAI.Probability.SLE.Geometry.MarkedSurvivingBall - -namespace OAI - -/-! Successful physical prefixes leave a definite ball around the marked -point in the actual surviving domain, hence finish before a fixed radius level. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma good_physicalFit_ball_survives {s : ℝ} (hs : 0 < s) (w : ℂ) - (hwy : s ≤ w.im) (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) - (γ : ℝ≥0 → ℂ) (τ : WithTop ℝ≥0) (ε : ℝ) - (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) - (physicalFitChart s w m) ε) : - ball w (s / 20) ⊆ survivingDomain γ (τ.untopD 0) := by - apply marked_ball_survives γ (τ.untopD 0) w (by positivity) (by linarith) - rintro q ⟨t, ht, rfl⟩ - obtain ⟨z, ⟨hz, _⟩, he⟩ := hpref.2.1 ht - change physicalFitChart s w m z = γ t at he - rw [← he] - rcases physicalFitChart_low_or_right hs w hwy hm hz.2 with hlo | hright - · left - linarith - · right - rw [abs_of_nonneg hwr] at hright - linarith - -theorem good_physicalExperiment_ball_survives {s : ℝ} (hs : 0 < s) (w : ℂ) - (hwy : s ≤ w.im) {m : ℕ} (hm : 0 < m) - (γ : ℝ≥0 → ℂ) (τ : WithTop ℝ≥0) (ε : ℝ) - (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) - (physicalExperimentChart s w m) ε) : - ball w (s / 20) ⊆ survivingDomain γ (τ.untopD 0) := by - by_cases hwr : 0 ≤ w.re - · have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by - funext z - simp only [physicalExperimentChart, ite_eq_left hwr] - apply good_physicalFit_ball_survives hs w hwy hwr hm γ τ ε - simpa only [hchart] using hpref - · have hneg : w.re < 0 := lt_of_not_ge hwr - have hp : () ∈ GoodRibbonPrefix (fun _ : Unit => (loewnerMirror ∘ γ)) (fun _ => τ) - (physicalFitChart s (loewnerMirror w) m) ε := by - rw [goodRibbonPrefix_physical_negative _ _ hneg m ε] at hpref - simpa only [Function.comp_def] using hpref - have hb := good_physicalFit_ball_survives hs (loewnerMirror w) - (by simpa only [loewnerMirror_im] using hwy) - (by simp only [loewnerMirror_re]; linarith) hm (loewnerMirror ∘ γ) τ ε hp - intro z hz - apply (loewnerMirror_surviving γ (τ.untopD 0) z).mp - apply hb - simpa only [mem_ball, loewnerMirror_dist] using hz - -def experimentRadiusLevel (s : ℝ) : ℝ := -Real.log (s / 80) - -lemma experimentRadiusLevel_nonneg {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : - 0 ≤ experimentRadiusLevel s := by - unfold experimentRadiusLevel - exact neg_nonneg.mpr (Real.log_nonpos (by positivity) (by linarith)) - -lemma experimentRadiusLevel_cutoff {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) : - (2 * w.im) * Real.exp (-experimentRadiusLevel s) < s / 20 := by - have hy := (Complex.im_le_norm w).trans hw - rw [experimentRadiusLevel, neg_neg, Real.exp_log (by positivity : 0 < s / 80)] - nlinarith - -theorem good_physicalExperiment_before_radius - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) - {s : ℝ} (hs : 0 < s) (w : ℂ) (hwn : ‖w‖ ≤ 1) (hwy : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (τ : WithTop ℝ≥0) (ε : ℝ) - (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) - (physicalExperimentChart s w m) ε) : - τ < radiusLevelTime γ G w (experimentRadiusLevel s) ∧ - w ∈ survivingDomain γ (τ.untopD 0) := by - have hb := good_physicalExperiment_ball_survives hs w hwy hm γ τ ε hpref - have hlive := hb (mem_ball_self (by positivity : 0 < s / 20)) - refine ⟨?_, hlive⟩ - have hh := completion_before_radiusLevelTime_of_prefix_lower hU hw (τ.untopD 0) w - (experimentRadiusLevel s) (s / 20) (fun t ht => by - have hbt := hb.trans (survivingDomain_antitone γ ht) - exact ⟨hbt (mem_ball_self (by positivity)), - fullRadius_lower_of_surviving_ball hU hw t w (by positivity) hbt⟩) - (experimentRadiusLevel_cutoff hs w hwn) - cases he : τ with - | top => exact (hpref.1 he).elim - | coe T => - have hh' : (T : ℝ≥0∞) < radiusLevelTime γ G w (experimentRadiusLevel s) := by - simpa only [he, WithTop.untopD_coe] using hh - exact_mod_cast hh' - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineActualGuide +import OAI.Probability.SLE.Gauge.SerpentineMarkedReflection +import OAI.Probability.SLE.Gauge.ExperimentClearance +import OAI.Probability.SLE.Geometry.MarkedSurvivingBall + +namespace OAI + +/-! Successful physical prefixes leave a definite ball around the marked +point in the actual surviving domain, hence finish before a fixed radius level. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma good_physicalFit_ball_survives {s : ℝ} (hs : 0 < s) (w : ℂ) + (hwy : s ≤ w.im) (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) + (γ : ℝ≥0 → ℂ) (τ : WithTop ℝ≥0) (ε : ℝ) + (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) + (physicalFitChart s w m) ε) : + ball w (s / 20) ⊆ survivingDomain γ (τ.untopD 0) := by + apply marked_ball_survives γ (τ.untopD 0) w (by positivity) (by linarith) + rintro q ⟨t, ht, rfl⟩ + obtain ⟨z, ⟨hz, _⟩, he⟩ := hpref.2.1 ht + change physicalFitChart s w m z = γ t at he + rw [← he] + rcases physicalFitChart_low_or_right hs w hwy hm hz.2 with hlo | hright + · left + linarith + · right + rw [abs_of_nonneg hwr] at hright + linarith + +theorem good_physicalExperiment_ball_survives {s : ℝ} (hs : 0 < s) (w : ℂ) + (hwy : s ≤ w.im) {m : ℕ} (hm : 0 < m) + (γ : ℝ≥0 → ℂ) (τ : WithTop ℝ≥0) (ε : ℝ) + (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) + (physicalExperimentChart s w m) ε) : + ball w (s / 20) ⊆ survivingDomain γ (τ.untopD 0) := by + by_cases hwr : 0 ≤ w.re + · have hchart : physicalExperimentChart s w m = physicalFitChart s w m := by + funext z + simp only [physicalExperimentChart, ite_eq_left hwr] + apply good_physicalFit_ball_survives hs w hwy hwr hm γ τ ε + simpa only [hchart] using hpref + · have hneg : w.re < 0 := lt_of_not_ge hwr + have hp : () ∈ GoodRibbonPrefix (fun _ : Unit => (loewnerMirror ∘ γ)) (fun _ => τ) + (physicalFitChart s (loewnerMirror w) m) ε := by + rw [goodRibbonPrefix_physical_negative _ _ hneg m ε] at hpref + simpa only [Function.comp_def] using hpref + have hb := good_physicalFit_ball_survives hs (loewnerMirror w) + (by simpa only [loewnerMirror_im] using hwy) + (by simp only [loewnerMirror_re]; linarith) hm (loewnerMirror ∘ γ) τ ε hp + intro z hz + apply (loewnerMirror_surviving γ (τ.untopD 0) z).mp + apply hb + simpa only [mem_ball, loewnerMirror_dist] using hz + +def experimentRadiusLevel (s : ℝ) : ℝ := -Real.log (s / 80) + +lemma experimentRadiusLevel_nonneg {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : + 0 ≤ experimentRadiusLevel s := by + unfold experimentRadiusLevel + exact neg_nonneg.mpr (Real.log_nonpos (by positivity) (by linarith)) + +lemma experimentRadiusLevel_cutoff {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) : + (2 * w.im) * Real.exp (-experimentRadiusLevel s) < s / 20 := by + have hy := (Complex.im_le_norm w).trans hw + rw [experimentRadiusLevel, neg_neg, Real.exp_log (by positivity : 0 < s / 80)] + nlinarith + +theorem good_physicalExperiment_before_radius + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) + {s : ℝ} (hs : 0 < s) (w : ℂ) (hwn : ‖w‖ ≤ 1) (hwy : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (τ : WithTop ℝ≥0) (ε : ℝ) + (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ) (fun _ => τ) + (physicalExperimentChart s w m) ε) : + τ < radiusLevelTime γ G w (experimentRadiusLevel s) ∧ + w ∈ survivingDomain γ (τ.untopD 0) := by + have hb := good_physicalExperiment_ball_survives hs w hwy hm γ τ ε hpref + have hlive := hb (mem_ball_self (by positivity : 0 < s / 20)) + refine ⟨?_, hlive⟩ + have hh := completion_before_radiusLevelTime_of_prefix_lower hU hw (τ.untopD 0) w + (experimentRadiusLevel s) (s / 20) (fun t ht => by + have hbt := hb.trans (survivingDomain_antitone γ ht) + exact ⟨hbt (mem_ball_self (by positivity)), + fullRadius_lower_of_surviving_ball hU hw t w (by positivity) hbt⟩) + (experimentRadiusLevel_cutoff hs w hwn) + cases he : τ with + | top => exact (hpref.1 he).elim + | coe T => + have hh' : (T : ℝ≥0∞) < radiusLevelTime γ G w (experimentRadiusLevel s) := by + simpa only [he, WithTop.untopD_coe] using hh + exact_mod_cast hh' + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineArrayEmbedding.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineArrayEmbedding.lean index 343260cba6948a2aec700ee53b30ee84d9823a51..0ccd8d2be9eacc4914767d436cf71a9d28947629 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineArrayEmbedding.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineArrayEmbedding.lean @@ -1,198 +1,198 @@ -import OAI.Probability.SLE.Gauge.SerpentineCurve -import OAI.Probability.SLE.Gauge.SerpentineEmbedding - -namespace OAI - -/-! Global normal coordinates for the actual finite serpentine array. -Every parameter belongs to a straight or circular piece, and the only -possible normal-band contacts are the prescribed common end sections. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -def arrayEnd (m j : ℕ) : ℝ := columnEntry 0 m j + 200 * m - -lemma array_pieces (m j : ℕ) {t : ℝ} (ht : t ∈ Icc 0 (arrayEnd m j)) : - (∃ i ≤ j, ∃ u ∈ Icc (0 : ℝ) (200 * m), - t = columnEntry 0 m i + u ∧ - arraySpine m j t = straightPiece m i u ∧ arrayTangent m j t = straightTangent i) ∨ - (∃ i < j, ∃ u ∈ Icc (0 : ℝ) (50 * Real.pi), - t = columnEntry 0 m i + 200 * m + u ∧ - arraySpine m j t = circularPiece m i u ∧ arrayTangent m j t = circularTangent i u) := by - induction j with - | zero => - exact Or.inl ⟨0, le_rfl, t, ⟨ht.1, by simpa only [arrayEnd, columnEntry, Nat.cast_zero, zero_mul, add_zero, zero_add] using ht.2⟩, - by simp [columnEntry], rfl, rfl⟩ - | succ j ih => - have ha : arrayEnd m j < columnEntry 0 m (j + 1) := by - rw [arrayEnd, columnEntry_succ] - linarith [Real.pi_pos] - by_cases hta : t ≤ arrayEnd m j - · change t ≤ columnEntry 0 m j + 200 * m at hta - rcases ih ⟨ht.1, hta⟩ with ⟨i, hi, u, hu, htu, hs, hT⟩ | ⟨i, hi, u, hu, htu, hs, hT⟩ - · refine Or.inl ⟨i, hi.trans (Nat.le_succ j), u, hu, htu, ?_, ?_⟩ - · simpa only [arraySpine, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hs - · simpa only [arrayTangent, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hT - · refine Or.inr ⟨i, hi.trans_le (Nat.le_succ j), u, hu, htu, ?_, ?_⟩ - · simpa only [arraySpine, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hs - · simpa only [arrayTangent, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hT - · have hat : columnEntry 0 m j + 200 * m < t := lt_of_not_ge hta - by_cases htb : t ≤ columnEntry 0 m (j + 1) - · refine Or.inr ⟨j, Nat.lt_succ_self j, t - arrayEnd m j, ?_, ?_, ?_, ?_⟩ - · constructor - · linarith - · rw [columnEntry_succ] at htb - dsimp [arrayEnd] - linarith - · dsimp [arrayEnd]; ring - · simp only [arraySpine, curveJoin_of_le htb, curveJoin_of_gt hat, arrayEnd] - · simp only [arrayTangent, curveJoin_of_le htb, curveJoin_of_gt hat, arrayEnd] - · have hbt : columnEntry 0 m (j + 1) < t := lt_of_not_ge htb - refine Or.inl ⟨j + 1, le_rfl, t - columnEntry 0 m (j + 1), ?_, by ring, ?_, ?_⟩ - · constructor - · linarith - · dsimp only [arrayEnd] at ht - linarith [ht.2] - · exact arraySpine_tail m (j + 1) hbt.le - · exact arrayTangent_tail m (j + 1) hbt.le - -lemma normalized_width {m : ℕ} (hm : 0 < m) : ribbonWidth (100 * m) m = 1 := by - unfold ribbonWidth - have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' - field_simp - -lemma normalized_spacing {m : ℕ} (hm : 0 < m) : spacing (100 * m) m = 100 := by - rw [spacing_eq_width, normalized_width hm, mul_one] - -lemma normalized_radius {m : ℕ} (hm : 0 < m) : turnRadius (100 * m) m = 50 := by - rw [turnRadius_eq_width, normalized_width hm, mul_one] - -lemma normal_straight_eq {m : ℕ} (hm : 0 < m) (j : ℕ) (u a : ℝ) : - straightPiece m j u + (a : ℂ) * Complex.I * straightTangent j = - columnPoint 0 (100 * m) m j (side j * (-100 * m + u)) a := by - apply Complex.ext <;> - simp only [straightPiece, straightTangent, columnPoint, columnOffset, - normalized_spacing hm, normalized_width hm, Complex.add_re, Complex.add_im, - Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, Complex.zero_re, Complex.zero_im] <;> ring - -lemma normal_circular_eq {m : ℕ} (hm : 0 < m) (j : ℕ) (u a : ℝ) : - circularPiece m j u + (a : ℂ) * Complex.I * circularTangent j u = - turnPoint 0 (100 * m) m j (u / 50) a := by - apply Complex.ext - · simp only [circularPiece, circularTangent, turnPoint, columnOffset, - normalized_spacing hm, normalized_radius hm, normalized_width hm, - Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - Complex.zero_re] - ring - · simp only [circularPiece, circularTangent, turnPoint, columnOffset, - normalized_spacing hm, normalized_radius hm, normalized_width hm, - Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - Complex.zero_im] - ring_nf - rw [side_sq] - ring - -lemma straight_height_bound (m j : ℕ) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) (200 * m)) : - |side j * (-100 * m + u)| ≤ 100 * m := by - rw [abs_mul, side_abs, one_mul] - exact abs_le.mpr ⟨by linarith [hu.1], by linarith [hu.2]⟩ - -lemma circular_angle_bound {u : ℝ} (hu : u ∈ Icc (0 : ℝ) (50 * Real.pi)) : - u / 50 ∈ Icc (0 : ℝ) Real.pi := by - constructor <;> linarith [hu.1, hu.2] - -lemma turn_column_time {m : ℕ} (hm : 0 < m) (j k : ℕ) {θ y a b : ℝ} - (hθ : θ ∈ Icc (0 : ℝ) Real.pi) (hy : |y| ≤ 100 * m) - (ha : |a| ≤ 1) (hb : |b| ≤ 1) - (he : turnPoint 0 (100 * m) m j θ a = columnPoint 0 (100 * m) m k y b) : - columnEntry 0 m j + 200 * m + 50 * θ = - columnEntry 0 m k + (side k * y + 100 * m) ∧ a = b := by - have hh : (0 : ℝ) < 100 * m := by positivity - rcases turnPoint_column_gluing hh hm 0 hθ hy ha hb he with - ⟨hθ, hk, hy, hab⟩ | ⟨hθ, hk, hy, hab⟩ - · subst θ - subst k - subst y - refine ⟨?_, hab⟩ - have hs : side j * (side j * (100 * (m : ℝ))) = 100 * m := by - rw [← mul_assoc, ← pow_two, side_sq, one_mul] - rw [hs] - ring - · subst θ - subst k - subst y - refine ⟨?_, hab⟩ - have hs : side (j + 1) * (side j * (100 * (m : ℝ))) = -(100 * m) := by - rw [side_succ, neg_mul, ← mul_assoc, ← pow_two, side_sq, one_mul] - rw [hs, columnEntry_succ] - ring - -/-- Normal coordinates over the complete finite array are injective, -including the endpoints shared by adjacent straight and circular pieces. -/ -theorem array_normal_injective {m : ℕ} (hm : 0 < m) (last : ℕ) : - InjOn (normalRibbon (arraySpine m last) (arrayTangent m last)) - {z : ℂ | z.re ∈ Icc (0 : ℝ) (arrayEnd m last) ∧ |z.im| ≤ 1} := by - intro z hz w hw he - have hh : (0 : ℝ) < 100 * m := by positivity - rcases array_pieces m last hz.1 with ⟨i, _, u, hu, hzu, hzs, hzT⟩ | - ⟨i, _, u, hu, hzu, hzs, hzT⟩ <;> - rcases array_pieces m last hw.1 with ⟨j, _, v, hv, hwv, hws, hwT⟩ | - ⟨j, _, v, hv, hwv, hws, hwT⟩ - · simp only [normalRibbon] at he - rw [hzs, hzT, hws, hwT, - normal_straight_eq hm, normal_straight_eq hm] at he - have hij := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 hw.2 he - subst j - obtain ⟨hy, hab⟩ := columnPoint_injective hh hm 0 i he - have hs : side i ≠ 0 := by - intro h - have hh := side_abs i - rw [h, abs_zero] at hh - norm_num at hh - have huv : u = v := by - have h := mul_left_cancel₀ hs hy - linarith - exact Complex.ext (by rw [hzu, hwv, huv]) hab - · simp only [normalRibbon] at he - rw [hzs, hzT, hws, hwT, - normal_straight_eq hm, normal_circular_eq hm] at he - obtain ⟨ht, hab⟩ := turn_column_time hm j i (circular_angle_bound hv) - (straight_height_bound m i hu) hw.2 hz.2 he.symm - have hs : side i * (side i * (-100 * (m : ℝ) + u)) = -100 * m + u := by - rw [← mul_assoc, ← pow_two, side_sq, one_mul] - rw [hs] at ht - exact Complex.ext (by rw [hzu, hwv]; linarith) hab.symm - · simp only [normalRibbon] at he - rw [hzs, hzT, hws, hwT, - normal_circular_eq hm, normal_straight_eq hm] at he - obtain ⟨ht, hab⟩ := turn_column_time hm i j (circular_angle_bound hu) - (straight_height_bound m j hv) hz.2 hw.2 he - have hs : side j * (side j * (-100 * (m : ℝ) + v)) = -100 * m + v := by - rw [← mul_assoc, ← pow_two, side_sq, one_mul] - rw [hs] at ht - exact Complex.ext (by rw [hzu, hwv]; linarith) hab - · simp only [normalRibbon] at he - rw [hzs, hzT, hws, hwT, - normal_circular_eq hm, normal_circular_eq hm] at he - have hij : i = j := by - by_contra hij - rcases lt_or_gt_of_ne hij with hij | hji - · exact turnPoint_ne hh hm hij 0 (circular_angle_bound hu) (circular_angle_bound hv) - hz.2 hw.2 he - · exact turnPoint_ne hh hm hji 0 (circular_angle_bound hv) (circular_angle_bound hu) - hw.2 hz.2 he.symm - subst j - obtain ⟨ht, hab⟩ := turnPoint_injective hh hm 0 i (circular_angle_bound hu) - (circular_angle_bound hv) hz.2 hw.2 he - exact Complex.ext (by rw [hzu, hwv]; linarith) hab - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineCurve +import OAI.Probability.SLE.Gauge.SerpentineEmbedding + +namespace OAI + +/-! Global normal coordinates for the actual finite serpentine array. +Every parameter belongs to a straight or circular piece, and the only +possible normal-band contacts are the prescribed common end sections. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +def arrayEnd (m j : ℕ) : ℝ := columnEntry 0 m j + 200 * m + +lemma array_pieces (m j : ℕ) {t : ℝ} (ht : t ∈ Icc 0 (arrayEnd m j)) : + (∃ i ≤ j, ∃ u ∈ Icc (0 : ℝ) (200 * m), + t = columnEntry 0 m i + u ∧ + arraySpine m j t = straightPiece m i u ∧ arrayTangent m j t = straightTangent i) ∨ + (∃ i < j, ∃ u ∈ Icc (0 : ℝ) (50 * Real.pi), + t = columnEntry 0 m i + 200 * m + u ∧ + arraySpine m j t = circularPiece m i u ∧ arrayTangent m j t = circularTangent i u) := by + induction j with + | zero => + exact Or.inl ⟨0, le_rfl, t, ⟨ht.1, by simpa only [arrayEnd, columnEntry, Nat.cast_zero, zero_mul, add_zero, zero_add] using ht.2⟩, + by simp [columnEntry], rfl, rfl⟩ + | succ j ih => + have ha : arrayEnd m j < columnEntry 0 m (j + 1) := by + rw [arrayEnd, columnEntry_succ] + linarith [Real.pi_pos] + by_cases hta : t ≤ arrayEnd m j + · change t ≤ columnEntry 0 m j + 200 * m at hta + rcases ih ⟨ht.1, hta⟩ with ⟨i, hi, u, hu, htu, hs, hT⟩ | ⟨i, hi, u, hu, htu, hs, hT⟩ + · refine Or.inl ⟨i, hi.trans (Nat.le_succ j), u, hu, htu, ?_, ?_⟩ + · simpa only [arraySpine, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hs + · simpa only [arrayTangent, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hT + · refine Or.inr ⟨i, hi.trans_le (Nat.le_succ j), u, hu, htu, ?_, ?_⟩ + · simpa only [arraySpine, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hs + · simpa only [arrayTangent, curveJoin_of_le (hta.trans ha.le), curveJoin_of_le hta] using hT + · have hat : columnEntry 0 m j + 200 * m < t := lt_of_not_ge hta + by_cases htb : t ≤ columnEntry 0 m (j + 1) + · refine Or.inr ⟨j, Nat.lt_succ_self j, t - arrayEnd m j, ?_, ?_, ?_, ?_⟩ + · constructor + · linarith + · rw [columnEntry_succ] at htb + dsimp [arrayEnd] + linarith + · dsimp [arrayEnd]; ring + · simp only [arraySpine, curveJoin_of_le htb, curveJoin_of_gt hat, arrayEnd] + · simp only [arrayTangent, curveJoin_of_le htb, curveJoin_of_gt hat, arrayEnd] + · have hbt : columnEntry 0 m (j + 1) < t := lt_of_not_ge htb + refine Or.inl ⟨j + 1, le_rfl, t - columnEntry 0 m (j + 1), ?_, by ring, ?_, ?_⟩ + · constructor + · linarith + · dsimp only [arrayEnd] at ht + linarith [ht.2] + · exact arraySpine_tail m (j + 1) hbt.le + · exact arrayTangent_tail m (j + 1) hbt.le + +lemma normalized_width {m : ℕ} (hm : 0 < m) : ribbonWidth (100 * m) m = 1 := by + unfold ribbonWidth + have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' + field_simp + +lemma normalized_spacing {m : ℕ} (hm : 0 < m) : spacing (100 * m) m = 100 := by + rw [spacing_eq_width, normalized_width hm, mul_one] + +lemma normalized_radius {m : ℕ} (hm : 0 < m) : turnRadius (100 * m) m = 50 := by + rw [turnRadius_eq_width, normalized_width hm, mul_one] + +lemma normal_straight_eq {m : ℕ} (hm : 0 < m) (j : ℕ) (u a : ℝ) : + straightPiece m j u + (a : ℂ) * Complex.I * straightTangent j = + columnPoint 0 (100 * m) m j (side j * (-100 * m + u)) a := by + apply Complex.ext <;> + simp only [straightPiece, straightTangent, columnPoint, columnOffset, + normalized_spacing hm, normalized_width hm, Complex.add_re, Complex.add_im, + Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, Complex.zero_re, Complex.zero_im] <;> ring + +lemma normal_circular_eq {m : ℕ} (hm : 0 < m) (j : ℕ) (u a : ℝ) : + circularPiece m j u + (a : ℂ) * Complex.I * circularTangent j u = + turnPoint 0 (100 * m) m j (u / 50) a := by + apply Complex.ext + · simp only [circularPiece, circularTangent, turnPoint, columnOffset, + normalized_spacing hm, normalized_radius hm, normalized_width hm, + Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + Complex.zero_re] + ring + · simp only [circularPiece, circularTangent, turnPoint, columnOffset, + normalized_spacing hm, normalized_radius hm, normalized_width hm, + Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + Complex.zero_im] + ring_nf + rw [side_sq] + ring + +lemma straight_height_bound (m j : ℕ) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) (200 * m)) : + |side j * (-100 * m + u)| ≤ 100 * m := by + rw [abs_mul, side_abs, one_mul] + exact abs_le.mpr ⟨by linarith [hu.1], by linarith [hu.2]⟩ + +lemma circular_angle_bound {u : ℝ} (hu : u ∈ Icc (0 : ℝ) (50 * Real.pi)) : + u / 50 ∈ Icc (0 : ℝ) Real.pi := by + constructor <;> linarith [hu.1, hu.2] + +lemma turn_column_time {m : ℕ} (hm : 0 < m) (j k : ℕ) {θ y a b : ℝ} + (hθ : θ ∈ Icc (0 : ℝ) Real.pi) (hy : |y| ≤ 100 * m) + (ha : |a| ≤ 1) (hb : |b| ≤ 1) + (he : turnPoint 0 (100 * m) m j θ a = columnPoint 0 (100 * m) m k y b) : + columnEntry 0 m j + 200 * m + 50 * θ = + columnEntry 0 m k + (side k * y + 100 * m) ∧ a = b := by + have hh : (0 : ℝ) < 100 * m := by positivity + rcases turnPoint_column_gluing hh hm 0 hθ hy ha hb he with + ⟨hθ, hk, hy, hab⟩ | ⟨hθ, hk, hy, hab⟩ + · subst θ + subst k + subst y + refine ⟨?_, hab⟩ + have hs : side j * (side j * (100 * (m : ℝ))) = 100 * m := by + rw [← mul_assoc, ← pow_two, side_sq, one_mul] + rw [hs] + ring + · subst θ + subst k + subst y + refine ⟨?_, hab⟩ + have hs : side (j + 1) * (side j * (100 * (m : ℝ))) = -(100 * m) := by + rw [side_succ, neg_mul, ← mul_assoc, ← pow_two, side_sq, one_mul] + rw [hs, columnEntry_succ] + ring + +/-- Normal coordinates over the complete finite array are injective, +including the endpoints shared by adjacent straight and circular pieces. -/ +theorem array_normal_injective {m : ℕ} (hm : 0 < m) (last : ℕ) : + InjOn (normalRibbon (arraySpine m last) (arrayTangent m last)) + {z : ℂ | z.re ∈ Icc (0 : ℝ) (arrayEnd m last) ∧ |z.im| ≤ 1} := by + intro z hz w hw he + have hh : (0 : ℝ) < 100 * m := by positivity + rcases array_pieces m last hz.1 with ⟨i, _, u, hu, hzu, hzs, hzT⟩ | + ⟨i, _, u, hu, hzu, hzs, hzT⟩ <;> + rcases array_pieces m last hw.1 with ⟨j, _, v, hv, hwv, hws, hwT⟩ | + ⟨j, _, v, hv, hwv, hws, hwT⟩ + · simp only [normalRibbon] at he + rw [hzs, hzT, hws, hwT, + normal_straight_eq hm, normal_straight_eq hm] at he + have hij := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 hw.2 he + subst j + obtain ⟨hy, hab⟩ := columnPoint_injective hh hm 0 i he + have hs : side i ≠ 0 := by + intro h + have hh := side_abs i + rw [h, abs_zero] at hh + norm_num at hh + have huv : u = v := by + have h := mul_left_cancel₀ hs hy + linarith + exact Complex.ext (by rw [hzu, hwv, huv]) hab + · simp only [normalRibbon] at he + rw [hzs, hzT, hws, hwT, + normal_straight_eq hm, normal_circular_eq hm] at he + obtain ⟨ht, hab⟩ := turn_column_time hm j i (circular_angle_bound hv) + (straight_height_bound m i hu) hw.2 hz.2 he.symm + have hs : side i * (side i * (-100 * (m : ℝ) + u)) = -100 * m + u := by + rw [← mul_assoc, ← pow_two, side_sq, one_mul] + rw [hs] at ht + exact Complex.ext (by rw [hzu, hwv]; linarith) hab.symm + · simp only [normalRibbon] at he + rw [hzs, hzT, hws, hwT, + normal_circular_eq hm, normal_straight_eq hm] at he + obtain ⟨ht, hab⟩ := turn_column_time hm i j (circular_angle_bound hu) + (straight_height_bound m j hv) hz.2 hw.2 he + have hs : side j * (side j * (-100 * (m : ℝ) + v)) = -100 * m + v := by + rw [← mul_assoc, ← pow_two, side_sq, one_mul] + rw [hs] at ht + exact Complex.ext (by rw [hzu, hwv]; linarith) hab + · simp only [normalRibbon] at he + rw [hzs, hzT, hws, hwT, + normal_circular_eq hm, normal_circular_eq hm] at he + have hij : i = j := by + by_contra hij + rcases lt_or_gt_of_ne hij with hij | hji + · exact turnPoint_ne hh hm hij 0 (circular_angle_bound hu) (circular_angle_bound hv) + hz.2 hw.2 he + · exact turnPoint_ne hh hm hji 0 (circular_angle_bound hv) (circular_angle_bound hu) + hw.2 hz.2 he.symm + subst j + obtain ⟨ht, hab⟩ := turnPoint_injective hh hm 0 i (circular_angle_bound hu) + (circular_angle_bound hv) hz.2 hw.2 he + exact Complex.ext (by rw [hzu, hwv]; linarith) hab + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineArrayRegion.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineArrayRegion.lean index e5ddf47a040806fd9fe972e3a6eeeb81a47bf1b7..259868b474f4972bb056ba7bbd8b325bbaa0092f 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineArrayRegion.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineArrayRegion.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Gauge.SerpentineAttachments - -namespace OAI - -/-! Horizontal confinement of the whole infinite array. This keeps -every part of the completed experiment away from the marked-point shadow. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma array_normal_re_lower {m : ℕ} (hm : 0 < m) {z : ℂ} - (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : - -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).re := by - have hh : (0 : ℝ) < 100 * m := by positivity - by_cases he : z.re ≤ arrayEnd m (2 * m) - · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] - rw [hs, hT, normal_straight_eq hm] - have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 - rw [dist_zero_right] at hb - have hi := Complex.abs_re_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) - have hl := neg_abs_le (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im).re - linarith - · simp only [normalRibbon] - rw [hs, hT, normal_circular_eq hm] - have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 - rw [dist_zero_right] at hb - have hi := Complex.abs_re_le_norm (turnPoint 0 (100 * m) m i (u / 50) z.im) - have hl := neg_abs_le (turnPoint 0 (100 * m) m i (u / 50) z.im).re - linarith - · have hE : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge he - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have ht := array_normal_terminal hm (a := z.im) hE - rw [Complex.re_add_im z] at ht - rw [ht, columnPoint_re] - simp only [columnOffset, normalized_spacing hm, normalized_width hm, side_even, - one_mul, Complex.zero_re, zero_add, Nat.cast_mul, Nat.cast_ofNat] - have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm - linarith [(abs_le.mp hz.2).2] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineAttachments + +namespace OAI + +/-! Horizontal confinement of the whole infinite array. This keeps +every part of the completed experiment away from the marked-point shadow. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma array_normal_re_lower {m : ℕ} (hm : 0 < m) {z : ℂ} + (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : + -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).re := by + have hh : (0 : ℝ) < 100 * m := by positivity + by_cases he : z.re ≤ arrayEnd m (2 * m) + · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] + rw [hs, hT, normal_straight_eq hm] + have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 + rw [dist_zero_right] at hb + have hi := Complex.abs_re_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) + have hl := neg_abs_le (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im).re + linarith + · simp only [normalRibbon] + rw [hs, hT, normal_circular_eq hm] + have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 + rw [dist_zero_right] at hb + have hi := Complex.abs_re_le_norm (turnPoint 0 (100 * m) m i (u / 50) z.im) + have hl := neg_abs_le (turnPoint 0 (100 * m) m i (u / 50) z.im).re + linarith + · have hE : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge he + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have ht := array_normal_terminal hm (a := z.im) hE + rw [Complex.re_add_im z] at ht + rw [ht, columnPoint_re] + simp only [columnOffset, normalized_spacing hm, normalized_width hm, side_even, + one_mul, Complex.zero_re, zero_add, Nat.cast_mul, Nat.cast_ofNat] + have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm + linarith [(abs_le.mp hz.2).2] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineAttachments.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineAttachments.lean index a78615ce9a46ea85a9ac3a78ba276e4e2eca35a1..9762683328323a3059c73732460aef627eb0aa7b 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineAttachments.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineAttachments.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Gauge.SerpentineInfiniteArray - -namespace OAI - -/-! Clearance below the array and the unique attachment to its first -column. These facts separate the low connector from the array. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma array_normal_im_lower {m : ℕ} (hm : 0 < m) {z : ℂ} - (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : - -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).im := by - have hh : (0 : ℝ) < 100 * m := by positivity - by_cases he : z.re ≤ arrayEnd m (2 * m) - · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] - rw [hs, hT, normal_straight_eq hm] - have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 - rw [dist_zero_right] at hb - have hi := Complex.abs_im_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) - have hl := neg_abs_le (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im).im - linarith - · simp only [normalRibbon] - rw [hs, hT, normal_circular_eq hm] - have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 - rw [dist_zero_right] at hb - have hi := Complex.abs_im_le_norm (turnPoint 0 (100 * m) m i (u / 50) z.im) - have hl := neg_abs_le (turnPoint 0 (100 * m) m i (u / 50) z.im).im - linarith - · have hE : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge he - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have ht := array_normal_terminal hm (a := z.im) hE - rw [Complex.re_add_im z] at ht - rw [ht, columnPoint_im] - have h := lt_of_not_ge he - dsimp only [arrayEnd] at h - simp only [Complex.zero_im, zero_add] - linarith - -/-- The downward extension of the first column meets the array only at -its initial cross-section, with identical normal coordinate. -/ -theorem array_downward_attachment {m : ℕ} (hm : 0 < m) {z : ℂ} - (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) {y a : ℝ} (hy : y ≤ -(100 * (m : ℝ))) - (ha : |a| ≤ 1) - (he : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z = - columnPoint 0 (100 * m) m 0 y a) : - z.re = 0 ∧ y = -(100 * (m : ℝ)) ∧ z.im = a := by - have hh : (0 : ℝ) < 100 * m := by positivity - have hs0 : side 0 = 1 := by simp [side] - by_cases hE : z.re ≤ arrayEnd m (2 * m) - · rcases array_pieces m (2 * m) ⟨hz.1, hE⟩ with ⟨i, _, u, hu, hzu, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] at he - rw [hs, hT, normal_straight_eq hm] at he - have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 ha he - subst i - obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm 0 0 he - rw [hs0, one_mul] at hy' - have hu0 : u = 0 := by linarith [hu.1] - subst u - exact ⟨by simpa [columnEntry] using hzu, by linarith, ha'⟩ - · simp only [normalRibbon] at he - rw [hs, hT, normal_circular_eq hm] at he - by_cases hi0 : i = 0 - · subst i - have him := turn_oriented_height hh hm 0 0 (circular_angle_bound hu) hz.2 - rw [hs0, one_mul, he, columnPoint_im] at him - simp only [Complex.zero_im, zero_add] at him - linarith - · exact (turnPoint_column_ne_nonadjacent hh hm (Or.inl (Nat.pos_of_ne_zero hi0)) - 0 (u / 50) y hz.2 ha he).elim - · have hze : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge hE - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have ht := array_normal_terminal hm (a := z.im) hze - rw [Complex.re_add_im z] at ht - rw [ht] at he - have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 ha he - omega - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineInfiniteArray + +namespace OAI + +/-! Clearance below the array and the unique attachment to its first +column. These facts separate the low connector from the array. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma array_normal_im_lower {m : ℕ} (hm : 0 < m) {z : ℂ} + (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : + -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).im := by + have hh : (0 : ℝ) < 100 * m := by positivity + by_cases he : z.re ≤ arrayEnd m (2 * m) + · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] + rw [hs, hT, normal_straight_eq hm] + have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 + rw [dist_zero_right] at hb + have hi := Complex.abs_im_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) + have hl := neg_abs_le (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im).im + linarith + · simp only [normalRibbon] + rw [hs, hT, normal_circular_eq hm] + have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 + rw [dist_zero_right] at hb + have hi := Complex.abs_im_le_norm (turnPoint 0 (100 * m) m i (u / 50) z.im) + have hl := neg_abs_le (turnPoint 0 (100 * m) m i (u / 50) z.im).im + linarith + · have hE : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge he + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have ht := array_normal_terminal hm (a := z.im) hE + rw [Complex.re_add_im z] at ht + rw [ht, columnPoint_im] + have h := lt_of_not_ge he + dsimp only [arrayEnd] at h + simp only [Complex.zero_im, zero_add] + linarith + +/-- The downward extension of the first column meets the array only at +its initial cross-section, with identical normal coordinate. -/ +theorem array_downward_attachment {m : ℕ} (hm : 0 < m) {z : ℂ} + (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) {y a : ℝ} (hy : y ≤ -(100 * (m : ℝ))) + (ha : |a| ≤ 1) + (he : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z = + columnPoint 0 (100 * m) m 0 y a) : + z.re = 0 ∧ y = -(100 * (m : ℝ)) ∧ z.im = a := by + have hh : (0 : ℝ) < 100 * m := by positivity + have hs0 : side 0 = 1 := by simp [side] + by_cases hE : z.re ≤ arrayEnd m (2 * m) + · rcases array_pieces m (2 * m) ⟨hz.1, hE⟩ with ⟨i, _, u, hu, hzu, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] at he + rw [hs, hT, normal_straight_eq hm] at he + have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 ha he + subst i + obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm 0 0 he + rw [hs0, one_mul] at hy' + have hu0 : u = 0 := by linarith [hu.1] + subst u + exact ⟨by simpa [columnEntry] using hzu, by linarith, ha'⟩ + · simp only [normalRibbon] at he + rw [hs, hT, normal_circular_eq hm] at he + by_cases hi0 : i = 0 + · subst i + have him := turn_oriented_height hh hm 0 0 (circular_angle_bound hu) hz.2 + rw [hs0, one_mul, he, columnPoint_im] at him + simp only [Complex.zero_im, zero_add] at him + linarith + · exact (turnPoint_column_ne_nonadjacent hh hm (Or.inl (Nat.pos_of_ne_zero hi0)) + 0 (u / 50) y hz.2 ha he).elim + · have hze : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge hE + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have ht := array_normal_terminal hm (a := z.im) hze + rw [Complex.re_add_im z] at ht + rw [ht] at he + have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 ha he + omega + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineCenteredMass.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineCenteredMass.lean index 745f4ad5e943abe6e996029fa0ca142e6000d96a..74cc19dbdfcbf92dd8d5db778814205d0d23a39a 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineCenteredMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineCenteredMass.lean @@ -1,132 +1,132 @@ -import OAI.Probability.SLE.Gauge.SerpentineActualRadius -import OAI.Probability.SLE.Gauge.SerpentineActualAngleReflection -import OAI.Probability.SLE.Gauge.ExtendedExperimentStop - -namespace OAI - -/-! The stopped physical experiment under the actual finite centered law. -Both its mass and its successful completion are observable at the front. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates Target FirstFrontPassage - -lemma experimentSteps_add_one {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : experimentSteps s w m + 1 = fitCompletion s w m := by - have hbound := (fitCompletion_bounds hs w hw hm).1 - have hEntry : 0 ≤ fitEntry s w m := - (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hw hm) - have hcol : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ - have hmR : (0 : ℝ) < m := by exact_mod_cast hm - have hk : (0 : ℝ) < fitCompletion s w m := by linarith - have hkN : 0 < fitCompletion s w m := by exact_mod_cast hk - change fitCompletion s w m - 1 + 1 = fitCompletion s w m - omega - -lemma experiment_front_eq {Ω : Type*} {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) : - ribbonIntegerFront g γ (experimentSteps s w m) = - ribbonFrontStop g γ (fitCompletion s w m) := by - have hh : (experimentSteps s w m : ℝ) + 1 = fitCompletion s w m := by - exact_mod_cast experimentSteps_add_one hs w hw hm - simp only [ribbonIntegerFront, hh] - -lemma exponential_prefactor_quadratic_cost {p C : ℝ} (hp : 0 < p) (hC : 0 < C) : - ∃ D : ℝ, 0 < D ∧ ∀ m : ℕ, 0 < m → - Real.exp (-D * (m : ℝ) ^ 2) ≤ p * Real.exp (-C * (m : ℝ) ^ 2) := by - refine ⟨C + |Real.log p| + 1, by positivity, fun m hm => ?_⟩ - have hm1 : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hlog : -|Real.log p| ≤ Real.log p := neg_abs_le _ - have hmul := mul_nonneg (abs_nonneg (Real.log p)) - (show 0 ≤ (m : ℝ) ^ 2 - 1 by nlinarith) - calc - _ ≤ Real.exp (Real.log p + -C * (m : ℝ) ^ 2) := by - apply Real.exp_le_exp.mpr - nlinarith [abs_nonneg (Real.log p)] - _ = _ := by rw [Real.exp_add, Real.exp_log hp] - -theorem uniform_serpentine_centered_stopped_mass - {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport μ γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → s ≤ 1 → - ∃ C : ℝ, 0 < C ∧ ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → - ∀ v : ℝ, 0 < v → v < ribbonWidth (halfHeight s) m * v₀ → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → - ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ - finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel s) - (RibbonStepSuccess γ (fun _ => 0) (ribbonFrontStop g γ (fitCompletion s w m)) - (physicalExperimentChart s w m) ε κ v - (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) - (closedBall (center s w) (s / 20))) := by - have hm (t : ℝ≥0) : Measurable (B t) := - hBm.comp (measurable_const.prodMk measurable_id) - obtain ⟨c, v₀, hcm, hv₀, hprob⟩ := uniform_serpentine_physical_stopped_mass μ κ hκ hκ8 - B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall - refine ⟨c, v₀, hcm, hv₀, fun s hs hs1 => ?_⟩ - obtain ⟨C, hC, hprob⟩ := hprob s hs - obtain ⟨s₀, hs₀, hangle⟩ := uniform_serpentine_actual_angle hplanar hstraight hexit hwiener W hW hs hs1 - obtain ⟨D, hD, hcost⟩ := exponential_prefactor_quadratic_cost - (Real.rpow_pos_of_pos hs₀ (greenExponent κ)) hC - refine ⟨D, hD, fun w hwn hwy m hmp v hv hvcut g hg => ?_⟩ - let τ := ribbonFrontStop g γ (fitCompletion s w m) - let E := RibbonStepSuccess γ (fun _ => 0) τ (physicalExperimentChart s w m) ε κ v - (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) - (closedBall (center s w) (s / 20)) - let ℱ := Filtration.natural B - (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable) - have ha : Adapted ℱ (fun t ω => γ ω t) := - brownian_capacity_trace_adapted B hm hc κ γ G F hw - have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 - have hτ : IsStoppingTime ℱ τ := ribbonFrontStop_stopping ℱ g γ ha hγc _ - have hE : MeasurableSet[hτ.measurableSpace] E := - ribbonStepSuccess_measurable ℱ γ ha hγc (fun _ => 0) τ - (isStoppingTime_const ℱ 0) hτ (fun _ => bot_le) _ - (physicalExperimentChart_closedEmbedding hs w hwy hmp) ε κ v _ _ - have hordinary : ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ E := by - have hh := hprob w hwn hwy m hmp v hv hvcut g hg - rw [experiment_front_eq hs w hwy hmp g γ] at hh - simpa only [E, τ, RibbonStepSuccess, Set.ofPred_and, Set.ofPred_mem_eq, - WithTop.untopD_zero] using hh - have hcentered := finiteCenteredLaw_completed_experiment_lower_bound hB hBm hc hB0 κ hκ hκ8 - γ G F (Eventually.of_forall hw) w (show w ∈ upperHalfPlane from lt_of_lt_of_le hs hwy) - (experimentRadiusLevel s) (experimentRadiusLevel_nonneg hs hs1) τ hτ E hE s₀ hs₀.le (by - apply Eventually.of_forall - intro ω hω - have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ ω) (fun _ => τ ω) - (physicalExperimentChart s w m) ε := hω.1 - have hbefore := good_physicalExperiment_before_radius ((hc ω).const_mul _) (hw ω) - hs w hwn hwy hmp (τ ω) ε hpref - refine ⟨hpref.1, hbefore.1, hbefore.2, ?_⟩ - rw [Complex.sin_arg] - simpa only [τ, ribbonFrontStop, Complex.ofReal_mul] using - hangle _ _ _ _ (hw ω) ((hc ω).const_mul _) w hwn hwy m hmp g hg ε hεsmall hpref) - calc - _ ≤ ENNReal.ofReal (s₀ ^ greenExponent κ) * - ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) := by - rw [← ENNReal.ofReal_mul (Real.rpow_nonneg hs₀.le _)] - exact ENNReal.ofReal_le_ofReal (hcost m hmp) - _ ≤ ENNReal.ofReal (s₀ ^ greenExponent κ) * μ E := mul_le_mul' le_rfl hordinary - _ ≤ _ := hcentered - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineActualRadius +import OAI.Probability.SLE.Gauge.SerpentineActualAngleReflection +import OAI.Probability.SLE.Gauge.ExtendedExperimentStop + +namespace OAI + +/-! The stopped physical experiment under the actual finite centered law. +Both its mass and its successful completion are observable at the front. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates Target FirstFrontPassage + +lemma experimentSteps_add_one {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : experimentSteps s w m + 1 = fitCompletion s w m := by + have hbound := (fitCompletion_bounds hs w hw hm).1 + have hEntry : 0 ≤ fitEntry s w m := + (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hw hm) + have hcol : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ + have hmR : (0 : ℝ) < m := by exact_mod_cast hm + have hk : (0 : ℝ) < fitCompletion s w m := by linarith + have hkN : 0 < fitCompletion s w m := by exact_mod_cast hk + change fitCompletion s w m - 1 + 1 = fitCompletion s w m + omega + +lemma experiment_front_eq {Ω : Type*} {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ) : + ribbonIntegerFront g γ (experimentSteps s w m) = + ribbonFrontStop g γ (fitCompletion s w m) := by + have hh : (experimentSteps s w m : ℝ) + 1 = fitCompletion s w m := by + exact_mod_cast experimentSteps_add_one hs w hw hm + simp only [ribbonIntegerFront, hh] + +lemma exponential_prefactor_quadratic_cost {p C : ℝ} (hp : 0 < p) (hC : 0 < C) : + ∃ D : ℝ, 0 < D ∧ ∀ m : ℕ, 0 < m → + Real.exp (-D * (m : ℝ) ^ 2) ≤ p * Real.exp (-C * (m : ℝ) ^ 2) := by + refine ⟨C + |Real.log p| + 1, by positivity, fun m hm => ?_⟩ + have hm1 : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hlog : -|Real.log p| ≤ Real.log p := neg_abs_le _ + have hmul := mul_nonneg (abs_nonneg (Real.log p)) + (show 0 ≤ (m : ℝ) ^ 2 - 1 by nlinarith) + calc + _ ≤ Real.exp (Real.log p + -C * (m : ℝ) ^ 2) := by + apply Real.exp_le_exp.mpr + nlinarith [abs_nonneg (Real.log p)] + _ = _ := by rw [Real.exp_add, Real.exp_log hp] + +theorem uniform_serpentine_centered_stopped_mass + {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport μ γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → s ≤ 1 → + ∃ C : ℝ, 0 < C ∧ ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → + ∀ v : ℝ, 0 < v → v < ribbonWidth (halfHeight s) m * v₀ → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → + ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ + finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel s) + (RibbonStepSuccess γ (fun _ => 0) (ribbonFrontStop g γ (fitCompletion s w m)) + (physicalExperimentChart s w m) ε κ v + (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) + (closedBall (center s w) (s / 20))) := by + have hm (t : ℝ≥0) : Measurable (B t) := + hBm.comp (measurable_const.prodMk measurable_id) + obtain ⟨c, v₀, hcm, hv₀, hprob⟩ := uniform_serpentine_physical_stopped_mass μ κ hκ hκ8 + B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall + refine ⟨c, v₀, hcm, hv₀, fun s hs hs1 => ?_⟩ + obtain ⟨C, hC, hprob⟩ := hprob s hs + obtain ⟨s₀, hs₀, hangle⟩ := uniform_serpentine_actual_angle hplanar hstraight hexit hwiener W hW hs hs1 + obtain ⟨D, hD, hcost⟩ := exponential_prefactor_quadratic_cost + (Real.rpow_pos_of_pos hs₀ (greenExponent κ)) hC + refine ⟨D, hD, fun w hwn hwy m hmp v hv hvcut g hg => ?_⟩ + let τ := ribbonFrontStop g γ (fitCompletion s w m) + let E := RibbonStepSuccess γ (fun _ => 0) τ (physicalExperimentChart s w m) ε κ v + (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) + (closedBall (center s w) (s / 20)) + let ℱ := Filtration.natural B + (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable) + have ha : Adapted ℱ (fun t ω => γ ω t) := + brownian_capacity_trace_adapted B hm hc κ γ G F hw + have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 + have hτ : IsStoppingTime ℱ τ := ribbonFrontStop_stopping ℱ g γ ha hγc _ + have hE : MeasurableSet[hτ.measurableSpace] E := + ribbonStepSuccess_measurable ℱ γ ha hγc (fun _ => 0) τ + (isStoppingTime_const ℱ 0) hτ (fun _ => bot_le) _ + (physicalExperimentChart_closedEmbedding hs w hwy hmp) ε κ v _ _ + have hordinary : ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ E := by + have hh := hprob w hwn hwy m hmp v hv hvcut g hg + rw [experiment_front_eq hs w hwy hmp g γ] at hh + simpa only [E, τ, RibbonStepSuccess, Set.ofPred_and, Set.ofPred_mem_eq, + WithTop.untopD_zero] using hh + have hcentered := finiteCenteredLaw_completed_experiment_lower_bound hB hBm hc hB0 κ hκ hκ8 + γ G F (Eventually.of_forall hw) w (show w ∈ upperHalfPlane from lt_of_lt_of_le hs hwy) + (experimentRadiusLevel s) (experimentRadiusLevel_nonneg hs hs1) τ hτ E hE s₀ hs₀.le (by + apply Eventually.of_forall + intro ω hω + have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ ω) (fun _ => τ ω) + (physicalExperimentChart s w m) ε := hω.1 + have hbefore := good_physicalExperiment_before_radius ((hc ω).const_mul _) (hw ω) + hs w hwn hwy hmp (τ ω) ε hpref + refine ⟨hpref.1, hbefore.1, hbefore.2, ?_⟩ + rw [Complex.sin_arg] + simpa only [τ, ribbonFrontStop, Complex.ofReal_mul] using + hangle _ _ _ _ (hw ω) ((hc ω).const_mul _) w hwn hwy m hmp g hg ε hεsmall hpref) + calc + _ ≤ ENNReal.ofReal (s₀ ^ greenExponent κ) * + ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) := by + rw [← ENNReal.ofReal_mul (Real.rpow_nonneg hs₀.le _)] + exact ENNReal.ofReal_le_ofReal (hcost m hmp) + _ ≤ ENNReal.ofReal (s₀ ^ greenExponent κ) * μ E := mul_le_mul' le_rfl hordinary + _ ≤ _ := hcentered + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteCurve.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteCurve.lean index eebee0d9f3698e9df724ae5b0fcbfce025d04e3b..ca7fc40c4cc3aa6628002fdeba9241e719bb53de 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteCurve.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteCurve.lean @@ -1,112 +1,112 @@ -import OAI.Probability.SLE.Gauge.SerpentineInfiniteArray -import OAI.Probability.SLE.Geometry.RoundedConnectorGeometry - -namespace OAI - -/-! Arclength gluing of the low rounded connector and the upward-continued -serpentine array. -/ - -noncomputable section -open Set Metric -open scoped NNReal - -namespace SLEExactGauge.Serpentine - -lemma arraySpine_zero (m j : ℕ) : - arraySpine m j 0 = ((-100 * (m : ℝ) : ℝ) : ℂ) + - ((-100 * (m : ℝ) : ℝ) : ℂ) * Complex.I := by - induction j with - | zero => simp [arraySpine, straightPiece, side] - | succ j ih => - have hb : (0 : ℝ) ≤ columnEntry 0 m (j + 1) := by unfold columnEntry; positivity - have ha : (0 : ℝ) ≤ columnEntry 0 m j + 200 * m := by unfold columnEntry; positivity - simpa only [arraySpine, curveJoin_of_le hb, curveJoin_of_le ha] using ih - -lemma arrayTangent_zero (m j : ℕ) : arrayTangent m j 0 = Complex.I := by - induction j with - | zero => simp [arrayTangent, straightTangent, side] - | succ j ih => - have hb : (0 : ℝ) ≤ columnEntry 0 m (j + 1) := by unfold columnEntry; positivity - have ha : (0 : ℝ) ≤ columnEntry 0 m j + 200 * m := by unfold columnEntry; positivity - simpa only [arrayTangent, curveJoin_of_le hb, curveJoin_of_le ha] using ih - -lemma lowConnector_after {H r X t : ℝ} - (ht : connectorFourth H r X < t) : - lowConnector H r X t = connectorTail H r X t ∧ - lowConnectorTangent H r X t = Complex.I := by - simp only [lowConnector, lowConnectorTangent, curveJoin_of_gt ht, and_self] - -def connectorEntry (H r X Y : ℝ) : ℝ := connectorFourth H r X + Y - H - 2 * r - -def arrayCenter (m : ℕ) (X Y : ℝ) : ℂ := - ((X + 100 * m : ℝ) : ℂ) + ((Y + 100 * m : ℝ) : ℂ) * Complex.I - -def completeSpine (m : ℕ) (H r X Y : ℝ) : ℝ → ℂ := - curveJoin (connectorEntry H r X Y) (lowConnector H r X) - (fun t => arrayCenter m X Y + arraySpine m (2 * m) (t - connectorEntry H r X Y)) - -def completeTangent (m : ℕ) (H r X Y : ℝ) : ℝ → ℂ := - curveJoin (connectorEntry H r X Y) (lowConnectorTangent H r X) - (fun t => arrayTangent m (2 * m) (t - connectorEntry H r X Y)) - -lemma completeSpine_hasDerivAt {H r X Y : ℝ} (m : ℕ) (hr : 0 < r) - (hX : 2 * r < X) (hY : H + 2 * r < Y) (t : ℝ) : - HasDerivAt (completeSpine m H r X Y) (completeTangent m H r X Y t) t := by - have hentry : connectorFourth H r X < connectorEntry H r X Y := by - unfold connectorEntry - linarith - have hp : lowConnector H r X (connectorEntry H r X Y) = - arrayCenter m X Y + arraySpine m (2 * m) - (connectorEntry H r X Y - connectorEntry H r X Y) := by - rw [(lowConnector_after hentry).1, sub_self, arraySpine_zero] - simp only [connectorTail, connectorEntry, arrayCenter] - push_cast - ring - have hT : lowConnectorTangent H r X (connectorEntry H r X Y) = - arrayTangent m (2 * m) (connectorEntry H r X Y - connectorEntry H r X Y) := by - rw [(lowConnector_after hentry).2, sub_self, arrayTangent_zero] - exact curveJoin_hasDerivAt _ _ _ _ _ (lowConnector_hasDerivAt hr hX) - (translate_curve_hasDerivAt _ _ (arraySpine_hasDerivAt m (2 * m)) _ _) hp hT t - -lemma completeTangent_norm (m : ℕ) (H r X Y t : ℝ) : ‖completeTangent m H r X Y t‖ = 1 := by - unfold completeTangent curveJoin - split - · exact lowConnectorTangent_norm _ _ _ _ - · exact arrayTangent_norm _ _ _ - -lemma completeTangent_lipschitz {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) - (hX : 2 * r < X) (hY : H + 2 * r < Y) : - LipschitzWith (1 / 50) (completeTangent m H r X Y) := by - have hr0 : 0 < r := by linarith - have hentry : connectorFourth H r X < connectorEntry H r X Y := by - unfold connectorEntry - linarith - apply curveJoin_lipschitz (lowConnectorTangent_lipschitz hr hX) - (shifted_lipschitz (arrayTangent_lipschitz m (2 * m)) _) - rw [(lowConnector_after hentry).2, sub_self, arrayTangent_zero] - -lemma complete_normal_patch_bounds {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) - (hX : 2 * r < X) (hY : H + 2 * r < Y) (k : ℝ) : - LipschitzOnWith 2 (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) - (RibbonCoordinates.ribbonStepPatch k) ∧ - ∀ z ∈ RibbonCoordinates.ribbonStepPatch k, - ∀ w ∈ RibbonCoordinates.ribbonStepPatch k, - dist z w ≤ 2 * dist (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z) - (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) w) := by - have hr0 : 0 < r := by linarith - have hbound z (hz : z ∈ RibbonCoordinates.ribbonStepPatch k) - w (hw : w ∈ RibbonCoordinates.ribbonStepPatch k) := - normalRibbon_patch_bounds (completeSpine m H r X Y) (completeTangent m H r X Y) (k - 1) (k + 3) - (by ring_nf; norm_num) - (fun t _ => (completeSpine_hasDerivAt m hr0 hX hY t).hasDerivWithinAt) - (fun t _ => completeTangent_norm m H r X Y t) - (completeTangent_lipschitz m hr hX hY).lipschitzOnWith - z w ⟨⟨hz.1, hz.2.1⟩, hz.2.2⟩ ⟨⟨hw.1, hw.2.1⟩, hw.2.2⟩ - exact ⟨lipschitzOnWith_iff_dist_le_mul.mpr (fun z hz w hw => (hbound z hz w hw).1), - fun z hz w hw => (hbound z hz w hw).2⟩ - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineInfiniteArray +import OAI.Probability.SLE.Geometry.RoundedConnectorGeometry + +namespace OAI + +/-! Arclength gluing of the low rounded connector and the upward-continued +serpentine array. -/ + +noncomputable section +open Set Metric +open scoped NNReal + +namespace SLEExactGauge.Serpentine + +lemma arraySpine_zero (m j : ℕ) : + arraySpine m j 0 = ((-100 * (m : ℝ) : ℝ) : ℂ) + + ((-100 * (m : ℝ) : ℝ) : ℂ) * Complex.I := by + induction j with + | zero => simp [arraySpine, straightPiece, side] + | succ j ih => + have hb : (0 : ℝ) ≤ columnEntry 0 m (j + 1) := by unfold columnEntry; positivity + have ha : (0 : ℝ) ≤ columnEntry 0 m j + 200 * m := by unfold columnEntry; positivity + simpa only [arraySpine, curveJoin_of_le hb, curveJoin_of_le ha] using ih + +lemma arrayTangent_zero (m j : ℕ) : arrayTangent m j 0 = Complex.I := by + induction j with + | zero => simp [arrayTangent, straightTangent, side] + | succ j ih => + have hb : (0 : ℝ) ≤ columnEntry 0 m (j + 1) := by unfold columnEntry; positivity + have ha : (0 : ℝ) ≤ columnEntry 0 m j + 200 * m := by unfold columnEntry; positivity + simpa only [arrayTangent, curveJoin_of_le hb, curveJoin_of_le ha] using ih + +lemma lowConnector_after {H r X t : ℝ} + (ht : connectorFourth H r X < t) : + lowConnector H r X t = connectorTail H r X t ∧ + lowConnectorTangent H r X t = Complex.I := by + simp only [lowConnector, lowConnectorTangent, curveJoin_of_gt ht, and_self] + +def connectorEntry (H r X Y : ℝ) : ℝ := connectorFourth H r X + Y - H - 2 * r + +def arrayCenter (m : ℕ) (X Y : ℝ) : ℂ := + ((X + 100 * m : ℝ) : ℂ) + ((Y + 100 * m : ℝ) : ℂ) * Complex.I + +def completeSpine (m : ℕ) (H r X Y : ℝ) : ℝ → ℂ := + curveJoin (connectorEntry H r X Y) (lowConnector H r X) + (fun t => arrayCenter m X Y + arraySpine m (2 * m) (t - connectorEntry H r X Y)) + +def completeTangent (m : ℕ) (H r X Y : ℝ) : ℝ → ℂ := + curveJoin (connectorEntry H r X Y) (lowConnectorTangent H r X) + (fun t => arrayTangent m (2 * m) (t - connectorEntry H r X Y)) + +lemma completeSpine_hasDerivAt {H r X Y : ℝ} (m : ℕ) (hr : 0 < r) + (hX : 2 * r < X) (hY : H + 2 * r < Y) (t : ℝ) : + HasDerivAt (completeSpine m H r X Y) (completeTangent m H r X Y t) t := by + have hentry : connectorFourth H r X < connectorEntry H r X Y := by + unfold connectorEntry + linarith + have hp : lowConnector H r X (connectorEntry H r X Y) = + arrayCenter m X Y + arraySpine m (2 * m) + (connectorEntry H r X Y - connectorEntry H r X Y) := by + rw [(lowConnector_after hentry).1, sub_self, arraySpine_zero] + simp only [connectorTail, connectorEntry, arrayCenter] + push_cast + ring + have hT : lowConnectorTangent H r X (connectorEntry H r X Y) = + arrayTangent m (2 * m) (connectorEntry H r X Y - connectorEntry H r X Y) := by + rw [(lowConnector_after hentry).2, sub_self, arrayTangent_zero] + exact curveJoin_hasDerivAt _ _ _ _ _ (lowConnector_hasDerivAt hr hX) + (translate_curve_hasDerivAt _ _ (arraySpine_hasDerivAt m (2 * m)) _ _) hp hT t + +lemma completeTangent_norm (m : ℕ) (H r X Y t : ℝ) : ‖completeTangent m H r X Y t‖ = 1 := by + unfold completeTangent curveJoin + split + · exact lowConnectorTangent_norm _ _ _ _ + · exact arrayTangent_norm _ _ _ + +lemma completeTangent_lipschitz {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) + (hX : 2 * r < X) (hY : H + 2 * r < Y) : + LipschitzWith (1 / 50) (completeTangent m H r X Y) := by + have hr0 : 0 < r := by linarith + have hentry : connectorFourth H r X < connectorEntry H r X Y := by + unfold connectorEntry + linarith + apply curveJoin_lipschitz (lowConnectorTangent_lipschitz hr hX) + (shifted_lipschitz (arrayTangent_lipschitz m (2 * m)) _) + rw [(lowConnector_after hentry).2, sub_self, arrayTangent_zero] + +lemma complete_normal_patch_bounds {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) + (hX : 2 * r < X) (hY : H + 2 * r < Y) (k : ℝ) : + LipschitzOnWith 2 (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) + (RibbonCoordinates.ribbonStepPatch k) ∧ + ∀ z ∈ RibbonCoordinates.ribbonStepPatch k, + ∀ w ∈ RibbonCoordinates.ribbonStepPatch k, + dist z w ≤ 2 * dist (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z) + (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) w) := by + have hr0 : 0 < r := by linarith + have hbound z (hz : z ∈ RibbonCoordinates.ribbonStepPatch k) + w (hw : w ∈ RibbonCoordinates.ribbonStepPatch k) := + normalRibbon_patch_bounds (completeSpine m H r X Y) (completeTangent m H r X Y) (k - 1) (k + 3) + (by ring_nf; norm_num) + (fun t _ => (completeSpine_hasDerivAt m hr0 hX hY t).hasDerivWithinAt) + (fun t _ => completeTangent_norm m H r X Y t) + (completeTangent_lipschitz m hr hX hY).lipschitzOnWith + z w ⟨⟨hz.1, hz.2.1⟩, hz.2.2⟩ ⟨⟨hw.1, hw.2.1⟩, hw.2.2⟩ + exact ⟨lipschitzOnWith_iff_dist_le_mul.mpr (fun z hz w hw => (hbound z hz w hw).1), + fun z hz w hw => (hbound z hz w hw).2⟩ + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteEmbedding.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteEmbedding.lean index 86e92d3ff75989fa0d3930e7449dc6433e5500a8..4c6e3365807a92e1566864bdd2fcb341c7b0e8aa 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteEmbedding.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineCompleteEmbedding.lean @@ -1,110 +1,110 @@ -import OAI.Probability.SLE.Gauge.SerpentineCompleteCurve -import OAI.Probability.SLE.Gauge.SerpentineAttachments -import OAI.Probability.SLE.Geometry.RoundedConnectorRegion - -namespace OAI - -/-! The low connector and the complete infinite array have one common -normal cross-section. Hence the entire prescribed ribbon is embedded. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma complete_normal_of_le (m : ℕ) (H r X Y : ℝ) {z : ℂ} - (hz : z.re ≤ connectorEntry H r X Y) : - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = - normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z := by - simp only [normalRibbon, completeSpine, completeTangent, curveJoin_of_le hz] - -lemma complete_normal_of_gt (m : ℕ) (H r X Y : ℝ) {z : ℂ} - (hz : connectorEntry H r X Y < z.re) : - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = - arrayCenter m X Y + normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) - (z - (connectorEntry H r X Y : ℂ)) := by - simp only [normalRibbon, completeSpine, completeTangent, curveJoin_of_gt hz, - Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - abel - -lemma complete_normal_cross_disjoint {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} - (hr : 50 ≤ r) (hX : 2 * r < X) (hY : H + 2 * r + 1 + 300 * m < Y) - {z w : ℂ} (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) (hwa : |w.im| ≤ 1) - (hzA : z.re ≤ connectorEntry H r X Y) (hwA : connectorEntry H r X Y < w.re) : - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z ≠ - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) w := by - intro he - rw [complete_normal_of_le m H r X Y hzA, complete_normal_of_gt m H r X Y hwA] at he - let w' := w - (connectorEntry H r X Y : ℂ) - have hw' : 0 ≤ w'.re ∧ |w'.im| ≤ 1 := by - dsimp [w'] - simp only [sub_zero] - exact ⟨by linarith, hwa⟩ - have hr1 : 1 < r := by linarith - by_cases hz4 : z.re ≤ connectorFourth H r X - · have hlo := lowConnector_normal_im_upper hr1 hX hz4 hz.2 - have hhi := array_normal_im_lower hm hw' - have him := congrArg Complex.im he - simp only [arrayCenter, Complex.add_im, Complex.ofReal_im, Complex.mul_im, - Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] at him - change -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w').im at hhi - change (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).im = - Y + 100 * m + (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w').im at him - linarith - · have hz4' : connectorFourth H r X < z.re := lt_of_not_ge hz4 - let y := H + 2 * r + (z.re - connectorFourth H r X) - Y - 100 * m - have hy : y ≤ -(100 * (m : ℝ)) := by - dsimp [y] - dsimp only [connectorEntry] at hzA - linarith - have hpoint : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w' = - columnPoint 0 (100 * m) m 0 y z.im := by - have hs := congrArg (fun q : ℂ => q - arrayCenter m X Y) he - simp only [add_sub_cancel_left] at hs - rw [normalRibbon, (lowConnector_after hz4').1, (lowConnector_after hz4').2] at hs - have heq : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w' = - connectorTail H r X z.re + (z.im : ℂ) * Complex.I * Complex.I - arrayCenter m X Y := by - exact hs.symm - rw [heq] - have hs0 : side 0 = 1 := by simp [side] - apply Complex.ext <;> - simp only [connectorTail, arrayCenter, columnPoint, columnOffset, Nat.cast_zero, - zero_mul, hs0, one_mul, normalized_width hm, Complex.add_re, Complex.add_im, - Complex.sub_re, Complex.sub_im, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.zero_re, Complex.zero_im, Complex.I_re, Complex.I_im, y] <;> ring - have hzero := (array_downward_attachment hm hw' hy hz.2 hpoint).1 - dsimp [w'] at hzero - linarith - -/-- Global embedding of the whole explicit normal ribbon, including its -real-boundary connector and infinite final vertical continuation. -/ -theorem complete_normal_injective {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} - (hr : 50 ≤ r) (hX : 2 * r < X) (hY : H + 2 * r + 1 + 300 * m < Y) : - InjOn (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) - {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by - intro z hz w hw he - by_cases hzA : z.re ≤ connectorEntry H r X Y - · by_cases hwA : w.re ≤ connectorEntry H r X Y - · rw [complete_normal_of_le m H r X Y hzA, complete_normal_of_le m H r X Y hwA] at he - exact lowConnector_normal_injective hr hX hz hw he - · exact (complete_normal_cross_disjoint hm hr hX hY hz hw.2 hzA (lt_of_not_ge hwA) he).elim - · by_cases hwA : w.re ≤ connectorEntry H r X Y - · exact (complete_normal_cross_disjoint hm hr hX hY hw hz.2 hwA (lt_of_not_ge hzA) he.symm).elim - · rw [complete_normal_of_gt m H r X Y (lt_of_not_ge hzA), - complete_normal_of_gt m H r X Y (lt_of_not_ge hwA)] at he - have he' := add_left_cancel he - have hz' : 0 ≤ (z - (connectorEntry H r X Y : ℂ)).re ∧ - |(z - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by linarith, hz.2⟩ - have hw' : 0 ≤ (w - (connectorEntry H r X Y : ℂ)).re ∧ - |(w - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by linarith, hw.2⟩ - exact sub_left_injective (array_normal_injective_infinite hm hz' hw' he') - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineCompleteCurve +import OAI.Probability.SLE.Gauge.SerpentineAttachments +import OAI.Probability.SLE.Geometry.RoundedConnectorRegion + +namespace OAI + +/-! The low connector and the complete infinite array have one common +normal cross-section. Hence the entire prescribed ribbon is embedded. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma complete_normal_of_le (m : ℕ) (H r X Y : ℝ) {z : ℂ} + (hz : z.re ≤ connectorEntry H r X Y) : + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = + normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z := by + simp only [normalRibbon, completeSpine, completeTangent, curveJoin_of_le hz] + +lemma complete_normal_of_gt (m : ℕ) (H r X Y : ℝ) {z : ℂ} + (hz : connectorEntry H r X Y < z.re) : + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = + arrayCenter m X Y + normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) + (z - (connectorEntry H r X Y : ℂ)) := by + simp only [normalRibbon, completeSpine, completeTangent, curveJoin_of_gt hz, + Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + abel + +lemma complete_normal_cross_disjoint {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} + (hr : 50 ≤ r) (hX : 2 * r < X) (hY : H + 2 * r + 1 + 300 * m < Y) + {z w : ℂ} (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) (hwa : |w.im| ≤ 1) + (hzA : z.re ≤ connectorEntry H r X Y) (hwA : connectorEntry H r X Y < w.re) : + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z ≠ + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) w := by + intro he + rw [complete_normal_of_le m H r X Y hzA, complete_normal_of_gt m H r X Y hwA] at he + let w' := w - (connectorEntry H r X Y : ℂ) + have hw' : 0 ≤ w'.re ∧ |w'.im| ≤ 1 := by + dsimp [w'] + simp only [sub_zero] + exact ⟨by linarith, hwa⟩ + have hr1 : 1 < r := by linarith + by_cases hz4 : z.re ≤ connectorFourth H r X + · have hlo := lowConnector_normal_im_upper hr1 hX hz4 hz.2 + have hhi := array_normal_im_lower hm hw' + have him := congrArg Complex.im he + simp only [arrayCenter, Complex.add_im, Complex.ofReal_im, Complex.mul_im, + Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] at him + change -(400 * (m : ℝ)) ≤ (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w').im at hhi + change (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).im = + Y + 100 * m + (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w').im at him + linarith + · have hz4' : connectorFourth H r X < z.re := lt_of_not_ge hz4 + let y := H + 2 * r + (z.re - connectorFourth H r X) - Y - 100 * m + have hy : y ≤ -(100 * (m : ℝ)) := by + dsimp [y] + dsimp only [connectorEntry] at hzA + linarith + have hpoint : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w' = + columnPoint 0 (100 * m) m 0 y z.im := by + have hs := congrArg (fun q : ℂ => q - arrayCenter m X Y) he + simp only [add_sub_cancel_left] at hs + rw [normalRibbon, (lowConnector_after hz4').1, (lowConnector_after hz4').2] at hs + have heq : normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w' = + connectorTail H r X z.re + (z.im : ℂ) * Complex.I * Complex.I - arrayCenter m X Y := by + exact hs.symm + rw [heq] + have hs0 : side 0 = 1 := by simp [side] + apply Complex.ext <;> + simp only [connectorTail, arrayCenter, columnPoint, columnOffset, Nat.cast_zero, + zero_mul, hs0, one_mul, normalized_width hm, Complex.add_re, Complex.add_im, + Complex.sub_re, Complex.sub_im, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.zero_re, Complex.zero_im, Complex.I_re, Complex.I_im, y] <;> ring + have hzero := (array_downward_attachment hm hw' hy hz.2 hpoint).1 + dsimp [w'] at hzero + linarith + +/-- Global embedding of the whole explicit normal ribbon, including its +real-boundary connector and infinite final vertical continuation. -/ +theorem complete_normal_injective {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} + (hr : 50 ≤ r) (hX : 2 * r < X) (hY : H + 2 * r + 1 + 300 * m < Y) : + InjOn (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) + {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by + intro z hz w hw he + by_cases hzA : z.re ≤ connectorEntry H r X Y + · by_cases hwA : w.re ≤ connectorEntry H r X Y + · rw [complete_normal_of_le m H r X Y hzA, complete_normal_of_le m H r X Y hwA] at he + exact lowConnector_normal_injective hr hX hz hw he + · exact (complete_normal_cross_disjoint hm hr hX hY hz hw.2 hzA (lt_of_not_ge hwA) he).elim + · by_cases hwA : w.re ≤ connectorEntry H r X Y + · exact (complete_normal_cross_disjoint hm hr hX hY hw hz.2 hwA (lt_of_not_ge hzA) he.symm).elim + · rw [complete_normal_of_gt m H r X Y (lt_of_not_ge hzA), + complete_normal_of_gt m H r X Y (lt_of_not_ge hwA)] at he + have he' := add_left_cancel he + have hz' : 0 ≤ (z - (connectorEntry H r X Y : ℂ)).re ∧ + |(z - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by linarith, hz.2⟩ + have hw' : 0 ≤ (w - (connectorEntry H r X Y : ℂ)).re ∧ + |(w - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by linarith, hw.2⟩ + exact sub_left_injective (array_normal_injective_infinite hm hz' hw' he') + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineConformalGeometry.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineConformalGeometry.lean index d400ef1f8ebd4cc34dcb45cdb261f6cd6805aa64..ac92d0872ecffe39909959c792030dc6345a5209 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineConformalGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineConformalGeometry.lean @@ -1,170 +1,170 @@ -import OAI.Probability.SLE.Gauge.RestrictedConformalSausage -import OAI.Probability.SLE.Geometry.NormalizedConformalBounds -import OAI.Probability.SLE.Gauge.SerpentineLayout - -namespace OAI - -/-! A common small displaced experiment disk in normalized half-plane -coordinates. Its nearby trace witnesses stay in a controlled conformal -patch, and its image stays in one physical conformal-radius disk. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge.Serpentine - -def conformalExperimentScale (s C : ℝ) : ℝ := min (s / 10) (s / (100 * C)) - -lemma conformalExperimentScale_bounds {s C : ℝ} (hs : 0 < s) (hC : 0 < C) : - 0 < conformalExperimentScale s C ∧ conformalExperimentScale s C ≤ s / 10 ∧ - C * conformalExperimentScale s C ≤ s / 100 := by - refine ⟨lt_min (by positivity) (by positivity), min_le_left _ _, ?_⟩ - have hh := (le_div_iff₀ (by positivity : 0 < 100 * C)).mp - (min_le_right (s / 10) (s / (100 * C))) - dsimp only [conformalExperimentScale] - nlinarith - -lemma experiment_patch_subset_compact {s h : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (hh : h ≤ s / 10) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) : - closedBall w h ⊆ normalizedInteriorCompact s := by - intro z hz - have hn := norm_le_norm_add_const_of_dist_le hz - have hi := Complex.im_le_norm (w - z) - have hd : ‖w - z‖ ≤ h := by - simpa only [mem_closedBall, ← dist_eq_norm, dist_comm] using hz - rw [Complex.sub_im] at hi - change ‖z‖ ≤ 2 ∧ s / 2 ≤ z.im - rw [hwn] at hn - constructor <;> linarith - -lemma displaced_experiment_dist {h : ℝ} (hh : 0 ≤ h) (w z : ℂ) - (hz : z ∈ closedBall (center h w) (h / 20)) : - dist z w ≤ 3 * h / 10 := by - have hd := (dist_triangle z (center h w) w).trans - (add_le_add hz (center_dist hh w).le) - linarith - -lemma displaced_experiment_subset_patch {h : ℝ} (hh : 0 ≤ h) (w : ℂ) : - closedBall (center h w) (h / 20) ⊆ closedBall w h := by - intro z hz - exact (displaced_experiment_dist hh w z hz).trans (by linarith) - -lemma displaced_experiment_near_patch {h u : ℝ} (hh : 0 < h) (hu : u < h / 20) - (w z y : ℂ) (hz : z ∈ closedBall (center h w) (h / 20)) (hzy : dist z y < u) : - y ∈ closedBall w h := by - have hd := dist_triangle y z w - rw [dist_comm y z] at hd - have hz' := displaced_experiment_dist hh.le w z hz - change dist y w ≤ h - linarith - -/-- Converting any positive deterministic parameter cutoff into a -cutoff proportional to the physical conformal radius. -/ -lemma conformal_experiment_mesh_cutoff {C a q v : ℝ} - (hC : 0 < C) (ha : 0 < a) (hq : 0 < q) (w : ℂ) (hwi : w.im ≤ 1) - (hv : v < (C * q / 2) * (2 * w.im * a)) : - v / (C * a) < q := by - apply (div_lt_iff₀ (mul_pos hC ha)).mpr - have hm := mul_le_mul_of_nonneg_left hwi (show 0 ≤ C * q * a by positivity) - nlinarith - -theorem conformal_experiment_image_subset - {s h c C : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) (hh : 0 < h) - (hhs : h ≤ s / 10) (hCh : C * h ≤ s / 100) (hC : 0 < C) - (F : ℂ → ℂ) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hbound : ∀ z ∈ normalizedInteriorCompact s, - c * ‖deriv F w‖ ≤ ‖deriv F z‖ ∧ ‖deriv F z‖ ≤ C * ‖deriv F w‖) : - F '' closedBall (center h w) (h / 20) ⊆ - closedBall (F w) (2 * w.im * ‖deriv F w‖) := by - let B : ℝ≥0 := ⟨C * ‖deriv F w‖, mul_nonneg hC.le (norm_nonneg _)⟩ - have hK := experiment_patch_subset_compact hs hs1 hhs w hwn hwy - have hKH := hK.trans (normalizedInteriorCompact_subset_halfPlane hs) - have hLip : LipschitzOnWith B F (closedBall w h) := - (convex_closedBall w h).lipschitzOnWith_of_nnnorm_hasDerivWithin_le - (fun z hz => ((hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds (hKH hz))).hasDerivAt).hasDerivWithinAt) - (fun z hz => by exact_mod_cast (hbound z (hK hz)).2) - rintro _ ⟨z, hz, rfl⟩ - have hzK := displaced_experiment_subset_patch hh.le w hz - have hd := hLip.dist_le_mul z hzK w (mem_closedBall_self hh.le) - have hzdist := displaced_experiment_dist hh.le w z hz - have hnonneg := norm_nonneg (deriv F w) - have hmul := mul_le_mul_of_nonneg_right hCh hnonneg - have hdist := mul_le_mul_of_nonneg_left hzdist (show 0 ≤ C * ‖deriv F w‖ by positivity) - change dist (F z) (F w) ≤ 2 * w.im * ‖deriv F w‖ - change dist (F z) (F w) ≤ (C * ‖deriv F w‖) * dist z w at hd - nlinarith - -theorem conformal_experiment_restricted_transfer - (κ : ℝ) {s h c C : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (hh : 0 < h) (hhs : h ≤ s / 10) (hCh : C * h ≤ s / 100) - (hc : 0 < c) (hC : 0 < C) - (F : ℂ → ℂ) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) - (hF : DifferentiableOn ℂ F upperHalfPlane) (hi : InjOn F upperHalfPlane) - (hne : deriv F w ≠ 0) - (hbound : ∀ z ∈ normalizedInteriorCompact s, - c * ‖deriv F w‖ ≤ ‖deriv F z‖ ∧ ‖deriv F z‖ ≤ C * ‖deriv F w‖) - (γ η : ℝ≥0 → ℂ) (a b : ℝ≥0) - (heq : ∀ u ∈ Icc a b, γ u ∈ closedBall w h → η u = F (γ u)) - {v : ℝ} (hv : 0 < v) - (hvc : v < (C * h / 40) * (2 * w.im * ‖deriv F w‖)) : - ENNReal.ofReal ((c ^ 2 * (1 / C) ^ codimension κ) * ‖deriv F w‖ ^ dimension κ) * - segmentSausageCharge κ γ a b (v / (C * ‖deriv F w‖)) - (closedBall (center h w) (h / 20)) ≤ - segmentSausageCharge κ η a b v (closedBall (F w) (2 * w.im * ‖deriv F w‖)) := by - have hnorm : 0 < ‖deriv F w‖ := norm_pos_iff.mpr hne - have hCa : 0 < C * ‖deriv F w‖ := mul_pos hC hnorm - have hwi : w.im ≤ 1 := by simpa only [hwn] using Complex.im_le_norm w - have hmesh : v / (C * ‖deriv F w‖) < h / 20 := by - apply conformal_experiment_mesh_cutoff hC hnorm (by positivity) w hwi - convert hvc using 1 - ring - have hK := experiment_patch_subset_compact hs hs1 hhs w hwn hwy - have hKH := hK.trans (normalizedInteriorCompact_subset_halfPlane hs) - let B : ℝ≥0 := ⟨C * ‖deriv F w‖, hCa.le⟩ - have hstep : (1 / (C * ‖deriv F w‖)) * v = v / (C * ‖deriv F w‖) := by ring - have ht := restricted_segment_sausage_conformal_transfer κ F (deriv F) γ η a b - (closedBall (center h w) (h / 20)) (closedBall w h) - (closedBall (F w) (2 * w.im * ‖deriv F w‖)) isClosed_closedBall.measurableSet - (displaced_experiment_subset_patch hh.le w) (convex_closedBall w h) - (hi.mono hKH) - (fun z hz => (hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds (hKH hz))).hasDerivAt) - (mul_pos hc hnorm).le hv (one_div_pos.mpr hCa) - (fun z hz => (hbound z (hK (displaced_experiment_subset_patch hh.le w hz))).1) - B (fun z hz => by exact_mod_cast (hbound z (hK hz)).2) - (by change (C * ‖deriv F w‖) * (1 / (C * ‖deriv F w‖)) ≤ 1; rw [mul_one_div_cancel hCa.ne']) - (fun z hz y hd => displaced_experiment_near_patch hh hmesh w z y hz (by simpa only [hstep] using hd)) - heq (conformal_experiment_image_subset hs hs1 hh hhs hCh hC F w hwn hwy hF hbound) - rw [conformal_transfer_coefficient κ hC hnorm, hstep] at ht - exact ht - -theorem uniform_conformal_experiment_restricted_transfer - (κ : ℝ) (hcompact : PublishedHalfPlaneUnivalentCompactness) - {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : - ∃ h A C δ : ℝ, 0 < h ∧ h ≤ s ∧ 0 < A ∧ 0 < C ∧ 0 < δ ∧ - ∀ (F : ℂ → ℂ) (w : ℂ), DifferentiableOn ℂ F upperHalfPlane → - InjOn F upperHalfPlane → ‖w‖ = 1 → s ≤ w.im → deriv F w ≠ 0 → - ∀ (γ η : ℝ≥0 → ℂ) (a b : ℝ≥0), - (∀ u ∈ Icc a b, γ u ∈ closedBall w h → η u = F (γ u)) → - ∀ v : ℝ, 0 < v → v < δ * (2 * w.im * ‖deriv F w‖) → - ENNReal.ofReal (A * ‖deriv F w‖ ^ dimension κ) * - segmentSausageCharge κ γ a b (v / (C * ‖deriv F w‖)) - (closedBall (center h w) (h / 20)) ≤ - segmentSausageCharge κ η a b v - (closedBall (F w) (2 * w.im * ‖deriv F w‖)) := by - obtain ⟨c, C, hc, hC, hbounds⟩ := uniform_waypoint_derivative_ratios hcompact hs hs1 - let h := conformalExperimentScale s C - obtain ⟨hh, hhs, hCh⟩ := conformalExperimentScale_bounds hs hC - refine ⟨h, c ^ 2 * (1 / C) ^ codimension κ, C, C * h / 40, hh, - hhs.trans (by linarith), by positivity, hC, by positivity, ?_⟩ - intro F w hF hi hwn hwy hne γ η a b heq v hv hvc - exact conformal_experiment_restricted_transfer κ hs hs1 hh hhs hCh hc hC - F w hwn hwy hF hi hne (fun z hz => hbounds F w hF hi hwn hwy hne z hz.1 hz.2) - γ η a b heq hv hvc - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.RestrictedConformalSausage +import OAI.Probability.SLE.Geometry.NormalizedConformalBounds +import OAI.Probability.SLE.Gauge.SerpentineLayout + +namespace OAI + +/-! A common small displaced experiment disk in normalized half-plane +coordinates. Its nearby trace witnesses stay in a controlled conformal +patch, and its image stays in one physical conformal-radius disk. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge.Serpentine + +def conformalExperimentScale (s C : ℝ) : ℝ := min (s / 10) (s / (100 * C)) + +lemma conformalExperimentScale_bounds {s C : ℝ} (hs : 0 < s) (hC : 0 < C) : + 0 < conformalExperimentScale s C ∧ conformalExperimentScale s C ≤ s / 10 ∧ + C * conformalExperimentScale s C ≤ s / 100 := by + refine ⟨lt_min (by positivity) (by positivity), min_le_left _ _, ?_⟩ + have hh := (le_div_iff₀ (by positivity : 0 < 100 * C)).mp + (min_le_right (s / 10) (s / (100 * C))) + dsimp only [conformalExperimentScale] + nlinarith + +lemma experiment_patch_subset_compact {s h : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (hh : h ≤ s / 10) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) : + closedBall w h ⊆ normalizedInteriorCompact s := by + intro z hz + have hn := norm_le_norm_add_const_of_dist_le hz + have hi := Complex.im_le_norm (w - z) + have hd : ‖w - z‖ ≤ h := by + simpa only [mem_closedBall, ← dist_eq_norm, dist_comm] using hz + rw [Complex.sub_im] at hi + change ‖z‖ ≤ 2 ∧ s / 2 ≤ z.im + rw [hwn] at hn + constructor <;> linarith + +lemma displaced_experiment_dist {h : ℝ} (hh : 0 ≤ h) (w z : ℂ) + (hz : z ∈ closedBall (center h w) (h / 20)) : + dist z w ≤ 3 * h / 10 := by + have hd := (dist_triangle z (center h w) w).trans + (add_le_add hz (center_dist hh w).le) + linarith + +lemma displaced_experiment_subset_patch {h : ℝ} (hh : 0 ≤ h) (w : ℂ) : + closedBall (center h w) (h / 20) ⊆ closedBall w h := by + intro z hz + exact (displaced_experiment_dist hh w z hz).trans (by linarith) + +lemma displaced_experiment_near_patch {h u : ℝ} (hh : 0 < h) (hu : u < h / 20) + (w z y : ℂ) (hz : z ∈ closedBall (center h w) (h / 20)) (hzy : dist z y < u) : + y ∈ closedBall w h := by + have hd := dist_triangle y z w + rw [dist_comm y z] at hd + have hz' := displaced_experiment_dist hh.le w z hz + change dist y w ≤ h + linarith + +/-- Converting any positive deterministic parameter cutoff into a +cutoff proportional to the physical conformal radius. -/ +lemma conformal_experiment_mesh_cutoff {C a q v : ℝ} + (hC : 0 < C) (ha : 0 < a) (hq : 0 < q) (w : ℂ) (hwi : w.im ≤ 1) + (hv : v < (C * q / 2) * (2 * w.im * a)) : + v / (C * a) < q := by + apply (div_lt_iff₀ (mul_pos hC ha)).mpr + have hm := mul_le_mul_of_nonneg_left hwi (show 0 ≤ C * q * a by positivity) + nlinarith + +theorem conformal_experiment_image_subset + {s h c C : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) (hh : 0 < h) + (hhs : h ≤ s / 10) (hCh : C * h ≤ s / 100) (hC : 0 < C) + (F : ℂ → ℂ) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hbound : ∀ z ∈ normalizedInteriorCompact s, + c * ‖deriv F w‖ ≤ ‖deriv F z‖ ∧ ‖deriv F z‖ ≤ C * ‖deriv F w‖) : + F '' closedBall (center h w) (h / 20) ⊆ + closedBall (F w) (2 * w.im * ‖deriv F w‖) := by + let B : ℝ≥0 := ⟨C * ‖deriv F w‖, mul_nonneg hC.le (norm_nonneg _)⟩ + have hK := experiment_patch_subset_compact hs hs1 hhs w hwn hwy + have hKH := hK.trans (normalizedInteriorCompact_subset_halfPlane hs) + have hLip : LipschitzOnWith B F (closedBall w h) := + (convex_closedBall w h).lipschitzOnWith_of_nnnorm_hasDerivWithin_le + (fun z hz => ((hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds (hKH hz))).hasDerivAt).hasDerivWithinAt) + (fun z hz => by exact_mod_cast (hbound z (hK hz)).2) + rintro _ ⟨z, hz, rfl⟩ + have hzK := displaced_experiment_subset_patch hh.le w hz + have hd := hLip.dist_le_mul z hzK w (mem_closedBall_self hh.le) + have hzdist := displaced_experiment_dist hh.le w z hz + have hnonneg := norm_nonneg (deriv F w) + have hmul := mul_le_mul_of_nonneg_right hCh hnonneg + have hdist := mul_le_mul_of_nonneg_left hzdist (show 0 ≤ C * ‖deriv F w‖ by positivity) + change dist (F z) (F w) ≤ 2 * w.im * ‖deriv F w‖ + change dist (F z) (F w) ≤ (C * ‖deriv F w‖) * dist z w at hd + nlinarith + +theorem conformal_experiment_restricted_transfer + (κ : ℝ) {s h c C : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (hh : 0 < h) (hhs : h ≤ s / 10) (hCh : C * h ≤ s / 100) + (hc : 0 < c) (hC : 0 < C) + (F : ℂ → ℂ) (w : ℂ) (hwn : ‖w‖ = 1) (hwy : s ≤ w.im) + (hF : DifferentiableOn ℂ F upperHalfPlane) (hi : InjOn F upperHalfPlane) + (hne : deriv F w ≠ 0) + (hbound : ∀ z ∈ normalizedInteriorCompact s, + c * ‖deriv F w‖ ≤ ‖deriv F z‖ ∧ ‖deriv F z‖ ≤ C * ‖deriv F w‖) + (γ η : ℝ≥0 → ℂ) (a b : ℝ≥0) + (heq : ∀ u ∈ Icc a b, γ u ∈ closedBall w h → η u = F (γ u)) + {v : ℝ} (hv : 0 < v) + (hvc : v < (C * h / 40) * (2 * w.im * ‖deriv F w‖)) : + ENNReal.ofReal ((c ^ 2 * (1 / C) ^ codimension κ) * ‖deriv F w‖ ^ dimension κ) * + segmentSausageCharge κ γ a b (v / (C * ‖deriv F w‖)) + (closedBall (center h w) (h / 20)) ≤ + segmentSausageCharge κ η a b v (closedBall (F w) (2 * w.im * ‖deriv F w‖)) := by + have hnorm : 0 < ‖deriv F w‖ := norm_pos_iff.mpr hne + have hCa : 0 < C * ‖deriv F w‖ := mul_pos hC hnorm + have hwi : w.im ≤ 1 := by simpa only [hwn] using Complex.im_le_norm w + have hmesh : v / (C * ‖deriv F w‖) < h / 20 := by + apply conformal_experiment_mesh_cutoff hC hnorm (by positivity) w hwi + convert hvc using 1 + ring + have hK := experiment_patch_subset_compact hs hs1 hhs w hwn hwy + have hKH := hK.trans (normalizedInteriorCompact_subset_halfPlane hs) + let B : ℝ≥0 := ⟨C * ‖deriv F w‖, hCa.le⟩ + have hstep : (1 / (C * ‖deriv F w‖)) * v = v / (C * ‖deriv F w‖) := by ring + have ht := restricted_segment_sausage_conformal_transfer κ F (deriv F) γ η a b + (closedBall (center h w) (h / 20)) (closedBall w h) + (closedBall (F w) (2 * w.im * ‖deriv F w‖)) isClosed_closedBall.measurableSet + (displaced_experiment_subset_patch hh.le w) (convex_closedBall w h) + (hi.mono hKH) + (fun z hz => (hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds (hKH hz))).hasDerivAt) + (mul_pos hc hnorm).le hv (one_div_pos.mpr hCa) + (fun z hz => (hbound z (hK (displaced_experiment_subset_patch hh.le w hz))).1) + B (fun z hz => by exact_mod_cast (hbound z (hK hz)).2) + (by change (C * ‖deriv F w‖) * (1 / (C * ‖deriv F w‖)) ≤ 1; rw [mul_one_div_cancel hCa.ne']) + (fun z hz y hd => displaced_experiment_near_patch hh hmesh w z y hz (by simpa only [hstep] using hd)) + heq (conformal_experiment_image_subset hs hs1 hh hhs hCh hC F w hwn hwy hF hbound) + rw [conformal_transfer_coefficient κ hC hnorm, hstep] at ht + exact ht + +theorem uniform_conformal_experiment_restricted_transfer + (κ : ℝ) (hcompact : PublishedHalfPlaneUnivalentCompactness) + {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : + ∃ h A C δ : ℝ, 0 < h ∧ h ≤ s ∧ 0 < A ∧ 0 < C ∧ 0 < δ ∧ + ∀ (F : ℂ → ℂ) (w : ℂ), DifferentiableOn ℂ F upperHalfPlane → + InjOn F upperHalfPlane → ‖w‖ = 1 → s ≤ w.im → deriv F w ≠ 0 → + ∀ (γ η : ℝ≥0 → ℂ) (a b : ℝ≥0), + (∀ u ∈ Icc a b, γ u ∈ closedBall w h → η u = F (γ u)) → + ∀ v : ℝ, 0 < v → v < δ * (2 * w.im * ‖deriv F w‖) → + ENNReal.ofReal (A * ‖deriv F w‖ ^ dimension κ) * + segmentSausageCharge κ γ a b (v / (C * ‖deriv F w‖)) + (closedBall (center h w) (h / 20)) ≤ + segmentSausageCharge κ η a b v + (closedBall (F w) (2 * w.im * ‖deriv F w‖)) := by + obtain ⟨c, C, hc, hC, hbounds⟩ := uniform_waypoint_derivative_ratios hcompact hs hs1 + let h := conformalExperimentScale s C + obtain ⟨hh, hhs, hCh⟩ := conformalExperimentScale_bounds hs hC + refine ⟨h, c ^ 2 * (1 / C) ^ codimension κ, C, C * h / 40, hh, + hhs.trans (by linarith), by positivity, hC, by positivity, ?_⟩ + intro F w hF hi hwn hwy hne γ η a b heq v hv hvc + exact conformal_experiment_restricted_transfer κ hs hs1 hh hhs hCh hc hC + F w hwn hwy hF hi hne (fun z hz => hbounds F w hF hi hwn hwy hne z hz.1 hz.2) + γ η a b heq hv hvc + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineCurve.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineCurve.lean index d15d23efc6fa311f9448fef67929136b885655bc..5d9653bea51febe04a751e7a338cf2843f3f8e41 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineCurve.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineCurve.lean @@ -1,163 +1,163 @@ -import OAI.Probability.SLE.Gauge.SerpentinePieces -import OAI.Probability.SLE.Gauge.SerpentineStepCount - -namespace OAI - -/-! Explicit arclength parametrization of the finite serpentine array, -continued vertically along its last column. -/ - -noncomputable section -open Set Metric -open scoped NNReal - -namespace SLEExactGauge.Serpentine - -lemma columnEntry_succ (A : ℝ) (m j : ℕ) : - columnEntry A m (j + 1) = columnEntry A m j + 200 * m + 50 * Real.pi := by - simp only [columnEntry, Nat.cast_add, Nat.cast_one] - ring - -def arraySpine (m : ℕ) : ℕ → ℝ → ℂ - | 0 => straightPiece m 0 - | j + 1 => curveJoin (columnEntry 0 m (j + 1)) - (curveJoin (columnEntry 0 m j + 200 * m) (arraySpine m j) - (fun t => circularPiece m j (t - (columnEntry 0 m j + 200 * m)))) - (fun t => straightPiece m (j + 1) (t - columnEntry 0 m (j + 1))) - -def arrayTangent (m : ℕ) : ℕ → ℝ → ℂ - | 0 => fun _ => straightTangent 0 - | j + 1 => curveJoin (columnEntry 0 m (j + 1)) - (curveJoin (columnEntry 0 m j + 200 * m) (arrayTangent m j) - (fun t => circularTangent j (t - (columnEntry 0 m j + 200 * m)))) - (fun _ => straightTangent (j + 1)) - -lemma arraySpine_tail (m j : ℕ) {t : ℝ} (ht : columnEntry 0 m j ≤ t) : - arraySpine m j t = straightPiece m j (t - columnEntry 0 m j) := by - cases j with - | zero => simp [arraySpine, columnEntry] - | succ j => - by_cases hlt : columnEntry 0 m (j + 1) < t - · simp only [arraySpine, curveJoin_of_gt hlt] - · have he : t = columnEntry 0 m (j + 1) := le_antisymm (le_of_not_gt hlt) ht - subst t - have hcut : columnEntry 0 m j + 200 * m < columnEntry 0 m (j + 1) := by - rw [columnEntry_succ] - linarith [Real.pi_pos] - simp only [arraySpine, curveJoin_of_le le_rfl, curveJoin_of_gt hcut, sub_self] - rw [columnEntry_succ] - have he : columnEntry 0 m j + 200 * m + 50 * Real.pi - - (columnEntry 0 m j + 200 * m) = 50 * Real.pi := by ring - rw [he, circular_straight_position_match] - -lemma arrayTangent_tail (m j : ℕ) {t : ℝ} (ht : columnEntry 0 m j ≤ t) : - arrayTangent m j t = straightTangent j := by - cases j with - | zero => rfl - | succ j => - by_cases hlt : columnEntry 0 m (j + 1) < t - · simp only [arrayTangent, curveJoin_of_gt hlt] - · have he : t = columnEntry 0 m (j + 1) := le_antisymm (le_of_not_gt hlt) ht - subst t - have hcut : columnEntry 0 m j + 200 * m < columnEntry 0 m (j + 1) := by - rw [columnEntry_succ] - linarith [Real.pi_pos] - simp only [arrayTangent, curveJoin_of_le le_rfl, curveJoin_of_gt hcut] - rw [columnEntry_succ] - have he : columnEntry 0 m j + 200 * m + 50 * Real.pi - - (columnEntry 0 m j + 200 * m) = 50 * Real.pi := by ring - rw [he, circular_straight_tangent_match] - -lemma arrayTangent_norm (m j : ℕ) (t : ℝ) : ‖arrayTangent m j t‖ = 1 := by - induction j with - | zero => exact straightTangent_norm 0 - | succ j ih => - unfold arrayTangent curveJoin - split - · split - · exact ih - · exact circularTangent_norm _ _ - · exact straightTangent_norm _ - -lemma arraySpine_hasDerivAt (m j : ℕ) (t : ℝ) : - HasDerivAt (arraySpine m j) (arrayTangent m j t) t := by - induction j generalizing t with - | zero => exact straightPiece_hasDerivAt m 0 t - | succ j ih => - let a := columnEntry 0 m j + 200 * m - let b := columnEntry 0 m (j + 1) - have hc (t : ℝ) : HasDerivAt (fun t => circularPiece m j (t - a)) - (circularTangent j (t - a)) t := by - convert (circularPiece_hasDerivAt m j (t - a)).scomp t - ((hasDerivAt_id t).sub_const a) using 1 <;> first | rfl | simp only [one_smul] - have hs (t : ℝ) : HasDerivAt (fun t => straightPiece m (j + 1) (t - b)) - (straightTangent (j + 1)) t := by - convert (straightPiece_hasDerivAt m (j + 1) (t - b)).scomp t - ((hasDerivAt_id t).sub_const b) using 1 <;> first | rfl | simp only [one_smul] - have ha0 : columnEntry 0 m j ≤ a := by dsimp [a]; linarith [Nat.cast_nonneg (α := ℝ) m] - have hab : a < b := by dsimp [a, b]; rw [columnEntry_succ]; linarith [Real.pi_pos] - have habv : b - a = 50 * Real.pi := by dsimp [a, b]; rw [columnEntry_succ]; ring - have he₁ : arraySpine m j a = circularPiece m j (a - a) := by - rw [arraySpine_tail m j ha0] - have he : a - columnEntry 0 m j = 200 * m := by dsimp [a]; ring - rw [he, sub_self, straight_circular_position_match] - have hd₁ : arrayTangent m j a = circularTangent j (a - a) := by - rw [arrayTangent_tail m j ha0, sub_self, straight_circular_tangent_match] - have hfirst := curveJoin_hasDerivAt a (arraySpine m j) - (fun t => circularPiece m j (t - a)) (arrayTangent m j) - (fun t => circularTangent j (t - a)) ih hc he₁ hd₁ - have he₂ : curveJoin a (arraySpine m j) (fun t => circularPiece m j (t - a)) b = - straightPiece m (j + 1) (b - b) := by - rw [curveJoin_of_gt hab, habv, sub_self, circular_straight_position_match] - have hd₂ : curveJoin a (arrayTangent m j) (fun t => circularTangent j (t - a)) b = - straightTangent (j + 1) := by - rw [curveJoin_of_gt hab, habv, circular_straight_tangent_match] - exact curveJoin_hasDerivAt b _ _ _ _ hfirst hs he₂ hd₂ t - -lemma arrayTangent_lipschitz (m j : ℕ) : LipschitzWith (1 / 50) (arrayTangent m j) := by - induction j with - | zero => exact (LipschitzWith.const _).weaken (by norm_num) - | succ j ih => - let a := columnEntry 0 m j + 200 * m - let b := columnEntry 0 m (j + 1) - have hc : LipschitzWith (1 / 50) (fun t => circularTangent j (t - a)) := by - apply lipschitzWith_iff_dist_le_mul.mpr - intro u v - simpa only [dist_sub_right] using (circularTangent_lipschitz j).dist_le_mul (u - a) (v - a) - have hs : LipschitzWith (1 / 50) (fun _ : ℝ => straightTangent (j + 1)) := - (LipschitzWith.const _).weaken (by norm_num) - have ha0 : columnEntry 0 m j ≤ a := by dsimp [a]; linarith [Nat.cast_nonneg (α := ℝ) m] - have hab : a < b := by dsimp [a, b]; rw [columnEntry_succ]; linarith [Real.pi_pos] - have habv : b - a = 50 * Real.pi := by dsimp [a, b]; rw [columnEntry_succ]; ring - have hd₁ : arrayTangent m j a = circularTangent j (a - a) := by - rw [arrayTangent_tail m j ha0, sub_self, straight_circular_tangent_match] - have hfirst := curveJoin_lipschitz ih hc hd₁ - have hd₂ : curveJoin a (arrayTangent m j) (fun t => circularTangent j (t - a)) b = - straightTangent (j + 1) := by - rw [curveJoin_of_gt hab, habv, circular_straight_tangent_match] - exact curveJoin_lipschitz hfirst hs hd₂ - -/-- Every length-four normal coordinate patch of the explicit array has -uniform upper and inverse Lipschitz constants equal to two. -/ -theorem array_normal_patch_bounds (m j : ℕ) (k : ℝ) : - LipschitzOnWith 2 (normalRibbon (arraySpine m j) (arrayTangent m j)) - (RibbonCoordinates.ribbonStepPatch k) ∧ - ∀ z ∈ RibbonCoordinates.ribbonStepPatch k, - ∀ w ∈ RibbonCoordinates.ribbonStepPatch k, - dist z w ≤ 2 * dist (normalRibbon (arraySpine m j) (arrayTangent m j) z) - (normalRibbon (arraySpine m j) (arrayTangent m j) w) := by - have hbound z (hz : z ∈ RibbonCoordinates.ribbonStepPatch k) - w (hw : w ∈ RibbonCoordinates.ribbonStepPatch k) := - normalRibbon_patch_bounds (arraySpine m j) (arrayTangent m j) (k - 1) (k + 3) - (by ring_nf; norm_num) - (fun t _ => (arraySpine_hasDerivAt m j t).hasDerivWithinAt) - (fun t _ => arrayTangent_norm m j t) (arrayTangent_lipschitz m j).lipschitzOnWith - z w ⟨⟨hz.1, hz.2.1⟩, hz.2.2⟩ ⟨⟨hw.1, hw.2.1⟩, hw.2.2⟩ - exact ⟨lipschitzOnWith_iff_dist_le_mul.mpr (fun z hz w hw => (hbound z hz w hw).1), - fun z hz w hw => (hbound z hz w hw).2⟩ - - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentinePieces +import OAI.Probability.SLE.Gauge.SerpentineStepCount + +namespace OAI + +/-! Explicit arclength parametrization of the finite serpentine array, +continued vertically along its last column. -/ + +noncomputable section +open Set Metric +open scoped NNReal + +namespace SLEExactGauge.Serpentine + +lemma columnEntry_succ (A : ℝ) (m j : ℕ) : + columnEntry A m (j + 1) = columnEntry A m j + 200 * m + 50 * Real.pi := by + simp only [columnEntry, Nat.cast_add, Nat.cast_one] + ring + +def arraySpine (m : ℕ) : ℕ → ℝ → ℂ + | 0 => straightPiece m 0 + | j + 1 => curveJoin (columnEntry 0 m (j + 1)) + (curveJoin (columnEntry 0 m j + 200 * m) (arraySpine m j) + (fun t => circularPiece m j (t - (columnEntry 0 m j + 200 * m)))) + (fun t => straightPiece m (j + 1) (t - columnEntry 0 m (j + 1))) + +def arrayTangent (m : ℕ) : ℕ → ℝ → ℂ + | 0 => fun _ => straightTangent 0 + | j + 1 => curveJoin (columnEntry 0 m (j + 1)) + (curveJoin (columnEntry 0 m j + 200 * m) (arrayTangent m j) + (fun t => circularTangent j (t - (columnEntry 0 m j + 200 * m)))) + (fun _ => straightTangent (j + 1)) + +lemma arraySpine_tail (m j : ℕ) {t : ℝ} (ht : columnEntry 0 m j ≤ t) : + arraySpine m j t = straightPiece m j (t - columnEntry 0 m j) := by + cases j with + | zero => simp [arraySpine, columnEntry] + | succ j => + by_cases hlt : columnEntry 0 m (j + 1) < t + · simp only [arraySpine, curveJoin_of_gt hlt] + · have he : t = columnEntry 0 m (j + 1) := le_antisymm (le_of_not_gt hlt) ht + subst t + have hcut : columnEntry 0 m j + 200 * m < columnEntry 0 m (j + 1) := by + rw [columnEntry_succ] + linarith [Real.pi_pos] + simp only [arraySpine, curveJoin_of_le le_rfl, curveJoin_of_gt hcut, sub_self] + rw [columnEntry_succ] + have he : columnEntry 0 m j + 200 * m + 50 * Real.pi - + (columnEntry 0 m j + 200 * m) = 50 * Real.pi := by ring + rw [he, circular_straight_position_match] + +lemma arrayTangent_tail (m j : ℕ) {t : ℝ} (ht : columnEntry 0 m j ≤ t) : + arrayTangent m j t = straightTangent j := by + cases j with + | zero => rfl + | succ j => + by_cases hlt : columnEntry 0 m (j + 1) < t + · simp only [arrayTangent, curveJoin_of_gt hlt] + · have he : t = columnEntry 0 m (j + 1) := le_antisymm (le_of_not_gt hlt) ht + subst t + have hcut : columnEntry 0 m j + 200 * m < columnEntry 0 m (j + 1) := by + rw [columnEntry_succ] + linarith [Real.pi_pos] + simp only [arrayTangent, curveJoin_of_le le_rfl, curveJoin_of_gt hcut] + rw [columnEntry_succ] + have he : columnEntry 0 m j + 200 * m + 50 * Real.pi - + (columnEntry 0 m j + 200 * m) = 50 * Real.pi := by ring + rw [he, circular_straight_tangent_match] + +lemma arrayTangent_norm (m j : ℕ) (t : ℝ) : ‖arrayTangent m j t‖ = 1 := by + induction j with + | zero => exact straightTangent_norm 0 + | succ j ih => + unfold arrayTangent curveJoin + split + · split + · exact ih + · exact circularTangent_norm _ _ + · exact straightTangent_norm _ + +lemma arraySpine_hasDerivAt (m j : ℕ) (t : ℝ) : + HasDerivAt (arraySpine m j) (arrayTangent m j t) t := by + induction j generalizing t with + | zero => exact straightPiece_hasDerivAt m 0 t + | succ j ih => + let a := columnEntry 0 m j + 200 * m + let b := columnEntry 0 m (j + 1) + have hc (t : ℝ) : HasDerivAt (fun t => circularPiece m j (t - a)) + (circularTangent j (t - a)) t := by + convert (circularPiece_hasDerivAt m j (t - a)).scomp t + ((hasDerivAt_id t).sub_const a) using 1 <;> first | rfl | simp only [one_smul] + have hs (t : ℝ) : HasDerivAt (fun t => straightPiece m (j + 1) (t - b)) + (straightTangent (j + 1)) t := by + convert (straightPiece_hasDerivAt m (j + 1) (t - b)).scomp t + ((hasDerivAt_id t).sub_const b) using 1 <;> first | rfl | simp only [one_smul] + have ha0 : columnEntry 0 m j ≤ a := by dsimp [a]; linarith [Nat.cast_nonneg (α := ℝ) m] + have hab : a < b := by dsimp [a, b]; rw [columnEntry_succ]; linarith [Real.pi_pos] + have habv : b - a = 50 * Real.pi := by dsimp [a, b]; rw [columnEntry_succ]; ring + have he₁ : arraySpine m j a = circularPiece m j (a - a) := by + rw [arraySpine_tail m j ha0] + have he : a - columnEntry 0 m j = 200 * m := by dsimp [a]; ring + rw [he, sub_self, straight_circular_position_match] + have hd₁ : arrayTangent m j a = circularTangent j (a - a) := by + rw [arrayTangent_tail m j ha0, sub_self, straight_circular_tangent_match] + have hfirst := curveJoin_hasDerivAt a (arraySpine m j) + (fun t => circularPiece m j (t - a)) (arrayTangent m j) + (fun t => circularTangent j (t - a)) ih hc he₁ hd₁ + have he₂ : curveJoin a (arraySpine m j) (fun t => circularPiece m j (t - a)) b = + straightPiece m (j + 1) (b - b) := by + rw [curveJoin_of_gt hab, habv, sub_self, circular_straight_position_match] + have hd₂ : curveJoin a (arrayTangent m j) (fun t => circularTangent j (t - a)) b = + straightTangent (j + 1) := by + rw [curveJoin_of_gt hab, habv, circular_straight_tangent_match] + exact curveJoin_hasDerivAt b _ _ _ _ hfirst hs he₂ hd₂ t + +lemma arrayTangent_lipschitz (m j : ℕ) : LipschitzWith (1 / 50) (arrayTangent m j) := by + induction j with + | zero => exact (LipschitzWith.const _).weaken (by norm_num) + | succ j ih => + let a := columnEntry 0 m j + 200 * m + let b := columnEntry 0 m (j + 1) + have hc : LipschitzWith (1 / 50) (fun t => circularTangent j (t - a)) := by + apply lipschitzWith_iff_dist_le_mul.mpr + intro u v + simpa only [dist_sub_right] using (circularTangent_lipschitz j).dist_le_mul (u - a) (v - a) + have hs : LipschitzWith (1 / 50) (fun _ : ℝ => straightTangent (j + 1)) := + (LipschitzWith.const _).weaken (by norm_num) + have ha0 : columnEntry 0 m j ≤ a := by dsimp [a]; linarith [Nat.cast_nonneg (α := ℝ) m] + have hab : a < b := by dsimp [a, b]; rw [columnEntry_succ]; linarith [Real.pi_pos] + have habv : b - a = 50 * Real.pi := by dsimp [a, b]; rw [columnEntry_succ]; ring + have hd₁ : arrayTangent m j a = circularTangent j (a - a) := by + rw [arrayTangent_tail m j ha0, sub_self, straight_circular_tangent_match] + have hfirst := curveJoin_lipschitz ih hc hd₁ + have hd₂ : curveJoin a (arrayTangent m j) (fun t => circularTangent j (t - a)) b = + straightTangent (j + 1) := by + rw [curveJoin_of_gt hab, habv, circular_straight_tangent_match] + exact curveJoin_lipschitz hfirst hs hd₂ + +/-- Every length-four normal coordinate patch of the explicit array has +uniform upper and inverse Lipschitz constants equal to two. -/ +theorem array_normal_patch_bounds (m j : ℕ) (k : ℝ) : + LipschitzOnWith 2 (normalRibbon (arraySpine m j) (arrayTangent m j)) + (RibbonCoordinates.ribbonStepPatch k) ∧ + ∀ z ∈ RibbonCoordinates.ribbonStepPatch k, + ∀ w ∈ RibbonCoordinates.ribbonStepPatch k, + dist z w ≤ 2 * dist (normalRibbon (arraySpine m j) (arrayTangent m j) z) + (normalRibbon (arraySpine m j) (arrayTangent m j) w) := by + have hbound z (hz : z ∈ RibbonCoordinates.ribbonStepPatch k) + w (hw : w ∈ RibbonCoordinates.ribbonStepPatch k) := + normalRibbon_patch_bounds (arraySpine m j) (arrayTangent m j) (k - 1) (k + 3) + (by ring_nf; norm_num) + (fun t _ => (arraySpine_hasDerivAt m j t).hasDerivWithinAt) + (fun t _ => arrayTangent_norm m j t) (arrayTangent_lipschitz m j).lipschitzOnWith + z w ⟨⟨hz.1, hz.2.1⟩, hz.2.2⟩ ⟨⟨hw.1, hw.2.1⟩, hw.2.2⟩ + exact ⟨lipschitzOnWith_iff_dist_le_mul.mpr (fun z hz w hw => (hbound z hz w hw).1), + fun z hz w hw => (hbound z hz w hw).2⟩ + + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineEmbedding.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineEmbedding.lean index 729999bee93aad573898ea69da5ce47bda61d490..29a0a18f73a430c461301f3912a492d54f782ff8 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineEmbedding.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineEmbedding.lean @@ -1,160 +1,160 @@ -import OAI.Probability.SLE.Gauge.SerpentineLayout - -namespace OAI - -/-! The only contacts between the straight and circular normal bands are -their prescribed common end sections. These geometric statements support -the global embedding of the concatenated serpentine. -/ - -noncomputable section -open Set - -namespace SLEExactGauge.Serpentine - -@[simp] lemma turnPoint_im (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : - (turnPoint c h m j t a).im = c.im + side j * h + - (turnRadius h m + side j * ribbonWidth h m * a) * (side j * Real.sin t) := by - simp only [turnPoint, Complex.add_re, Complex.add_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, - mul_zero, zero_mul, sub_zero, mul_one, add_zero, zero_add] - -lemma turnPoint_oriented_im (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : - side j * ((turnPoint c h m j t a).im - c.im) = - h + (turnRadius h m + side j * ribbonWidth h m * a) * Real.sin t := by - rw [turnPoint_im] - let R := turnRadius h m + side j * ribbonWidth h m * a - change side j * (c.im + side j * h + R * (side j * Real.sin t) - c.im) = h + R * Real.sin t - calc - _ = side j ^ 2 * (h + R * Real.sin t) := by ring - _ = _ := by rw [side_sq, one_mul] - -lemma columnPoint_equal_forces_index {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) - (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) - (he : columnPoint c h m i y a = columnPoint c h m j y' b) : i = j := by - by_contra hn - rcases lt_or_gt_of_ne hn with hij | hji - · exact columnPoint_ne hh hm hij c y y' ha hb he - · exact columnPoint_ne hh hm hji c y' y hb ha he.symm - -lemma columnPoint_injective {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) - (c : ℂ) (j : ℕ) {y y' a b : ℝ} - (he : columnPoint c h m j y a = columnPoint c h m j y' b) : y = y' ∧ a = b := by - have hy := congrArg Complex.im he - simp only [columnPoint_im] at hy - have ha := congrArg Complex.re he - simp only [columnPoint_re] at ha - have hs : side j ≠ 0 := by - intro hz - have hh := side_abs j - rw [hz, abs_zero] at hh - norm_num at hh - refine ⟨by linarith, ?_⟩ - apply mul_left_cancel₀ (mul_ne_zero hs (ribbonWidth_pos hh hm).ne') - nlinarith - -/-- A circular normal band can meet a straight band only at angle zero -or pi; its interior lies strictly above or below all straight passes. -/ -theorem turnPoint_column_contact {h : ℝ} (hh : 0 < h) {m j k : ℕ} (hm : 0 < m) - (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) (hy : |y| ≤ h) - (ha : |a| ≤ 1) (he : turnPoint c h m j t a = columnPoint c h m k y b) : - t = 0 ∨ t = Real.pi := by - by_contra hn - push Not at hn - have hsin : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi - (lt_of_le_of_ne ht.1 (Ne.symm hn.1)) (lt_of_le_of_ne ht.2 hn.2) - have hor := turnPoint_oriented_im c h m j t a - have hcoeff := turn_radial_coefficient_pos hh hm j ha - have hy' : |(turnPoint c h m j t a).im - c.im| ≤ h := by - rw [he, columnPoint_im, add_sub_cancel_left] - exact hy - have hu : side j * ((turnPoint c h m j t a).im - c.im) ≤ h := by - calc - _ ≤ |side j * ((turnPoint c h m j t a).im - c.im)| := le_abs_self _ - _ = |(turnPoint c h m j t a).im - c.im| := by rw [abs_mul, side_abs, one_mul] - _ ≤ h := hy' - nlinarith [mul_pos hcoeff hsin] - -/-- Exact gluing sections, including the matching normal coordinate. -/ -theorem turnPoint_column_gluing {h : ℝ} (hh : 0 < h) {m j k : ℕ} (hm : 0 < m) - (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) (hy : |y| ≤ h) - (ha : |a| ≤ 1) (hb : |b| ≤ 1) - (he : turnPoint c h m j t a = columnPoint c h m k y b) : - (t = 0 ∧ k = j ∧ y = side j * h ∧ a = b) ∨ - (t = Real.pi ∧ k = j + 1 ∧ y = side j * h ∧ a = b) := by - rcases turnPoint_column_contact hh hm c ht hy ha he with rfl | rfl - · rw [turnPoint_zero] at he - have hk := columnPoint_equal_forces_index hh hm c (side j * h) y ha hb he - subst k - obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm c j he - exact Or.inl ⟨rfl, rfl, hy'.symm, ha'⟩ - · rw [turnPoint_pi] at he - have hk := columnPoint_equal_forces_index hh hm c (side j * h) y ha hb he - subst k - obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm c (j + 1) he - exact Or.inr ⟨rfl, rfl, hy'.symm, ha'⟩ - -@[simp] lemma turnPoint_re (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : - (turnPoint c h m j t a).re = c.re + columnOffset h m j + turnRadius h m - - (turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t := by - simp only [turnPoint, Complex.add_re, Complex.add_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, - mul_zero, zero_mul, sub_zero, mul_one, add_zero, zero_add] - ring - -lemma turn_horizontal_bound {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) - (c : ℂ) (j : ℕ) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : - |(turnPoint c h m j t a).re - (c.re + columnOffset h m j + turnRadius h m)| ≤ - turnRadius h m + ribbonWidth h m := by - have hr := (turnRadius_pos hh hm).le - have hw := (ribbonWidth_pos hh hm).le - have hb : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hw] - exact mul_le_of_le_one_right hw ha - have ha' := abs_add_le (turnRadius h m) (side j * ribbonWidth h m * a) - rw [abs_of_nonneg hr] at ha' - rw [turnPoint_re] - have he : c.re + columnOffset h m j + turnRadius h m - - (turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t - - (c.re + columnOffset h m j + turnRadius h m) = - -((turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t) := by ring - rw [he, abs_neg, abs_mul] - exact (mul_le_of_le_one_right (abs_nonneg _) (Real.abs_cos_le_one t)).trans (ha'.trans (by linarith)) - -lemma turn_oriented_height {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) - (c : ℂ) (j : ℕ) {t a : ℝ} (ht : t ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) : - h ≤ side j * ((turnPoint c h m j t a).im - c.im) := by - rw [turnPoint_oriented_im] - exact le_add_of_nonneg_right (mul_nonneg (turn_radial_coefficient_pos hh hm j ha).le - (Real.sin_nonneg_of_mem_Icc ht)) - -/-- Different circular turns have disjoint full normal bands. Neighboring -turns lie on opposite sides; all other turn pairs have horizontal clearance. -/ -theorem turnPoint_ne {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) - (hij : i < j) (c : ℂ) {t u a b : ℝ} (ht : t ∈ Icc 0 Real.pi) - (hu : u ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : - turnPoint c h m i t a ≠ turnPoint c h m j u b := by - intro he - by_cases hnext : j = i + 1 - · subst j - have h₁ := turn_oriented_height hh hm c i ht ha - have h₂ := turn_oriented_height hh hm c (i + 1) hu hb - rw [← he, side_succ] at h₂ - nlinarith - · have hfar : i + 2 ≤ j := by omega - have hfar' : (i : ℝ) + 2 ≤ j := by exact_mod_cast hfar - have hstep := mul_le_mul_of_nonneg_right hfar' (spacing_pos hh hm).le - have h₁ := abs_le.mp (turn_horizontal_bound hh hm c i t ha) - have h₂ := abs_le.mp (turn_horizontal_bound hh hm c j u hb) - rw [← he] at h₂ - simp only [columnOffset] at h₁ h₂ - rw [spacing_eq_width] at hstep h₁ h₂ - rw [turnRadius_eq_width] at h₁ h₂ - have hw := ribbonWidth_pos hh hm - nlinarith - - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineLayout + +namespace OAI + +/-! The only contacts between the straight and circular normal bands are +their prescribed common end sections. These geometric statements support +the global embedding of the concatenated serpentine. -/ + +noncomputable section +open Set + +namespace SLEExactGauge.Serpentine + +@[simp] lemma turnPoint_im (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : + (turnPoint c h m j t a).im = c.im + side j * h + + (turnRadius h m + side j * ribbonWidth h m * a) * (side j * Real.sin t) := by + simp only [turnPoint, Complex.add_re, Complex.add_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, + mul_zero, zero_mul, sub_zero, mul_one, add_zero, zero_add] + +lemma turnPoint_oriented_im (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : + side j * ((turnPoint c h m j t a).im - c.im) = + h + (turnRadius h m + side j * ribbonWidth h m * a) * Real.sin t := by + rw [turnPoint_im] + let R := turnRadius h m + side j * ribbonWidth h m * a + change side j * (c.im + side j * h + R * (side j * Real.sin t) - c.im) = h + R * Real.sin t + calc + _ = side j ^ 2 * (h + R * Real.sin t) := by ring + _ = _ := by rw [side_sq, one_mul] + +lemma columnPoint_equal_forces_index {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) + (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) + (he : columnPoint c h m i y a = columnPoint c h m j y' b) : i = j := by + by_contra hn + rcases lt_or_gt_of_ne hn with hij | hji + · exact columnPoint_ne hh hm hij c y y' ha hb he + · exact columnPoint_ne hh hm hji c y' y hb ha he.symm + +lemma columnPoint_injective {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) + (c : ℂ) (j : ℕ) {y y' a b : ℝ} + (he : columnPoint c h m j y a = columnPoint c h m j y' b) : y = y' ∧ a = b := by + have hy := congrArg Complex.im he + simp only [columnPoint_im] at hy + have ha := congrArg Complex.re he + simp only [columnPoint_re] at ha + have hs : side j ≠ 0 := by + intro hz + have hh := side_abs j + rw [hz, abs_zero] at hh + norm_num at hh + refine ⟨by linarith, ?_⟩ + apply mul_left_cancel₀ (mul_ne_zero hs (ribbonWidth_pos hh hm).ne') + nlinarith + +/-- A circular normal band can meet a straight band only at angle zero +or pi; its interior lies strictly above or below all straight passes. -/ +theorem turnPoint_column_contact {h : ℝ} (hh : 0 < h) {m j k : ℕ} (hm : 0 < m) + (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) (hy : |y| ≤ h) + (ha : |a| ≤ 1) (he : turnPoint c h m j t a = columnPoint c h m k y b) : + t = 0 ∨ t = Real.pi := by + by_contra hn + push Not at hn + have hsin : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi + (lt_of_le_of_ne ht.1 (Ne.symm hn.1)) (lt_of_le_of_ne ht.2 hn.2) + have hor := turnPoint_oriented_im c h m j t a + have hcoeff := turn_radial_coefficient_pos hh hm j ha + have hy' : |(turnPoint c h m j t a).im - c.im| ≤ h := by + rw [he, columnPoint_im, add_sub_cancel_left] + exact hy + have hu : side j * ((turnPoint c h m j t a).im - c.im) ≤ h := by + calc + _ ≤ |side j * ((turnPoint c h m j t a).im - c.im)| := le_abs_self _ + _ = |(turnPoint c h m j t a).im - c.im| := by rw [abs_mul, side_abs, one_mul] + _ ≤ h := hy' + nlinarith [mul_pos hcoeff hsin] + +/-- Exact gluing sections, including the matching normal coordinate. -/ +theorem turnPoint_column_gluing {h : ℝ} (hh : 0 < h) {m j k : ℕ} (hm : 0 < m) + (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) (hy : |y| ≤ h) + (ha : |a| ≤ 1) (hb : |b| ≤ 1) + (he : turnPoint c h m j t a = columnPoint c h m k y b) : + (t = 0 ∧ k = j ∧ y = side j * h ∧ a = b) ∨ + (t = Real.pi ∧ k = j + 1 ∧ y = side j * h ∧ a = b) := by + rcases turnPoint_column_contact hh hm c ht hy ha he with rfl | rfl + · rw [turnPoint_zero] at he + have hk := columnPoint_equal_forces_index hh hm c (side j * h) y ha hb he + subst k + obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm c j he + exact Or.inl ⟨rfl, rfl, hy'.symm, ha'⟩ + · rw [turnPoint_pi] at he + have hk := columnPoint_equal_forces_index hh hm c (side j * h) y ha hb he + subst k + obtain ⟨hy', ha'⟩ := columnPoint_injective hh hm c (j + 1) he + exact Or.inr ⟨rfl, rfl, hy'.symm, ha'⟩ + +@[simp] lemma turnPoint_re (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : + (turnPoint c h m j t a).re = c.re + columnOffset h m j + turnRadius h m - + (turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t := by + simp only [turnPoint, Complex.add_re, Complex.add_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, + mul_zero, zero_mul, sub_zero, mul_one, add_zero, zero_add] + ring + +lemma turn_horizontal_bound {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) + (c : ℂ) (j : ℕ) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : + |(turnPoint c h m j t a).re - (c.re + columnOffset h m j + turnRadius h m)| ≤ + turnRadius h m + ribbonWidth h m := by + have hr := (turnRadius_pos hh hm).le + have hw := (ribbonWidth_pos hh hm).le + have hb : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hw] + exact mul_le_of_le_one_right hw ha + have ha' := abs_add_le (turnRadius h m) (side j * ribbonWidth h m * a) + rw [abs_of_nonneg hr] at ha' + rw [turnPoint_re] + have he : c.re + columnOffset h m j + turnRadius h m - + (turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t - + (c.re + columnOffset h m j + turnRadius h m) = + -((turnRadius h m + side j * ribbonWidth h m * a) * Real.cos t) := by ring + rw [he, abs_neg, abs_mul] + exact (mul_le_of_le_one_right (abs_nonneg _) (Real.abs_cos_le_one t)).trans (ha'.trans (by linarith)) + +lemma turn_oriented_height {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) + (c : ℂ) (j : ℕ) {t a : ℝ} (ht : t ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) : + h ≤ side j * ((turnPoint c h m j t a).im - c.im) := by + rw [turnPoint_oriented_im] + exact le_add_of_nonneg_right (mul_nonneg (turn_radial_coefficient_pos hh hm j ha).le + (Real.sin_nonneg_of_mem_Icc ht)) + +/-- Different circular turns have disjoint full normal bands. Neighboring +turns lie on opposite sides; all other turn pairs have horizontal clearance. -/ +theorem turnPoint_ne {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) + (hij : i < j) (c : ℂ) {t u a b : ℝ} (ht : t ∈ Icc 0 Real.pi) + (hu : u ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : + turnPoint c h m i t a ≠ turnPoint c h m j u b := by + intro he + by_cases hnext : j = i + 1 + · subst j + have h₁ := turn_oriented_height hh hm c i ht ha + have h₂ := turn_oriented_height hh hm c (i + 1) hu hb + rw [← he, side_succ] at h₂ + nlinarith + · have hfar : i + 2 ≤ j := by omega + have hfar' : (i : ℝ) + 2 ≤ j := by exact_mod_cast hfar + have hstep := mul_le_mul_of_nonneg_right hfar' (spacing_pos hh hm).le + have h₁ := abs_le.mp (turn_horizontal_bound hh hm c i t ha) + have h₂ := abs_le.mp (turn_horizontal_bound hh hm c j u hb) + rw [← he] at h₂ + simp only [columnOffset] at h₁ h₂ + rw [spacing_eq_width] at hstep h₁ h₂ + rw [turnRadius_eq_width] at h₁ h₂ + have hw := ribbonWidth_pos hh hm + nlinarith + + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineEscape.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineEscape.lean index 95ec5cc1c034a37bfab048d176f25c6f6d2be02d..aeb5ee793ff8bbb9fed29668455ba2e95e901d43 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineEscape.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineEscape.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Gauge.SerpentineCompleteCurve - -namespace OAI - -/-! The final linear tail and bounded normal displacement of the -complete serpentine. These supply properness of the ribbon chart. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine - -lemma complete_linear_tail (m : ℕ) (H r X Y : ℝ) {t : ℝ} - (ht : connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1 ≤ t) : - completeSpine m H r X Y t = ((X + 200 * m : ℝ) : ℂ) + - ((Y + t - connectorEntry H r X Y - columnEntry 0 m (2 * m) : ℝ) : ℂ) * Complex.I ∧ - completeTangent m H r X Y t = Complex.I := by - have hc : 0 ≤ columnEntry 0 m (2 * m) := by unfold columnEntry; positivity - have htA : connectorEntry H r X Y < t := by linarith - have htC : columnEntry 0 m (2 * m) ≤ t - connectorEntry H r X Y := by linarith - simp only [completeSpine, completeTangent, curveJoin_of_gt htA] - rw [arraySpine_tail m (2 * m) htC, arrayTangent_tail m (2 * m) htC] - constructor - · apply Complex.ext <;> - simp only [arrayCenter, straightPiece, side_even, one_mul, Nat.cast_mul, Nat.cast_ofNat, - Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.I_re, Complex.I_im] <;> ring - · simp only [straightTangent, side_even, Complex.ofReal_one, one_mul] - -lemma complete_eventually_ray (m : ℕ) (H r X Y : ℝ) : - ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → - completeSpine m H r X Y t = completeSpine m H r X Y T + ((t - T : ℝ) : ℂ) * Complex.I := by - let T := max 0 (connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1) - have hT : connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1 ≤ T := le_max_right _ _ - refine ⟨T, le_max_left _ _, fun t ht => ?_⟩ - rw [(complete_linear_tail m H r X Y (hT.trans ht)).1, - (complete_linear_tail m H r X Y hT).1] - push_cast - ring - -lemma completeSpine_escape (m : ℕ) (H r X Y : ℝ) : - Tendsto (completeSpine m H r X Y) atTop (cocompact ℂ) := by - have him : Tendsto (fun t => (completeSpine m H r X Y t).im) atTop atTop := by - have hh : Tendsto (fun t : ℝ => Y + t - connectorEntry H r X Y - columnEntry 0 m (2 * m)) - atTop atTop := by - simpa only [sub_eq_add_neg, id_eq] using - tendsto_atTop_add_const_right atTop (-columnEntry 0 m (2 * m)) - (tendsto_atTop_add_const_right atTop (-connectorEntry H r X Y) - (tendsto_atTop_add_const_left atTop Y tendsto_id)) - apply hh.congr' - filter_upwards [eventually_ge_atTop - (connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1)] with t ht - rw [(complete_linear_tail m H r X Y ht).1] - simp - have hn : Tendsto (fun t => ‖completeSpine m H r X Y t‖) atTop atTop := by - apply tendsto_atTop_mono (fun t => ?_) him - exact (le_abs_self _).trans (Complex.abs_im_le_norm _) - apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 - simpa only [dist_zero_right] using hn - -lemma complete_normal_displacement (m : ℕ) (H r X Y : ℝ) {z : ℂ} (hz : |z.im| ≤ 1) : - dist (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z) - (completeSpine m H r X Y z.re) ≤ 1 := by - rw [normalRibbon, dist_self_add_left, norm_mul, norm_mul, Complex.norm_real, - Real.norm_eq_abs, Complex.norm_I, completeTangent_norm, mul_one, mul_one] - exact hz - -lemma complete_normal_width (m : ℕ) (H r X Y : ℝ) {z : ℂ} (hz : |z.im| ≤ 1) : - ‖normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z - - completeSpine m H r X Y z.re‖ ≤ 1 := by - simpa only [dist_eq_norm] using complete_normal_displacement m H r X Y hz - -lemma complete_normal_continuous {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) - (hX : 2 * r < X) (hY : H + 2 * r < Y) : - Continuous (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) := by - have hr0 : 0 < r := by linarith - have hσ : Continuous (completeSpine m H r X Y) := - continuous_iff_continuousAt.mpr (fun t => (completeSpine_hasDerivAt m hr0 hX hY t).continuousAt) - have hT : Continuous (completeTangent m H r X Y) := - (completeTangent_lipschitz m hr hX hY).continuous - exact (hσ.comp Complex.continuous_re).add - (((Complex.continuous_ofReal.comp Complex.continuous_im).mul continuous_const).mul - (hT.comp Complex.continuous_re)) - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineCompleteCurve + +namespace OAI + +/-! The final linear tail and bounded normal displacement of the +complete serpentine. These supply properness of the ribbon chart. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine + +lemma complete_linear_tail (m : ℕ) (H r X Y : ℝ) {t : ℝ} + (ht : connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1 ≤ t) : + completeSpine m H r X Y t = ((X + 200 * m : ℝ) : ℂ) + + ((Y + t - connectorEntry H r X Y - columnEntry 0 m (2 * m) : ℝ) : ℂ) * Complex.I ∧ + completeTangent m H r X Y t = Complex.I := by + have hc : 0 ≤ columnEntry 0 m (2 * m) := by unfold columnEntry; positivity + have htA : connectorEntry H r X Y < t := by linarith + have htC : columnEntry 0 m (2 * m) ≤ t - connectorEntry H r X Y := by linarith + simp only [completeSpine, completeTangent, curveJoin_of_gt htA] + rw [arraySpine_tail m (2 * m) htC, arrayTangent_tail m (2 * m) htC] + constructor + · apply Complex.ext <;> + simp only [arrayCenter, straightPiece, side_even, one_mul, Nat.cast_mul, Nat.cast_ofNat, + Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.I_re, Complex.I_im] <;> ring + · simp only [straightTangent, side_even, Complex.ofReal_one, one_mul] + +lemma complete_eventually_ray (m : ℕ) (H r X Y : ℝ) : + ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → + completeSpine m H r X Y t = completeSpine m H r X Y T + ((t - T : ℝ) : ℂ) * Complex.I := by + let T := max 0 (connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1) + have hT : connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1 ≤ T := le_max_right _ _ + refine ⟨T, le_max_left _ _, fun t ht => ?_⟩ + rw [(complete_linear_tail m H r X Y (hT.trans ht)).1, + (complete_linear_tail m H r X Y hT).1] + push_cast + ring + +lemma completeSpine_escape (m : ℕ) (H r X Y : ℝ) : + Tendsto (completeSpine m H r X Y) atTop (cocompact ℂ) := by + have him : Tendsto (fun t => (completeSpine m H r X Y t).im) atTop atTop := by + have hh : Tendsto (fun t : ℝ => Y + t - connectorEntry H r X Y - columnEntry 0 m (2 * m)) + atTop atTop := by + simpa only [sub_eq_add_neg, id_eq] using + tendsto_atTop_add_const_right atTop (-columnEntry 0 m (2 * m)) + (tendsto_atTop_add_const_right atTop (-connectorEntry H r X Y) + (tendsto_atTop_add_const_left atTop Y tendsto_id)) + apply hh.congr' + filter_upwards [eventually_ge_atTop + (connectorEntry H r X Y + columnEntry 0 m (2 * m) + 1)] with t ht + rw [(complete_linear_tail m H r X Y ht).1] + simp + have hn : Tendsto (fun t => ‖completeSpine m H r X Y t‖) atTop atTop := by + apply tendsto_atTop_mono (fun t => ?_) him + exact (le_abs_self _).trans (Complex.abs_im_le_norm _) + apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 + simpa only [dist_zero_right] using hn + +lemma complete_normal_displacement (m : ℕ) (H r X Y : ℝ) {z : ℂ} (hz : |z.im| ≤ 1) : + dist (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z) + (completeSpine m H r X Y z.re) ≤ 1 := by + rw [normalRibbon, dist_self_add_left, norm_mul, norm_mul, Complex.norm_real, + Real.norm_eq_abs, Complex.norm_I, completeTangent_norm, mul_one, mul_one] + exact hz + +lemma complete_normal_width (m : ℕ) (H r X Y : ℝ) {z : ℂ} (hz : |z.im| ≤ 1) : + ‖normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z - + completeSpine m H r X Y z.re‖ ≤ 1 := by + simpa only [dist_eq_norm] using complete_normal_displacement m H r X Y hz + +lemma complete_normal_continuous {H r X Y : ℝ} (m : ℕ) (hr : 50 ≤ r) + (hX : 2 * r < X) (hY : H + 2 * r < Y) : + Continuous (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y)) := by + have hr0 : 0 < r := by linarith + have hσ : Continuous (completeSpine m H r X Y) := + continuous_iff_continuousAt.mpr (fun t => (completeSpine_hasDerivAt m hr0 hX hY t).continuousAt) + have hT : Continuous (completeTangent m H r X Y) := + (completeTangent_lipschitz m hr hX hY).continuous + exact (hσ.comp Complex.continuous_re).add + (((Complex.continuous_ofReal.comp Complex.continuous_im).mul continuous_const).mul + (hT.comp Complex.continuous_re)) + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineExperimentGeometry.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineExperimentGeometry.lean index 7b8deb6f75833b640f5b3bfe3ef1d15703ee8deb..40ecc1f91ef9ebaa209218feef32eb9192bfd729 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineExperimentGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineExperimentGeometry.lean @@ -1,179 +1,179 @@ -import OAI.Probability.SLE.Gauge.SerpentineFittedShadow -import OAI.Probability.SLE.Gauge.SerpentineReflection -import OAI.Probability.SLE.Gauge.ProperRibbonCoordinates - -namespace OAI - -/-! The actual left-or-right displaced serpentine. Reflection is applied -to both the physical image and the normal coordinate, so the initial chart -is still `z ↦ I*z`. -/ - -noncomputable section -open Set Filter Metric ComplexConjugate -open scoped NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -def experimentChart (s : ℝ) (w : ℂ) (m : ℕ) : ℂ → ℂ := - if 0 ≤ w.re then fitChart s w m else reflectedChart (fitChart s w m) - -def experimentSpine (s : ℝ) (w : ℂ) (m : ℕ) (t : ℝ) : ℂ := - if 0 ≤ w.re then fitSpine s w m t else loewnerMirror (fitSpine s w m t) - -def physicalExperimentChart (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : ℂ := - if 0 ≤ w.re then physicalFitChart s w m z else loewnerMirror (physicalFitChart s w m (conj z)) - -lemma loewnerMirror_real_mul (c : ℝ) (z : ℂ) : - loewnerMirror ((c : ℂ) * z) = (c : ℂ) * loewnerMirror z := by - simp [loewnerMirror, mul_neg] - -lemma physicalExperimentChart_eq (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : - physicalExperimentChart s w m z = (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z := by - by_cases hw : 0 ≤ w.re - · simp only [physicalExperimentChart, experimentChart, ite_eq_left hw, physicalFitChart] - · simp only [physicalExperimentChart, experimentChart, ite_eq_right hw, physicalFitChart, - reflectedChart, loewnerMirror_real_mul] - -theorem experimentChart_geometry {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - Continuous (experimentChart s w m) ∧ InjOn (experimentChart s w m) closedRibbon ∧ - (∀ k : ℝ, LipschitzOnWith 2 (experimentChart s w m) (ribbonStepPatch k) ∧ - ∀ z ∈ ribbonStepPatch k, ∀ z' ∈ ribbonStepPatch k, - dist z z' ≤ 2 * dist (experimentChart s w m z) (experimentChart s w m z')) ∧ - (∀ z ∈ closedRibbon, 0 < z.re → experimentChart s w m z ∈ upperHalfPlane) ∧ - (∀ z ∈ closedRibbon, z.re ≤ 4 → experimentChart s w m z = Complex.I * z) := by - have hcont := fitChart_continuous hs w hw hm - have hinj := fitChart_injective hs w hw hm - have hpatch := fitChart_patch_bounds hs w hw hm - have hu := fitChart_upper hs w hw hm - have hi : ∀ z ∈ closedRibbon, z.re ≤ 4 → fitChart s w m z = Complex.I * z := by - intro z _ hz - exact fitChart_initial hs w hw hm (hz.trans (fit_parameters hs w hw hm).2.2.2) - by_cases hwr : 0 ≤ w.re - · simpa only [experimentChart, ite_eq_left hwr] using ⟨hcont, hinj, hpatch, hu, hi⟩ - · simp only [experimentChart, ite_eq_right hwr] - exact ⟨reflectedChart_continuous hcont, reflectedChart_injective hinj, - fun k => ⟨reflectedChart_patch_upper (hpatch k).1, - reflectedChart_patch_lower (hpatch k).2⟩, - reflectedChart_upper hu, reflectedChart_initial hi⟩ - -lemma experimentChart_width (s : ℝ) (w : ℂ) (m : ℕ) {z : ℂ} (hz : z ∈ closedRibbon) : - ‖experimentChart s w m z - experimentSpine s w m z.re‖ ≤ 1 := by - by_cases hwr : 0 ≤ w.re - · simpa only [experimentChart, experimentSpine, ite_eq_left hwr] using fitChart_width s w m hz - · simpa only [experimentChart, experimentSpine, ite_eq_right hwr] using - reflectedChart_width (P := fitChart s w m) (σ := fitSpine s w m) - (fun _ hz => fitChart_width s w m hz) z hz - -lemma experimentSpine_eventually_ray (s : ℝ) (w : ℂ) (m : ℕ) : - ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → - experimentSpine s w m t = experimentSpine s w m T + ((t - T : ℝ) : ℂ) * Complex.I := by - obtain ⟨T, hT, htail⟩ := fitSpine_eventually_ray s w m - refine ⟨T, hT, fun t ht => ?_⟩ - by_cases hwr : 0 ≤ w.re - · simpa only [experimentSpine, ite_eq_left hwr] using htail t ht - · simp only [experimentSpine, ite_eq_right hwr] - rw [htail t ht] - simp [loewnerMirror, mul_neg, add_comm] - -lemma experimentChart_real_norm (s : ℝ) (w : ℂ) (m : ℕ) (t : ℝ) : - ‖experimentChart s w m (t : ℂ)‖ = ‖fitSpine s w m t‖ := by - by_cases hwr : 0 ≤ w.re <;> - simp [experimentChart, hwr, reflectedChart, fitChart, normalRibbon] - -theorem experimentChart_escape (s : ℝ) (w : ℂ) (m : ℕ) : - Tendsto (fun t : ℝ => ‖experimentChart s w m (t : ℂ)‖) atTop atTop := by - simp only [experimentChart_real_norm] - exact tendsto_norm_cocompact_atTop.comp - (completeSpine_escape m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) - -theorem experimentChart_zero {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : experimentChart s w m 0 = 0 := by - have hh := (experimentChart_geometry hs w hw hm).2.2.2.2 0 - (by simp [closedRibbon]) (by norm_num) - simpa only [mul_zero] using hh - -theorem experimentChart_closedEmbedding {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - Topology.IsClosedEmbedding (fun z : closedRibbon => experimentChart s w m z) := by - obtain ⟨hc, hi, _⟩ := experimentChart_geometry hs w hw hm - obtain ⟨T, hT, htail⟩ := experimentSpine_eventually_ray s w m - exact ribbon_closedEmbedding_of_linear_tail _ hc.continuousOn hi _ T hT - (experimentSpine s w m T) htail (fun _ hz => experimentChart_width s w m hz) - -theorem experimentChart_global_progress {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - ∃ g : C(ℂ, ℝ), ∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re := - exists_global_ribbon_progress _ (experimentChart_closedEmbedding hs w hw hm) - -lemma reflected_positiveCenter {s : ℝ} {w : ℂ} (hw : w.re < 0) : - loewnerMirror (positiveCenter s w) = center s w := by - have hn : ¬ 0 ≤ w.re := not_le.mpr hw - apply Complex.ext - · simp only [loewnerMirror_re, positiveCenter, center, Complex.add_re, Complex.mul_re, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - mul_zero, zero_mul, sub_zero, add_zero] - rw [abs_of_neg hw] - simp only [displacementSign, ite_eq_right hn] - ring - · simp only [loewnerMirror_im, positiveCenter, center, Complex.add_im, Complex.mul_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - mul_zero, mul_one, add_zero, zero_add] - -theorem physicalExperimentChart_retained_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} - (hz : z.re ∈ Icc - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) - (ha : |z.im| ≤ 1) : - physicalExperimentChart s w m z ∈ closedBall (center s w) (s / 20) := by - by_cases hwr : 0 ≤ w.re - · simpa only [physicalExperimentChart, ite_eq_left hwr] using - physicalFitChart_retained_disk hs w hw hwr hm hj hi hz ha - · have hzc : (conj z).re ∈ Icc - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3) := by - simpa only [Complex.conj_re] using hz - have hac : |(conj z).im| ≤ 1 := by simpa only [Complex.conj_im, abs_neg] using ha - have hd := physicalFitChart_retained_distance hs w hw hm hj hi hzc hac - change dist _ _ ≤ _ - rw [physicalExperimentChart, ite_eq_right hwr, ← reflected_positiveCenter (lt_of_not_ge hwr), - loewnerMirror_dist] - linarith - -theorem physicalExperimentChart_avoids_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : - ¬ (|(physicalExperimentChart s w m z).re - w.re| < s / 100 ∧ - w.im - s / 100 < (physicalExperimentChart s w m z).im) := by - by_cases hwr : 0 ≤ w.re - · simpa only [physicalExperimentChart, ite_eq_left hwr, abs_of_nonneg hwr] using - physicalFitChart_avoids_shadow hs w hw hm ha - · have hneg : w.re < 0 := lt_of_not_ge hwr - have hac : |(conj z).im| ≤ 1 := by simpa only [Complex.conj_im, abs_neg] using ha - have hn := physicalFitChart_avoids_shadow hs w hw hm hac - have he : abs (-(physicalFitChart s w m (conj z)).re - w.re) = - abs ((physicalFitChart s w m (conj z)).re - |w.re|) := by - rw [abs_of_neg hneg] - have hh : -(physicalFitChart s w m (conj z)).re - w.re = - -((physicalFitChart s w m (conj z)).re - -w.re) := by ring - rw [hh, abs_neg] - simpa only [physicalExperimentChart, ite_eq_right hwr, loewnerMirror_re, loewnerMirror_im, he] using hn - -theorem selected_middle_regions_inside {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {ε : ℝ} (hε : ε ≤ 1) : - ∀ k ∈ retainedSteps (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m, - physicalMiddleRegion (physicalExperimentChart s w m) closedRibbon k ε ⊆ - closedBall (center s w) (s / 20) := by - intro k hk - obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk - rw [physicalMiddleRegion_eq_image_middleStrip _ (Nat.cast_nonneg _) hε] - rintro y ⟨z, hz, rfl⟩ - exact physicalExperimentChart_retained_disk hs w hw hm j.isLt i.isLt - ⟨by linarith [hz.1], by linarith [hz.2.1]⟩ (hz.2.2.le.trans hε) - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineFittedShadow +import OAI.Probability.SLE.Gauge.SerpentineReflection +import OAI.Probability.SLE.Gauge.ProperRibbonCoordinates + +namespace OAI + +/-! The actual left-or-right displaced serpentine. Reflection is applied +to both the physical image and the normal coordinate, so the initial chart +is still `z ↦ I*z`. -/ + +noncomputable section +open Set Filter Metric ComplexConjugate +open scoped NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +def experimentChart (s : ℝ) (w : ℂ) (m : ℕ) : ℂ → ℂ := + if 0 ≤ w.re then fitChart s w m else reflectedChart (fitChart s w m) + +def experimentSpine (s : ℝ) (w : ℂ) (m : ℕ) (t : ℝ) : ℂ := + if 0 ≤ w.re then fitSpine s w m t else loewnerMirror (fitSpine s w m t) + +def physicalExperimentChart (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : ℂ := + if 0 ≤ w.re then physicalFitChart s w m z else loewnerMirror (physicalFitChart s w m (conj z)) + +lemma loewnerMirror_real_mul (c : ℝ) (z : ℂ) : + loewnerMirror ((c : ℂ) * z) = (c : ℂ) * loewnerMirror z := by + simp [loewnerMirror, mul_neg] + +lemma physicalExperimentChart_eq (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : + physicalExperimentChart s w m z = (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z := by + by_cases hw : 0 ≤ w.re + · simp only [physicalExperimentChart, experimentChart, ite_eq_left hw, physicalFitChart] + · simp only [physicalExperimentChart, experimentChart, ite_eq_right hw, physicalFitChart, + reflectedChart, loewnerMirror_real_mul] + +theorem experimentChart_geometry {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + Continuous (experimentChart s w m) ∧ InjOn (experimentChart s w m) closedRibbon ∧ + (∀ k : ℝ, LipschitzOnWith 2 (experimentChart s w m) (ribbonStepPatch k) ∧ + ∀ z ∈ ribbonStepPatch k, ∀ z' ∈ ribbonStepPatch k, + dist z z' ≤ 2 * dist (experimentChart s w m z) (experimentChart s w m z')) ∧ + (∀ z ∈ closedRibbon, 0 < z.re → experimentChart s w m z ∈ upperHalfPlane) ∧ + (∀ z ∈ closedRibbon, z.re ≤ 4 → experimentChart s w m z = Complex.I * z) := by + have hcont := fitChart_continuous hs w hw hm + have hinj := fitChart_injective hs w hw hm + have hpatch := fitChart_patch_bounds hs w hw hm + have hu := fitChart_upper hs w hw hm + have hi : ∀ z ∈ closedRibbon, z.re ≤ 4 → fitChart s w m z = Complex.I * z := by + intro z _ hz + exact fitChart_initial hs w hw hm (hz.trans (fit_parameters hs w hw hm).2.2.2) + by_cases hwr : 0 ≤ w.re + · simpa only [experimentChart, ite_eq_left hwr] using ⟨hcont, hinj, hpatch, hu, hi⟩ + · simp only [experimentChart, ite_eq_right hwr] + exact ⟨reflectedChart_continuous hcont, reflectedChart_injective hinj, + fun k => ⟨reflectedChart_patch_upper (hpatch k).1, + reflectedChart_patch_lower (hpatch k).2⟩, + reflectedChart_upper hu, reflectedChart_initial hi⟩ + +lemma experimentChart_width (s : ℝ) (w : ℂ) (m : ℕ) {z : ℂ} (hz : z ∈ closedRibbon) : + ‖experimentChart s w m z - experimentSpine s w m z.re‖ ≤ 1 := by + by_cases hwr : 0 ≤ w.re + · simpa only [experimentChart, experimentSpine, ite_eq_left hwr] using fitChart_width s w m hz + · simpa only [experimentChart, experimentSpine, ite_eq_right hwr] using + reflectedChart_width (P := fitChart s w m) (σ := fitSpine s w m) + (fun _ hz => fitChart_width s w m hz) z hz + +lemma experimentSpine_eventually_ray (s : ℝ) (w : ℂ) (m : ℕ) : + ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → + experimentSpine s w m t = experimentSpine s w m T + ((t - T : ℝ) : ℂ) * Complex.I := by + obtain ⟨T, hT, htail⟩ := fitSpine_eventually_ray s w m + refine ⟨T, hT, fun t ht => ?_⟩ + by_cases hwr : 0 ≤ w.re + · simpa only [experimentSpine, ite_eq_left hwr] using htail t ht + · simp only [experimentSpine, ite_eq_right hwr] + rw [htail t ht] + simp [loewnerMirror, mul_neg, add_comm] + +lemma experimentChart_real_norm (s : ℝ) (w : ℂ) (m : ℕ) (t : ℝ) : + ‖experimentChart s w m (t : ℂ)‖ = ‖fitSpine s w m t‖ := by + by_cases hwr : 0 ≤ w.re <;> + simp [experimentChart, hwr, reflectedChart, fitChart, normalRibbon] + +theorem experimentChart_escape (s : ℝ) (w : ℂ) (m : ℕ) : + Tendsto (fun t : ℝ => ‖experimentChart s w m (t : ℂ)‖) atTop atTop := by + simp only [experimentChart_real_norm] + exact tendsto_norm_cocompact_atTop.comp + (completeSpine_escape m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) + +theorem experimentChart_zero {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : experimentChart s w m 0 = 0 := by + have hh := (experimentChart_geometry hs w hw hm).2.2.2.2 0 + (by simp [closedRibbon]) (by norm_num) + simpa only [mul_zero] using hh + +theorem experimentChart_closedEmbedding {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + Topology.IsClosedEmbedding (fun z : closedRibbon => experimentChart s w m z) := by + obtain ⟨hc, hi, _⟩ := experimentChart_geometry hs w hw hm + obtain ⟨T, hT, htail⟩ := experimentSpine_eventually_ray s w m + exact ribbon_closedEmbedding_of_linear_tail _ hc.continuousOn hi _ T hT + (experimentSpine s w m T) htail (fun _ hz => experimentChart_width s w m hz) + +theorem experimentChart_global_progress {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + ∃ g : C(ℂ, ℝ), ∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re := + exists_global_ribbon_progress _ (experimentChart_closedEmbedding hs w hw hm) + +lemma reflected_positiveCenter {s : ℝ} {w : ℂ} (hw : w.re < 0) : + loewnerMirror (positiveCenter s w) = center s w := by + have hn : ¬ 0 ≤ w.re := not_le.mpr hw + apply Complex.ext + · simp only [loewnerMirror_re, positiveCenter, center, Complex.add_re, Complex.mul_re, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + mul_zero, zero_mul, sub_zero, add_zero] + rw [abs_of_neg hw] + simp only [displacementSign, ite_eq_right hn] + ring + · simp only [loewnerMirror_im, positiveCenter, center, Complex.add_im, Complex.mul_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + mul_zero, mul_one, add_zero, zero_add] + +theorem physicalExperimentChart_retained_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} + (hz : z.re ∈ Icc + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) + (ha : |z.im| ≤ 1) : + physicalExperimentChart s w m z ∈ closedBall (center s w) (s / 20) := by + by_cases hwr : 0 ≤ w.re + · simpa only [physicalExperimentChart, ite_eq_left hwr] using + physicalFitChart_retained_disk hs w hw hwr hm hj hi hz ha + · have hzc : (conj z).re ∈ Icc + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3) := by + simpa only [Complex.conj_re] using hz + have hac : |(conj z).im| ≤ 1 := by simpa only [Complex.conj_im, abs_neg] using ha + have hd := physicalFitChart_retained_distance hs w hw hm hj hi hzc hac + change dist _ _ ≤ _ + rw [physicalExperimentChart, ite_eq_right hwr, ← reflected_positiveCenter (lt_of_not_ge hwr), + loewnerMirror_dist] + linarith + +theorem physicalExperimentChart_avoids_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : + ¬ (|(physicalExperimentChart s w m z).re - w.re| < s / 100 ∧ + w.im - s / 100 < (physicalExperimentChart s w m z).im) := by + by_cases hwr : 0 ≤ w.re + · simpa only [physicalExperimentChart, ite_eq_left hwr, abs_of_nonneg hwr] using + physicalFitChart_avoids_shadow hs w hw hm ha + · have hneg : w.re < 0 := lt_of_not_ge hwr + have hac : |(conj z).im| ≤ 1 := by simpa only [Complex.conj_im, abs_neg] using ha + have hn := physicalFitChart_avoids_shadow hs w hw hm hac + have he : abs (-(physicalFitChart s w m (conj z)).re - w.re) = + abs ((physicalFitChart s w m (conj z)).re - |w.re|) := by + rw [abs_of_neg hneg] + have hh : -(physicalFitChart s w m (conj z)).re - w.re = + -((physicalFitChart s w m (conj z)).re - -w.re) := by ring + rw [hh, abs_neg] + simpa only [physicalExperimentChart, ite_eq_right hwr, loewnerMirror_re, loewnerMirror_im, he] using hn + +theorem selected_middle_regions_inside {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {ε : ℝ} (hε : ε ≤ 1) : + ∀ k ∈ retainedSteps (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m, + physicalMiddleRegion (physicalExperimentChart s w m) closedRibbon k ε ⊆ + closedBall (center s w) (s / 20) := by + intro k hk + obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk + rw [physicalMiddleRegion_eq_image_middleStrip _ (Nat.cast_nonneg _) hε] + rintro y ⟨z, hz, rfl⟩ + exact physicalExperimentChart_retained_disk hs w hw hm j.isLt i.isLt + ⟨by linarith [hz.1], by linarith [hz.2.1]⟩ (hz.2.2.le.trans hε) + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineExponentialMass.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineExponentialMass.lean index abc4b152e4d5233d499f517f0e3849645ea3ec11..91ceb626285b7aa5aeebe7969dccc9783f395f9a 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineExponentialMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineExponentialMass.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Gauge.SerpentineNormalizedMass -import OAI.Probability.SLE.Gauge.DenseRibbonNumerics - -namespace OAI - -/-! The concrete stopped dense experiment costs at most exponentially -in m²; the constants are independent of the marked point and mesh. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates Target - -theorem uniform_serpentine_exponential_stopped_mass - {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport μ γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → - ∃ C : ℝ, 0 < C ∧ ∀ v : ℝ, 0 < v → v < v₀ → - ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → - ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ {ω | - ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) (experimentChart s w m) ε ∧ - ((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0 - ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) v (normalizedExperimentDisk s w m)} := by - obtain ⟨q₀, q, c, v₀, hq₀, hq₀1, hq, hq1, hcm, hv₀, hprob⟩ := - uniform_serpentine_normalized_stopped_mass μ κ hκ hκ8 B hB hm hc γ G F hw - hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall - have hq₀fin : q₀ ≠ ⊤ := ne_top_of_le_ne_top (by norm_num : (1 : ℝ≥0∞) ≠ ⊤) hq₀1 - have hp : 0 < q₀.toReal := ENNReal.toReal_pos hq₀.ne' hq₀fin - refine ⟨c, v₀, hcm, hv₀, fun s hs => ?_⟩ - let D := fitEntryBound s + 601 + 100 * Real.pi - have hD : 0 ≤ D := by dsimp only [D]; linarith [fitEntryBound_pos hs, Real.pi_pos] - obtain ⟨C, hC, hcost⟩ := ribbon_initial_and_steps_quadratic_cost hq hq1 hp hD - refine ⟨C, hC, fun v hv hvcut w hwn hwy m hmp g hg => ?_⟩ - have hcount : (experimentSteps s w m : ℝ) ≤ D * (m : ℝ) ^ 2 := by - apply le_trans (Nat.cast_le.mpr (Nat.sub_le _ 1)) - exact fit_completion_quadratic hs w hwn hwy hmp - have hreal := hcost m (experimentSteps s w m) hmp hcount - have hENN := ENNReal.ofReal_le_ofReal hreal - rw [ENNReal.ofReal_mul (pow_nonneg hq.le _), ENNReal.ofReal_pow hq.le, - ENNReal.ofReal_toReal hq₀fin] at hENN - exact hENN.trans (hprob v hv hvcut s hs w hwy m hmp g hg) - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineNormalizedMass +import OAI.Probability.SLE.Gauge.DenseRibbonNumerics + +namespace OAI + +/-! The concrete stopped dense experiment costs at most exponentially +in m²; the constants are independent of the marked point and mesh. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates Target + +theorem uniform_serpentine_exponential_stopped_mass + {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport μ γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → + ∃ C : ℝ, 0 < C ∧ ∀ v : ℝ, 0 < v → v < v₀ → + ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → + ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ {ω | + ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) (experimentChart s w m) ε ∧ + ((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0 + ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) v (normalizedExperimentDisk s w m)} := by + obtain ⟨q₀, q, c, v₀, hq₀, hq₀1, hq, hq1, hcm, hv₀, hprob⟩ := + uniform_serpentine_normalized_stopped_mass μ κ hκ hκ8 B hB hm hc γ G F hw + hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall + have hq₀fin : q₀ ≠ ⊤ := ne_top_of_le_ne_top (by norm_num : (1 : ℝ≥0∞) ≠ ⊤) hq₀1 + have hp : 0 < q₀.toReal := ENNReal.toReal_pos hq₀.ne' hq₀fin + refine ⟨c, v₀, hcm, hv₀, fun s hs => ?_⟩ + let D := fitEntryBound s + 601 + 100 * Real.pi + have hD : 0 ≤ D := by dsimp only [D]; linarith [fitEntryBound_pos hs, Real.pi_pos] + obtain ⟨C, hC, hcost⟩ := ribbon_initial_and_steps_quadratic_cost hq hq1 hp hD + refine ⟨C, hC, fun v hv hvcut w hwn hwy m hmp g hg => ?_⟩ + have hcount : (experimentSteps s w m : ℝ) ≤ D * (m : ℝ) ^ 2 := by + apply le_trans (Nat.cast_le.mpr (Nat.sub_le _ 1)) + exact fit_completion_quadratic hs w hwn hwy hmp + have hreal := hcost m (experimentSteps s w m) hmp hcount + have hENN := ENNReal.ofReal_le_ofReal hreal + rw [ENNReal.ofReal_mul (pow_nonneg hq.le _), ENNReal.ofReal_pow hq.le, + ENNReal.ofReal_toReal hq₀fin] at hENN + exact hENN.trans (hprob v hv hvcut s hs w hwy m hmp g hg) + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineExteriorBounds.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineExteriorBounds.lean index 397bb325a7f32569be2ea3b6e19201923eba98ec..9ada52e03ed0d9482e69e1d23b82f7b0677d2b27 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineExteriorBounds.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineExteriorBounds.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Gauge.SerpentineHorizontalBound - -namespace OAI - -/-! The physical experiment lies strictly between two fixed exterior -vertical lines. These lines provide unobstructed real-boundary exits. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma physicalFitChart_re_bounds {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) - {z : ℂ} (hz : z ∈ closedRibbon) : - -2 ≤ (physicalFitChart s w m z).re ∧ (physicalFitChart s w m z).re ≤ 2 := by - obtain ⟨hr, hx, _, _⟩ := fit_parameters hs w hw hm - have hH : 0 ≤ fitHeight m := by unfold fitHeight; positivity - have hℓ := (fit_width_pos hs hm).le - have hℓ1 := fit_width_le hs hm - have hX := fit_scaledX hs w hm - have hwr := (Complex.abs_re_le_norm w).trans hw1 - let A := connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - by_cases hA : z.re ≤ A - · have hb := lowConnector_normal_re_bounds hH (by linarith : 0 < fitRadius m) hx hz - have hlo := mul_le_mul_of_nonneg_left hb.1 hℓ - have hhi := mul_le_mul_of_nonneg_left hb.2 hℓ - change -2 ≤ ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ∧ - ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ≤ 2 - rw [complete_normal_of_le m _ _ _ _ hA] - simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero] - constructor <;> nlinarith - · have hz' : 0 ≤ (z - (A : ℂ)).re ∧ |(z - (A : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by linarith, hz.2⟩ - have hlo := mul_le_mul_of_nonneg_left (array_normal_re_lower hm hz') hℓ - have hhi := mul_le_mul_of_nonneg_left (array_normal_re_upper hm hz') hℓ - have hcenter := congrArg Complex.re (fit_scaled_arrayCenter hs w hm) - simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, - positiveCenter, Complex.add_re, Complex.mul_re, Complex.I_re, Complex.I_im, - mul_zero, sub_zero, add_zero] at hcenter - have hmℓ := fit_width_mul hs hm - change -2 ≤ ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ∧ - ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ≤ 2 - rw [complete_normal_of_gt m _ _ _ _ (lt_of_not_ge hA)] - simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, - Complex.add_re] - dsimp only [A] at hlo hhi - constructor <;> nlinarith [abs_nonneg w.re] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineHorizontalBound + +namespace OAI + +/-! The physical experiment lies strictly between two fixed exterior +vertical lines. These lines provide unobstructed real-boundary exits. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma physicalFitChart_re_bounds {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) + {z : ℂ} (hz : z ∈ closedRibbon) : + -2 ≤ (physicalFitChart s w m z).re ∧ (physicalFitChart s w m z).re ≤ 2 := by + obtain ⟨hr, hx, _, _⟩ := fit_parameters hs w hw hm + have hH : 0 ≤ fitHeight m := by unfold fitHeight; positivity + have hℓ := (fit_width_pos hs hm).le + have hℓ1 := fit_width_le hs hm + have hX := fit_scaledX hs w hm + have hwr := (Complex.abs_re_le_norm w).trans hw1 + let A := connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + by_cases hA : z.re ≤ A + · have hb := lowConnector_normal_re_bounds hH (by linarith : 0 < fitRadius m) hx hz + have hlo := mul_le_mul_of_nonneg_left hb.1 hℓ + have hhi := mul_le_mul_of_nonneg_left hb.2 hℓ + change -2 ≤ ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ∧ + ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ≤ 2 + rw [complete_normal_of_le m _ _ _ _ hA] + simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero] + constructor <;> nlinarith + · have hz' : 0 ≤ (z - (A : ℂ)).re ∧ |(z - (A : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by linarith, hz.2⟩ + have hlo := mul_le_mul_of_nonneg_left (array_normal_re_lower hm hz') hℓ + have hhi := mul_le_mul_of_nonneg_left (array_normal_re_upper hm hz') hℓ + have hcenter := congrArg Complex.re (fit_scaled_arrayCenter hs w hm) + simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, + positiveCenter, Complex.add_re, Complex.mul_re, Complex.I_re, Complex.I_im, + mul_zero, sub_zero, add_zero] at hcenter + have hmℓ := fit_width_mul hs hm + change -2 ≤ ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ∧ + ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re ≤ 2 + rw [complete_normal_of_gt m _ _ _ _ (lt_of_not_ge hA)] + simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, + Complex.add_re] + dsimp only [A] at hlo hhi + constructor <;> nlinarith [abs_nonneg w.re] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineFiniteHeight.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineFiniteHeight.lean index 2d2d5101af9a5fc2e08f900659376a67ba8480bd..76369610668f18f3eab48eff4010d9fa05f4d17a 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineFiniteHeight.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineFiniteHeight.lean @@ -1,122 +1,122 @@ -import OAI.Probability.SLE.Gauge.SerpentineTerminalGuide - -namespace OAI - -/-! Every point in the finite experiment, before its final integer front, -is below the height used to attach the marked hook. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma array_normal_im_upper_before_end {m : ℕ} (hm : 0 < m) {z : ℂ} - (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) (hze : z.re ≤ arrayEnd m (2 * m) + 1) : - (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).im ≤ 401 * m := by - have hh : (0 : ℝ) < 100 * m := by positivity - by_cases he : z.re ≤ arrayEnd m (2 * m) - · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] - rw [hs, hT, normal_straight_eq hm] - have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 - rw [dist_zero_right] at hb - exact (Complex.im_le_norm _).trans (by linarith) - · simp only [normalRibbon] - rw [hs, hT, normal_circular_eq hm] - have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 - rw [dist_zero_right] at hb - exact (Complex.im_le_norm _).trans (by linarith) - · have hE : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge he - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have ht := array_normal_terminal hm (a := z.im) hE - rw [Complex.re_add_im z] at ht - rw [ht, columnPoint_im] - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - dsimp only [arrayEnd] at hze - simp only [Complex.zero_im, zero_add] - linarith - -lemma lowConnector_im_mono {r X : ℝ} (hr : 0 < r) (hX : 2 * r < X) (H : ℝ) : - Monotone (fun t => (lowConnector H r X t).im) := - monotone_of_hasDerivAt_nonneg - (fun t => Complex.imCLM.hasFDerivAt.comp_hasDerivAt t (lowConnector_hasDerivAt hr hX t)) - (fun t => (lowConnectorTangent_nonneg (H := H) hr hX t).2) - -lemma physicalFitChart_before_entry_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) (hz : z.re ≤ fitEntry s w m) : - (physicalFitChart s w m z).im ≤ w.im := by - obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm - have hA4 : connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) < fitEntry s w m := by - unfold fitEntry connectorEntry - linarith [Nat.cast_nonneg (α := ℝ) m] - have hcenter := lowConnector_im_mono (by linarith : 0 < fitRadius m) hx (fitHeight m) hz - dsimp only at hcenter - rw [(lowConnector_after hA4).1] at hcenter - simp only [connectorTail, Complex.add_im, Complex.mul_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add] at hcenter - have hAeq : fitHeight m + 2 * fitRadius m + - (fitEntry s w m - connectorFourth (fitHeight m) (fitRadius m) (fitX s w m)) = fitY s w m := by - unfold fitEntry connectorEntry - ring - rw [hAeq] at hcenter - have hnormal : (normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) - (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z).im ≤ fitY s w m + 1 := by - have hb : ‖normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) - (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z - - lowConnector (fitHeight m) (fitRadius m) (fitX s w m) z.re‖ ≤ 1 := by - simp only [normalRibbon, add_sub_cancel_left, norm_mul, Complex.norm_real, - Real.norm_eq_abs, Complex.norm_I, lowConnectorTangent_norm, mul_one] - exact ha - have hi := (Complex.abs_im_le_norm _).trans hb - rw [Complex.sub_im] at hi - linarith [le_abs_self ((normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) - (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z).im - - (lowConnector (fitHeight m) (fitRadius m) (fitX s w m) z.re).im)] - have hh := mul_le_mul_of_nonneg_left hnormal (fit_width_pos hs hm).le - have hY := fit_scaledY hs w hm - have hwidth := fit_width_le hs hm - change ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im ≤ _ - rw [complete_normal_of_le m _ _ _ _ hz] - simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] - nlinarith - -lemma physicalFitChart_finite_height_lt_two {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) - {z : ℂ} (hz : z ∈ closedRibbon) (hzK : z.re ≤ fitCompletion s w m) : - (physicalFitChart s w m z).im < 2 := by - have hwi := (Complex.im_le_norm w).trans hw1 - by_cases hA : z.re ≤ fitEntry s w m - · exact (physicalFitChart_before_entry_height hs w hw hm hz.2 hA).trans_lt (by linarith) - · have hz' : 0 ≤ (z - (fitEntry s w m : ℂ)).re ∧ - |(z - (fitEntry s w m : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by linarith, hz.2⟩ - have hze : (z - (fitEntry s w m : ℂ)).re ≤ arrayEnd m (2 * m) + 1 := by - have hh := (fitCompletion_bounds hs w hw hm).2 - simp only [Complex.sub_re, Complex.ofReal_re, arrayEnd] - linarith - have hb := mul_le_mul_of_nonneg_left (array_normal_im_upper_before_end hm hz' hze) - (fit_width_pos hs hm).le - have hmℓ := fit_width_mul hs hm - have hcenter := congrArg Complex.im (fit_scaled_arrayCenter hs w hm) - simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero, - positiveCenter, Complex.add_im, Complex.mul_im, Complex.I_re, Complex.I_im, - mul_one, mul_zero, zero_add] at hcenter - change ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im < _ - rw [complete_normal_of_gt m _ _ _ _ (lt_of_not_ge hA)] - simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero, - Complex.add_im] - unfold fitEntry at hb - nlinarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineTerminalGuide + +namespace OAI + +/-! Every point in the finite experiment, before its final integer front, +is below the height used to attach the marked hook. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma array_normal_im_upper_before_end {m : ℕ} (hm : 0 < m) {z : ℂ} + (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) (hze : z.re ≤ arrayEnd m (2 * m) + 1) : + (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).im ≤ 401 * m := by + have hh : (0 : ℝ) < 100 * m := by positivity + by_cases he : z.re ≤ arrayEnd m (2 * m) + · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] + rw [hs, hT, normal_straight_eq hm] + have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 + rw [dist_zero_right] at hb + exact (Complex.im_le_norm _).trans (by linarith) + · simp only [normalRibbon] + rw [hs, hT, normal_circular_eq hm] + have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 + rw [dist_zero_right] at hb + exact (Complex.im_le_norm _).trans (by linarith) + · have hE : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge he + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have ht := array_normal_terminal hm (a := z.im) hE + rw [Complex.re_add_im z] at ht + rw [ht, columnPoint_im] + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + dsimp only [arrayEnd] at hze + simp only [Complex.zero_im, zero_add] + linarith + +lemma lowConnector_im_mono {r X : ℝ} (hr : 0 < r) (hX : 2 * r < X) (H : ℝ) : + Monotone (fun t => (lowConnector H r X t).im) := + monotone_of_hasDerivAt_nonneg + (fun t => Complex.imCLM.hasFDerivAt.comp_hasDerivAt t (lowConnector_hasDerivAt hr hX t)) + (fun t => (lowConnectorTangent_nonneg (H := H) hr hX t).2) + +lemma physicalFitChart_before_entry_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) (hz : z.re ≤ fitEntry s w m) : + (physicalFitChart s w m z).im ≤ w.im := by + obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm + have hA4 : connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) < fitEntry s w m := by + unfold fitEntry connectorEntry + linarith [Nat.cast_nonneg (α := ℝ) m] + have hcenter := lowConnector_im_mono (by linarith : 0 < fitRadius m) hx (fitHeight m) hz + dsimp only at hcenter + rw [(lowConnector_after hA4).1] at hcenter + simp only [connectorTail, Complex.add_im, Complex.mul_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add] at hcenter + have hAeq : fitHeight m + 2 * fitRadius m + + (fitEntry s w m - connectorFourth (fitHeight m) (fitRadius m) (fitX s w m)) = fitY s w m := by + unfold fitEntry connectorEntry + ring + rw [hAeq] at hcenter + have hnormal : (normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) + (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z).im ≤ fitY s w m + 1 := by + have hb : ‖normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) + (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z - + lowConnector (fitHeight m) (fitRadius m) (fitX s w m) z.re‖ ≤ 1 := by + simp only [normalRibbon, add_sub_cancel_left, norm_mul, Complex.norm_real, + Real.norm_eq_abs, Complex.norm_I, lowConnectorTangent_norm, mul_one] + exact ha + have hi := (Complex.abs_im_le_norm _).trans hb + rw [Complex.sub_im] at hi + linarith [le_abs_self ((normalRibbon (lowConnector (fitHeight m) (fitRadius m) (fitX s w m)) + (lowConnectorTangent (fitHeight m) (fitRadius m) (fitX s w m)) z).im - + (lowConnector (fitHeight m) (fitRadius m) (fitX s w m) z.re).im)] + have hh := mul_le_mul_of_nonneg_left hnormal (fit_width_pos hs hm).le + have hY := fit_scaledY hs w hm + have hwidth := fit_width_le hs hm + change ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im ≤ _ + rw [complete_normal_of_le m _ _ _ _ hz] + simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] + nlinarith + +lemma physicalFitChart_finite_height_lt_two {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) + {z : ℂ} (hz : z ∈ closedRibbon) (hzK : z.re ≤ fitCompletion s w m) : + (physicalFitChart s w m z).im < 2 := by + have hwi := (Complex.im_le_norm w).trans hw1 + by_cases hA : z.re ≤ fitEntry s w m + · exact (physicalFitChart_before_entry_height hs w hw hm hz.2 hA).trans_lt (by linarith) + · have hz' : 0 ≤ (z - (fitEntry s w m : ℂ)).re ∧ + |(z - (fitEntry s w m : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by linarith, hz.2⟩ + have hze : (z - (fitEntry s w m : ℂ)).re ≤ arrayEnd m (2 * m) + 1 := by + have hh := (fitCompletion_bounds hs w hw hm).2 + simp only [Complex.sub_re, Complex.ofReal_re, arrayEnd] + linarith + have hb := mul_le_mul_of_nonneg_left (array_normal_im_upper_before_end hm hz' hze) + (fit_width_pos hs hm).le + have hmℓ := fit_width_mul hs hm + have hcenter := congrArg Complex.im (fit_scaled_arrayCenter hs w hm) + simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero, + positiveCenter, Complex.add_im, Complex.mul_im, Complex.I_re, Complex.I_im, + mul_one, mul_zero, zero_add] at hcenter + change ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im < _ + rw [complete_normal_of_gt m _ _ _ _ (lt_of_not_ge hA)] + simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero, + Complex.add_im] + unfold fitEntry at hb + nlinarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedChart.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedChart.lean index d6ff403152a56a56e4490bf1cbd737eb65d29f86..6ccb7fd9be9c955f5f95193f238f116a159e996b 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedChart.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedChart.lean @@ -1,100 +1,100 @@ -import OAI.Probability.SLE.Gauge.SerpentineParameters -import OAI.Probability.SLE.Gauge.SerpentineUpperHalfPlane -import OAI.Probability.SLE.Gauge.SerpentineRetainedStrips - -namespace OAI - -/-! The concrete normalized chart, with the universal local constants -used by the ribbon iteration. The physical width is recorded separately. -/ - -noncomputable section -open Set Metric -open scoped NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -def fitSpine (s : ℝ) (w : ℂ) (m : ℕ) : ℝ → ℂ := - completeSpine m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - -def fitTangent (s : ℝ) (w : ℂ) (m : ℕ) : ℝ → ℂ := - completeTangent m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - -def fitChart (s : ℝ) (w : ℂ) (m : ℕ) : ℂ → ℂ := - normalRibbon (fitSpine s w m) (fitTangent s w m) - -def positiveCenter (s : ℝ) (w : ℂ) : ℂ := - ((|w.re| + s / 4 : ℝ) : ℂ) + (w.im : ℂ) * Complex.I - -lemma fitChart_continuous {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : Continuous (fitChart s w m) := by - obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm - exact complete_normal_continuous m hr hx (by linarith [Nat.cast_nonneg (α := ℝ) m]) - -lemma fitChart_injective {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : InjOn (fitChart s w m) closedRibbon := by - obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm - exact complete_normal_injective hm hr hx hy - -lemma fitChart_patch_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (k : ℝ) : - LipschitzOnWith 2 (fitChart s w m) (ribbonStepPatch k) ∧ - ∀ z ∈ ribbonStepPatch k, ∀ z' ∈ ribbonStepPatch k, - dist z z' ≤ 2 * dist (fitChart s w m z) (fitChart s w m z') := by - obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm - exact complete_normal_patch_bounds m hr hx (by linarith [Nat.cast_nonneg (α := ℝ) m]) k - -lemma fitChart_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - ∀ z ∈ closedRibbon, 0 < z.re → fitChart s w m z ∈ upperHalfPlane := by - obtain ⟨hr, hx, hy, hH⟩ := fit_parameters hs w hw hm - intro z hz hz0 - exact complete_normal_upper hm (by linarith) (by linarith) hx - (by linarith [Nat.cast_nonneg (α := ℝ) m]) hz0 hz.2 - -lemma fitChart_initial {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : z.re ≤ fitHeight m) : - fitChart s w m z = Complex.I * z := by - obtain ⟨hr, hx, _, _⟩ := fit_parameters hs w hw hm - have hA := fit_entry_lower hs w hw hm - change normalRibbon (completeSpine m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) - (completeTangent m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) z = _ - rw [complete_normal_of_le m _ _ _ _ (hz.trans hA)] - obtain ⟨hσ, hT⟩ := lowConnector_initial (by linarith : 0 < fitRadius m) hx hz - rw [normalRibbon, hσ, hT] - apply Complex.ext <;> - simp only [Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im] <;> ring - -lemma fitChart_width (s : ℝ) (w : ℂ) (m : ℕ) {z : ℂ} (hz : z ∈ closedRibbon) : - ‖fitChart s w m z - fitSpine s w m z.re‖ ≤ 1 := - complete_normal_width m _ _ _ _ hz.2 - -lemma fitSpine_eventually_ray (s : ℝ) (w : ℂ) (m : ℕ) : - ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → - fitSpine s w m t = fitSpine s w m T + ((t - T : ℝ) : ℂ) * Complex.I := - complete_eventually_ray m _ _ _ _ - -lemma fit_scaled_arrayCenter {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - (ribbonWidth (halfHeight s) m : ℂ) * arrayCenter m (fitX s w m) (fitY s w m) = - positiveCenter s w := by - have hℓ := (fit_width_pos hs hm).ne' - have hmul := fit_width_mul hs hm - apply Complex.ext <;> - simp only [arrayCenter, positiveCenter, Complex.add_re, Complex.add_im, - Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero, zero_add] - · unfold fitX - rw [mul_add, mul_div_cancel₀ _ hℓ] - unfold halfHeight at * - nlinarith - · unfold fitY - rw [mul_add, mul_div_cancel₀ _ hℓ] - unfold halfHeight at * - nlinarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineParameters +import OAI.Probability.SLE.Gauge.SerpentineUpperHalfPlane +import OAI.Probability.SLE.Gauge.SerpentineRetainedStrips + +namespace OAI + +/-! The concrete normalized chart, with the universal local constants +used by the ribbon iteration. The physical width is recorded separately. -/ + +noncomputable section +open Set Metric +open scoped NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +def fitSpine (s : ℝ) (w : ℂ) (m : ℕ) : ℝ → ℂ := + completeSpine m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + +def fitTangent (s : ℝ) (w : ℂ) (m : ℕ) : ℝ → ℂ := + completeTangent m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + +def fitChart (s : ℝ) (w : ℂ) (m : ℕ) : ℂ → ℂ := + normalRibbon (fitSpine s w m) (fitTangent s w m) + +def positiveCenter (s : ℝ) (w : ℂ) : ℂ := + ((|w.re| + s / 4 : ℝ) : ℂ) + (w.im : ℂ) * Complex.I + +lemma fitChart_continuous {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : Continuous (fitChart s w m) := by + obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm + exact complete_normal_continuous m hr hx (by linarith [Nat.cast_nonneg (α := ℝ) m]) + +lemma fitChart_injective {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : InjOn (fitChart s w m) closedRibbon := by + obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm + exact complete_normal_injective hm hr hx hy + +lemma fitChart_patch_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (k : ℝ) : + LipschitzOnWith 2 (fitChart s w m) (ribbonStepPatch k) ∧ + ∀ z ∈ ribbonStepPatch k, ∀ z' ∈ ribbonStepPatch k, + dist z z' ≤ 2 * dist (fitChart s w m z) (fitChart s w m z') := by + obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm + exact complete_normal_patch_bounds m hr hx (by linarith [Nat.cast_nonneg (α := ℝ) m]) k + +lemma fitChart_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + ∀ z ∈ closedRibbon, 0 < z.re → fitChart s w m z ∈ upperHalfPlane := by + obtain ⟨hr, hx, hy, hH⟩ := fit_parameters hs w hw hm + intro z hz hz0 + exact complete_normal_upper hm (by linarith) (by linarith) hx + (by linarith [Nat.cast_nonneg (α := ℝ) m]) hz0 hz.2 + +lemma fitChart_initial {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : z.re ≤ fitHeight m) : + fitChart s w m z = Complex.I * z := by + obtain ⟨hr, hx, _, _⟩ := fit_parameters hs w hw hm + have hA := fit_entry_lower hs w hw hm + change normalRibbon (completeSpine m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) + (completeTangent m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) z = _ + rw [complete_normal_of_le m _ _ _ _ (hz.trans hA)] + obtain ⟨hσ, hT⟩ := lowConnector_initial (by linarith : 0 < fitRadius m) hx hz + rw [normalRibbon, hσ, hT] + apply Complex.ext <;> + simp only [Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im] <;> ring + +lemma fitChart_width (s : ℝ) (w : ℂ) (m : ℕ) {z : ℂ} (hz : z ∈ closedRibbon) : + ‖fitChart s w m z - fitSpine s w m z.re‖ ≤ 1 := + complete_normal_width m _ _ _ _ hz.2 + +lemma fitSpine_eventually_ray (s : ℝ) (w : ℂ) (m : ℕ) : + ∃ T : ℝ, 0 ≤ T ∧ ∀ t : ℝ, T ≤ t → + fitSpine s w m t = fitSpine s w m T + ((t - T : ℝ) : ℂ) * Complex.I := + complete_eventually_ray m _ _ _ _ + +lemma fit_scaled_arrayCenter {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + (ribbonWidth (halfHeight s) m : ℂ) * arrayCenter m (fitX s w m) (fitY s w m) = + positiveCenter s w := by + have hℓ := (fit_width_pos hs hm).ne' + have hmul := fit_width_mul hs hm + apply Complex.ext <;> + simp only [arrayCenter, positiveCenter, Complex.add_re, Complex.add_im, + Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero, zero_add] + · unfold fitX + rw [mul_add, mul_div_cancel₀ _ hℓ] + unfold halfHeight at * + nlinarith + · unfold fitY + rw [mul_add, mul_div_cancel₀ _ hℓ] + unfold halfHeight at * + nlinarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedRegion.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedRegion.lean index 581f236546975789a689e8b42793b52bfb18bfe3..de7f4617591914a44ada2c309be502cf5d37c495 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedRegion.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedRegion.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Gauge.SerpentineFittedChart -import OAI.Probability.SLE.Gauge.SerpentineArrayRegion - -namespace OAI - -/-! The retained integer strips lie in the prescribed displaced disk -after multiplication by the physical width. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -def physicalFitChart (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : ℂ := - (ribbonWidth (halfHeight s) m : ℂ) * fitChart s w m z - -lemma fitChart_retained_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} - (hz : z.re ∈ Icc - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) - (ha : |z.im| ≤ 1) : - dist (fitChart s w m z) (arrayCenter m (fitX s w m) (fitY s w m)) ≤ 300 * m := by - have hA : 0 ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := - (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hw hm) - have hmid := retained_middle_in_straight hA hi hz - have he : columnEntry (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j = - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + columnEntry 0 m j := by - unfold columnEntry - ring - rw [he] at hmid - have hu : z.re - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - - columnEntry 0 m j ∈ Icc (0 : ℝ) (200 * m) := ⟨by linarith [hmid.1], by linarith [hmid.2]⟩ - unfold fitChart fitSpine fitTangent - rw [retained_strip_on_column hm hj hi _ _ _ _ hA hz, dist_self_add_left] - have hb := columnPoint_bound (j := j) (by positivity : (0 : ℝ) < 100 * m) hm (by omega) 0 - (straight_height_bound m j hu) ha - rw [dist_zero_right] at hb - linarith - -lemma physicalFitChart_retained_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} - (hz : z.re ∈ Icc - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) - (ha : |z.im| ≤ 1) : - dist (physicalFitChart s w m z) (positiveCenter s w) ≤ 3 * s / 200 := by - have hd := fitChart_retained_distance hs w hw hm hj hi hz ha - have hℓ := fit_width_pos hs hm - have hmul := fit_width_mul hs hm - rw [physicalFitChart, ← fit_scaled_arrayCenter hs w hm, dist_eq_norm, ← mul_sub, norm_mul, - Complex.norm_real, Real.norm_eq_abs, abs_of_pos hℓ] - have h := mul_le_mul_of_nonneg_left hd hℓ.le - rw [dist_eq_norm] at h - nlinarith - -lemma positiveCenter_eq_center {s : ℝ} {w : ℂ} (hw : 0 ≤ w.re) : - positiveCenter s w = center s w := by - apply Complex.ext <;> - simp [positiveCenter, center, displacementSign, hw, abs_of_nonneg hw] - -lemma physicalFitChart_retained_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} - (hz : z.re ∈ Icc - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) - (ha : |z.im| ≤ 1) : - physicalFitChart s w m z ∈ closedBall (center s w) (s / 20) := by - have hd := physicalFitChart_retained_distance hs w hw hm hj hi hz ha - rw [positiveCenter_eq_center hwr] at hd - change dist _ _ ≤ _ - linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineFittedChart +import OAI.Probability.SLE.Gauge.SerpentineArrayRegion + +namespace OAI + +/-! The retained integer strips lie in the prescribed displaced disk +after multiplication by the physical width. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +def physicalFitChart (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : ℂ := + (ribbonWidth (halfHeight s) m : ℂ) * fitChart s w m z + +lemma fitChart_retained_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} + (hz : z.re ∈ Icc + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) + (ha : |z.im| ≤ 1) : + dist (fitChart s w m z) (arrayCenter m (fitX s w m) (fitY s w m)) ≤ 300 * m := by + have hA : 0 ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := + (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hw hm) + have hmid := retained_middle_in_straight hA hi hz + have he : columnEntry (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j = + connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + columnEntry 0 m j := by + unfold columnEntry + ring + rw [he] at hmid + have hu : z.re - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - + columnEntry 0 m j ∈ Icc (0 : ℝ) (200 * m) := ⟨by linarith [hmid.1], by linarith [hmid.2]⟩ + unfold fitChart fitSpine fitTangent + rw [retained_strip_on_column hm hj hi _ _ _ _ hA hz, dist_self_add_left] + have hb := columnPoint_bound (j := j) (by positivity : (0 : ℝ) < 100 * m) hm (by omega) 0 + (straight_height_bound m j hu) ha + rw [dist_zero_right] at hb + linarith + +lemma physicalFitChart_retained_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} + (hz : z.re ∈ Icc + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) + (ha : |z.im| ≤ 1) : + dist (physicalFitChart s w m z) (positiveCenter s w) ≤ 3 * s / 200 := by + have hd := fitChart_retained_distance hs w hw hm hj hi hz ha + have hℓ := fit_width_pos hs hm + have hmul := fit_width_mul hs hm + rw [physicalFitChart, ← fit_scaled_arrayCenter hs w hm, dist_eq_norm, ← mul_sub, norm_mul, + Complex.norm_real, Real.norm_eq_abs, abs_of_pos hℓ] + have h := mul_le_mul_of_nonneg_left hd hℓ.le + rw [dist_eq_norm] at h + nlinarith + +lemma positiveCenter_eq_center {s : ℝ} {w : ℂ} (hw : 0 ≤ w.re) : + positiveCenter s w = center s w := by + apply Complex.ext <;> + simp [positiveCenter, center, displacementSign, hw, abs_of_nonneg hw] + +lemma physicalFitChart_retained_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) {z : ℂ} + (hz : z.re ∈ Icc + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m j i : ℝ) + 2 / 3)) + (ha : |z.im| ≤ 1) : + physicalFitChart s w m z ∈ closedBall (center s w) (s / 20) := by + have hd := physicalFitChart_retained_distance hs w hw hm hj hi hz ha + rw [positiveCenter_eq_center hwr] at hd + change dist _ _ ≤ _ + linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedShadow.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedShadow.lean index 1e13aa1d121f454849d4183ed481c8f543dcccfe..425b6b2929ef983d7aeebaba8f5fcdefea8112fa 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineFittedShadow.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineFittedShadow.lean @@ -1,108 +1,108 @@ -import OAI.Probability.SLE.Gauge.SerpentineFittedRegion - -namespace OAI - -/-! The complete fitted ribbon avoids a fixed upward shadow of the -marked point, including the connector and the infinite final tail. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma fit_width_le {s : ℝ} (hs : 0 < s) {m : ℕ} (hm : 0 < m) : - ribbonWidth (halfHeight s) m ≤ s / 20000 := by - have hmul := fit_width_mul hs hm - have hp := (fit_width_pos hs hm).le - have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm - nlinarith - -lemma fit_scaledX {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - ribbonWidth (halfHeight s) m * fitX s w m = |w.re| + s / 4 - s / 200 := by - unfold fitX halfHeight - exact mul_div_cancel₀ _ (fit_width_pos hs hm).ne' - -lemma physicalFitChart_low {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) - (hz : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m)) : - (physicalFitChart s w m z).im ≤ s / 2 := by - obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm - have hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := by - have hY : fitHeight m + 2 * fitRadius m < fitY s w m := by - linarith [Nat.cast_nonneg (α := ℝ) m] - unfold connectorEntry - linarith - have hlow := lowConnector_normal_im_upper (by linarith : 1 < fitRadius m) hx hz ha - have hℓ := (fit_width_pos hs hm).le - have hmul := fit_width_mul hs hm - have hle := fit_width_le hs hm - have hscaled := mul_le_mul_of_nonneg_left hlow hℓ - change ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im ≤ _ - rw [complete_normal_of_le m _ _ _ _ hA] - simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] - dsimp only [fitHeight, fitRadius] at hscaled ⊢ - nlinarith - -lemma physicalFitChart_tail_right {s : ℝ} (hs : 0 < s) (w : ℂ) (_hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) - (hz : connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) < z.re) - (hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) : - s / 5 ≤ (physicalFitChart s w m z).re - |w.re| := by - have hℓ := (fit_width_pos hs hm).le - have hx := fit_scaledX hs w hm - have hle := fit_width_le hs hm - have hza := mul_le_mul_of_nonneg_left (abs_le.mp ha).2 hℓ - change ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re - _ ≥ _ - rw [complete_normal_of_le m _ _ _ _ hA, normalRibbon, - (lowConnector_after hz).1, (lowConnector_after hz).2] - simp only [connectorTail, Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - mul_zero, zero_mul, mul_one, sub_zero, add_zero, zero_add] - nlinarith - -lemma physicalFitChart_array_right {s : ℝ} (hs : 0 < s) (w : ℂ) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) - (hA : connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) < z.re) : - s / 5 ≤ (physicalFitChart s w m z).re - |w.re| := by - have hℓ := (fit_width_pos hs hm).le - have hmul := fit_width_mul hs hm - let A := connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - have hz' : 0 ≤ (z - (A : ℂ)).re ∧ |(z - (A : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by dsimp [A]; linarith, ha⟩ - have hbound := mul_le_mul_of_nonneg_left (array_normal_re_lower hm hz') hℓ - have hcenter := congrArg Complex.re (fit_scaled_arrayCenter hs w hm) - simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, - positiveCenter, Complex.add_re, Complex.mul_re, Complex.I_re, Complex.I_im, - mul_zero, sub_zero, add_zero] at hcenter - change ((ribbonWidth (halfHeight s) m : ℂ) * - normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re - _ ≥ _ - rw [complete_normal_of_gt m _ _ _ _ hA] - simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, - Complex.add_re] - dsimp [A] at hbound - nlinarith - -theorem physicalFitChart_avoids_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : - ¬ (abs ((physicalFitChart s w m z).re - |w.re|) < s / 100 ∧ - w.im - s / 100 < (physicalFitChart s w m z).im) := by - intro h - by_cases h4 : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) - · have hh := physicalFitChart_low hs w hw hm ha h4 - linarith [h.2] - · by_cases hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - · have hh := physicalFitChart_tail_right hs w hw hm ha (lt_of_not_ge h4) hA - have hl := le_abs_self ((physicalFitChart s w m z).re - |w.re|) - linarith [h.1] - · have hh := physicalFitChart_array_right hs w hm ha (lt_of_not_ge hA) - have hl := le_abs_self ((physicalFitChart s w m z).re - |w.re|) - linarith [h.1] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineFittedRegion + +namespace OAI + +/-! The complete fitted ribbon avoids a fixed upward shadow of the +marked point, including the connector and the infinite final tail. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma fit_width_le {s : ℝ} (hs : 0 < s) {m : ℕ} (hm : 0 < m) : + ribbonWidth (halfHeight s) m ≤ s / 20000 := by + have hmul := fit_width_mul hs hm + have hp := (fit_width_pos hs hm).le + have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm + nlinarith + +lemma fit_scaledX {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + ribbonWidth (halfHeight s) m * fitX s w m = |w.re| + s / 4 - s / 200 := by + unfold fitX halfHeight + exact mul_div_cancel₀ _ (fit_width_pos hs hm).ne' + +lemma physicalFitChart_low {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) + (hz : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m)) : + (physicalFitChart s w m z).im ≤ s / 2 := by + obtain ⟨hr, hx, hy, _⟩ := fit_parameters hs w hw hm + have hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := by + have hY : fitHeight m + 2 * fitRadius m < fitY s w m := by + linarith [Nat.cast_nonneg (α := ℝ) m] + unfold connectorEntry + linarith + have hlow := lowConnector_normal_im_upper (by linarith : 1 < fitRadius m) hx hz ha + have hℓ := (fit_width_pos hs hm).le + have hmul := fit_width_mul hs hm + have hle := fit_width_le hs hm + have hscaled := mul_le_mul_of_nonneg_left hlow hℓ + change ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).im ≤ _ + rw [complete_normal_of_le m _ _ _ _ hA] + simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] + dsimp only [fitHeight, fitRadius] at hscaled ⊢ + nlinarith + +lemma physicalFitChart_tail_right {s : ℝ} (hs : 0 < s) (w : ℂ) (_hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) + (hz : connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) < z.re) + (hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) : + s / 5 ≤ (physicalFitChart s w m z).re - |w.re| := by + have hℓ := (fit_width_pos hs hm).le + have hx := fit_scaledX hs w hm + have hle := fit_width_le hs hm + have hza := mul_le_mul_of_nonneg_left (abs_le.mp ha).2 hℓ + change ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re - _ ≥ _ + rw [complete_normal_of_le m _ _ _ _ hA, normalRibbon, + (lowConnector_after hz).1, (lowConnector_after hz).2] + simp only [connectorTail, Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + mul_zero, zero_mul, mul_one, sub_zero, add_zero, zero_add] + nlinarith + +lemma physicalFitChart_array_right {s : ℝ} (hs : 0 < s) (w : ℂ) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) + (hA : connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) < z.re) : + s / 5 ≤ (physicalFitChart s w m z).re - |w.re| := by + have hℓ := (fit_width_pos hs hm).le + have hmul := fit_width_mul hs hm + let A := connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + have hz' : 0 ≤ (z - (A : ℂ)).re ∧ |(z - (A : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by dsimp [A]; linarith, ha⟩ + have hbound := mul_le_mul_of_nonneg_left (array_normal_re_lower hm hz') hℓ + have hcenter := congrArg Complex.re (fit_scaled_arrayCenter hs w hm) + simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, + positiveCenter, Complex.add_re, Complex.mul_re, Complex.I_re, Complex.I_im, + mul_zero, sub_zero, add_zero] at hcenter + change ((ribbonWidth (halfHeight s) m : ℂ) * + normalRibbon (completeSpine m _ _ _ _) (completeTangent m _ _ _ _) z).re - _ ≥ _ + rw [complete_normal_of_gt m _ _ _ _ hA] + simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, + Complex.add_re] + dsimp [A] at hbound + nlinarith + +theorem physicalFitChart_avoids_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : + ¬ (abs ((physicalFitChart s w m z).re - |w.re|) < s / 100 ∧ + w.im - s / 100 < (physicalFitChart s w m z).im) := by + intro h + by_cases h4 : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) + · have hh := physicalFitChart_low hs w hw hm ha h4 + linarith [h.2] + · by_cases hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + · have hh := physicalFitChart_tail_right hs w hw hm ha (lt_of_not_ge h4) hA + have hl := le_abs_self ((physicalFitChart s w m z).re - |w.re|) + linarith [h.1] + · have hh := physicalFitChart_array_right hs w hm ha (lt_of_not_ge hA) + have hl := le_abs_self ((physicalFitChart s w m z).re - |w.re|) + linarith [h.1] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineHorizontalBound.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineHorizontalBound.lean index f75706b71314cc62e7fe8b183b29cd5bb30aad1e..03ae8fefe89398338e3b06acec3b94010b3e9d31 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineHorizontalBound.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineHorizontalBound.lean @@ -1,85 +1,85 @@ -import OAI.Probability.SLE.Gauge.SerpentineExperimentGeometry - -namespace OAI - -/-! The whole completed physical ribbon stays between the exterior -vertical lines x=-2 and x=2. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma array_normal_re_upper {m : ℕ} (hm : 0 < m) {z : ℂ} - (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : - (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).re ≤ 400 * m := by - have hh : (0 : ℝ) < 100 * m := by positivity - by_cases he : z.re ≤ arrayEnd m (2 * m) - · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] - rw [hs, hT, normal_straight_eq hm] - have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 - rw [dist_zero_right] at hb - have hi := Complex.re_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) - linarith - · simp only [normalRibbon] - rw [hs, hT, normal_circular_eq hm] - have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 - rw [dist_zero_right] at hb - exact (Complex.re_le_norm _).trans (by linarith) - · have hE : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge he - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have ht := array_normal_terminal hm (a := z.im) hE - rw [Complex.re_add_im z] at ht - rw [ht, columnPoint_re] - simp only [columnOffset, normalized_spacing hm, normalized_width hm, side_even, - one_mul, Complex.zero_re, zero_add, Nat.cast_mul, Nat.cast_ofNat] - have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm - linarith [(abs_le.mp hz.2).1] - -lemma lowConnector_re_bounds {H r X : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (hX : 2 * r < X) - {t : ℝ} (ht : 0 ≤ t) : - 0 ≤ (lowConnector H r X t).re ∧ (lowConnector H r X t).re ≤ X := by - have hmono : Monotone (fun t => (lowConnector H r X t).re) := - monotone_of_hasDerivAt_nonneg - (fun t => Complex.reCLM.hasFDerivAt.comp_hasDerivAt t (lowConnector_hasDerivAt hr hX t)) - (fun t => (lowConnectorTangent_nonneg (H := H) hr hX t).1) - have hzero := hmono ht - dsimp only at hzero - rw [(lowConnector_initial hr hX hH).1] at hzero - simp only [Complex.ofReal_zero, zero_mul, Complex.zero_re] at hzero - let u := max t (connectorFourth H r X + 1) - have hu : connectorFourth H r X < u := by - have h := le_max_right t (connectorFourth H r X + 1) - dsimp only [u] - linarith - have hup := hmono (le_max_left t (connectorFourth H r X + 1)) - dsimp only at hup - rw [(lowConnector_after hu).1] at hup - have hup' : (lowConnector H r X t).re ≤ X := by - simpa only [connectorTail, Complex.add_re, Complex.mul_re, Complex.ofReal_re, - Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, add_zero] using hup - exact ⟨hzero, hup'⟩ - -lemma lowConnector_normal_re_bounds {H r X : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (hX : 2 * r < X) - {z : ℂ} (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : - -1 ≤ (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).re ∧ - (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).re ≤ X + 1 := by - have hcenter := lowConnector_re_bounds hH hr hX hz.1 - have hdist : ‖normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z - - lowConnector H r X z.re‖ ≤ 1 := by - simp only [normalRibbon, add_sub_cancel_left, norm_mul, Complex.norm_real, - Real.norm_eq_abs, Complex.norm_I, lowConnectorTangent_norm, mul_one] - exact hz.2 - have hab := (Complex.abs_re_le_norm _).trans hdist - rw [Complex.sub_re, abs_le] at hab - constructor <;> linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineExperimentGeometry + +namespace OAI + +/-! The whole completed physical ribbon stays between the exterior +vertical lines x=-2 and x=2. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma array_normal_re_upper {m : ℕ} (hm : 0 < m) {z : ℂ} + (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : + (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z).re ≤ 400 * m := by + have hh : (0 : ℝ) < 100 * m := by positivity + by_cases he : z.re ≤ arrayEnd m (2 * m) + · rcases array_pieces m (2 * m) ⟨hz.1, he⟩ with ⟨i, hi, u, hu, _, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] + rw [hs, hT, normal_straight_eq hm] + have hb := columnPoint_bound hh hm hi 0 (straight_height_bound m i hu) hz.2 + rw [dist_zero_right] at hb + have hi := Complex.re_le_norm (columnPoint 0 (100 * m) m i (side i * (-100 * m + u)) z.im) + linarith + · simp only [normalRibbon] + rw [hs, hT, normal_circular_eq hm] + have hb := turnPoint_bound hh hm (show i + 1 ≤ 2 * m by omega) 0 (u / 50) hz.2 + rw [dist_zero_right] at hb + exact (Complex.re_le_norm _).trans (by linarith) + · have hE : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge he + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have ht := array_normal_terminal hm (a := z.im) hE + rw [Complex.re_add_im z] at ht + rw [ht, columnPoint_re] + simp only [columnOffset, normalized_spacing hm, normalized_width hm, side_even, + one_mul, Complex.zero_re, zero_add, Nat.cast_mul, Nat.cast_ofNat] + have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm + linarith [(abs_le.mp hz.2).1] + +lemma lowConnector_re_bounds {H r X : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (hX : 2 * r < X) + {t : ℝ} (ht : 0 ≤ t) : + 0 ≤ (lowConnector H r X t).re ∧ (lowConnector H r X t).re ≤ X := by + have hmono : Monotone (fun t => (lowConnector H r X t).re) := + monotone_of_hasDerivAt_nonneg + (fun t => Complex.reCLM.hasFDerivAt.comp_hasDerivAt t (lowConnector_hasDerivAt hr hX t)) + (fun t => (lowConnectorTangent_nonneg (H := H) hr hX t).1) + have hzero := hmono ht + dsimp only at hzero + rw [(lowConnector_initial hr hX hH).1] at hzero + simp only [Complex.ofReal_zero, zero_mul, Complex.zero_re] at hzero + let u := max t (connectorFourth H r X + 1) + have hu : connectorFourth H r X < u := by + have h := le_max_right t (connectorFourth H r X + 1) + dsimp only [u] + linarith + have hup := hmono (le_max_left t (connectorFourth H r X + 1)) + dsimp only at hup + rw [(lowConnector_after hu).1] at hup + have hup' : (lowConnector H r X t).re ≤ X := by + simpa only [connectorTail, Complex.add_re, Complex.mul_re, Complex.ofReal_re, + Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, add_zero] using hup + exact ⟨hzero, hup'⟩ + +lemma lowConnector_normal_re_bounds {H r X : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (hX : 2 * r < X) + {z : ℂ} (hz : 0 ≤ z.re ∧ |z.im| ≤ 1) : + -1 ≤ (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).re ∧ + (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).re ≤ X + 1 := by + have hcenter := lowConnector_re_bounds hH hr hX hz.1 + have hdist : ‖normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z - + lowConnector H r X z.re‖ ≤ 1 := by + simp only [normalRibbon, add_sub_cancel_left, norm_mul, Complex.norm_real, + Real.norm_eq_abs, Complex.norm_I, lowConnectorTangent_norm, mul_one] + exact hz.2 + have hab := (Complex.abs_re_le_norm _).trans hdist + rw [Complex.sub_re, abs_le] at hab + constructor <;> linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineInfiniteArray.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineInfiniteArray.lean index 426b89ea6d8d09309a616691e232cea1fd0d228f..c56a20cf3894aa004dd5ed83aaa5b08019978f09 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineInfiniteArray.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineInfiniteArray.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Gauge.SerpentineArrayEmbedding -import OAI.Probability.SLE.Gauge.SerpentineTerminal - -namespace OAI - -/-! The finite serpentine remains embedded after continuing its last -column vertically to infinity. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma array_normal_terminal {m : ℕ} (hm : 0 < m) {t a : ℝ} - (ht : columnEntry 0 m (2 * m) ≤ t) : - normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) (t + a * Complex.I) = - columnPoint 0 (100 * m) m (2 * m) (-100 * m + (t - columnEntry 0 m (2 * m))) a := by - simp only [normalRibbon, Complex.add_re, Complex.add_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, - mul_zero, sub_zero, mul_one, zero_add, add_zero] - rw [arraySpine_tail m (2 * m) ht, arrayTangent_tail m (2 * m) ht, normal_straight_eq hm, - side_even, one_mul] - -lemma array_normal_finite_terminal_disjoint {m : ℕ} (hm : 0 < m) {z w : ℂ} - (hz : z.re ∈ Icc (0 : ℝ) (arrayEnd m (2 * m)) ∧ |z.im| ≤ 1) - (hw : arrayEnd m (2 * m) < w.re) (hwa : |w.im| ≤ 1) : - normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z ≠ - normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w := by - intro he - have hh : (0 : ℝ) < 100 * m := by positivity - have hwe : columnEntry 0 m (2 * m) ≤ w.re := by - dsimp only [arrayEnd] at hw - linarith [Nat.cast_nonneg (α := ℝ) m] - have hwy : 100 * (m : ℝ) < -100 * m + (w.re - columnEntry 0 m (2 * m)) := by - dsimp only [arrayEnd] at hw - linarith - have htw := array_normal_terminal hm (a := w.im) hwe - rw [Complex.re_add_im w] at htw - rw [htw] at he - rcases array_pieces m (2 * m) hz.1 with ⟨i, _, u, hu, _, hs, hT⟩ | - ⟨i, hi, u, hu, _, hs, hT⟩ - · simp only [normalRibbon] at he - rw [hs, hT, normal_straight_eq hm] at he - have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 hwa he - subst i - have hy := congrArg Complex.im he - simp only [columnPoint_im, side_even, one_mul] at hy - have hsmall := (abs_le.mp (straight_height_bound m (2 * m) hu)).2 - rw [side_even, one_mul] at hsmall - linarith - · simp only [normalRibbon] at he - rw [hs, hT, normal_circular_eq hm] at he - exact turnPoint_terminal_column_ne hh hm hi 0 (circular_angle_bound hu) hwy hz.2 hwa he - -/-- The complete upward-continued array is embedded on the whole closed -width-one normal strip. -/ -theorem array_normal_injective_infinite {m : ℕ} (hm : 0 < m) : - InjOn (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m))) - {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by - intro z hz w hw he - by_cases hzE : z.re ≤ arrayEnd m (2 * m) - · by_cases hwE : w.re ≤ arrayEnd m (2 * m) - · exact array_normal_injective hm (2 * m) ⟨⟨hz.1, hzE⟩, hz.2⟩ ⟨⟨hw.1, hwE⟩, hw.2⟩ he - · exact (array_normal_finite_terminal_disjoint hm ⟨⟨hz.1, hzE⟩, hz.2⟩ - (lt_of_not_ge hwE) hw.2 he).elim - · by_cases hwE : w.re ≤ arrayEnd m (2 * m) - · exact (array_normal_finite_terminal_disjoint hm ⟨⟨hw.1, hwE⟩, hw.2⟩ - (lt_of_not_ge hzE) hz.2 he.symm).elim - · have hze : columnEntry 0 m (2 * m) ≤ z.re := by - have h := lt_of_not_ge hzE - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have hwe : columnEntry 0 m (2 * m) ≤ w.re := by - have h := lt_of_not_ge hwE - dsimp only [arrayEnd] at h - linarith [Nat.cast_nonneg (α := ℝ) m] - have htz := array_normal_terminal hm (a := z.im) hze - have htw := array_normal_terminal hm (a := w.im) hwe - rw [Complex.re_add_im z] at htz - rw [Complex.re_add_im w] at htw - rw [htz, htw] at he - have hh : (0 : ℝ) < 100 * m := by positivity - obtain ⟨hy, ha⟩ := columnPoint_injective hh hm 0 (2 * m) he - exact Complex.ext (by linarith) ha - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineArrayEmbedding +import OAI.Probability.SLE.Gauge.SerpentineTerminal + +namespace OAI + +/-! The finite serpentine remains embedded after continuing its last +column vertically to infinity. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma array_normal_terminal {m : ℕ} (hm : 0 < m) {t a : ℝ} + (ht : columnEntry 0 m (2 * m) ≤ t) : + normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) (t + a * Complex.I) = + columnPoint 0 (100 * m) m (2 * m) (-100 * m + (t - columnEntry 0 m (2 * m))) a := by + simp only [normalRibbon, Complex.add_re, Complex.add_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, + mul_zero, sub_zero, mul_one, zero_add, add_zero] + rw [arraySpine_tail m (2 * m) ht, arrayTangent_tail m (2 * m) ht, normal_straight_eq hm, + side_even, one_mul] + +lemma array_normal_finite_terminal_disjoint {m : ℕ} (hm : 0 < m) {z w : ℂ} + (hz : z.re ∈ Icc (0 : ℝ) (arrayEnd m (2 * m)) ∧ |z.im| ≤ 1) + (hw : arrayEnd m (2 * m) < w.re) (hwa : |w.im| ≤ 1) : + normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) z ≠ + normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m)) w := by + intro he + have hh : (0 : ℝ) < 100 * m := by positivity + have hwe : columnEntry 0 m (2 * m) ≤ w.re := by + dsimp only [arrayEnd] at hw + linarith [Nat.cast_nonneg (α := ℝ) m] + have hwy : 100 * (m : ℝ) < -100 * m + (w.re - columnEntry 0 m (2 * m)) := by + dsimp only [arrayEnd] at hw + linarith + have htw := array_normal_terminal hm (a := w.im) hwe + rw [Complex.re_add_im w] at htw + rw [htw] at he + rcases array_pieces m (2 * m) hz.1 with ⟨i, _, u, hu, _, hs, hT⟩ | + ⟨i, hi, u, hu, _, hs, hT⟩ + · simp only [normalRibbon] at he + rw [hs, hT, normal_straight_eq hm] at he + have hii := columnPoint_equal_forces_index hh hm 0 _ _ hz.2 hwa he + subst i + have hy := congrArg Complex.im he + simp only [columnPoint_im, side_even, one_mul] at hy + have hsmall := (abs_le.mp (straight_height_bound m (2 * m) hu)).2 + rw [side_even, one_mul] at hsmall + linarith + · simp only [normalRibbon] at he + rw [hs, hT, normal_circular_eq hm] at he + exact turnPoint_terminal_column_ne hh hm hi 0 (circular_angle_bound hu) hwy hz.2 hwa he + +/-- The complete upward-continued array is embedded on the whole closed +width-one normal strip. -/ +theorem array_normal_injective_infinite {m : ℕ} (hm : 0 < m) : + InjOn (normalRibbon (arraySpine m (2 * m)) (arrayTangent m (2 * m))) + {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by + intro z hz w hw he + by_cases hzE : z.re ≤ arrayEnd m (2 * m) + · by_cases hwE : w.re ≤ arrayEnd m (2 * m) + · exact array_normal_injective hm (2 * m) ⟨⟨hz.1, hzE⟩, hz.2⟩ ⟨⟨hw.1, hwE⟩, hw.2⟩ he + · exact (array_normal_finite_terminal_disjoint hm ⟨⟨hz.1, hzE⟩, hz.2⟩ + (lt_of_not_ge hwE) hw.2 he).elim + · by_cases hwE : w.re ≤ arrayEnd m (2 * m) + · exact (array_normal_finite_terminal_disjoint hm ⟨⟨hw.1, hwE⟩, hw.2⟩ + (lt_of_not_ge hzE) hz.2 he.symm).elim + · have hze : columnEntry 0 m (2 * m) ≤ z.re := by + have h := lt_of_not_ge hzE + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have hwe : columnEntry 0 m (2 * m) ≤ w.re := by + have h := lt_of_not_ge hwE + dsimp only [arrayEnd] at h + linarith [Nat.cast_nonneg (α := ℝ) m] + have htz := array_normal_terminal hm (a := z.im) hze + have htw := array_normal_terminal hm (a := w.im) hwe + rw [Complex.re_add_im z] at htz + rw [Complex.re_add_im w] at htw + rw [htz, htw] at he + have hh : (0 : ℝ) < 100 * m := by positivity + obtain ⟨hy, ha⟩ := columnPoint_injective hh hm 0 (2 * m) he + exact Complex.ext (by linarith) ha + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineInitialPrefix.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineInitialPrefix.lean index f63d46ff699842fb8321481b5aa32830404aa60d..28a93eef1fd5e1246d59ee53199276f2734c1fb8 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineInitialPrefix.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineInitialPrefix.lean @@ -1,149 +1,149 @@ -import OAI.Probability.SLE.Geometry.FirstHeightSupport -import OAI.Probability.SLE.Gauge.RibbonGoodPrefix -import OAI.Probability.SLE.Gauge.SerpentineExperimentGeometry - -namespace OAI - -/-! A single vertical support event starts every fitted serpentine at -integer front one. Its probability is independent of the marked point -and of the multiplicity. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal unitInterval - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates FirstFrontPassage FirstHeightSupport - -def initialRiseStrip (ε : ℝ) : Set ℂ := {z | |z.re| < ε / 4} - -def initialRiseTarget : Set ℂ := {z | 1 < z.im} - -def InitialRise {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (ε : ℝ) : Set Ω := - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (initialRiseStrip ε) ∧ - γ ω T ∈ initialRiseTarget} - -lemma initialRise_positive {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (γ : Ω → ℝ≥0 → ℂ) (hsupport : PublishedGeometricTubeSupport P γ) - {ε : ℝ} (hε : 0 < ε) : 0 < P (InitialRise γ ε) := by - apply hsupport _ (verticalPath 4) (verticalPath_injective (by norm_num)) - · intro t ht - change 0 < (verticalPath 4 t).im - simp only [verticalPath_apply, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero] - have ht0 : 0 < (t : ℝ) := by - by_contra hn - exact ht (Subtype.ext (le_antisymm (le_of_not_gt hn) t.property.1)) - linarith - · exact isOpen_lt Complex.continuous_re.abs continuous_const - · exact isOpen_lt continuous_const Complex.continuous_im - · rintro z ⟨t, rfl⟩ - change |(verticalPath 4 t).re| < ε / 4 - simpa only [verticalPath_apply, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, abs_zero] using - (show (0 : ℝ) < ε / 4 by positivity) - · change 1 < (((4 / 2 : ℝ) : ℂ) * Complex.I).im - norm_num - -lemma initial_prefix_coordinates (γ : ℝ≥0 → ℂ) (P : ℂ → ℂ) - {ε : ℝ} (hε : ε ≤ 1) - (hP : ∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) - {t : ℝ≥0} (hheight : 0 ≤ (γ t).im ∧ (γ t).im ≤ 1) - (hstrip : |(γ t).re| < ε / 4) : - ∃ z ∈ closedRibbon, |z.im| ≤ ε / 4 ∧ z.re = (γ t).im ∧ P z = γ t := by - let z := -Complex.I * γ t - have hzre : z.re = (γ t).im := by simp [z] - have hzim : z.im = -(γ t).re := by simp [z] - have hz : z ∈ closedRibbon := by - change 0 ≤ z.re ∧ |z.im| ≤ 1 - rw [hzre, hzim, abs_neg] - exact ⟨hheight.1, by linarith⟩ - refine ⟨z, hz, ?_, hzre, ?_⟩ - · rw [hzim, abs_neg] - exact hstrip.le - · rw [hP z hz (by rw [hzre]; linarith)] - dsimp [z] - rw [← mul_assoc] - norm_num - -theorem initialRise_gives_goodPrefix - (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) (hγ0 : γ 0 = 0) - (hγH : ∀ t, 0 ≤ (γ t).im) - (P : ℂ → ℂ) (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re) - {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) - (hP : ∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) - (hrise : () ∈ InitialRise (fun _ : Unit => γ) ε) : - () ∈ GoodRibbonPrefix (fun _ : Unit => γ) - (fun _ => firstFrontTime (fun t => g (γ t)) 1) P ε := by - obtain ⟨T, _, hstrip, htarget⟩ := hrise - obtain ⟨τ, _, _, hτT, hhit, hprior⟩ := - exists_first_height γ hγ hγ0 zero_lt_one htarget - have hheight (t : ℝ≥0) (ht : t ≤ τ) : 0 ≤ (γ t).im ∧ (γ t).im ≤ 1 := by - refine ⟨hγH t, ?_⟩ - rcases eq_or_lt_of_le ht with rfl | ht - · exact hhit.le - · exact (hprior t ht).le - have hcoord (t : ℝ≥0) (ht : t ≤ τ) := initial_prefix_coordinates γ P hε1 hP - (hheight t ht) (hstrip ⟨bot_le, ht.trans hτT.le⟩) - have hgheight (t : ℝ≥0) (ht : t ≤ τ) : g (γ t) = (γ t).im := by - obtain ⟨z, hz, _, hzre, he⟩ := hcoord t ht - exact (congrArg g he.symm).trans ((hg z hz).trans hzre) - have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ g (γ t) - 1 ∈ ({0} : Set ℝ)} τ := by - refine ⟨⟨bot_le, ?_⟩, ?_⟩ - · simp only [hgheight τ le_rfl, hhit, sub_self, mem_singleton_iff] - · intro t ht - by_contra hn - have htτ : t < τ := lt_of_not_ge hn - have he : g (γ t) = 1 := sub_eq_zero.mp ht.2 - rw [hgheight t htτ.le] at he - exact (ne_of_lt (hprior t htτ)) he - have hfront : firstFrontTime (fun t => g (γ t)) 1 = (τ : WithTop ℝ≥0) := by - unfold firstFrontTime hittingAfter - rw [ite_eq_left ⟨τ, hleast.1⟩, hleast.csInf_eq] - change firstFrontTime (fun t => g (γ t)) 1 ≠ ⊤ ∧ _ - rw [hfront] - refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ - · intro t ht - obtain ⟨z, hz, hzi, _, he⟩ := hcoord t ht.2 - exact ⟨z, ⟨hz, hzi.trans (by linarith)⟩, he⟩ - · obtain ⟨z, hz, hzi, _, he⟩ := hcoord τ le_rfl - exact ⟨z, ⟨hz, hzi⟩, he⟩ - -/-- The same positive lower bound works for every chart which is the -initial vertical strip, and for every global extension of its progress. -/ -theorem uniform_initial_goodPrefix_probability - {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ μ) - (hsupport : PublishedGeometricTubeSupport μ γ) - {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ P : ℂ → ℂ, ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (P z) = z.re) → - (∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) → - q ≤ μ (GoodRibbonPrefix γ (ribbonFrontStop g γ 1) P ε) := by - refine ⟨μ (InitialRise γ ε), initialRise_positive μ γ hsupport hε, ?_⟩ - intro P g hg hP - apply measure_mono_ae - obtain ⟨_, _, htrace⟩ := hSLE.2 - filter_upwards [htrace] with ω hω - intro hrise - obtain ⟨G, F, hw⟩ := hω - exact initialRise_gives_goodPrefix (γ ω) hw.1 hw.2.1 hw.2.2.1 P g hg hε hε1 hP hrise - -theorem uniform_experiment_initial_goodPrefix_probability - {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ μ) - (hsupport : PublishedGeometricTubeSupport μ γ) - {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ s : ℝ, 0 < s → ∀ w : ℂ, s ≤ w.im → - ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → - q ≤ μ (GoodRibbonPrefix γ (ribbonFrontStop g γ 1) (experimentChart s w m) ε) := by - obtain ⟨q, hq, hp⟩ := uniform_initial_goodPrefix_probability μ κ γ hSLE hsupport hε hε1 - refine ⟨q, hq, fun s hs w hw m hm g hg => ?_⟩ - exact hp _ g hg (experimentChart_geometry hs w hw hm).2.2.2.2 - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Geometry.FirstHeightSupport +import OAI.Probability.SLE.Gauge.RibbonGoodPrefix +import OAI.Probability.SLE.Gauge.SerpentineExperimentGeometry + +namespace OAI + +/-! A single vertical support event starts every fitted serpentine at +integer front one. Its probability is independent of the marked point +and of the multiplicity. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal unitInterval + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates FirstFrontPassage FirstHeightSupport + +def initialRiseStrip (ε : ℝ) : Set ℂ := {z | |z.re| < ε / 4} + +def initialRiseTarget : Set ℂ := {z | 1 < z.im} + +def InitialRise {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (ε : ℝ) : Set Ω := + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (initialRiseStrip ε) ∧ + γ ω T ∈ initialRiseTarget} + +lemma initialRise_positive {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (γ : Ω → ℝ≥0 → ℂ) (hsupport : PublishedGeometricTubeSupport P γ) + {ε : ℝ} (hε : 0 < ε) : 0 < P (InitialRise γ ε) := by + apply hsupport _ (verticalPath 4) (verticalPath_injective (by norm_num)) + · intro t ht + change 0 < (verticalPath 4 t).im + simp only [verticalPath_apply, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero] + have ht0 : 0 < (t : ℝ) := by + by_contra hn + exact ht (Subtype.ext (le_antisymm (le_of_not_gt hn) t.property.1)) + linarith + · exact isOpen_lt Complex.continuous_re.abs continuous_const + · exact isOpen_lt continuous_const Complex.continuous_im + · rintro z ⟨t, rfl⟩ + change |(verticalPath 4 t).re| < ε / 4 + simpa only [verticalPath_apply, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, abs_zero] using + (show (0 : ℝ) < ε / 4 by positivity) + · change 1 < (((4 / 2 : ℝ) : ℂ) * Complex.I).im + norm_num + +lemma initial_prefix_coordinates (γ : ℝ≥0 → ℂ) (P : ℂ → ℂ) + {ε : ℝ} (hε : ε ≤ 1) + (hP : ∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) + {t : ℝ≥0} (hheight : 0 ≤ (γ t).im ∧ (γ t).im ≤ 1) + (hstrip : |(γ t).re| < ε / 4) : + ∃ z ∈ closedRibbon, |z.im| ≤ ε / 4 ∧ z.re = (γ t).im ∧ P z = γ t := by + let z := -Complex.I * γ t + have hzre : z.re = (γ t).im := by simp [z] + have hzim : z.im = -(γ t).re := by simp [z] + have hz : z ∈ closedRibbon := by + change 0 ≤ z.re ∧ |z.im| ≤ 1 + rw [hzre, hzim, abs_neg] + exact ⟨hheight.1, by linarith⟩ + refine ⟨z, hz, ?_, hzre, ?_⟩ + · rw [hzim, abs_neg] + exact hstrip.le + · rw [hP z hz (by rw [hzre]; linarith)] + dsimp [z] + rw [← mul_assoc] + norm_num + +theorem initialRise_gives_goodPrefix + (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) (hγ0 : γ 0 = 0) + (hγH : ∀ t, 0 ≤ (γ t).im) + (P : ℂ → ℂ) (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re) + {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) + (hP : ∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) + (hrise : () ∈ InitialRise (fun _ : Unit => γ) ε) : + () ∈ GoodRibbonPrefix (fun _ : Unit => γ) + (fun _ => firstFrontTime (fun t => g (γ t)) 1) P ε := by + obtain ⟨T, _, hstrip, htarget⟩ := hrise + obtain ⟨τ, _, _, hτT, hhit, hprior⟩ := + exists_first_height γ hγ hγ0 zero_lt_one htarget + have hheight (t : ℝ≥0) (ht : t ≤ τ) : 0 ≤ (γ t).im ∧ (γ t).im ≤ 1 := by + refine ⟨hγH t, ?_⟩ + rcases eq_or_lt_of_le ht with rfl | ht + · exact hhit.le + · exact (hprior t ht).le + have hcoord (t : ℝ≥0) (ht : t ≤ τ) := initial_prefix_coordinates γ P hε1 hP + (hheight t ht) (hstrip ⟨bot_le, ht.trans hτT.le⟩) + have hgheight (t : ℝ≥0) (ht : t ≤ τ) : g (γ t) = (γ t).im := by + obtain ⟨z, hz, _, hzre, he⟩ := hcoord t ht + exact (congrArg g he.symm).trans ((hg z hz).trans hzre) + have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ g (γ t) - 1 ∈ ({0} : Set ℝ)} τ := by + refine ⟨⟨bot_le, ?_⟩, ?_⟩ + · simp only [hgheight τ le_rfl, hhit, sub_self, mem_singleton_iff] + · intro t ht + by_contra hn + have htτ : t < τ := lt_of_not_ge hn + have he : g (γ t) = 1 := sub_eq_zero.mp ht.2 + rw [hgheight t htτ.le] at he + exact (ne_of_lt (hprior t htτ)) he + have hfront : firstFrontTime (fun t => g (γ t)) 1 = (τ : WithTop ℝ≥0) := by + unfold firstFrontTime hittingAfter + rw [ite_eq_left ⟨τ, hleast.1⟩, hleast.csInf_eq] + change firstFrontTime (fun t => g (γ t)) 1 ≠ ⊤ ∧ _ + rw [hfront] + refine ⟨WithTop.coe_ne_top, ?_, ?_⟩ + · intro t ht + obtain ⟨z, hz, hzi, _, he⟩ := hcoord t ht.2 + exact ⟨z, ⟨hz, hzi.trans (by linarith)⟩, he⟩ + · obtain ⟨z, hz, hzi, _, he⟩ := hcoord τ le_rfl + exact ⟨z, ⟨hz, hzi⟩, he⟩ + +/-- The same positive lower bound works for every chart which is the +initial vertical strip, and for every global extension of its progress. -/ +theorem uniform_initial_goodPrefix_probability + {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ μ) + (hsupport : PublishedGeometricTubeSupport μ γ) + {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ P : ℂ → ℂ, ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (P z) = z.re) → + (∀ z ∈ closedRibbon, z.re ≤ 4 → P z = Complex.I * z) → + q ≤ μ (GoodRibbonPrefix γ (ribbonFrontStop g γ 1) P ε) := by + refine ⟨μ (InitialRise γ ε), initialRise_positive μ γ hsupport hε, ?_⟩ + intro P g hg hP + apply measure_mono_ae + obtain ⟨_, _, htrace⟩ := hSLE.2 + filter_upwards [htrace] with ω hω + intro hrise + obtain ⟨G, F, hw⟩ := hω + exact initialRise_gives_goodPrefix (γ ω) hw.1 hw.2.1 hw.2.2.1 P g hg hε hε1 hP hrise + +theorem uniform_experiment_initial_goodPrefix_probability + {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ μ) + (hsupport : PublishedGeometricTubeSupport μ γ) + {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ s : ℝ, 0 < s → ∀ w : ℂ, s ≤ w.im → + ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → + q ≤ μ (GoodRibbonPrefix γ (ribbonFrontStop g γ 1) (experimentChart s w m) ε) := by + obtain ⟨q, hq, hp⟩ := uniform_initial_goodPrefix_probability μ κ γ hSLE hsupport hε hε1 + refine ⟨q, hq, fun s hs w hw m hm g hg => ?_⟩ + exact hp _ g hg (experimentChart_geometry hs w hw hm).2.2.2.2 + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineLayout.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineLayout.lean index 257fa57b59cd9629f928e151499256105a444fed..eb1cd57b2d8e1ac17e7f75ef2cb9b0a78815addf 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineLayout.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineLayout.lean @@ -1,326 +1,326 @@ -import OAI.Probability.SLE.Gauge.RibbonMiddleMass -import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv - -namespace OAI - -/-! Explicit columns and circular turns for the dense-visit serpentine. -The array and its normal bands fit in the displaced disk and stay a fixed -distance from the marked point. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine - -def side (j : ℕ) : ℝ := if Even j then 1 else -1 - -lemma side_abs (j : ℕ) : |side j| = 1 := by - unfold side - split <;> norm_num - -lemma side_sq (j : ℕ) : side j ^ 2 = 1 := by - unfold side - split <;> norm_num - -def displacementSign (w : ℂ) : ℝ := if 0 ≤ w.re then 1 else -1 - -def center (s : ℝ) (w : ℂ) : ℂ := w + ((displacementSign w * (s / 4) : ℝ) : ℂ) - -def halfHeight (s : ℝ) : ℝ := s / 200 - -def spacing (h : ℝ) (m : ℕ) : ℝ := h / m - -def turnRadius (h : ℝ) (m : ℕ) : ℝ := h / (2 * m) - -def ribbonWidth (h : ℝ) (m : ℕ) : ℝ := h / (100 * m) - -def columnOffset (h : ℝ) (m j : ℕ) : ℝ := -h + j * spacing h m - -/-- The closed width-one band over one straight pass, in physical scale. -/ -def columnPoint (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : ℂ := - c + ((columnOffset h m j - side j * ribbonWidth h m * a : ℝ) : ℂ) + - (y : ℂ) * Complex.I - -/-- A circular connector, with its actual normal ribbon coordinate `a`. -The parameter runs from zero to pi, alternating above and below. -/ -def turnPoint (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : ℂ := - c + ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + - (side j * h : ℂ) * Complex.I + - ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * - ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I) - -lemma center_dist {s : ℝ} (hs : 0 ≤ s) (w : ℂ) : dist (center s w) w = s / 4 := by - rw [center, dist_self_add_left, Complex.norm_real, Real.norm_eq_abs, abs_mul, - abs_of_nonneg (div_nonneg hs (by norm_num))] - have hh : |displacementSign w| = 1 := by unfold displacementSign; split <;> norm_num - rw [hh, one_mul] - -lemma center_im (s : ℝ) (w : ℂ) : (center s w).im = w.im := by - simp [center] - -lemma displaced_disk_inside - {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) : - closedBall (center s w) (s / 20) ⊆ ball w (w.im / 2) := by - intro z hz - have hd := (dist_triangle z (center s w) w).trans - (add_le_add hz (le_of_eq (center_dist hs.le w))) - change dist z w < w.im / 2 - linarith - -lemma displaced_disk_separated - {s : ℝ} (hs : 0 < s) (w z : ℂ) (hz : z ∈ closedBall (center s w) (s / 20)) : - s / 5 ≤ dist z w := by - have hh := dist_triangle (center s w) z w - rw [center_dist hs.le w, dist_comm (center s w) z] at hh - change dist z (center s w) ≤ s / 20 at hz - linarith - -lemma spacing_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < spacing h m := by - exact div_pos hh (by exact_mod_cast hm) - -lemma turnRadius_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < turnRadius h m := by - unfold turnRadius - positivity - -lemma ribbonWidth_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < ribbonWidth h m := by - unfold ribbonWidth - positivity - -lemma spacing_eq_width (h : ℝ) (m : ℕ) : spacing h m = 100 * ribbonWidth h m := by - unfold spacing ribbonWidth - by_cases hm : m = 0 - · simp [hm] - · have hmr : (m : ℝ) ≠ 0 := by exact_mod_cast hm - field_simp - -lemma turnRadius_eq_width (h : ℝ) (m : ℕ) : turnRadius h m = 50 * ribbonWidth h m := by - unfold turnRadius ribbonWidth - by_cases hm : m = 0 - · simp [hm] - · have hmr : (m : ℝ) ≠ 0 := by exact_mod_cast hm - field_simp - ring - -lemma columnOffset_bounds {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) - (hj : j ≤ 2 * m) : |columnOffset h m j| ≤ h := by - have hmr : (0 : ℝ) < m := by exact_mod_cast hm - have hjr : (j : ℝ) ≤ 2 * m := by exact_mod_cast hj - have hj0 : (0 : ℝ) ≤ j := Nat.cast_nonneg j - have hq := spacing_pos hh hm - have hmul : (j : ℝ) * spacing h m ≤ 2 * h := by - unfold spacing - rw [← mul_div_assoc] - exact (div_le_iff₀ hmr).mpr (by nlinarith) - rw [columnOffset] - exact abs_le.mpr ⟨by nlinarith [mul_nonneg hj0 hq.le], by linarith⟩ - -lemma width_le_halfHeight {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : - ribbonWidth h m ≤ h / 100 := by - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - unfold ribbonWidth - apply div_le_div_of_nonneg_left hh.le (by norm_num) - nlinarith - -lemma radius_le_halfHeight {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : - turnRadius h m ≤ h / 2 := by - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - unfold turnRadius - apply div_le_div_of_nonneg_left hh.le (by norm_num) - nlinarith - -lemma columnPoint_bound {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) - (hj : j ≤ 2 * m) (c : ℂ) {y a : ℝ} (hy : |y| ≤ h) (ha : |a| ≤ 1) : - dist (columnPoint c h m j y a) c ≤ 3 * h := by - have ho := columnOffset_bounds hh hm hj - have he := width_le_halfHeight hh hm - have hwp := (ribbonWidth_pos hh hm).le - have hprod : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hwp] - exact mul_le_of_le_one_right hwp ha - have hx : |columnOffset h m j - side j * ribbonWidth h m * a| ≤ - |columnOffset h m j| + |side j * ribbonWidth h m * a| := by - simpa only [sub_eq_add_neg, abs_neg] using - abs_add_le (columnOffset h m j) (-(side j * ribbonWidth h m * a)) - have hn := Complex.norm_le_abs_re_add_abs_im - (((columnOffset h m j - side j * ribbonWidth h m * a : ℝ) : ℂ) + - (y : ℂ) * Complex.I) - simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, - Complex.mul_im, mul_one, zero_add] at hn - rw [columnPoint, add_assoc, dist_self_add_left] - linarith - -lemma turnPoint_zero (c : ℂ) (h : ℝ) (m j : ℕ) (a : ℝ) : - turnPoint c h m j 0 a = columnPoint c h m j (side j * h) a := by - apply Complex.ext - · simp [turnPoint, columnPoint] - ring - · simp [turnPoint, columnPoint] - -lemma side_succ (j : ℕ) : side (j + 1) = -side j := by - by_cases h : Even j <;> simp [side, Nat.even_add_one, h] - -lemma turnPoint_pi (c : ℂ) (h : ℝ) (m j : ℕ) (a : ℝ) : - turnPoint c h m j Real.pi a = columnPoint c h m (j + 1) (side j * h) a := by - have he : spacing h m = 2 * turnRadius h m := by - rw [spacing_eq_width, turnRadius_eq_width] - ring - apply Complex.ext - · simp [turnPoint, columnPoint, columnOffset, side_succ, Nat.cast_add, Nat.cast_one] - nlinarith - · simp [turnPoint, columnPoint, columnOffset, side_succ, Nat.cast_add, Nat.cast_one] - -lemma turn_direction_norm (j : ℕ) (t : ℝ) : - ‖((-Real.cos t : ℝ) : ℂ) + (side j * Real.sin t : ℂ) * Complex.I‖ = 1 := by - apply (sq_eq_sq₀ (norm_nonneg _) zero_le_one).mp - rw [Complex.sq_norm, Complex.normSq_apply] - simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, - Complex.mul_im, mul_one, zero_add] - calc - _ = Real.cos t ^ 2 + side j ^ 2 * Real.sin t ^ 2 := by ring - _ = 1 ^ 2 := by rw [side_sq, one_mul]; nlinarith [Real.sin_sq_add_cos_sq t] - -lemma turnPoint_bound {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) - (hj : j + 1 ≤ 2 * m) (c : ℂ) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : - dist (turnPoint c h m j t a) c ≤ 4 * h := by - have ho := columnOffset_bounds hh hm (show j ≤ 2 * m by omega) - have hr := radius_le_halfHeight hh hm - have hw := width_le_halfHeight hh hm - have hrp := (turnRadius_pos hh hm).le - have hwp := (ribbonWidth_pos hh hm).le - have hprod : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hwp] - exact mul_le_of_le_one_right hwp ha - have hadd := abs_add_le (turnRadius h m) (side j * ribbonWidth h m * a) - rw [abs_of_nonneg hrp] at hadd - have hcen := abs_add_le (columnOffset h m j) (turnRadius h m) - rw [abs_of_nonneg hrp] at hcen - have hn : ‖((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + - (side j * h : ℂ) * Complex.I + - ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * - ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I)‖ ≤ - |columnOffset h m j + turnRadius h m| + h + - |turnRadius h m + side j * ribbonWidth h m * a| := by - calc - _ ≤ (‖((columnOffset h m j + turnRadius h m : ℝ) : ℂ)‖ + - ‖(side j * h : ℂ) * Complex.I‖) + - ‖((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * - ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I)‖ := - (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) - _ = _ := by - simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, - mul_one, side_abs, one_mul, abs_of_pos hh, turn_direction_norm] - have he : turnPoint c h m j t a - c = - ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + - (side j * h : ℂ) * Complex.I + - ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * - ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I) := by - unfold turnPoint - ring - rw [dist_eq_norm, he] - linarith - -lemma column_inside_displaced_disk {s : ℝ} (hs : 0 < s) (w : ℂ) - {m j : ℕ} (hm : 0 < m) (hj : j ≤ 2 * m) {y a : ℝ} - (hy : |y| ≤ halfHeight s) (ha : |a| ≤ 1) : - columnPoint (center s w) (halfHeight s) m j y a ∈ closedBall (center s w) (s / 20) := by - have hh : 0 < halfHeight s := by unfold halfHeight; positivity - have hb := columnPoint_bound hh hm hj (center s w) hy ha - change dist _ _ ≤ s / 20 - dsimp only [halfHeight] at hb ⊢ - linarith - - -@[simp] lemma columnPoint_re (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : - (columnPoint c h m j y a).re = c.re + columnOffset h m j - side j * ribbonWidth h m * a := by - simp [columnPoint] - ring - -@[simp] lemma columnPoint_im (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : - (columnPoint c h m j y a).im = c.im + y := by simp [columnPoint] - -lemma column_separation {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) - (hij : i < j) (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) : - 98 * ribbonWidth h m ≤ - dist (columnPoint c h m i y a) (columnPoint c h m j y' b) := by - have hepos := (ribbonWidth_pos hh hm).le - have hnoise (n : ℕ) (u : ℝ) (hu : |u| ≤ 1) : - |side n * ribbonWidth h m * u| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hepos] - exact mul_le_of_le_one_right hepos hu - have hai := abs_le.mp (hnoise i a ha) - have hbj := abs_le.mp (hnoise j b hb) - have hijr : (i : ℝ) + 1 ≤ j := by exact_mod_cast Nat.succ_le_of_lt hij - have hmul := mul_le_mul_of_nonneg_right hijr (spacing_pos hh hm).le - have hre : 98 * ribbonWidth h m ≤ - (columnPoint c h m j y' b).re - (columnPoint c h m i y a).re := by - simp only [columnPoint_re, columnOffset] at * - rw [spacing_eq_width] at hmul ⊢ - nlinarith - calc - _ ≤ (columnPoint c h m j y' b).re - (columnPoint c h m i y a).re := hre - _ ≤ |(columnPoint c h m j y' b).re - (columnPoint c h m i y a).re| := le_abs_self _ - _ ≤ dist (columnPoint c h m i y a) (columnPoint c h m j y' b) := by - simpa only [dist_eq_norm, Complex.sub_re, abs_sub_comm] using - Complex.abs_re_le_norm (columnPoint c h m i y a - columnPoint c h m j y' b) - -lemma columnPoint_ne {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) - (hij : i < j) (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) : - columnPoint c h m i y a ≠ columnPoint c h m j y' b := by - intro he - have hd := column_separation hh hm hij c y y' ha hb - rw [he, dist_self] at hd - have hp := ribbonWidth_pos hh hm - linarith - - -lemma turn_radial_coefficient_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) - (j : ℕ) {a : ℝ} (ha : |a| ≤ 1) : - 0 < turnRadius h m + side j * ribbonWidth h m * a := by - have he := ribbonWidth_pos hh hm - have hb : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_pos he] - exact mul_le_of_le_one_right he.le ha - rw [turnRadius_eq_width] - linarith [(abs_le.mp hb).1] - -/-- Each circular turn, with its full normal band, is embedded. -/ -theorem turnPoint_injective {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) - (c : ℂ) (j : ℕ) {t u a b : ℝ} (ht : t ∈ Icc 0 Real.pi) - (hu : u ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) (hb : |b| ≤ 1) - (he : turnPoint c h m j t a = turnPoint c h m j u b) : t = u ∧ a = b := by - let z := c + ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + - (side j * h : ℂ) * Complex.I - have hn (v q : ℝ) (hq : |q| ≤ 1) : dist (turnPoint c h m j v q) z = - turnRadius h m + side j * ribbonWidth h m * q := by - change dist (z + _) z = _ - rw [dist_self_add_left, norm_mul, Complex.norm_real, Real.norm_eq_abs, - turn_direction_norm, mul_one, abs_of_pos (turn_radial_coefficient_pos hh hm j hq)] - have hcoef := congrArg (fun w => dist w z) he - rw [hn t a ha, hn u b hb] at hcoef - have hs : side j ≠ 0 := by - intro hz - have hn := side_abs j - rw [hz, abs_zero] at hn - norm_num at hn - have hab : a = b := by - apply mul_left_cancel₀ (ribbonWidth_pos hh hm).ne' - apply mul_left_cancel₀ hs - nlinarith [hcoef] - subst b - have hre := congrArg Complex.re he - simp only [turnPoint, Complex.add_re, Complex.ofReal_re, Complex.mul_re, - Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, sub_zero, - zero_sub, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add] at hre - have hcos : Real.cos t = Real.cos u := by - nlinarith [turn_radial_coefficient_pos hh hm j ha] - exact ⟨Real.injOn_cos ht hu hcos, rfl⟩ - - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonMiddleMass +import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + +namespace OAI + +/-! Explicit columns and circular turns for the dense-visit serpentine. +The array and its normal bands fit in the displaced disk and stay a fixed +distance from the marked point. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine + +def side (j : ℕ) : ℝ := if Even j then 1 else -1 + +lemma side_abs (j : ℕ) : |side j| = 1 := by + unfold side + split <;> norm_num + +lemma side_sq (j : ℕ) : side j ^ 2 = 1 := by + unfold side + split <;> norm_num + +def displacementSign (w : ℂ) : ℝ := if 0 ≤ w.re then 1 else -1 + +def center (s : ℝ) (w : ℂ) : ℂ := w + ((displacementSign w * (s / 4) : ℝ) : ℂ) + +def halfHeight (s : ℝ) : ℝ := s / 200 + +def spacing (h : ℝ) (m : ℕ) : ℝ := h / m + +def turnRadius (h : ℝ) (m : ℕ) : ℝ := h / (2 * m) + +def ribbonWidth (h : ℝ) (m : ℕ) : ℝ := h / (100 * m) + +def columnOffset (h : ℝ) (m j : ℕ) : ℝ := -h + j * spacing h m + +/-- The closed width-one band over one straight pass, in physical scale. -/ +def columnPoint (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : ℂ := + c + ((columnOffset h m j - side j * ribbonWidth h m * a : ℝ) : ℂ) + + (y : ℂ) * Complex.I + +/-- A circular connector, with its actual normal ribbon coordinate `a`. +The parameter runs from zero to pi, alternating above and below. -/ +def turnPoint (c : ℂ) (h : ℝ) (m j : ℕ) (t a : ℝ) : ℂ := + c + ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + + (side j * h : ℂ) * Complex.I + + ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * + ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I) + +lemma center_dist {s : ℝ} (hs : 0 ≤ s) (w : ℂ) : dist (center s w) w = s / 4 := by + rw [center, dist_self_add_left, Complex.norm_real, Real.norm_eq_abs, abs_mul, + abs_of_nonneg (div_nonneg hs (by norm_num))] + have hh : |displacementSign w| = 1 := by unfold displacementSign; split <;> norm_num + rw [hh, one_mul] + +lemma center_im (s : ℝ) (w : ℂ) : (center s w).im = w.im := by + simp [center] + +lemma displaced_disk_inside + {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) : + closedBall (center s w) (s / 20) ⊆ ball w (w.im / 2) := by + intro z hz + have hd := (dist_triangle z (center s w) w).trans + (add_le_add hz (le_of_eq (center_dist hs.le w))) + change dist z w < w.im / 2 + linarith + +lemma displaced_disk_separated + {s : ℝ} (hs : 0 < s) (w z : ℂ) (hz : z ∈ closedBall (center s w) (s / 20)) : + s / 5 ≤ dist z w := by + have hh := dist_triangle (center s w) z w + rw [center_dist hs.le w, dist_comm (center s w) z] at hh + change dist z (center s w) ≤ s / 20 at hz + linarith + +lemma spacing_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < spacing h m := by + exact div_pos hh (by exact_mod_cast hm) + +lemma turnRadius_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < turnRadius h m := by + unfold turnRadius + positivity + +lemma ribbonWidth_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : 0 < ribbonWidth h m := by + unfold ribbonWidth + positivity + +lemma spacing_eq_width (h : ℝ) (m : ℕ) : spacing h m = 100 * ribbonWidth h m := by + unfold spacing ribbonWidth + by_cases hm : m = 0 + · simp [hm] + · have hmr : (m : ℝ) ≠ 0 := by exact_mod_cast hm + field_simp + +lemma turnRadius_eq_width (h : ℝ) (m : ℕ) : turnRadius h m = 50 * ribbonWidth h m := by + unfold turnRadius ribbonWidth + by_cases hm : m = 0 + · simp [hm] + · have hmr : (m : ℝ) ≠ 0 := by exact_mod_cast hm + field_simp + ring + +lemma columnOffset_bounds {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) + (hj : j ≤ 2 * m) : |columnOffset h m j| ≤ h := by + have hmr : (0 : ℝ) < m := by exact_mod_cast hm + have hjr : (j : ℝ) ≤ 2 * m := by exact_mod_cast hj + have hj0 : (0 : ℝ) ≤ j := Nat.cast_nonneg j + have hq := spacing_pos hh hm + have hmul : (j : ℝ) * spacing h m ≤ 2 * h := by + unfold spacing + rw [← mul_div_assoc] + exact (div_le_iff₀ hmr).mpr (by nlinarith) + rw [columnOffset] + exact abs_le.mpr ⟨by nlinarith [mul_nonneg hj0 hq.le], by linarith⟩ + +lemma width_le_halfHeight {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : + ribbonWidth h m ≤ h / 100 := by + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + unfold ribbonWidth + apply div_le_div_of_nonneg_left hh.le (by norm_num) + nlinarith + +lemma radius_le_halfHeight {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) : + turnRadius h m ≤ h / 2 := by + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + unfold turnRadius + apply div_le_div_of_nonneg_left hh.le (by norm_num) + nlinarith + +lemma columnPoint_bound {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) + (hj : j ≤ 2 * m) (c : ℂ) {y a : ℝ} (hy : |y| ≤ h) (ha : |a| ≤ 1) : + dist (columnPoint c h m j y a) c ≤ 3 * h := by + have ho := columnOffset_bounds hh hm hj + have he := width_le_halfHeight hh hm + have hwp := (ribbonWidth_pos hh hm).le + have hprod : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hwp] + exact mul_le_of_le_one_right hwp ha + have hx : |columnOffset h m j - side j * ribbonWidth h m * a| ≤ + |columnOffset h m j| + |side j * ribbonWidth h m * a| := by + simpa only [sub_eq_add_neg, abs_neg] using + abs_add_le (columnOffset h m j) (-(side j * ribbonWidth h m * a)) + have hn := Complex.norm_le_abs_re_add_abs_im + (((columnOffset h m j - side j * ribbonWidth h m * a : ℝ) : ℂ) + + (y : ℂ) * Complex.I) + simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, + Complex.mul_im, mul_one, zero_add] at hn + rw [columnPoint, add_assoc, dist_self_add_left] + linarith + +lemma turnPoint_zero (c : ℂ) (h : ℝ) (m j : ℕ) (a : ℝ) : + turnPoint c h m j 0 a = columnPoint c h m j (side j * h) a := by + apply Complex.ext + · simp [turnPoint, columnPoint] + ring + · simp [turnPoint, columnPoint] + +lemma side_succ (j : ℕ) : side (j + 1) = -side j := by + by_cases h : Even j <;> simp [side, Nat.even_add_one, h] + +lemma turnPoint_pi (c : ℂ) (h : ℝ) (m j : ℕ) (a : ℝ) : + turnPoint c h m j Real.pi a = columnPoint c h m (j + 1) (side j * h) a := by + have he : spacing h m = 2 * turnRadius h m := by + rw [spacing_eq_width, turnRadius_eq_width] + ring + apply Complex.ext + · simp [turnPoint, columnPoint, columnOffset, side_succ, Nat.cast_add, Nat.cast_one] + nlinarith + · simp [turnPoint, columnPoint, columnOffset, side_succ, Nat.cast_add, Nat.cast_one] + +lemma turn_direction_norm (j : ℕ) (t : ℝ) : + ‖((-Real.cos t : ℝ) : ℂ) + (side j * Real.sin t : ℂ) * Complex.I‖ = 1 := by + apply (sq_eq_sq₀ (norm_nonneg _) zero_le_one).mp + rw [Complex.sq_norm, Complex.normSq_apply] + simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, + Complex.mul_im, mul_one, zero_add] + calc + _ = Real.cos t ^ 2 + side j ^ 2 * Real.sin t ^ 2 := by ring + _ = 1 ^ 2 := by rw [side_sq, one_mul]; nlinarith [Real.sin_sq_add_cos_sq t] + +lemma turnPoint_bound {h : ℝ} (hh : 0 < h) {m j : ℕ} (hm : 0 < m) + (hj : j + 1 ≤ 2 * m) (c : ℂ) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : + dist (turnPoint c h m j t a) c ≤ 4 * h := by + have ho := columnOffset_bounds hh hm (show j ≤ 2 * m by omega) + have hr := radius_le_halfHeight hh hm + have hw := width_le_halfHeight hh hm + have hrp := (turnRadius_pos hh hm).le + have hwp := (ribbonWidth_pos hh hm).le + have hprod : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hwp] + exact mul_le_of_le_one_right hwp ha + have hadd := abs_add_le (turnRadius h m) (side j * ribbonWidth h m * a) + rw [abs_of_nonneg hrp] at hadd + have hcen := abs_add_le (columnOffset h m j) (turnRadius h m) + rw [abs_of_nonneg hrp] at hcen + have hn : ‖((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + + (side j * h : ℂ) * Complex.I + + ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * + ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I)‖ ≤ + |columnOffset h m j + turnRadius h m| + h + + |turnRadius h m + side j * ribbonWidth h m * a| := by + calc + _ ≤ (‖((columnOffset h m j + turnRadius h m : ℝ) : ℂ)‖ + + ‖(side j * h : ℂ) * Complex.I‖) + + ‖((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * + ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I)‖ := + (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) + _ = _ := by + simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, + mul_one, side_abs, one_mul, abs_of_pos hh, turn_direction_norm] + have he : turnPoint c h m j t a - c = + ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + + (side j * h : ℂ) * Complex.I + + ((turnRadius h m + side j * ribbonWidth h m * a : ℝ) : ℂ) * + ((-Real.cos t : ℝ) + (side j * Real.sin t : ℂ) * Complex.I) := by + unfold turnPoint + ring + rw [dist_eq_norm, he] + linarith + +lemma column_inside_displaced_disk {s : ℝ} (hs : 0 < s) (w : ℂ) + {m j : ℕ} (hm : 0 < m) (hj : j ≤ 2 * m) {y a : ℝ} + (hy : |y| ≤ halfHeight s) (ha : |a| ≤ 1) : + columnPoint (center s w) (halfHeight s) m j y a ∈ closedBall (center s w) (s / 20) := by + have hh : 0 < halfHeight s := by unfold halfHeight; positivity + have hb := columnPoint_bound hh hm hj (center s w) hy ha + change dist _ _ ≤ s / 20 + dsimp only [halfHeight] at hb ⊢ + linarith + + +@[simp] lemma columnPoint_re (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : + (columnPoint c h m j y a).re = c.re + columnOffset h m j - side j * ribbonWidth h m * a := by + simp [columnPoint] + ring + +@[simp] lemma columnPoint_im (c : ℂ) (h : ℝ) (m j : ℕ) (y a : ℝ) : + (columnPoint c h m j y a).im = c.im + y := by simp [columnPoint] + +lemma column_separation {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) + (hij : i < j) (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) : + 98 * ribbonWidth h m ≤ + dist (columnPoint c h m i y a) (columnPoint c h m j y' b) := by + have hepos := (ribbonWidth_pos hh hm).le + have hnoise (n : ℕ) (u : ℝ) (hu : |u| ≤ 1) : + |side n * ribbonWidth h m * u| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_nonneg hepos] + exact mul_le_of_le_one_right hepos hu + have hai := abs_le.mp (hnoise i a ha) + have hbj := abs_le.mp (hnoise j b hb) + have hijr : (i : ℝ) + 1 ≤ j := by exact_mod_cast Nat.succ_le_of_lt hij + have hmul := mul_le_mul_of_nonneg_right hijr (spacing_pos hh hm).le + have hre : 98 * ribbonWidth h m ≤ + (columnPoint c h m j y' b).re - (columnPoint c h m i y a).re := by + simp only [columnPoint_re, columnOffset] at * + rw [spacing_eq_width] at hmul ⊢ + nlinarith + calc + _ ≤ (columnPoint c h m j y' b).re - (columnPoint c h m i y a).re := hre + _ ≤ |(columnPoint c h m j y' b).re - (columnPoint c h m i y a).re| := le_abs_self _ + _ ≤ dist (columnPoint c h m i y a) (columnPoint c h m j y' b) := by + simpa only [dist_eq_norm, Complex.sub_re, abs_sub_comm] using + Complex.abs_re_le_norm (columnPoint c h m i y a - columnPoint c h m j y' b) + +lemma columnPoint_ne {h : ℝ} (hh : 0 < h) {m i j : ℕ} (hm : 0 < m) + (hij : i < j) (c : ℂ) (y y' : ℝ) {a b : ℝ} (ha : |a| ≤ 1) (hb : |b| ≤ 1) : + columnPoint c h m i y a ≠ columnPoint c h m j y' b := by + intro he + have hd := column_separation hh hm hij c y y' ha hb + rw [he, dist_self] at hd + have hp := ribbonWidth_pos hh hm + linarith + + +lemma turn_radial_coefficient_pos {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) + (j : ℕ) {a : ℝ} (ha : |a| ≤ 1) : + 0 < turnRadius h m + side j * ribbonWidth h m * a := by + have he := ribbonWidth_pos hh hm + have hb : |side j * ribbonWidth h m * a| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_pos he] + exact mul_le_of_le_one_right he.le ha + rw [turnRadius_eq_width] + linarith [(abs_le.mp hb).1] + +/-- Each circular turn, with its full normal band, is embedded. -/ +theorem turnPoint_injective {h : ℝ} (hh : 0 < h) {m : ℕ} (hm : 0 < m) + (c : ℂ) (j : ℕ) {t u a b : ℝ} (ht : t ∈ Icc 0 Real.pi) + (hu : u ∈ Icc 0 Real.pi) (ha : |a| ≤ 1) (hb : |b| ≤ 1) + (he : turnPoint c h m j t a = turnPoint c h m j u b) : t = u ∧ a = b := by + let z := c + ((columnOffset h m j + turnRadius h m : ℝ) : ℂ) + + (side j * h : ℂ) * Complex.I + have hn (v q : ℝ) (hq : |q| ≤ 1) : dist (turnPoint c h m j v q) z = + turnRadius h m + side j * ribbonWidth h m * q := by + change dist (z + _) z = _ + rw [dist_self_add_left, norm_mul, Complex.norm_real, Real.norm_eq_abs, + turn_direction_norm, mul_one, abs_of_pos (turn_radial_coefficient_pos hh hm j hq)] + have hcoef := congrArg (fun w => dist w z) he + rw [hn t a ha, hn u b hb] at hcoef + have hs : side j ≠ 0 := by + intro hz + have hn := side_abs j + rw [hz, abs_zero] at hn + norm_num at hn + have hab : a = b := by + apply mul_left_cancel₀ (ribbonWidth_pos hh hm).ne' + apply mul_left_cancel₀ hs + nlinarith [hcoef] + subst b + have hre := congrArg Complex.re he + simp only [turnPoint, Complex.add_re, Complex.ofReal_re, Complex.mul_re, + Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, sub_zero, + zero_sub, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add] at hre + have hcos : Real.cos t = Real.cos u := by + nlinarith [turn_radial_coefficient_pos hh hm j ha] + exact ⟨Real.injOn_cos ht hu hcos, rfl⟩ + + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedClearance.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedClearance.lean index 1703a53367ed7032b19a4fc5111e4435863097b7..85050c22539e299b1dfa07052d8ab02c7972afad 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedClearance.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedClearance.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Gauge.SerpentineExteriorBounds -import OAI.Probability.SLE.Gauge.SerpentineMarkedRegion -import OAI.Probability.SLE.Geometry.MarkedRouteClearance - -namespace OAI - -/-! Actual fitted ribbon points satisfy the geometric separation -conditions for the two marked-point exit routes. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -theorem physicalFitChart_markedRoutes_clearance {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : z ∈ closedRibbon) - (hheight : (physicalFitChart s w m z).im ≤ 2) (t : ℝ) : - s / 20 ≤ dist (markedLeftRoute w (s / 20) t) (physicalFitChart s w m z) ∧ - s / 20 ≤ dist (markedRightRoute w (s / 20) t) (physicalFitChart s w m z) := by - apply markedRoutes_past_clearance (by linarith) - ((Complex.re_le_norm w).trans hw1) ((Complex.im_le_norm w).trans hw1) - (by positivity) (by linarith) - (physicalFitChart_re_bounds hs hs1 w hw hw1 hm hz) hheight - rcases physicalFitChart_low_or_right hs w hw hm hz.2 with hlo | hright - · left - linarith - · right - rw [abs_of_nonneg hwr] at hright - linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineExteriorBounds +import OAI.Probability.SLE.Gauge.SerpentineMarkedRegion +import OAI.Probability.SLE.Geometry.MarkedRouteClearance + +namespace OAI + +/-! Actual fitted ribbon points satisfy the geometric separation +conditions for the two marked-point exit routes. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +theorem physicalFitChart_markedRoutes_clearance {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : z ∈ closedRibbon) + (hheight : (physicalFitChart s w m z).im ≤ 2) (t : ℝ) : + s / 20 ≤ dist (markedLeftRoute w (s / 20) t) (physicalFitChart s w m z) ∧ + s / 20 ≤ dist (markedRightRoute w (s / 20) t) (physicalFitChart s w m z) := by + apply markedRoutes_past_clearance (by linarith) + ((Complex.re_le_norm w).trans hw1) ((Complex.im_le_norm w).trans hw1) + (by positivity) (by linarith) + (physicalFitChart_re_bounds hs hs1 w hw hw1 hm hz) hheight + rcases physicalFitChart_low_or_right hs w hw hm hz.2 with hlo | hright + · left + linarith + · right + rw [abs_of_nonneg hwr] at hright + linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedGuide.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedGuide.lean index cbdd8f90bae4e951efb38824c4ff620e6fb5c69e..1b3f43a18ddcb2605610916b89e77ccde369f634 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedGuide.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedGuide.lean @@ -1,125 +1,125 @@ -import OAI.Probability.SLE.Gauge.SerpentineFiniteHeight -import OAI.Probability.SLE.Geometry.MarkedHookLocal - -namespace OAI - -/-! The explicit complete physical access guide from the final ribbon -tip, through the marked point, and then vertically to infinity. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -def terminalAccess (s : ℝ) (w : ℂ) (m : ℕ) (x : ℝ) (t : ℝ) : ℂ := - physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t) - -def completeMarkedAccess (s : ℝ) (w : ℂ) (m : ℕ) (x : ℝ) : ℝ → ℂ := - appendMarkedHook (terminalAccess s w m x) (fitHighTime s w m) w - (fitTerminalX s w) 2 (s / 20) - -lemma terminalAccess_continuous {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (x : ℝ) : Continuous (terminalAccess s w m x) := by - have hp : Continuous (physicalFitChart s w m) := - continuous_const.mul (fitChart_continuous hs w hw hm) - exact hp.comp (guideCoordinate_continuous _ _) - -lemma terminalAccess_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {x t : ℝ} (hx : |x| ≤ 1) (ht : 0 ≤ t) : - w.re + s / 5 ≤ (terminalAccess s w m x t).re := by - have hc := (fitCompletion_bounds hs w hw hm).1 - have hentry : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hA : fitEntry s w m < (guideCoordinate (fitCompletion s w m) x t).re := by - rw [guideCoordinate_re] - linarith - have him := (guideCoordinate_im_bound (k := (fitCompletion s w m : ℝ)) (x := x) ht).trans hx - have hh := physicalFitChart_array_right hs w hm him hA - rw [abs_of_nonneg hwr] at hh - exact by dsimp [terminalAccess]; linarith - -lemma completeMarkedAccess_continuous {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) (x : ℝ) : Continuous (completeMarkedAccess s w m x) := by - have hwi := (Complex.im_le_norm w).trans hw1 - apply appendMarkedHook_continuous _ (terminalAccess_continuous hs w hw hm x) - · unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - · linarith - · positivity - · exact physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x - -lemma completeMarkedAccess_injective {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) : - InjOn (completeMarkedAccess s w m x) (Ici 0) := by - have hT := fitHighTime_large hs hs1 w hw hw1 hm - have hwi := (Complex.im_le_norm w).trans hw1 - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - apply appendMarkedHook_injective _ (by linarith) - ((physicalFitChart_terminal_guide_injective hs w hw hm x).mono Icc_subset_Ici_self) - w hR (show w.im < (2 : ℝ) by linarith) (by positivity) - (physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x) - intro t ht - refine ⟨physicalFitChart_terminal_guide_strict_height hs w hw hm x ht, Or.inr ?_⟩ - have hr := terminalAccess_right hs w hw hwr hm hx ht.1 - dsimp only [terminalAccess] at hr - linarith - -lemma completeMarkedAccess_zero {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) (x : ℝ) : - completeMarkedAccess s w m x 0 = physicalFitChart s w m ((fitCompletion s w m : ℂ) + (x : ℂ) * Complex.I) := by - have hT := fitHighTime_large hs hs1 w hw hw1 hm - rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le (by linarith)] - simp [terminalAccess, guideCoordinate] - -lemma completeMarkedAccess_hits_marked {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) (x : ℝ) : - ∃ t : ℝ, 0 < t ∧ completeMarkedAccess s w m x t = w := by - have hT := fitHighTime_large hs hs1 w hw hw1 hm - have hwi := (Complex.im_le_norm w).trans hw1 - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - refine ⟨fitHighTime s w m + ((fitTerminalX s w - w.re) + (2 - w.im)), by linarith, ?_⟩ - exact appendMarkedHook_passes_marked _ _ w hR (by linarith) - -lemma completeMarkedAccess_norm_tendsto {s : ℝ} (hs : 0 < s) - (w : ℂ) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) (m : ℕ) (x : ℝ) : - Tendsto (fun t : ℝ => ‖completeMarkedAccess s w m x t‖) atTop atTop := by - have hwi : w.im < 2 := lt_of_le_of_lt ((Complex.im_le_norm w).trans hw1) (by norm_num) - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - let D := fitHighTime s w m + ((fitTerminalX s w - w.re) + (2 - w.im) + s / 20) - have hlin : Tendsto (fun t : ℝ => t - D + w.im) atTop atTop := by - simpa only [sub_eq_add_neg, add_assoc, id_eq] using - tendsto_atTop_add_const_right atTop (-D + w.im) tendsto_id - have him : Tendsto (fun t : ℝ => (completeMarkedAccess s w m x t).im) atTop atTop := by - apply hlin.congr' - filter_upwards [eventually_ge_atTop (D + 1)] with t ht - have ht' : fitHighTime s w m < t := by dsimp only [D] at ht; linarith - have hc : (fitTerminalX s w - w.re) + (2 - w.im) + s / 20 < - t - fitHighTime s w m := by dsimp only [D] at ht; linarith - rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt ht', - markedHook_tail w hwi (by positivity) hc] - simp only [hookUp, Complex.add_im, Complex.mul_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add] - dsimp only [D] - ring - exact tendsto_atTop_mono (fun t => Complex.im_le_norm _) him - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineFiniteHeight +import OAI.Probability.SLE.Geometry.MarkedHookLocal + +namespace OAI + +/-! The explicit complete physical access guide from the final ribbon +tip, through the marked point, and then vertically to infinity. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +def terminalAccess (s : ℝ) (w : ℂ) (m : ℕ) (x : ℝ) (t : ℝ) : ℂ := + physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t) + +def completeMarkedAccess (s : ℝ) (w : ℂ) (m : ℕ) (x : ℝ) : ℝ → ℂ := + appendMarkedHook (terminalAccess s w m x) (fitHighTime s w m) w + (fitTerminalX s w) 2 (s / 20) + +lemma terminalAccess_continuous {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (x : ℝ) : Continuous (terminalAccess s w m x) := by + have hp : Continuous (physicalFitChart s w m) := + continuous_const.mul (fitChart_continuous hs w hw hm) + exact hp.comp (guideCoordinate_continuous _ _) + +lemma terminalAccess_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {x t : ℝ} (hx : |x| ≤ 1) (ht : 0 ≤ t) : + w.re + s / 5 ≤ (terminalAccess s w m x t).re := by + have hc := (fitCompletion_bounds hs w hw hm).1 + have hentry : 0 ≤ columnEntry 0 m (2 * m) := columnEntry_nonneg le_rfl _ _ + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hA : fitEntry s w m < (guideCoordinate (fitCompletion s w m) x t).re := by + rw [guideCoordinate_re] + linarith + have him := (guideCoordinate_im_bound (k := (fitCompletion s w m : ℝ)) (x := x) ht).trans hx + have hh := physicalFitChart_array_right hs w hm him hA + rw [abs_of_nonneg hwr] at hh + exact by dsimp [terminalAccess]; linarith + +lemma completeMarkedAccess_continuous {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) (x : ℝ) : Continuous (completeMarkedAccess s w m x) := by + have hwi := (Complex.im_le_norm w).trans hw1 + apply appendMarkedHook_continuous _ (terminalAccess_continuous hs w hw hm x) + · unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + · linarith + · positivity + · exact physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x + +lemma completeMarkedAccess_injective {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) : + InjOn (completeMarkedAccess s w m x) (Ici 0) := by + have hT := fitHighTime_large hs hs1 w hw hw1 hm + have hwi := (Complex.im_le_norm w).trans hw1 + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + apply appendMarkedHook_injective _ (by linarith) + ((physicalFitChart_terminal_guide_injective hs w hw hm x).mono Icc_subset_Ici_self) + w hR (show w.im < (2 : ℝ) by linarith) (by positivity) + (physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x) + intro t ht + refine ⟨physicalFitChart_terminal_guide_strict_height hs w hw hm x ht, Or.inr ?_⟩ + have hr := terminalAccess_right hs w hw hwr hm hx ht.1 + dsimp only [terminalAccess] at hr + linarith + +lemma completeMarkedAccess_zero {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) (x : ℝ) : + completeMarkedAccess s w m x 0 = physicalFitChart s w m ((fitCompletion s w m : ℂ) + (x : ℂ) * Complex.I) := by + have hT := fitHighTime_large hs hs1 w hw hw1 hm + rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le (by linarith)] + simp [terminalAccess, guideCoordinate] + +lemma completeMarkedAccess_hits_marked {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) (x : ℝ) : + ∃ t : ℝ, 0 < t ∧ completeMarkedAccess s w m x t = w := by + have hT := fitHighTime_large hs hs1 w hw hw1 hm + have hwi := (Complex.im_le_norm w).trans hw1 + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + refine ⟨fitHighTime s w m + ((fitTerminalX s w - w.re) + (2 - w.im)), by linarith, ?_⟩ + exact appendMarkedHook_passes_marked _ _ w hR (by linarith) + +lemma completeMarkedAccess_norm_tendsto {s : ℝ} (hs : 0 < s) + (w : ℂ) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) (m : ℕ) (x : ℝ) : + Tendsto (fun t : ℝ => ‖completeMarkedAccess s w m x t‖) atTop atTop := by + have hwi : w.im < 2 := lt_of_le_of_lt ((Complex.im_le_norm w).trans hw1) (by norm_num) + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + let D := fitHighTime s w m + ((fitTerminalX s w - w.re) + (2 - w.im) + s / 20) + have hlin : Tendsto (fun t : ℝ => t - D + w.im) atTop atTop := by + simpa only [sub_eq_add_neg, add_assoc, id_eq] using + tendsto_atTop_add_const_right atTop (-D + w.im) tendsto_id + have him : Tendsto (fun t : ℝ => (completeMarkedAccess s w m x t).im) atTop atTop := by + apply hlin.congr' + filter_upwards [eventually_ge_atTop (D + 1)] with t ht + have ht' : fitHighTime s w m < t := by dsimp only [D] at ht; linarith + have hc : (fitTerminalX s w - w.re) + (2 - w.im) + s / 20 < + t - fitHighTime s w m := by dsimp only [D] at ht; linarith + rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt ht', + markedHook_tail w hwi (by positivity) hc] + simp only [hookUp, Complex.add_im, Complex.mul_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add] + dsimp only [D] + ring + exact tendsto_atTop_mono (fun t => Complex.im_le_norm _) him + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedLocal.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedLocal.lean index 7587be9d9f4e2d067cd7cd3d4eb831c9ac8ec205..331e2845ff4e185b1c42f3cb7af0d266fec92fc0 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedLocal.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedLocal.lean @@ -1,104 +1,104 @@ -import OAI.Probability.SLE.Gauge.SerpentineMarkedGuide - -namespace OAI - -/-! The explicit marked access guide meets a small corner-chart disk -exactly along its real diameter. The distant terminal prefix is excluded -quantitatively, so there are no nonlocal intersections. -/ - -noncomputable section -open Set Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma terminalAccess_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {x t : ℝ} (hx : |x| ≤ 1) (ht : 0 ≤ t) : - s / 5 ≤ dist (terminalAccess s w m x t) w := by - have hr := terminalAccess_right hs w hw hwr hm hx ht - have hb := Complex.abs_re_le_norm (terminalAccess s w m x t - w) - rw [Complex.sub_re, ← dist_eq_norm] at hb - linarith [le_abs_self ((terminalAccess s w m x t).re - w.re)] - -theorem completeMarkedAccess_corner_iff {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) {z : ℂ} - (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : - (∃ t : ℝ, 0 < t ∧ completeMarkedAccess s w m x t = markedCornerChart w z) ↔ z.im = 0 := by - have hn : ‖z‖ < (s / 20) / 12 := by simpa only [mem_ball, dist_zero_right] using hz - have hwi : w.im ≤ 1 := (Complex.im_le_norm w).trans hw1 - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - have hT := fitHighTime_large hs hs1 w hw hw1 hm - have hcorner := markedHook_small_corner_iff (w := w) (z := z) hR - (show w.im < 2 by linarith) (show 0 < s / 20 by positivity) - (show 0 < (s / 20) / 12 by positivity) - (show 3 * ((s / 20) / 12) < 2 - w.im by linarith) - (show 3 * ((s / 20) / 12) < s / 20 by linarith) hn - constructor - · rintro ⟨t, ht, he⟩ - by_cases htT : t ≤ fitHighTime s w m - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le htT] at he - have hd := terminalAccess_distance hs w hw hwr hm hx ht.le - rw [he] at hd - have hb := markedCornerChart_norm w z - linarith - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge htT)] at he - exact hcorner.mp ⟨t - fitHighTime s w m, by linarith, he⟩ - · intro him - obtain ⟨u, hu, he⟩ := hcorner.mpr him - refine ⟨fitHighTime s w m + u, by linarith, ?_⟩ - by_cases hu0 : u = 0 - · subst u - rw [add_zero, completeMarkedAccess, appendMarkedHook, curveJoin_of_le le_rfl] - change physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x (fitHighTime s w m)) = _ - rw [physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x] - simpa [markedHook_zero w hR] using he - · have hup : 0 < u := lt_of_le_of_ne hu (Ne.symm hu0) - rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (by linarith), add_sub_cancel_left] - exact he - -theorem completeMarkedAccess_corner_nnreal_iff {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) {z : ℂ} - (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : - (∃ t : ℝ≥0, 0 < t ∧ completeMarkedAccess s w m x t = markedCornerChart w z) ↔ z.im = 0 := by - rw [← completeMarkedAccess_corner_iff hs hs1 w hw hw1 hwr hm hx hz] - constructor - · rintro ⟨t, ht, he⟩ - exact ⟨t, by exact_mod_cast ht, he⟩ - · rintro ⟨t, ht, he⟩ - exact ⟨⟨t, ht.le⟩, by exact_mod_cast ht, he⟩ - -lemma completeMarkedAccess_upper {s : ℝ} (hs : 0 < s) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x u : ℝ} (hu : 0 ≤ u) : - completeMarkedAccess s w m x u ∈ upperHalfPlane := by - have hwi := (Complex.im_le_norm w).trans hw1 - change 0 < (completeMarkedAccess s w m x u).im - by_cases huT : u ≤ fitHighTime s w m - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] - change 0 < (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x u)).im - rw [physicalFitChart_terminal_guide hs w hw hm x hu] - simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] - have hb := (fitTerminalHeight_bounds hs w hw hm).1 - have hp := mul_nonneg (fit_width_pos hs hm).le hu - linarith - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - have hh := markedHook_im_lower hR (show w.im < 2 by linarith) - (show 0 < s / 20 by positivity) (show 0 ≤ u - fitHighTime s w m by linarith) - linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineMarkedGuide + +namespace OAI + +/-! The explicit marked access guide meets a small corner-chart disk +exactly along its real diameter. The distant terminal prefix is excluded +quantitatively, so there are no nonlocal intersections. -/ + +noncomputable section +open Set Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma terminalAccess_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {x t : ℝ} (hx : |x| ≤ 1) (ht : 0 ≤ t) : + s / 5 ≤ dist (terminalAccess s w m x t) w := by + have hr := terminalAccess_right hs w hw hwr hm hx ht + have hb := Complex.abs_re_le_norm (terminalAccess s w m x t - w) + rw [Complex.sub_re, ← dist_eq_norm] at hb + linarith [le_abs_self ((terminalAccess s w m x t).re - w.re)] + +theorem completeMarkedAccess_corner_iff {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) {z : ℂ} + (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : + (∃ t : ℝ, 0 < t ∧ completeMarkedAccess s w m x t = markedCornerChart w z) ↔ z.im = 0 := by + have hn : ‖z‖ < (s / 20) / 12 := by simpa only [mem_ball, dist_zero_right] using hz + have hwi : w.im ≤ 1 := (Complex.im_le_norm w).trans hw1 + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + have hT := fitHighTime_large hs hs1 w hw hw1 hm + have hcorner := markedHook_small_corner_iff (w := w) (z := z) hR + (show w.im < 2 by linarith) (show 0 < s / 20 by positivity) + (show 0 < (s / 20) / 12 by positivity) + (show 3 * ((s / 20) / 12) < 2 - w.im by linarith) + (show 3 * ((s / 20) / 12) < s / 20 by linarith) hn + constructor + · rintro ⟨t, ht, he⟩ + by_cases htT : t ≤ fitHighTime s w m + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le htT] at he + have hd := terminalAccess_distance hs w hw hwr hm hx ht.le + rw [he] at hd + have hb := markedCornerChart_norm w z + linarith + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge htT)] at he + exact hcorner.mp ⟨t - fitHighTime s w m, by linarith, he⟩ + · intro him + obtain ⟨u, hu, he⟩ := hcorner.mpr him + refine ⟨fitHighTime s w m + u, by linarith, ?_⟩ + by_cases hu0 : u = 0 + · subst u + rw [add_zero, completeMarkedAccess, appendMarkedHook, curveJoin_of_le le_rfl] + change physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x (fitHighTime s w m)) = _ + rw [physicalFitChart_terminal_guide_endpoint hs hs1 w hw hw1 hm x] + simpa [markedHook_zero w hR] using he + · have hup : 0 < u := lt_of_le_of_ne hu (Ne.symm hu0) + rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (by linarith), add_sub_cancel_left] + exact he + +theorem completeMarkedAccess_corner_nnreal_iff {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x : ℝ} (hx : |x| ≤ 1) {z : ℂ} + (hz : z ∈ ball (0 : ℂ) ((s / 20) / 12)) : + (∃ t : ℝ≥0, 0 < t ∧ completeMarkedAccess s w m x t = markedCornerChart w z) ↔ z.im = 0 := by + rw [← completeMarkedAccess_corner_iff hs hs1 w hw hw1 hwr hm hx hz] + constructor + · rintro ⟨t, ht, he⟩ + exact ⟨t, by exact_mod_cast ht, he⟩ + · rintro ⟨t, ht, he⟩ + exact ⟨⟨t, ht.le⟩, by exact_mod_cast ht, he⟩ + +lemma completeMarkedAccess_upper {s : ℝ} (hs : 0 < s) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x u : ℝ} (hu : 0 ≤ u) : + completeMarkedAccess s w m x u ∈ upperHalfPlane := by + have hwi := (Complex.im_le_norm w).trans hw1 + change 0 < (completeMarkedAccess s w m x u).im + by_cases huT : u ≤ fitHighTime s w m + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] + change 0 < (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x u)).im + rw [physicalFitChart_terminal_guide hs w hw hm x hu] + simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] + have hb := (fitTerminalHeight_bounds hs w hw hm).1 + have hp := mul_nonneg (fit_width_pos hs hm).le hu + linarith + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + have hh := markedHook_im_lower hR (show w.im < 2 by linarith) + (show 0 < s / 20 by positivity) (show 0 ≤ u - fitHighTime s w m by linarith) + linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedReflection.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedReflection.lean index fde1a329baf47d5c126967a135018e86f091463d..3821bfdabe31fbbc1fddd10b0167d9a53718ba54 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedReflection.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedReflection.lean @@ -1,129 +1,129 @@ -import OAI.Probability.SLE.Gauge.SerpentineMarkedRouteClearance -import OAI.Probability.SLE.Gauge.RibbonGoodPrefix - -namespace OAI - -/-! Reflection of the concrete experiment and its actual narrow-prefix -certificate. It reduces a marked point on the left to the right-hand layout. -/ - -noncomputable section -open Set Metric ComplexConjugate -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -@[simp] lemma fitX_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitX s (loewnerMirror w) m = fitX s w m := by simp [fitX] - -@[simp] lemma fitY_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitY s (loewnerMirror w) m = fitY s w m := by simp [fitY] - -@[simp] lemma fitChart_mirror (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : - fitChart s (loewnerMirror w) m z = fitChart s w m z := by - simp only [fitChart, fitSpine, fitTangent, fitX_mirror, fitY_mirror] - -@[simp] lemma physicalFitChart_mirror (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : - physicalFitChart s (loewnerMirror w) m z = physicalFitChart s w m z := by - simp only [physicalFitChart, fitChart_mirror] - -@[simp] lemma fitEntry_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitEntry s (loewnerMirror w) m = fitEntry s w m := by - simp only [fitEntry, fitX_mirror, fitY_mirror] - -@[simp] lemma fitCompletion_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitCompletion s (loewnerMirror w) m = fitCompletion s w m := by - simp only [fitCompletion, fitEntry_mirror] - -@[simp] lemma fitTerminalX_mirror (s : ℝ) (w : ℂ) : - fitTerminalX s (loewnerMirror w) = fitTerminalX s w := by simp [fitTerminalX] - -@[simp] lemma fitTerminalHeight_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitTerminalHeight s (loewnerMirror w) m = fitTerminalHeight s w m := by - simp only [fitTerminalHeight, fitY_mirror, fitCompletion_mirror, fitEntry_mirror] - -@[simp] lemma fitHighTime_mirror (s : ℝ) (w : ℂ) (m : ℕ) : - fitHighTime s (loewnerMirror w) m = fitHighTime s w m := by - simp only [fitHighTime, fitTerminalHeight_mirror] - -lemma physicalExperimentChart_negative {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) : - physicalExperimentChart s w m = reflectedChart (physicalFitChart s (loewnerMirror w) m) := by - funext z - simp only [physicalExperimentChart, ite_eq_right (not_le_of_gt hw), reflectedChart, - physicalFitChart_mirror] - -lemma experimentChart_negative {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) : - experimentChart s w m = reflectedChart (fitChart s (loewnerMirror w) m) := by - funext z - simp only [experimentChart, ite_eq_right (not_le_of_gt hw), reflectedChart, fitChart_mirror] - -lemma ribbonCore_reflected_iff (P : ℂ → ℂ) (ε : ℝ) (z : ℂ) : - z ∈ ribbonCore (reflectedChart P) ε ↔ loewnerMirror z ∈ ribbonCore P ε := by - constructor - · rintro ⟨u, ⟨hu, hε⟩, he⟩ - refine ⟨conj u, ⟨conj_mem_closedRibbon hu, ?_⟩, ?_⟩ - · simpa only [Complex.conj_im, abs_neg] using hε - · rw [← he] - simp only [reflectedChart, loewnerMirror_twice] - · rintro ⟨u, ⟨hu, hε⟩, he⟩ - refine ⟨conj u, ⟨conj_mem_closedRibbon hu, ?_⟩, ?_⟩ - · simpa only [Complex.conj_im, abs_neg] using hε - · simpa only [reflectedChart, Complex.conj_conj, loewnerMirror_twice] using - congrArg loewnerMirror he - -lemma goodRibbonPrefix_reflected {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) - (τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε : ℝ) : - GoodRibbonPrefix γ τ (reflectedChart P) ε = - GoodRibbonPrefix (fun ω t => loewnerMirror (γ ω t)) τ P ε := by - ext ω - change (_ ∧ _ ∧ _) ↔ (_ ∧ _ ∧ _) - constructor - · rintro ⟨hτ, hp, he⟩ - exact ⟨hτ, fun t ht => (ribbonCore_reflected_iff P ε _).mp (hp ht), - (ribbonCore_reflected_iff P (ε / 4) _).mp he⟩ - · rintro ⟨hτ, hp, he⟩ - exact ⟨hτ, fun t ht => (ribbonCore_reflected_iff P ε _).mpr (hp ht), - (ribbonCore_reflected_iff P (ε / 4) _).mpr he⟩ - -lemma goodRibbonPrefix_physical_negative {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) - (τ : Ω → WithTop ℝ≥0) {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) (ε : ℝ) : - GoodRibbonPrefix γ τ (physicalExperimentChart s w m) ε = - GoodRibbonPrefix (fun ω t => loewnerMirror (γ ω t)) τ - (physicalFitChart s (loewnerMirror w) m) ε := by - rw [physicalExperimentChart_negative hw m] - exact goodRibbonPrefix_reflected γ τ _ ε - -lemma physicalExperimentChart_reflect_point {s : ℝ} {w : ℂ} (hw : w.re < 0) - (m : ℕ) (z : ℂ) : - loewnerMirror (physicalExperimentChart s w m z) = - physicalFitChart s (loewnerMirror w) m (conj z) := by - rw [physicalExperimentChart_negative hw m] - simp only [reflectedChart, loewnerMirror_twice] - -def mirroredProgress (g : C(ℂ, ℝ)) : C(ℂ, ℝ) := - ⟨fun z => g (loewnerMirror z), g.continuous.comp loewnerMirror.continuous⟩ - -lemma mirroredProgress_trace (g : C(ℂ, ℝ)) (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : - mirroredProgress g ((loewnerMirror ∘ γ) t) = g (γ t) := by - simp [mirroredProgress] - -lemma mirroredProgress_chart (g : C(ℂ, ℝ)) (P : ℂ → ℂ) - (hg : ∀ z ∈ closedRibbon, g (P z) = z.re) : - ∀ z ∈ closedRibbon, mirroredProgress g (reflectedChart P z) = z.re := by - intro z hz - change g (loewnerMirror (loewnerMirror (P (conj z)))) = z.re - rw [loewnerMirror_twice] - simpa only [Complex.conj_re] using hg (conj z) (conj_mem_closedRibbon hz) - -lemma mirroredProgress_firstFrontTime (g : C(ℂ, ℝ)) (γ : ℝ≥0 → ℂ) (k : ℝ) : - FirstFrontPassage.firstFrontTime (fun t => mirroredProgress g ((loewnerMirror ∘ γ) t)) k = - FirstFrontPassage.firstFrontTime (fun t => g (γ t)) k := by - congr 1 - funext t - exact mirroredProgress_trace g γ t - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineMarkedRouteClearance +import OAI.Probability.SLE.Gauge.RibbonGoodPrefix + +namespace OAI + +/-! Reflection of the concrete experiment and its actual narrow-prefix +certificate. It reduces a marked point on the left to the right-hand layout. -/ + +noncomputable section +open Set Metric ComplexConjugate +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +@[simp] lemma fitX_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitX s (loewnerMirror w) m = fitX s w m := by simp [fitX] + +@[simp] lemma fitY_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitY s (loewnerMirror w) m = fitY s w m := by simp [fitY] + +@[simp] lemma fitChart_mirror (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : + fitChart s (loewnerMirror w) m z = fitChart s w m z := by + simp only [fitChart, fitSpine, fitTangent, fitX_mirror, fitY_mirror] + +@[simp] lemma physicalFitChart_mirror (s : ℝ) (w : ℂ) (m : ℕ) (z : ℂ) : + physicalFitChart s (loewnerMirror w) m z = physicalFitChart s w m z := by + simp only [physicalFitChart, fitChart_mirror] + +@[simp] lemma fitEntry_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitEntry s (loewnerMirror w) m = fitEntry s w m := by + simp only [fitEntry, fitX_mirror, fitY_mirror] + +@[simp] lemma fitCompletion_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitCompletion s (loewnerMirror w) m = fitCompletion s w m := by + simp only [fitCompletion, fitEntry_mirror] + +@[simp] lemma fitTerminalX_mirror (s : ℝ) (w : ℂ) : + fitTerminalX s (loewnerMirror w) = fitTerminalX s w := by simp [fitTerminalX] + +@[simp] lemma fitTerminalHeight_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitTerminalHeight s (loewnerMirror w) m = fitTerminalHeight s w m := by + simp only [fitTerminalHeight, fitY_mirror, fitCompletion_mirror, fitEntry_mirror] + +@[simp] lemma fitHighTime_mirror (s : ℝ) (w : ℂ) (m : ℕ) : + fitHighTime s (loewnerMirror w) m = fitHighTime s w m := by + simp only [fitHighTime, fitTerminalHeight_mirror] + +lemma physicalExperimentChart_negative {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) : + physicalExperimentChart s w m = reflectedChart (physicalFitChart s (loewnerMirror w) m) := by + funext z + simp only [physicalExperimentChart, ite_eq_right (not_le_of_gt hw), reflectedChart, + physicalFitChart_mirror] + +lemma experimentChart_negative {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) : + experimentChart s w m = reflectedChart (fitChart s (loewnerMirror w) m) := by + funext z + simp only [experimentChart, ite_eq_right (not_le_of_gt hw), reflectedChart, fitChart_mirror] + +lemma ribbonCore_reflected_iff (P : ℂ → ℂ) (ε : ℝ) (z : ℂ) : + z ∈ ribbonCore (reflectedChart P) ε ↔ loewnerMirror z ∈ ribbonCore P ε := by + constructor + · rintro ⟨u, ⟨hu, hε⟩, he⟩ + refine ⟨conj u, ⟨conj_mem_closedRibbon hu, ?_⟩, ?_⟩ + · simpa only [Complex.conj_im, abs_neg] using hε + · rw [← he] + simp only [reflectedChart, loewnerMirror_twice] + · rintro ⟨u, ⟨hu, hε⟩, he⟩ + refine ⟨conj u, ⟨conj_mem_closedRibbon hu, ?_⟩, ?_⟩ + · simpa only [Complex.conj_im, abs_neg] using hε + · simpa only [reflectedChart, Complex.conj_conj, loewnerMirror_twice] using + congrArg loewnerMirror he + +lemma goodRibbonPrefix_reflected {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) + (τ : Ω → WithTop ℝ≥0) (P : ℂ → ℂ) (ε : ℝ) : + GoodRibbonPrefix γ τ (reflectedChart P) ε = + GoodRibbonPrefix (fun ω t => loewnerMirror (γ ω t)) τ P ε := by + ext ω + change (_ ∧ _ ∧ _) ↔ (_ ∧ _ ∧ _) + constructor + · rintro ⟨hτ, hp, he⟩ + exact ⟨hτ, fun t ht => (ribbonCore_reflected_iff P ε _).mp (hp ht), + (ribbonCore_reflected_iff P (ε / 4) _).mp he⟩ + · rintro ⟨hτ, hp, he⟩ + exact ⟨hτ, fun t ht => (ribbonCore_reflected_iff P ε _).mpr (hp ht), + (ribbonCore_reflected_iff P (ε / 4) _).mpr he⟩ + +lemma goodRibbonPrefix_physical_negative {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) + (τ : Ω → WithTop ℝ≥0) {s : ℝ} {w : ℂ} (hw : w.re < 0) (m : ℕ) (ε : ℝ) : + GoodRibbonPrefix γ τ (physicalExperimentChart s w m) ε = + GoodRibbonPrefix (fun ω t => loewnerMirror (γ ω t)) τ + (physicalFitChart s (loewnerMirror w) m) ε := by + rw [physicalExperimentChart_negative hw m] + exact goodRibbonPrefix_reflected γ τ _ ε + +lemma physicalExperimentChart_reflect_point {s : ℝ} {w : ℂ} (hw : w.re < 0) + (m : ℕ) (z : ℂ) : + loewnerMirror (physicalExperimentChart s w m z) = + physicalFitChart s (loewnerMirror w) m (conj z) := by + rw [physicalExperimentChart_negative hw m] + simp only [reflectedChart, loewnerMirror_twice] + +def mirroredProgress (g : C(ℂ, ℝ)) : C(ℂ, ℝ) := + ⟨fun z => g (loewnerMirror z), g.continuous.comp loewnerMirror.continuous⟩ + +lemma mirroredProgress_trace (g : C(ℂ, ℝ)) (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : + mirroredProgress g ((loewnerMirror ∘ γ) t) = g (γ t) := by + simp [mirroredProgress] + +lemma mirroredProgress_chart (g : C(ℂ, ℝ)) (P : ℂ → ℂ) + (hg : ∀ z ∈ closedRibbon, g (P z) = z.re) : + ∀ z ∈ closedRibbon, mirroredProgress g (reflectedChart P z) = z.re := by + intro z hz + change g (loewnerMirror (loewnerMirror (P (conj z)))) = z.re + rw [loewnerMirror_twice] + simpa only [Complex.conj_re] using hg (conj z) (conj_mem_closedRibbon hz) + +lemma mirroredProgress_firstFrontTime (g : C(ℂ, ℝ)) (γ : ℝ≥0 → ℂ) (k : ℝ) : + FirstFrontPassage.firstFrontTime (fun t => mirroredProgress g ((loewnerMirror ∘ γ) t)) k = + FirstFrontPassage.firstFrontTime (fun t => g (γ t)) k := by + congr 1 + funext t + exact mirroredProgress_trace g γ t + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRegion.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRegion.lean index 1de789f83a0f176d953a3a2ecbca0d505e94d7bf..e28d59c264e15d76b74891502eab72f4cb7d7458 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRegion.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRegion.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Gauge.SerpentineFittedShadow - -namespace OAI - -/-! The connector lies below the marked point, and every later part of -the rightward serpentine lies strictly to its right. This is the geometric -separation used to append the marked-point access hook. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -theorem physicalFitChart_low_or_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : - (physicalFitChart s w m z).im ≤ s / 2 ∨ - |w.re| + s / 5 ≤ (physicalFitChart s w m z).re := by - by_cases h4 : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) - · exact Or.inl (physicalFitChart_low hs w hw hm ha h4) - · right - by_cases hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - · have hh := physicalFitChart_tail_right hs w hw hm ha (lt_of_not_ge h4) hA - linarith - · have hh := physicalFitChart_array_right hs w hm ha (lt_of_not_ge hA) - linarith - -theorem physicalFitChart_marked_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : - s / 5 ≤ dist (physicalFitChart s w m z) w := by - rw [dist_eq_norm] - rcases physicalFitChart_low_or_right hs w hw hm ha with hlow | hright - · have hh := Complex.abs_im_le_norm (physicalFitChart s w m z - w) - rw [Complex.sub_im] at hh - have hl := neg_le_abs ((physicalFitChart s w m z).im - w.im) - linarith - · rw [abs_of_nonneg hwr] at hright - have hh := Complex.abs_re_le_norm (physicalFitChart s w m z - w) - rw [Complex.sub_re] at hh - have hl := le_abs_self ((physicalFitChart s w m z).re - w.re) - linarith - -theorem physicalFitChart_hook_region {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : - (physicalFitChart s w m z).im < w.im ∨ w.re < (physicalFitChart s w m z).re := by - rcases physicalFitChart_low_or_right hs w hw hm ha with hh | hh - · left - linarith - · right - rw [abs_of_nonneg hwr] at hh - linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineFittedShadow + +namespace OAI + +/-! The connector lies below the marked point, and every later part of +the rightward serpentine lies strictly to its right. This is the geometric +separation used to append the marked-point access hook. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +theorem physicalFitChart_low_or_right {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : + (physicalFitChart s w m z).im ≤ s / 2 ∨ + |w.re| + s / 5 ≤ (physicalFitChart s w m z).re := by + by_cases h4 : z.re ≤ connectorFourth (fitHeight m) (fitRadius m) (fitX s w m) + · exact Or.inl (physicalFitChart_low hs w hw hm ha h4) + · right + by_cases hA : z.re ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + · have hh := physicalFitChart_tail_right hs w hw hm ha (lt_of_not_ge h4) hA + linarith + · have hh := physicalFitChart_array_right hs w hm ha (lt_of_not_ge hA) + linarith + +theorem physicalFitChart_marked_distance {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : + s / 5 ≤ dist (physicalFitChart s w m z) w := by + rw [dist_eq_norm] + rcases physicalFitChart_low_or_right hs w hw hm ha with hlow | hright + · have hh := Complex.abs_im_le_norm (physicalFitChart s w m z - w) + rw [Complex.sub_im] at hh + have hl := neg_le_abs ((physicalFitChart s w m z).im - w.im) + linarith + · rw [abs_of_nonneg hwr] at hright + have hh := Complex.abs_re_le_norm (physicalFitChart s w m z - w) + rw [Complex.sub_re] at hh + have hl := le_abs_self ((physicalFitChart s w m z).re - w.re) + linarith + +theorem physicalFitChart_hook_region {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + (hwr : 0 ≤ w.re) {m : ℕ} (hm : 0 < m) {z : ℂ} (ha : |z.im| ≤ 1) : + (physicalFitChart s w m z).im < w.im ∨ w.re < (physicalFitChart s w m z).re := by + rcases physicalFitChart_low_or_right hs w hw hm ha with hh | hh + · left + linarith + · right + rw [abs_of_nonneg hwr] at hh + linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRouteClearance.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRouteClearance.lean index 91dc7920ce69fb3e920dbd5efd3d3672f2a89f56..08fb795cd02e82826eccfa9f3aeb41e67170c1c2 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRouteClearance.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineMarkedRouteClearance.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Gauge.SerpentineMarkedLocal -import OAI.Probability.SLE.Geometry.MarkedRouteEarlyClearance - -namespace OAI - -/-! The two exit routes avoid every point of the complete marked access -guide by one uniform margin, including its distant initial prefix. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -lemma terminalAccess_bounds {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x u : ℝ} (hx : |x| ≤ 1) - (hu : u ∈ Icc (0 : ℝ) (fitHighTime s w m)) : - -2 ≤ (terminalAccess s w m x u).re ∧ (terminalAccess s w m x u).re ≤ 2 ∧ - (terminalAccess s w m x u).im ≤ 2 ∧ w.re + s / 5 ≤ (terminalAccess s w m x u).re := by - have hz : guideCoordinate (fitCompletion s w m) x u ∈ closedRibbon := by - refine ⟨?_, (guideCoordinate_im_bound hu.1).trans hx⟩ - rw [guideCoordinate_re] - exact add_nonneg (Nat.cast_nonneg _) hu.1 - have hre := physicalFitChart_re_bounds hs hs1 w hw hw1 hm hz - exact ⟨hre.1, hre.2, physicalFitChart_terminal_guide_height hs w hw hm x hu, - terminalAccess_right hs w hw hwr hm hx hu.1⟩ - -/-- A margin independent of the number of serpentine columns. -/ -theorem completeMarkedAccess_route_clearance {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) - {m : ℕ} (hm : 0 < m) {x t u : ℝ} (hx : |x| ≤ 1) (ht : 1 / 64 ≤ t) (hu : 0 ≤ u) : - (s / 20) / 1000 ≤ dist (markedLeftRoute w (s / 20) t) (completeMarkedAccess s w m x u) ∧ - (s / 20) / 1000 ≤ dist (markedRightRoute w (s / 20) t) (completeMarkedAccess s w m x u) := by - have hwr1 := (Complex.re_le_norm w).trans hw1 - have hwi1 := (Complex.im_le_norm w).trans hw1 - have ha : 0 < s / 20 := by positivity - have ha1 : s / 20 ≤ 1 := by linarith - by_cases huT : u ≤ fitHighTime s w m - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] - have hb := terminalAccess_bounds hs hs1 w hw hw1 hwr hm hx ⟨hu, huT⟩ - have hq : w.re + 2 * (s / 20) ≤ (terminalAccess s w m x u).re := by linarith [hb.2.2.2] - have hh := markedRoutes_past_clearance (w := w) (q := terminalAccess s w m x u) - (t := t) (by linarith) hwr1 hwi1 ha.le ha1 ⟨hb.1, hb.2.1⟩ hb.2.2.1 (Or.inr hq) - constructor <;> linarith [hh.1, hh.2] - · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] - have hR : w.re < fitTerminalX s w := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - have hR2 : fitTerminalX s w ≤ 2 := by - unfold fitTerminalX - rw [abs_of_nonneg hwr] - linarith - have hh := markedRoutes_hook_clearance_early hwr hwr1 hwi1 hR hR2 ha ha1 ht - (show 0 ≤ u - fitHighTime s w m by linarith) - constructor <;> linarith [hh.1, hh.2] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineMarkedLocal +import OAI.Probability.SLE.Geometry.MarkedRouteEarlyClearance + +namespace OAI + +/-! The two exit routes avoid every point of the complete marked access +guide by one uniform margin, including its distant initial prefix. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +lemma terminalAccess_bounds {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x u : ℝ} (hx : |x| ≤ 1) + (hu : u ∈ Icc (0 : ℝ) (fitHighTime s w m)) : + -2 ≤ (terminalAccess s w m x u).re ∧ (terminalAccess s w m x u).re ≤ 2 ∧ + (terminalAccess s w m x u).im ≤ 2 ∧ w.re + s / 5 ≤ (terminalAccess s w m x u).re := by + have hz : guideCoordinate (fitCompletion s w m) x u ∈ closedRibbon := by + refine ⟨?_, (guideCoordinate_im_bound hu.1).trans hx⟩ + rw [guideCoordinate_re] + exact add_nonneg (Nat.cast_nonneg _) hu.1 + have hre := physicalFitChart_re_bounds hs hs1 w hw hw1 hm hz + exact ⟨hre.1, hre.2, physicalFitChart_terminal_guide_height hs w hw hm x hu, + terminalAccess_right hs w hw hwr hm hx hu.1⟩ + +/-- A margin independent of the number of serpentine columns. -/ +theorem completeMarkedAccess_route_clearance {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) (hwr : 0 ≤ w.re) + {m : ℕ} (hm : 0 < m) {x t u : ℝ} (hx : |x| ≤ 1) (ht : 1 / 64 ≤ t) (hu : 0 ≤ u) : + (s / 20) / 1000 ≤ dist (markedLeftRoute w (s / 20) t) (completeMarkedAccess s w m x u) ∧ + (s / 20) / 1000 ≤ dist (markedRightRoute w (s / 20) t) (completeMarkedAccess s w m x u) := by + have hwr1 := (Complex.re_le_norm w).trans hw1 + have hwi1 := (Complex.im_le_norm w).trans hw1 + have ha : 0 < s / 20 := by positivity + have ha1 : s / 20 ≤ 1 := by linarith + by_cases huT : u ≤ fitHighTime s w m + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_le huT] + have hb := terminalAccess_bounds hs hs1 w hw hw1 hwr hm hx ⟨hu, huT⟩ + have hq : w.re + 2 * (s / 20) ≤ (terminalAccess s w m x u).re := by linarith [hb.2.2.2] + have hh := markedRoutes_past_clearance (w := w) (q := terminalAccess s w m x u) + (t := t) (by linarith) hwr1 hwi1 ha.le ha1 ⟨hb.1, hb.2.1⟩ hb.2.2.1 (Or.inr hq) + constructor <;> linarith [hh.1, hh.2] + · rw [completeMarkedAccess, appendMarkedHook, curveJoin_of_gt (lt_of_not_ge huT)] + have hR : w.re < fitTerminalX s w := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + have hR2 : fitTerminalX s w ≤ 2 := by + unfold fitTerminalX + rw [abs_of_nonneg hwr] + linarith + have hh := markedRoutes_hook_clearance_early hwr hwr1 hwi1 hR hR2 ha ha1 ht + (show 0 ≤ u - fitHighTime s w m by linarith) + constructor <;> linarith [hh.1, hh.2] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineNormalizedMass.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineNormalizedMass.lean index fd1f5aa4371c98cd1ebacd80d9fbac7b44cd8dff..c46a9b26216a51a40d6daff750700c74de95bf23 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineNormalizedMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineNormalizedMass.lean @@ -1,111 +1,111 @@ -import OAI.Probability.SLE.Gauge.ActualRibbonStoppedMass -import OAI.Probability.SLE.Gauge.SerpentineInitialPrefix -import OAI.Probability.SLE.Gauge.SerpentineRetentionCount - -namespace OAI - -/-! The concrete connector and serpentine instantiate the actual stopped -iteration. All m² selected strip masses are present by completion, with -one positive initial probability and universal one-step constants. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates Target - -def experimentEntry (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - -def experimentSteps (s : ℝ) (w : ℂ) (m : ℕ) : ℕ := completionStep (experimentEntry s w m) m - 1 - -def normalizedExperimentDisk (s : ℝ) (w : ℂ) (m : ℕ) : Set ℂ := - (fun z : ℂ => (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' closedBall (center s w) (s / 20) - -lemma retained_middle_in_normalized_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {ε : ℝ} (hε : ε ≤ 1) : - ∀ k ∈ retainedSteps (experimentEntry s w m) m, - physicalMiddleRegion (experimentChart s w m) closedRibbon k ε ⊆ normalizedExperimentDisk s w m := by - intro k hk z hz - obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk - obtain ⟨a, ⟨ha, hl, hu, hi⟩, rfl⟩ := hz - change (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m a ∈ closedBall (center s w) (s / 20) - rw [← physicalExperimentChart_eq] - dsimp only [experimentEntry] at hl hu - exact physicalExperimentChart_retained_disk hs w hw hm j.isLt i.isLt - ⟨by linarith, by linarith⟩ (hi.le.trans hε) - -/-- The actual normalized dense experiment, conditional only on the -recorded published inputs. Completion is finite on success and all mass -belongs to the stopped prefix at the prescribed common mesh. -/ -theorem uniform_serpentine_normalized_stopped_mass - {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport μ γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ q₀ : ℝ≥0∞, ∃ q c v₀ : ℝ, 0 < q₀ ∧ q₀ ≤ 1 ∧ 0 < q ∧ q ≤ 1 ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ v : ℝ, 0 < v → v < v₀ → ∀ s : ℝ, 0 < s → ∀ w : ℂ, s ≤ w.im → - ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → - ENNReal.ofReal q ^ experimentSteps s w m * q₀ ≤ μ {ω | - ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) (experimentChart s w m) ε ∧ - ((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0 - ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) v (normalizedExperimentDisk s w m)} := by - obtain ⟨q, c, v₀, hq, hq1, hcm, hv₀, hiter⟩ := uniform_actual_ribbon_iteration μ κ hκ hκ8 - B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW - 2 2 (by norm_num) (by norm_num) ε hε hεsmall - have ha := brownian_capacity_trace_adapted B hm hc κ γ G F hw - have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 - have hSLE : IsOrdinaryChordalSLE κ γ μ := - ⟨fun t => (ha t).mono (Filtration.le _ t) le_rfl, B, hB, - Filter.Eventually.of_forall (fun ω => ⟨G ω, F ω, hw ω⟩)⟩ - have hε1 : ε ≤ 1 := by linarith - obtain ⟨q₀, hq₀, hinit⟩ := uniform_experiment_initial_goodPrefix_probability μ κ γ hSLE hsupport hε hε1 - refine ⟨min q₀ 1, q, c, v₀, lt_min hq₀ (by norm_num), min_le_right _ _, hq, hq1, hcm, hv₀, ?_⟩ - intro v hv hvcut s hs w hwy m hmp g hg - obtain ⟨hP, hi, hpatch, hH, hinitial⟩ := experimentChart_geometry hs w hwy hmp - have hemb := experimentChart_closedEmbedding hs w hwy hmp - have hP0 : experimentChart s w m 0 = 0 := by - simpa only [mul_zero] using hinitial 0 (by constructor <;> norm_num) (by norm_num) - have hg0 (ω : Ω) : g (γ ω 0) = 0 := by - rw [(hw ω).2.1, ← hP0] - exact hg 0 (by constructor <;> norm_num) - have hA : 0 ≤ experimentEntry s w m := - (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hmp) - have hb := hiter v hv hvcut _ hemb hP.continuousOn hi hP0 hH - (experimentChart_escape s w m) (fun k => (hpatch k).1) (fun k => (hpatch k).2) - g hg (experimentSteps s w m) - calc - _ ≤ ENNReal.ofReal q ^ experimentSteps s w m * - μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) (experimentChart s w m) ε) := by - apply mul_le_mul' le_rfl - apply (min_le_left q₀ 1).trans - simpa only [ribbonIntegerFront, Nat.cast_zero, zero_add] using hinit s hs w hwy m hmp g hg - _ ≤ _ := hb.trans (measure_mono (by - intro ω hω - have hI : ∀ i ∈ retainedSteps (experimentEntry s w m) m, 0 < i ∧ i ≤ experimentSteps s w m := by - intro i hiI - obtain ⟨hi0, hic⟩ := retainedSteps_bounds hA hmp i hiI - exact ⟨hi0, by dsimp only [experimentSteps]; omega⟩ - have hh := actual_successes_force_stopped_mass γ hγc g hg0 _ hplanar hP.continuousOn hi - ε κ v c hε1 (experimentSteps s w m) (retainedSteps (experimentEntry s w m) m) hI _ - (retained_middle_in_normalized_disk hs w hwy hmp hε1) ω hω - simpa only [mem_ofPred_eq, retainedSteps_card hA hmp] using hh)) - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.ActualRibbonStoppedMass +import OAI.Probability.SLE.Gauge.SerpentineInitialPrefix +import OAI.Probability.SLE.Gauge.SerpentineRetentionCount + +namespace OAI + +/-! The concrete connector and serpentine instantiate the actual stopped +iteration. All m² selected strip masses are present by completion, with +one positive initial probability and universal one-step constants. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates Target + +def experimentEntry (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + +def experimentSteps (s : ℝ) (w : ℂ) (m : ℕ) : ℕ := completionStep (experimentEntry s w m) m - 1 + +def normalizedExperimentDisk (s : ℝ) (w : ℂ) (m : ℕ) : Set ℂ := + (fun z : ℂ => (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' closedBall (center s w) (s / 20) + +lemma retained_middle_in_normalized_disk {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {ε : ℝ} (hε : ε ≤ 1) : + ∀ k ∈ retainedSteps (experimentEntry s w m) m, + physicalMiddleRegion (experimentChart s w m) closedRibbon k ε ⊆ normalizedExperimentDisk s w m := by + intro k hk z hz + obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk + obtain ⟨a, ⟨ha, hl, hu, hi⟩, rfl⟩ := hz + change (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m a ∈ closedBall (center s w) (s / 20) + rw [← physicalExperimentChart_eq] + dsimp only [experimentEntry] at hl hu + exact physicalExperimentChart_retained_disk hs w hw hm j.isLt i.isLt + ⟨by linarith, by linarith⟩ (hi.le.trans hε) + +/-- The actual normalized dense experiment, conditional only on the +recorded published inputs. Completion is finite on success and all mass +belongs to the stopped prefix at the prescribed common mesh. -/ +theorem uniform_serpentine_normalized_stopped_mass + {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport μ γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ q₀ : ℝ≥0∞, ∃ q c v₀ : ℝ, 0 < q₀ ∧ q₀ ≤ 1 ∧ 0 < q ∧ q ≤ 1 ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ v : ℝ, 0 < v → v < v₀ → ∀ s : ℝ, 0 < s → ∀ w : ℂ, s ≤ w.im → + ∀ m : ℕ, 0 < m → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (experimentChart s w m z) = z.re) → + ENNReal.ofReal q ^ experimentSteps s w m * q₀ ≤ μ {ω | + ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) (experimentChart s w m) ε ∧ + ((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0 + ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) v (normalizedExperimentDisk s w m)} := by + obtain ⟨q, c, v₀, hq, hq1, hcm, hv₀, hiter⟩ := uniform_actual_ribbon_iteration μ κ hκ hκ8 + B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW + 2 2 (by norm_num) (by norm_num) ε hε hεsmall + have ha := brownian_capacity_trace_adapted B hm hc κ γ G F hw + have hγc : ∀ ω, Continuous (γ ω) := fun ω => (hw ω).1 + have hSLE : IsOrdinaryChordalSLE κ γ μ := + ⟨fun t => (ha t).mono (Filtration.le _ t) le_rfl, B, hB, + Filter.Eventually.of_forall (fun ω => ⟨G ω, F ω, hw ω⟩)⟩ + have hε1 : ε ≤ 1 := by linarith + obtain ⟨q₀, hq₀, hinit⟩ := uniform_experiment_initial_goodPrefix_probability μ κ γ hSLE hsupport hε hε1 + refine ⟨min q₀ 1, q, c, v₀, lt_min hq₀ (by norm_num), min_le_right _ _, hq, hq1, hcm, hv₀, ?_⟩ + intro v hv hvcut s hs w hwy m hmp g hg + obtain ⟨hP, hi, hpatch, hH, hinitial⟩ := experimentChart_geometry hs w hwy hmp + have hemb := experimentChart_closedEmbedding hs w hwy hmp + have hP0 : experimentChart s w m 0 = 0 := by + simpa only [mul_zero] using hinitial 0 (by constructor <;> norm_num) (by norm_num) + have hg0 (ω : Ω) : g (γ ω 0) = 0 := by + rw [(hw ω).2.1, ← hP0] + exact hg 0 (by constructor <;> norm_num) + have hA : 0 ≤ experimentEntry s w m := + (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hmp) + have hb := hiter v hv hvcut _ hemb hP.continuousOn hi hP0 hH + (experimentChart_escape s w m) (fun k => (hpatch k).1) (fun k => (hpatch k).2) + g hg (experimentSteps s w m) + calc + _ ≤ ENNReal.ofReal q ^ experimentSteps s w m * + μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) (experimentChart s w m) ε) := by + apply mul_le_mul' le_rfl + apply (min_le_left q₀ 1).trans + simpa only [ribbonIntegerFront, Nat.cast_zero, zero_add] using hinit s hs w hwy m hmp g hg + _ ≤ _ := hb.trans (measure_mono (by + intro ω hω + have hI : ∀ i ∈ retainedSteps (experimentEntry s w m) m, 0 < i ∧ i ≤ experimentSteps s w m := by + intro i hiI + obtain ⟨hi0, hic⟩ := retainedSteps_bounds hA hmp i hiI + exact ⟨hi0, by dsimp only [experimentSteps]; omega⟩ + have hh := actual_successes_force_stopped_mass γ hγc g hg0 _ hplanar hP.continuousOn hi + ε κ v c hε1 (experimentSteps s w m) (retainedSteps (experimentEntry s w m) m) hI _ + (retained_middle_in_normalized_disk hs w hwy hmp hε1) ω hω + simpa only [mem_ofPred_eq, retainedSteps_card hA hmp] using hh)) + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineParameters.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineParameters.lean index adcb0acfe246227c0d290111776725375968b1d3..7d36221ffdf5b92f23f7833c7deced396d10ea4d 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineParameters.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineParameters.lean @@ -1,150 +1,150 @@ -import OAI.Probability.SLE.Gauge.SerpentineEscape -import OAI.Probability.SLE.Gauge.SerpentineShadow - -namespace OAI - -/-! Concrete normalized connector parameters for the dense-visit layout. -All constants depend only on the fixed positive angle lower bound. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -def fitHeight (m : ℕ) : ℝ := 2500 * m - -def fitRadius (m : ℕ) : ℝ := 625 * m - -def fitX (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - (|w.re| + s / 4 - halfHeight s) / ribbonWidth (halfHeight s) m - -def fitY (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - (w.im - halfHeight s) / ribbonWidth (halfHeight s) m - -def fitEntryBound (s : ℝ) : ℝ := 40000 / s + 10000 + 1000 * Real.pi - -lemma fit_width_pos {s : ℝ} (hs : 0 < s) {m : ℕ} (hm : 0 < m) : - 0 < ribbonWidth (halfHeight s) m := by - apply ribbonWidth_pos (by unfold halfHeight; positivity) hm - -lemma fit_width_eq (s : ℝ) (m : ℕ) : - ribbonWidth (halfHeight s) m = s / (20000 * m) := by - unfold ribbonWidth halfHeight - ring - -lemma fit_width_mul {s : ℝ} (_hs : 0 < s) {m : ℕ} (hm : 0 < m) : - ribbonWidth (halfHeight s) m * m = s / 20000 := by - rw [fit_width_eq] - have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' - field_simp - -lemma fitX_eq {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - fitX s w m = (20000 * |w.re| / s + 4900) * m := by - unfold fitX - rw [fit_width_eq] - unfold halfHeight - have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' - field_simp [hs.ne', hm'] - ring - -lemma fitY_eq {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - fitY s w m = (20000 * w.im / s - 100) * m := by - unfold fitY - rw [fit_width_eq] - unfold halfHeight - have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' - field_simp [hs.ne', hm'] - ring - -lemma fitX_lower {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - 4900 * (m : ℝ) ≤ fitX s w m := by - rw [fitX_eq hs w hm] - have h := div_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 20000) (abs_nonneg w.re)) hs.le - nlinarith [Nat.cast_nonneg (α := ℝ) m] - -lemma fitY_lower {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) {m : ℕ} (hm : 0 < m) : - 19900 * (m : ℝ) ≤ fitY s w m := by - rw [fitY_eq hs w hm] - have h : (20000 : ℝ) ≤ 20000 * w.im / s := (le_div_iff₀ hs).mpr (by nlinarith) - nlinarith [Nat.cast_nonneg (α := ℝ) m] - -lemma fitX_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : - fitX s w m ≤ (20000 / s + 4900) * m := by - rw [fitX_eq hs w hm] - have hre : |w.re| ≤ 1 := (Complex.abs_re_le_norm w).trans hw - have h : 20000 * |w.re| / s ≤ 20000 / s := by - apply div_le_div_of_nonneg_right _ hs.le - nlinarith - exact mul_le_mul_of_nonneg_right (by linarith) (Nat.cast_nonneg m) - -lemma fitY_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : - fitY s w m ≤ (20000 / s) * m := by - rw [fitY_eq hs w hm] - have him : w.im ≤ 1 := (Complex.im_le_norm w).trans hw - have h : 20000 * w.im / s ≤ 20000 / s := by - apply div_le_div_of_nonneg_right _ hs.le - nlinarith - exact mul_le_mul_of_nonneg_right (by linarith) (Nat.cast_nonneg m) - -/-- The fixed numerical parameters satisfy all proved global embedding -and local ribbon estimates, with ample strict separation. -/ -theorem fit_parameters {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - 50 ≤ fitRadius m ∧ 2 * fitRadius m < fitX s w m ∧ - fitHeight m + 2 * fitRadius m + 1 + 300 * m < fitY s w m ∧ - 4 ≤ fitHeight m := by - have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hx := fitX_lower hs w hm - have hy := fitY_lower hs w hw hm - dsimp only [fitRadius, fitHeight] - constructor - · nlinarith - constructor - · nlinarith - constructor <;> nlinarith - -lemma fitEntryBound_pos {s : ℝ} (hs : 0 < s) : 0 < fitEntryBound s := by - unfold fitEntryBound - positivity - -lemma connectorEntry_eq (H r X Y : ℝ) : - connectorEntry H r X Y = Real.pi * r + X + Y - 4 * r := by - unfold connectorEntry connectorFourth connectorThird connectorSecond quarterLength - ring - -lemma fit_entry_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) - {m : ℕ} (hm : 0 < m) : - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) ≤ fitEntryBound s * m := by - have hx := fitX_upper hs w hw hm - have hy := fitY_upper hs w hw hm - rw [connectorEntry_eq] - dsimp only [fitRadius, fitEntryBound] - have hdiv : 40000 / s = 2 * (20000 / s) := by ring - rw [hdiv] - nlinarith [mul_nonneg Real.pi_pos.le (Nat.cast_nonneg (α := ℝ) m)] - -lemma fit_entry_lower {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - fitHeight m ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := by - have hx := fitX_lower hs w hm - have hy := fitY_lower hs w hw hm - have hm' : (0 : ℝ) ≤ m := Nat.cast_nonneg m - rw [connectorEntry_eq] - dsimp only [fitHeight, fitRadius] - nlinarith [Real.pi_pos] - -lemma fit_completion_quadratic {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) - (hwy : s ≤ w.im) {m : ℕ} (hm : 0 < m) : - (completionStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m : ℝ) ≤ - (fitEntryBound s + 601 + 100 * Real.pi) * (m : ℝ) ^ 2 := by - apply completionStep_quadratic_bound - · exact (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hm) - · exact (fitEntryBound_pos hs).le - · exact hm - · exact fit_entry_upper hs w hw hm - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineEscape +import OAI.Probability.SLE.Gauge.SerpentineShadow + +namespace OAI + +/-! Concrete normalized connector parameters for the dense-visit layout. +All constants depend only on the fixed positive angle lower bound. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +def fitHeight (m : ℕ) : ℝ := 2500 * m + +def fitRadius (m : ℕ) : ℝ := 625 * m + +def fitX (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + (|w.re| + s / 4 - halfHeight s) / ribbonWidth (halfHeight s) m + +def fitY (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + (w.im - halfHeight s) / ribbonWidth (halfHeight s) m + +def fitEntryBound (s : ℝ) : ℝ := 40000 / s + 10000 + 1000 * Real.pi + +lemma fit_width_pos {s : ℝ} (hs : 0 < s) {m : ℕ} (hm : 0 < m) : + 0 < ribbonWidth (halfHeight s) m := by + apply ribbonWidth_pos (by unfold halfHeight; positivity) hm + +lemma fit_width_eq (s : ℝ) (m : ℕ) : + ribbonWidth (halfHeight s) m = s / (20000 * m) := by + unfold ribbonWidth halfHeight + ring + +lemma fit_width_mul {s : ℝ} (_hs : 0 < s) {m : ℕ} (hm : 0 < m) : + ribbonWidth (halfHeight s) m * m = s / 20000 := by + rw [fit_width_eq] + have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' + field_simp + +lemma fitX_eq {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + fitX s w m = (20000 * |w.re| / s + 4900) * m := by + unfold fitX + rw [fit_width_eq] + unfold halfHeight + have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' + field_simp [hs.ne', hm'] + ring + +lemma fitY_eq {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + fitY s w m = (20000 * w.im / s - 100) * m := by + unfold fitY + rw [fit_width_eq] + unfold halfHeight + have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast hm.ne' + field_simp [hs.ne', hm'] + ring + +lemma fitX_lower {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + 4900 * (m : ℝ) ≤ fitX s w m := by + rw [fitX_eq hs w hm] + have h := div_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 20000) (abs_nonneg w.re)) hs.le + nlinarith [Nat.cast_nonneg (α := ℝ) m] + +lemma fitY_lower {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) {m : ℕ} (hm : 0 < m) : + 19900 * (m : ℝ) ≤ fitY s w m := by + rw [fitY_eq hs w hm] + have h : (20000 : ℝ) ≤ 20000 * w.im / s := (le_div_iff₀ hs).mpr (by nlinarith) + nlinarith [Nat.cast_nonneg (α := ℝ) m] + +lemma fitX_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : + fitX s w m ≤ (20000 / s + 4900) * m := by + rw [fitX_eq hs w hm] + have hre : |w.re| ≤ 1 := (Complex.abs_re_le_norm w).trans hw + have h : 20000 * |w.re| / s ≤ 20000 / s := by + apply div_le_div_of_nonneg_right _ hs.le + nlinarith + exact mul_le_mul_of_nonneg_right (by linarith) (Nat.cast_nonneg m) + +lemma fitY_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : + fitY s w m ≤ (20000 / s) * m := by + rw [fitY_eq hs w hm] + have him : w.im ≤ 1 := (Complex.im_le_norm w).trans hw + have h : 20000 * w.im / s ≤ 20000 / s := by + apply div_le_div_of_nonneg_right _ hs.le + nlinarith + exact mul_le_mul_of_nonneg_right (by linarith) (Nat.cast_nonneg m) + +/-- The fixed numerical parameters satisfy all proved global embedding +and local ribbon estimates, with ample strict separation. -/ +theorem fit_parameters {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + 50 ≤ fitRadius m ∧ 2 * fitRadius m < fitX s w m ∧ + fitHeight m + 2 * fitRadius m + 1 + 300 * m < fitY s w m ∧ + 4 ≤ fitHeight m := by + have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hx := fitX_lower hs w hm + have hy := fitY_lower hs w hw hm + dsimp only [fitRadius, fitHeight] + constructor + · nlinarith + constructor + · nlinarith + constructor <;> nlinarith + +lemma fitEntryBound_pos {s : ℝ} (hs : 0 < s) : 0 < fitEntryBound s := by + unfold fitEntryBound + positivity + +lemma connectorEntry_eq (H r X Y : ℝ) : + connectorEntry H r X Y = Real.pi * r + X + Y - 4 * r := by + unfold connectorEntry connectorFourth connectorThird connectorSecond quarterLength + ring + +lemma fit_entry_upper {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) + {m : ℕ} (hm : 0 < m) : + connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) ≤ fitEntryBound s * m := by + have hx := fitX_upper hs w hw hm + have hy := fitY_upper hs w hw hm + rw [connectorEntry_eq] + dsimp only [fitRadius, fitEntryBound] + have hdiv : 40000 / s = 2 * (20000 / s) := by ring + rw [hdiv] + nlinarith [mul_nonneg Real.pi_pos.le (Nat.cast_nonneg (α := ℝ) m)] + +lemma fit_entry_lower {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + fitHeight m ≤ connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) := by + have hx := fitX_lower hs w hm + have hy := fitY_lower hs w hw hm + have hm' : (0 : ℝ) ≤ m := Nat.cast_nonneg m + rw [connectorEntry_eq] + dsimp only [fitHeight, fitRadius] + nlinarith [Real.pi_pos] + +lemma fit_completion_quadratic {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : ‖w‖ ≤ 1) + (hwy : s ≤ w.im) {m : ℕ} (hm : 0 < m) : + (completionStep (connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) m : ℝ) ≤ + (fitEntryBound s + 601 + 100 * Real.pi) * (m : ℝ) ^ 2 := by + apply completionStep_quadratic_bound + · exact (by unfold fitHeight; positivity : (0 : ℝ) ≤ fitHeight m).trans (fit_entry_lower hs w hwy hm) + · exact (fitEntryBound_pos hs).le + · exact hm + · exact fit_entry_upper hs w hw hm + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalGeometry.lean b/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalGeometry.lean index 3c54082530a84d155e9d98cdfa93b13234fa89c8..1cd9fcb01ae38d0af824d6c4bde3c098d72576c9 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalGeometry.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalGeometry.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Gauge.SerpentinePhysicalMass - -namespace OAI - -/-! Properness and half-plane geometry of the physical fitted chart. -Only the normalized chart is used in the uniform probabilistic iteration. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -theorem physicalFitChart_geometry {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - Continuous (physicalFitChart s w m) ∧ InjOn (physicalFitChart s w m) closedRibbon ∧ - (∀ z ∈ closedRibbon, 0 < z.re → physicalFitChart s w m z ∈ upperHalfPlane) ∧ - Tendsto (fun t : ℝ => ‖physicalFitChart s w m (t : ℂ)‖) atTop atTop := by - have hℓ := fit_width_pos hs hm - refine ⟨continuous_const.mul (fitChart_continuous hs w hw hm), ?_, ?_, ?_⟩ - · intro z hz z' hz' he - apply fitChart_injective hs w hw hm hz hz' - exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr hℓ.ne') he - · intro z hz hz0 - have hh : 0 < (fitChart s w m z).im := fitChart_upper hs w hw hm z hz hz0 - change 0 < ((ribbonWidth (halfHeight s) m : ℂ) * fitChart s w m z).im - simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] using mul_pos hℓ hh - · have hn : Tendsto (fun t : ℝ => ‖fitChart s w m (t : ℂ)‖) atTop atTop := by - simpa only [fitChart, normalRibbon, Complex.ofReal_im, Complex.ofReal_re, - Complex.ofReal_zero, zero_mul, zero_add, add_zero, fitSpine, Function.comp_def] using - tendsto_norm_cocompact_atTop.comp - (completeSpine_escape m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) - have hh := hn.const_mul_atTop hℓ - simpa only [physicalFitChart, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hℓ] using hh - -lemma physicalFitChart_zero {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : physicalFitChart s w m 0 = 0 := by - have he := fitChart_initial hs w hw hm - (z := 0) (show (0 : ℂ).re ≤ fitHeight m by unfold fitHeight; positivity) - simp only [physicalFitChart, he, mul_zero] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentinePhysicalMass + +namespace OAI + +/-! Properness and half-plane geometry of the physical fitted chart. +Only the normalized chart is used in the uniform probabilistic iteration. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +theorem physicalFitChart_geometry {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + Continuous (physicalFitChart s w m) ∧ InjOn (physicalFitChart s w m) closedRibbon ∧ + (∀ z ∈ closedRibbon, 0 < z.re → physicalFitChart s w m z ∈ upperHalfPlane) ∧ + Tendsto (fun t : ℝ => ‖physicalFitChart s w m (t : ℂ)‖) atTop atTop := by + have hℓ := fit_width_pos hs hm + refine ⟨continuous_const.mul (fitChart_continuous hs w hw hm), ?_, ?_, ?_⟩ + · intro z hz z' hz' he + apply fitChart_injective hs w hw hm hz hz' + exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr hℓ.ne') he + · intro z hz hz0 + have hh : 0 < (fitChart s w m z).im := fitChart_upper hs w hw hm z hz hz0 + change 0 < ((ribbonWidth (halfHeight s) m : ℂ) * fitChart s w m z).im + simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero] using mul_pos hℓ hh + · have hn : Tendsto (fun t : ℝ => ‖fitChart s w m (t : ℂ)‖) atTop atTop := by + simpa only [fitChart, normalRibbon, Complex.ofReal_im, Complex.ofReal_re, + Complex.ofReal_zero, zero_mul, zero_add, add_zero, fitSpine, Function.comp_def] using + tendsto_norm_cocompact_atTop.comp + (completeSpine_escape m (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m)) + have hh := hn.const_mul_atTop hℓ + simpa only [physicalFitChart, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hℓ] using hh + +lemma physicalFitChart_zero {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : physicalFitChart s w m 0 = 0 := by + have he := fitChart_initial hs w hw hm + (z := 0) (show (0 : ℂ).re ≤ fitHeight m by unfold fitHeight; positivity) + simp only [physicalFitChart, he, mul_zero] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalMass.lean b/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalMass.lean index 2d5a9b895dc93574a076bce5614f8c2e04a9c35c..c70084747368588cc92afaf05b0b7f86e55b17a1 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalMass.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentinePhysicalMass.lean @@ -1,136 +1,136 @@ -import OAI.Probability.SLE.Gauge.SerpentineExponentialMass -import OAI.Probability.SLE.Gauge.RibbonStoppedMassLaw - -namespace OAI - -/-! The stopped dense serpentine experiment in the original half-plane -coordinates, with one common physical sausage mesh and its exact m^(2-d) -mass threshold. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates Target ScaleTraceLaw RibbonPassageScaling - -local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -lemma physicalExperimentChart_closedEmbedding {s : ℝ} (hs : 0 < s) (w : ℂ) - (hw : s ≤ w.im) {m : ℕ} (hm : 0 < m) : - Topology.IsClosedEmbedding (fun z : closedRibbon => physicalExperimentChart s w m z) := by - have hℓ := fit_width_pos hs hm - have hh := (Homeomorph.mulLeft₀ (ribbonWidth (halfHeight s) m : ℂ) - (Complex.ofReal_ne_zero.mpr hℓ.ne')).isClosedEmbedding.comp - (experimentChart_closedEmbedding hs w hw hm) - change Topology.IsClosedEmbedding (fun z : closedRibbon => - (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z) at hh - simpa only [Function.comp_def, physicalExperimentChart_eq] using hh - -lemma normalizedExperimentDisk_scaled {s : ℝ} (hs : 0 < s) (w : ℂ) - {m : ℕ} (hm : 0 < m) : - affine 0 (ribbonWidth (halfHeight s) m) '' normalizedExperimentDisk s w m = - closedBall (center s w) (s / 20) := by - have hℓ := fit_width_pos hs hm - change (fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) '' - ((fun z : ℂ => (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' closedBall (center s w) (s / 20)) = _ - have hh := Set.image_preimage_eq (closedBall (center s w) (s / 20)) (affine_bijective 0 hℓ).surjective - change (fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) '' - ((fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' - closedBall (center s w) (s / 20)) = _ at hh - simpa only [zero_add] using hh - -lemma physical_ribbon_mass_threshold (κ c s : ℝ) (hs : 0 < s) - (m : ℕ) (hm : 0 < m) : - ENNReal.ofReal (ribbonWidth (halfHeight s) m ^ dimension κ) * - (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) = - ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by - have he : ribbonWidth (halfHeight s) m = (s / 20000) / (m : ℝ) := by - rw [fit_width_eq] - rw [div_div] - rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (by positivity : (0 : ℝ) ≤ (m ^ 2 : ℕ)), - ← ENNReal.ofReal_mul (Real.rpow_nonneg (fit_width_pos hs hm).le _)] - congr 1 - rw [he] - simp only [Nat.cast_pow] - exact ribbon_mass_after_width_scaling κ c (s / 20000) (by positivity) m hm - -theorem uniform_serpentine_physical_stopped_mass - {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport μ γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → - ∃ C : ℝ, 0 < C ∧ ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → - ∀ v : ℝ, 0 < v → v < ribbonWidth (halfHeight s) m * v₀ → ∀ g : C(ℂ, ℝ), - (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → - ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ {ω | - ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) - (physicalExperimentChart s w m) ε ∧ - ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ - segmentSausageCharge κ (γ ω) 0 - ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) - v (closedBall (center s w) (s / 20))} := by - obtain ⟨c, v₀, hcm, hv₀, hprob⟩ := uniform_serpentine_exponential_stopped_mass μ κ hκ hκ8 - B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall - let f := sourceTracePath γ (fun ω => (hw ω).1) - have ha := brownian_capacity_trace_adapted B hm hc κ γ G F hw - have hfm : Measurable f := source_path_measurable _ - (fun t => (ha t).mono (Filtration.le _ t) le_rfl) - refine ⟨c, v₀, hcm, hv₀, fun s hs => ?_⟩ - obtain ⟨C, hC, hprob⟩ := hprob s hs - refine ⟨C, hC, fun w hwn hwy m hmp v hv hvcut g hg => ?_⟩ - let ℓ := ribbonWidth (halfHeight s) m - have hℓ : 0 < ℓ := fit_width_pos hs hmp - let ell : ℝ≥0 := ⟨ℓ, hℓ.le⟩ - have hell : 0 < ell := hℓ - let gn : C(ℂ, ℝ) := g.comp ⟨fun z : ℂ => (ℓ : ℂ) * z, continuous_const.mul continuous_id⟩ - have hgg (z : ℂ) : g ((ell : ℂ) * z) = gn z := rfl - have hgn (z : ℂ) (hz : z ∈ closedRibbon) : gn (experimentChart s w m z) = z.re := by - change g ((ℓ : ℂ) * experimentChart s w m z) = z.re - rw [← physicalExperimentChart_eq] - exact hg z hz - have hvn : 0 < v / ℓ := div_pos hv hℓ - have hvncut : v / ℓ < v₀ := (div_lt_iff₀ hℓ).mpr (by simpa only [mul_comm] using hvcut) - have hstart := hprob (v / ℓ) hvn hvncut w hwn hwy m hmp gn hgn - have he : Topology.IsClosedEmbedding (fun z : closedRibbon => (ell : ℂ) * experimentChart s w m z) := by - have he' := physicalExperimentChart_closedEmbedding hs w hwy hmp - simp only [physicalExperimentChart_eq] at he' - change Topology.IsClosedEmbedding (fun z : closedRibbon => - (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z) - exact he' - have htrans := ribbonStoppedMass_scale_probability μ f hfm ell hell - (scaled_trace_path_law κ hκ hκ8 B hB hm hc γ G F hw ell hell) - (experimentChart s w m) he gn g hgg ((experimentSteps s w m : ℝ) + 1) ε κ hvn - (normalizedExperimentDisk s w m) (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) - have hmass := physical_ribbon_mass_threshold κ c s hs m hmp - have hmesh : (ell : ℝ) * (v / ℓ) = v := mul_div_cancel₀ v hℓ.ne' - have hregion : affine 0 (ell : ℝ) '' normalizedExperimentDisk s w m = - closedBall (center s w) (s / 20) := normalizedExperimentDisk_scaled hs w hmp - apply hstart.trans - have hchart : (fun z : ℂ => (ell : ℂ) * experimentChart s w m z) = physicalExperimentChart s w m := by - funext z - exact (physicalExperimentChart_eq s w m z).symm - have hmass' : ENNReal.ofReal ((ell : ℝ) ^ dimension κ) * - (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) = - ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := hmass - simpa only [ribbonStoppedMassEvent, GoodRibbonPrefix, mem_ofPred_eq, f, sourceTracePath, - ContinuousMap.coe_mk, ribbonIntegerFront, ribbonFrontStop, hmesh, hregion, hmass', hchart] using htrans - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineExponentialMass +import OAI.Probability.SLE.Gauge.RibbonStoppedMassLaw + +namespace OAI + +/-! The stopped dense serpentine experiment in the original half-plane +coordinates, with one common physical sausage mesh and its exact m^(2-d) +mass threshold. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates Target ScaleTraceLaw RibbonPassageScaling + +local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +lemma physicalExperimentChart_closedEmbedding {s : ℝ} (hs : 0 < s) (w : ℂ) + (hw : s ≤ w.im) {m : ℕ} (hm : 0 < m) : + Topology.IsClosedEmbedding (fun z : closedRibbon => physicalExperimentChart s w m z) := by + have hℓ := fit_width_pos hs hm + have hh := (Homeomorph.mulLeft₀ (ribbonWidth (halfHeight s) m : ℂ) + (Complex.ofReal_ne_zero.mpr hℓ.ne')).isClosedEmbedding.comp + (experimentChart_closedEmbedding hs w hw hm) + change Topology.IsClosedEmbedding (fun z : closedRibbon => + (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z) at hh + simpa only [Function.comp_def, physicalExperimentChart_eq] using hh + +lemma normalizedExperimentDisk_scaled {s : ℝ} (hs : 0 < s) (w : ℂ) + {m : ℕ} (hm : 0 < m) : + affine 0 (ribbonWidth (halfHeight s) m) '' normalizedExperimentDisk s w m = + closedBall (center s w) (s / 20) := by + have hℓ := fit_width_pos hs hm + change (fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) '' + ((fun z : ℂ => (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' closedBall (center s w) (s / 20)) = _ + have hh := Set.image_preimage_eq (closedBall (center s w) (s / 20)) (affine_bijective 0 hℓ).surjective + change (fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) '' + ((fun z : ℂ => 0 + (ribbonWidth (halfHeight s) m : ℂ) * z) ⁻¹' + closedBall (center s w) (s / 20)) = _ at hh + simpa only [zero_add] using hh + +lemma physical_ribbon_mass_threshold (κ c s : ℝ) (hs : 0 < s) + (m : ℕ) (hm : 0 < m) : + ENNReal.ofReal (ribbonWidth (halfHeight s) m ^ dimension κ) * + (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) = + ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by + have he : ribbonWidth (halfHeight s) m = (s / 20000) / (m : ℝ) := by + rw [fit_width_eq] + rw [div_div] + rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (by positivity : (0 : ℝ) ≤ (m ^ 2 : ℕ)), + ← ENNReal.ofReal_mul (Real.rpow_nonneg (fit_width_pos hs hm).le _)] + congr 1 + rw [he] + simp only [Nat.cast_pow] + exact ribbon_mass_after_width_scaling κ c (s / 20000) (by positivity) m hm + +theorem uniform_serpentine_physical_stopped_mass + {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport μ γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ c v₀ : ℝ, 0 < c ∧ 0 < v₀ ∧ ∀ s : ℝ, 0 < s → + ∃ C : ℝ, 0 < C ∧ ∀ w : ℂ, ‖w‖ ≤ 1 → s ≤ w.im → ∀ m : ℕ, 0 < m → + ∀ v : ℝ, 0 < v → v < ribbonWidth (halfHeight s) m * v₀ → ∀ g : C(ℂ, ℝ), + (∀ z ∈ closedRibbon, g (physicalExperimentChart s w m z) = z.re) → + ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤ μ {ω | + ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ (experimentSteps s w m)) + (physicalExperimentChart s w m) ε ∧ + ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) ≤ + segmentSausageCharge κ (γ ω) 0 + ((ribbonIntegerFront g γ (experimentSteps s w m) ω).untopD 0) + v (closedBall (center s w) (s / 20))} := by + obtain ⟨c, v₀, hcm, hv₀, hprob⟩ := uniform_serpentine_exponential_stopped_mass μ κ hκ hκ8 + B hB hm hc γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall + let f := sourceTracePath γ (fun ω => (hw ω).1) + have ha := brownian_capacity_trace_adapted B hm hc κ γ G F hw + have hfm : Measurable f := source_path_measurable _ + (fun t => (ha t).mono (Filtration.le _ t) le_rfl) + refine ⟨c, v₀, hcm, hv₀, fun s hs => ?_⟩ + obtain ⟨C, hC, hprob⟩ := hprob s hs + refine ⟨C, hC, fun w hwn hwy m hmp v hv hvcut g hg => ?_⟩ + let ℓ := ribbonWidth (halfHeight s) m + have hℓ : 0 < ℓ := fit_width_pos hs hmp + let ell : ℝ≥0 := ⟨ℓ, hℓ.le⟩ + have hell : 0 < ell := hℓ + let gn : C(ℂ, ℝ) := g.comp ⟨fun z : ℂ => (ℓ : ℂ) * z, continuous_const.mul continuous_id⟩ + have hgg (z : ℂ) : g ((ell : ℂ) * z) = gn z := rfl + have hgn (z : ℂ) (hz : z ∈ closedRibbon) : gn (experimentChart s w m z) = z.re := by + change g ((ℓ : ℂ) * experimentChart s w m z) = z.re + rw [← physicalExperimentChart_eq] + exact hg z hz + have hvn : 0 < v / ℓ := div_pos hv hℓ + have hvncut : v / ℓ < v₀ := (div_lt_iff₀ hℓ).mpr (by simpa only [mul_comm] using hvcut) + have hstart := hprob (v / ℓ) hvn hvncut w hwn hwy m hmp gn hgn + have he : Topology.IsClosedEmbedding (fun z : closedRibbon => (ell : ℂ) * experimentChart s w m z) := by + have he' := physicalExperimentChart_closedEmbedding hs w hwy hmp + simp only [physicalExperimentChart_eq] at he' + change Topology.IsClosedEmbedding (fun z : closedRibbon => + (ribbonWidth (halfHeight s) m : ℂ) * experimentChart s w m z) + exact he' + have htrans := ribbonStoppedMass_scale_probability μ f hfm ell hell + (scaled_trace_path_law κ hκ hκ8 B hB hm hc γ G F hw ell hell) + (experimentChart s w m) he gn g hgg ((experimentSteps s w m : ℝ) + 1) ε κ hvn + (normalizedExperimentDisk s w m) (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) + have hmass := physical_ribbon_mass_threshold κ c s hs m hmp + have hmesh : (ell : ℝ) * (v / ℓ) = v := mul_div_cancel₀ v hℓ.ne' + have hregion : affine 0 (ell : ℝ) '' normalizedExperimentDisk s w m = + closedBall (center s w) (s / 20) := normalizedExperimentDisk_scaled hs w hmp + apply hstart.trans + have hchart : (fun z : ℂ => (ell : ℂ) * experimentChart s w m z) = physicalExperimentChart s w m := by + funext z + exact (physicalExperimentChart_eq s w m z).symm + have hmass' : ENNReal.ofReal ((ell : ℝ) ^ dimension κ) * + (((m ^ 2 : ℕ) : ℝ≥0∞) * ENNReal.ofReal c) = + ENNReal.ofReal (c * (s / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := hmass + simpa only [ribbonStoppedMassEvent, GoodRibbonPrefix, mem_ofPred_eq, f, sourceTracePath, + ContinuousMap.coe_mk, ribbonIntegerFront, ribbonFrontStop, hmesh, hregion, hmass', hchart] using htrans + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentinePieces.lean b/lean/OAI/Probability/SLE/Gauge/SerpentinePieces.lean index 267f44a70970e6f8f0a6c90ab52c0caee68475bb..ce87fc57a75893cbf54770d860241188419ccd4d 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentinePieces.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentinePieces.lean @@ -1,117 +1,117 @@ -import OAI.Probability.SLE.Geometry.CurveJoin -import OAI.Probability.SLE.Gauge.SerpentineLayout - -namespace OAI - -/-! Unit-speed straight and semicircular pieces in ribbon-width units. -Their tangents match at each join and have the uniform Lipschitz bound -1/50 required by the fixed local ribbon constants. -/ - -noncomputable section -open Set Metric -open scoped NNReal - -namespace SLEExactGauge.Serpentine - -def straightPiece (m j : ℕ) (t : ℝ) : ℂ := - ((-100 * (m : ℝ) + 100 * j : ℝ) : ℂ) + - ((side j * (-100 * (m : ℝ) + t) : ℝ) : ℂ) * Complex.I - -def straightTangent (j : ℕ) : ℂ := (side j : ℂ) * Complex.I - -def circularPiece (m j : ℕ) (t : ℝ) : ℂ := - ((-100 * (m : ℝ) + 100 * j + 50 - 50 * Real.cos (t / 50) : ℝ) : ℂ) + - ((side j * (100 * (m : ℝ) + 50 * Real.sin (t / 50)) : ℝ) : ℂ) * Complex.I - -def circularTangent (j : ℕ) (t : ℝ) : ℂ := - (Real.sin (t / 50) : ℂ) + ((side j * Real.cos (t / 50) : ℝ) : ℂ) * Complex.I - -lemma straightPiece_hasDerivAt (m j : ℕ) (t : ℝ) : - HasDerivAt (straightPiece m j) (straightTangent j) t := by - have hd := (((hasDerivAt_id t).const_add (-100 * (m : ℝ))).const_mul (side j)).ofReal_comp - convert (hd.mul_const Complex.I).const_add (((-100 * (m : ℝ) + 100 * j : ℝ) : ℂ)) using 1 <;> try rfl - simp only [straightTangent, mul_one] - -lemma circularPiece_hasDerivAt (m j : ℕ) (t : ℝ) : - HasDerivAt (circularPiece m j) (circularTangent j t) t := by - have ht : HasDerivAt (fun u : ℝ => u / 50) (1 / 50) t := - (hasDerivAt_id t).div_const 50 - have hx := (ht.cos.const_mul 50).const_sub (-100 * (m : ℝ) + 100 * j + 50) - have hy := ((ht.sin.const_mul 50).const_add (100 * (m : ℝ))).const_mul (side j) - convert hx.ofReal_comp.add (hy.ofReal_comp.mul_const Complex.I) using 1 <;> try rfl - all_goals simp only [circularTangent] - all_goals push_cast - all_goals ring - -lemma straightTangent_norm (j : ℕ) : ‖straightTangent j‖ = 1 := by - simp only [straightTangent, norm_mul, Complex.norm_real, Real.norm_eq_abs, side_abs, - Complex.norm_I, mul_one] - -lemma circularTangent_norm (j : ℕ) (t : ℝ) : ‖circularTangent j t‖ = 1 := by - apply (sq_eq_sq₀ (norm_nonneg _) zero_le_one).mp - rw [circularTangent, Complex.sq_norm, Complex.normSq_apply] - simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, - Complex.mul_im, mul_one, zero_add] - calc - _ = Real.sin (t / 50) ^ 2 + side j ^ 2 * Real.cos (t / 50) ^ 2 := by ring - _ = 1 ^ 2 := by rw [side_sq, one_mul]; nlinarith [Real.sin_sq_add_cos_sq (t / 50)] - -lemma circularTangent_hasDerivAt (j : ℕ) (t : ℝ) : - HasDerivAt (circularTangent j) - (((Real.cos (t / 50) / 50 : ℝ) : ℂ) - ((side j * Real.sin (t / 50) / 50 : ℝ) : ℂ) * Complex.I) t := by - have ht : HasDerivAt (fun u : ℝ => u / 50) (1 / 50) t := - (hasDerivAt_id t).div_const 50 - convert ht.sin.ofReal_comp.add ((ht.cos.const_mul (side j)).ofReal_comp.mul_const Complex.I) - using 1 <;> try rfl - all_goals push_cast - all_goals ring - -lemma circularTangent_deriv_norm (j : ℕ) (t : ℝ) : - ‖((Real.cos (t / 50) / 50 : ℝ) : ℂ) - - ((side j * Real.sin (t / 50) / 50 : ℝ) : ℂ) * Complex.I‖ = (1 / 50 : ℝ) := by - apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1 / 50)).mp - rw [Complex.sq_norm, Complex.normSq_apply] - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, - Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, - mul_zero, sub_zero, zero_sub, mul_one, add_zero] - calc - _ = (Real.cos (t / 50) ^ 2 + side j ^ 2 * Real.sin (t / 50) ^ 2) / 50 ^ 2 := by ring - _ = (1 / 50 : ℝ) ^ 2 := by - rw [side_sq, one_mul] - nlinarith [Real.sin_sq_add_cos_sq (t / 50)] - -lemma circularTangent_lipschitz (j : ℕ) : LipschitzWith (1 / 50) (circularTangent j) := by - apply lipschitzWith_of_nnnorm_deriv_le (fun t => (circularTangent_hasDerivAt j t).differentiableAt) - intro t - rw [(circularTangent_hasDerivAt j t).deriv] - exact_mod_cast (circularTangent_deriv_norm j t).le - -lemma straight_circular_position_match (m j : ℕ) : - straightPiece m j (200 * m) = circularPiece m j 0 := by - apply Complex.ext - · simp [straightPiece, circularPiece] - · simp only [straightPiece, circularPiece, Complex.add_im, Complex.ofReal_im, - Complex.mul_im, Complex.I_re, Complex.I_im, Complex.ofReal_re, - mul_zero, mul_one, add_zero, zero_add, zero_div, Real.sin_zero] - ring - -lemma circular_straight_position_match (m j : ℕ) : - circularPiece m j (50 * Real.pi) = straightPiece m (j + 1) 0 := by - apply Complex.ext - · simp [straightPiece, circularPiece, side_succ] - ring - · simp [straightPiece, circularPiece, side_succ] - -lemma straight_circular_tangent_match (j : ℕ) : - straightTangent j = circularTangent j 0 := by simp [straightTangent, circularTangent] - -lemma circular_straight_tangent_match (j : ℕ) : - circularTangent j (50 * Real.pi) = straightTangent (j + 1) := by - simp [straightTangent, circularTangent, side_succ] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Geometry.CurveJoin +import OAI.Probability.SLE.Gauge.SerpentineLayout + +namespace OAI + +/-! Unit-speed straight and semicircular pieces in ribbon-width units. +Their tangents match at each join and have the uniform Lipschitz bound +1/50 required by the fixed local ribbon constants. -/ + +noncomputable section +open Set Metric +open scoped NNReal + +namespace SLEExactGauge.Serpentine + +def straightPiece (m j : ℕ) (t : ℝ) : ℂ := + ((-100 * (m : ℝ) + 100 * j : ℝ) : ℂ) + + ((side j * (-100 * (m : ℝ) + t) : ℝ) : ℂ) * Complex.I + +def straightTangent (j : ℕ) : ℂ := (side j : ℂ) * Complex.I + +def circularPiece (m j : ℕ) (t : ℝ) : ℂ := + ((-100 * (m : ℝ) + 100 * j + 50 - 50 * Real.cos (t / 50) : ℝ) : ℂ) + + ((side j * (100 * (m : ℝ) + 50 * Real.sin (t / 50)) : ℝ) : ℂ) * Complex.I + +def circularTangent (j : ℕ) (t : ℝ) : ℂ := + (Real.sin (t / 50) : ℂ) + ((side j * Real.cos (t / 50) : ℝ) : ℂ) * Complex.I + +lemma straightPiece_hasDerivAt (m j : ℕ) (t : ℝ) : + HasDerivAt (straightPiece m j) (straightTangent j) t := by + have hd := (((hasDerivAt_id t).const_add (-100 * (m : ℝ))).const_mul (side j)).ofReal_comp + convert (hd.mul_const Complex.I).const_add (((-100 * (m : ℝ) + 100 * j : ℝ) : ℂ)) using 1 <;> try rfl + simp only [straightTangent, mul_one] + +lemma circularPiece_hasDerivAt (m j : ℕ) (t : ℝ) : + HasDerivAt (circularPiece m j) (circularTangent j t) t := by + have ht : HasDerivAt (fun u : ℝ => u / 50) (1 / 50) t := + (hasDerivAt_id t).div_const 50 + have hx := (ht.cos.const_mul 50).const_sub (-100 * (m : ℝ) + 100 * j + 50) + have hy := ((ht.sin.const_mul 50).const_add (100 * (m : ℝ))).const_mul (side j) + convert hx.ofReal_comp.add (hy.ofReal_comp.mul_const Complex.I) using 1 <;> try rfl + all_goals simp only [circularTangent] + all_goals push_cast + all_goals ring + +lemma straightTangent_norm (j : ℕ) : ‖straightTangent j‖ = 1 := by + simp only [straightTangent, norm_mul, Complex.norm_real, Real.norm_eq_abs, side_abs, + Complex.norm_I, mul_one] + +lemma circularTangent_norm (j : ℕ) (t : ℝ) : ‖circularTangent j t‖ = 1 := by + apply (sq_eq_sq₀ (norm_nonneg _) zero_le_one).mp + rw [circularTangent, Complex.sq_norm, Complex.normSq_apply] + simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, sub_zero, add_zero, Complex.add_im, + Complex.mul_im, mul_one, zero_add] + calc + _ = Real.sin (t / 50) ^ 2 + side j ^ 2 * Real.cos (t / 50) ^ 2 := by ring + _ = 1 ^ 2 := by rw [side_sq, one_mul]; nlinarith [Real.sin_sq_add_cos_sq (t / 50)] + +lemma circularTangent_hasDerivAt (j : ℕ) (t : ℝ) : + HasDerivAt (circularTangent j) + (((Real.cos (t / 50) / 50 : ℝ) : ℂ) - ((side j * Real.sin (t / 50) / 50 : ℝ) : ℂ) * Complex.I) t := by + have ht : HasDerivAt (fun u : ℝ => u / 50) (1 / 50) t := + (hasDerivAt_id t).div_const 50 + convert ht.sin.ofReal_comp.add ((ht.cos.const_mul (side j)).ofReal_comp.mul_const Complex.I) + using 1 <;> try rfl + all_goals push_cast + all_goals ring + +lemma circularTangent_deriv_norm (j : ℕ) (t : ℝ) : + ‖((Real.cos (t / 50) / 50 : ℝ) : ℂ) - + ((side j * Real.sin (t / 50) / 50 : ℝ) : ℂ) * Complex.I‖ = (1 / 50 : ℝ) := by + apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1 / 50)).mp + rw [Complex.sq_norm, Complex.normSq_apply] + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, + Complex.ofReal_im, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im, + mul_zero, sub_zero, zero_sub, mul_one, add_zero] + calc + _ = (Real.cos (t / 50) ^ 2 + side j ^ 2 * Real.sin (t / 50) ^ 2) / 50 ^ 2 := by ring + _ = (1 / 50 : ℝ) ^ 2 := by + rw [side_sq, one_mul] + nlinarith [Real.sin_sq_add_cos_sq (t / 50)] + +lemma circularTangent_lipschitz (j : ℕ) : LipschitzWith (1 / 50) (circularTangent j) := by + apply lipschitzWith_of_nnnorm_deriv_le (fun t => (circularTangent_hasDerivAt j t).differentiableAt) + intro t + rw [(circularTangent_hasDerivAt j t).deriv] + exact_mod_cast (circularTangent_deriv_norm j t).le + +lemma straight_circular_position_match (m j : ℕ) : + straightPiece m j (200 * m) = circularPiece m j 0 := by + apply Complex.ext + · simp [straightPiece, circularPiece] + · simp only [straightPiece, circularPiece, Complex.add_im, Complex.ofReal_im, + Complex.mul_im, Complex.I_re, Complex.I_im, Complex.ofReal_re, + mul_zero, mul_one, add_zero, zero_add, zero_div, Real.sin_zero] + ring + +lemma circular_straight_position_match (m j : ℕ) : + circularPiece m j (50 * Real.pi) = straightPiece m (j + 1) 0 := by + apply Complex.ext + · simp [straightPiece, circularPiece, side_succ] + ring + · simp [straightPiece, circularPiece, side_succ] + +lemma straight_circular_tangent_match (j : ℕ) : + straightTangent j = circularTangent j 0 := by simp [straightTangent, circularTangent] + +lemma circular_straight_tangent_match (j : ℕ) : + circularTangent j (50 * Real.pi) = straightTangent (j + 1) := by + simp [straightTangent, circularTangent, side_succ] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineReflection.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineReflection.lean index 26506e4d0afc1862eff3ed40722a1344867a8851..9fb9ca97415251cdf6af5e4ff47df79e23fce2a0 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineReflection.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineReflection.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding -import OAI.Probability.SLE.LoewnerReflection - -namespace OAI - -/-! Reflecting the deterministic ribbon to the left of the imaginary -axis preserves its chart geometry. No stochastic symmetry is used. -/ - -noncomputable section -open Set Metric ComplexConjugate -open scoped NNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -def reflectedChart (P : ℂ → ℂ) (z : ℂ) : ℂ := loewnerMirror (P (conj z)) - -lemma conj_mem_closedRibbon {z : ℂ} (hz : z ∈ closedRibbon) : conj z ∈ closedRibbon := by - simpa only [closedRibbon, mem_ofPred_eq, Complex.conj_re, Complex.conj_im, abs_neg] using hz - -lemma conj_mem_stepPatch {k : ℝ} {z : ℂ} (hz : z ∈ ribbonStepPatch k) : - conj z ∈ ribbonStepPatch k := by - simpa only [ribbonStepPatch, mem_ofPred_eq, Complex.conj_re, Complex.conj_im, abs_neg] using hz - -lemma reflectedChart_continuous {P : ℂ → ℂ} (hP : Continuous P) : Continuous (reflectedChart P) := - loewnerMirror.continuous.comp (hP.comp Complex.continuous_conj) - -lemma reflectedChart_injective {P : ℂ → ℂ} (hP : InjOn P closedRibbon) : - InjOn (reflectedChart P) closedRibbon := by - intro z hz w hw he - have hh := hP (conj_mem_closedRibbon hz) (conj_mem_closedRibbon hw) (loewnerMirror.injective he) - exact Complex.conjLIE.injective hh - -lemma reflectedChart_patch_upper {P : ℂ → ℂ} {C : ℝ≥0} {k : ℝ} - (hP : LipschitzOnWith C P (ribbonStepPatch k)) : - LipschitzOnWith C (reflectedChart P) (ribbonStepPatch k) := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro z hz w hw - simpa only [reflectedChart, loewnerMirror_dist, Complex.dist_conj_conj] using - hP.dist_le_mul _ (conj_mem_stepPatch hz) _ (conj_mem_stepPatch hw) - -lemma reflectedChart_patch_lower {P : ℂ → ℂ} {L k : ℝ} - (hP : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, dist z w ≤ L * dist (P z) (P w)) : - ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, - dist z w ≤ L * dist (reflectedChart P z) (reflectedChart P w) := by - intro z hz w hw - simpa only [reflectedChart, loewnerMirror_dist, Complex.dist_conj_conj] using - hP (conj z) (conj_mem_stepPatch hz) (conj w) (conj_mem_stepPatch hw) - -lemma reflectedChart_upper {P : ℂ → ℂ} - (hP : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) : - ∀ z ∈ closedRibbon, 0 < z.re → reflectedChart P z ∈ upperHalfPlane := by - intro z hz hz0 - have hh := hP (conj z) (conj_mem_closedRibbon hz) (by simpa using hz0) - simpa only [reflectedChart, upperHalfPlane, mem_ofPred_eq, loewnerMirror_im] using hh - -lemma reflectedChart_initial {P : ℂ → ℂ} {H : ℝ} - (hP : ∀ z ∈ closedRibbon, z.re ≤ H → P z = Complex.I * z) : - ∀ z ∈ closedRibbon, z.re ≤ H → reflectedChart P z = Complex.I * z := by - intro z hz hzH - rw [reflectedChart, hP (conj z) (conj_mem_closedRibbon hz) (by simpa using hzH)] - simp [loewnerMirror] - -lemma reflectedChart_width {P : ℂ → ℂ} {σ : ℝ → ℂ} - (hwidth : ∀ z ∈ closedRibbon, ‖P z - σ z.re‖ ≤ 1) : - ∀ z ∈ closedRibbon, ‖reflectedChart P z - loewnerMirror (σ z.re)‖ ≤ 1 := by - intro z hz - have hh := hwidth (conj z) (conj_mem_closedRibbon hz) - rw [← dist_eq_norm, reflectedChart, loewnerMirror_dist] - simpa only [dist_eq_norm, Complex.conj_re] using hh - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding +import OAI.Probability.SLE.LoewnerReflection + +namespace OAI + +/-! Reflecting the deterministic ribbon to the left of the imaginary +axis preserves its chart geometry. No stochastic symmetry is used. -/ + +noncomputable section +open Set Metric ComplexConjugate +open scoped NNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +def reflectedChart (P : ℂ → ℂ) (z : ℂ) : ℂ := loewnerMirror (P (conj z)) + +lemma conj_mem_closedRibbon {z : ℂ} (hz : z ∈ closedRibbon) : conj z ∈ closedRibbon := by + simpa only [closedRibbon, mem_ofPred_eq, Complex.conj_re, Complex.conj_im, abs_neg] using hz + +lemma conj_mem_stepPatch {k : ℝ} {z : ℂ} (hz : z ∈ ribbonStepPatch k) : + conj z ∈ ribbonStepPatch k := by + simpa only [ribbonStepPatch, mem_ofPred_eq, Complex.conj_re, Complex.conj_im, abs_neg] using hz + +lemma reflectedChart_continuous {P : ℂ → ℂ} (hP : Continuous P) : Continuous (reflectedChart P) := + loewnerMirror.continuous.comp (hP.comp Complex.continuous_conj) + +lemma reflectedChart_injective {P : ℂ → ℂ} (hP : InjOn P closedRibbon) : + InjOn (reflectedChart P) closedRibbon := by + intro z hz w hw he + have hh := hP (conj_mem_closedRibbon hz) (conj_mem_closedRibbon hw) (loewnerMirror.injective he) + exact Complex.conjLIE.injective hh + +lemma reflectedChart_patch_upper {P : ℂ → ℂ} {C : ℝ≥0} {k : ℝ} + (hP : LipschitzOnWith C P (ribbonStepPatch k)) : + LipschitzOnWith C (reflectedChart P) (ribbonStepPatch k) := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro z hz w hw + simpa only [reflectedChart, loewnerMirror_dist, Complex.dist_conj_conj] using + hP.dist_le_mul _ (conj_mem_stepPatch hz) _ (conj_mem_stepPatch hw) + +lemma reflectedChart_patch_lower {P : ℂ → ℂ} {L k : ℝ} + (hP : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, dist z w ≤ L * dist (P z) (P w)) : + ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k, + dist z w ≤ L * dist (reflectedChart P z) (reflectedChart P w) := by + intro z hz w hw + simpa only [reflectedChart, loewnerMirror_dist, Complex.dist_conj_conj] using + hP (conj z) (conj_mem_stepPatch hz) (conj w) (conj_mem_stepPatch hw) + +lemma reflectedChart_upper {P : ℂ → ℂ} + (hP : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) : + ∀ z ∈ closedRibbon, 0 < z.re → reflectedChart P z ∈ upperHalfPlane := by + intro z hz hz0 + have hh := hP (conj z) (conj_mem_closedRibbon hz) (by simpa using hz0) + simpa only [reflectedChart, upperHalfPlane, mem_ofPred_eq, loewnerMirror_im] using hh + +lemma reflectedChart_initial {P : ℂ → ℂ} {H : ℝ} + (hP : ∀ z ∈ closedRibbon, z.re ≤ H → P z = Complex.I * z) : + ∀ z ∈ closedRibbon, z.re ≤ H → reflectedChart P z = Complex.I * z := by + intro z hz hzH + rw [reflectedChart, hP (conj z) (conj_mem_closedRibbon hz) (by simpa using hzH)] + simp [loewnerMirror] + +lemma reflectedChart_width {P : ℂ → ℂ} {σ : ℝ → ℂ} + (hwidth : ∀ z ∈ closedRibbon, ‖P z - σ z.re‖ ≤ 1) : + ∀ z ∈ closedRibbon, ‖reflectedChart P z - loewnerMirror (σ z.re)‖ ≤ 1 := by + intro z hz + have hh := hwidth (conj z) (conj_mem_closedRibbon hz) + rw [← dist_eq_norm, reflectedChart, loewnerMirror_dist] + simpa only [dist_eq_norm, Complex.conj_re] using hh + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineRetainedStrips.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineRetainedStrips.lean index 95e1d6c495ee5977d5d059532c519343ebd0eec0..a68c582e45fb63144cd4139ed64906ae808efc6d 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineRetainedStrips.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineRetainedStrips.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Gauge.SerpentineParameters -import OAI.Probability.SLE.Gauge.SerpentineStepCount -import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding - -namespace OAI - -/-! The prescribed integer middle strips are actual straight portions -of the joined serpentine, with exactly m² distinct retained indices. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma columnEntry_mono (A : ℝ) (m : ℕ) {i j : ℕ} (hij : i ≤ j) : - columnEntry A m i ≤ columnEntry A m j := by - have hij' : (i : ℝ) ≤ j := by exact_mod_cast hij - have hD : 0 ≤ 200 * (m : ℝ) + 50 * Real.pi := by positivity - unfold columnEntry - nlinarith - -lemma array_preserves_column {m i last : ℕ} (hi : i ≤ last) {t : ℝ} - (ht : t ≤ arrayEnd m i) : - arraySpine m last t = arraySpine m i t ∧ arrayTangent m last t = arrayTangent m i t := by - induction last with - | zero => - have hi0 : i = 0 := by omega - subst i - exact ⟨rfl, rfl⟩ - | succ last ih => - by_cases hil : i = last + 1 - · subst i; exact ⟨rfl, rfl⟩ - · have hil' : i ≤ last := by omega - have ht' : t ≤ columnEntry 0 m last + 200 * m := by - have he := columnEntry_mono 0 m hil' - dsimp only [arrayEnd] at ht - linarith - have hc : t ≤ columnEntry 0 m (last + 1) := by - rw [columnEntry_succ] - linarith [Real.pi_pos] - obtain ⟨hs, hT⟩ := ih hil' - constructor - · simpa only [arraySpine, curveJoin_of_le hc, curveJoin_of_le ht'] using hs - · simpa only [arrayTangent, curveJoin_of_le hc, curveJoin_of_le ht'] using hT - -lemma array_on_column {m i last : ℕ} (hi : i ≤ last) {t : ℝ} - (ht : t ∈ Icc (columnEntry 0 m i) (arrayEnd m i)) : - arraySpine m last t = straightPiece m i (t - columnEntry 0 m i) ∧ - arrayTangent m last t = straightTangent i := by - obtain ⟨hs, hT⟩ := array_preserves_column hi ht.2 - rw [hs, hT, arraySpine_tail m i ht.1, arrayTangent_tail m i ht.1] - exact ⟨rfl, rfl⟩ - -lemma complete_normal_on_column {m i : ℕ} (hm : 0 < m) (hi : i ≤ 2 * m) - (H r X Y : ℝ) {z : ℂ} - (hz : z.re - connectorEntry H r X Y ∈ - Ioo (columnEntry 0 m i) (arrayEnd m i)) : - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = - arrayCenter m X Y + columnPoint 0 (100 * m) m i - (side i * (-100 * m + (z.re - connectorEntry H r X Y - columnEntry 0 m i))) z.im := by - have hentry : connectorEntry H r X Y < z.re := by - have he : 0 ≤ columnEntry 0 m i := by unfold columnEntry; positivity - linarith [hz.1] - rw [complete_normal_of_gt m H r X Y hentry] - have hpart := array_on_column hi ⟨hz.1.le, hz.2.le⟩ - simp only [normalRibbon, Complex.sub_re, Complex.sub_im, Complex.ofReal_re, - Complex.ofReal_im, sub_zero] - rw [hpart.1, hpart.2, normal_straight_eq hm] - -lemma retained_strip_on_column {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) - (H r X Y : ℝ) (hA : 0 ≤ connectorEntry H r X Y) {z : ℂ} - (hz : z.re ∈ Icc - ((retainedStep (connectorEntry H r X Y) m j i : ℝ) + 1 / 3) - ((retainedStep (connectorEntry H r X Y) m j i : ℝ) + 2 / 3)) : - normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = - arrayCenter m X Y + columnPoint 0 (100 * m) m j - (side j * (-100 * m + (z.re - connectorEntry H r X Y - columnEntry 0 m j))) z.im := by - apply complete_normal_on_column hm (by omega) - have he := retained_middle_in_straight hA hi hz - have hc : columnEntry (connectorEntry H r X Y) m j = - connectorEntry H r X Y + columnEntry 0 m j := by unfold columnEntry; ring - rw [hc] at he - dsimp only [arrayEnd] - constructor <;> linarith [he.1, he.2] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineParameters +import OAI.Probability.SLE.Gauge.SerpentineStepCount +import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding + +namespace OAI + +/-! The prescribed integer middle strips are actual straight portions +of the joined serpentine, with exactly m² distinct retained indices. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma columnEntry_mono (A : ℝ) (m : ℕ) {i j : ℕ} (hij : i ≤ j) : + columnEntry A m i ≤ columnEntry A m j := by + have hij' : (i : ℝ) ≤ j := by exact_mod_cast hij + have hD : 0 ≤ 200 * (m : ℝ) + 50 * Real.pi := by positivity + unfold columnEntry + nlinarith + +lemma array_preserves_column {m i last : ℕ} (hi : i ≤ last) {t : ℝ} + (ht : t ≤ arrayEnd m i) : + arraySpine m last t = arraySpine m i t ∧ arrayTangent m last t = arrayTangent m i t := by + induction last with + | zero => + have hi0 : i = 0 := by omega + subst i + exact ⟨rfl, rfl⟩ + | succ last ih => + by_cases hil : i = last + 1 + · subst i; exact ⟨rfl, rfl⟩ + · have hil' : i ≤ last := by omega + have ht' : t ≤ columnEntry 0 m last + 200 * m := by + have he := columnEntry_mono 0 m hil' + dsimp only [arrayEnd] at ht + linarith + have hc : t ≤ columnEntry 0 m (last + 1) := by + rw [columnEntry_succ] + linarith [Real.pi_pos] + obtain ⟨hs, hT⟩ := ih hil' + constructor + · simpa only [arraySpine, curveJoin_of_le hc, curveJoin_of_le ht'] using hs + · simpa only [arrayTangent, curveJoin_of_le hc, curveJoin_of_le ht'] using hT + +lemma array_on_column {m i last : ℕ} (hi : i ≤ last) {t : ℝ} + (ht : t ∈ Icc (columnEntry 0 m i) (arrayEnd m i)) : + arraySpine m last t = straightPiece m i (t - columnEntry 0 m i) ∧ + arrayTangent m last t = straightTangent i := by + obtain ⟨hs, hT⟩ := array_preserves_column hi ht.2 + rw [hs, hT, arraySpine_tail m i ht.1, arrayTangent_tail m i ht.1] + exact ⟨rfl, rfl⟩ + +lemma complete_normal_on_column {m i : ℕ} (hm : 0 < m) (hi : i ≤ 2 * m) + (H r X Y : ℝ) {z : ℂ} + (hz : z.re - connectorEntry H r X Y ∈ + Ioo (columnEntry 0 m i) (arrayEnd m i)) : + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = + arrayCenter m X Y + columnPoint 0 (100 * m) m i + (side i * (-100 * m + (z.re - connectorEntry H r X Y - columnEntry 0 m i))) z.im := by + have hentry : connectorEntry H r X Y < z.re := by + have he : 0 ≤ columnEntry 0 m i := by unfold columnEntry; positivity + linarith [hz.1] + rw [complete_normal_of_gt m H r X Y hentry] + have hpart := array_on_column hi ⟨hz.1.le, hz.2.le⟩ + simp only [normalRibbon, Complex.sub_re, Complex.sub_im, Complex.ofReal_re, + Complex.ofReal_im, sub_zero] + rw [hpart.1, hpart.2, normal_straight_eq hm] + +lemma retained_strip_on_column {m j i : ℕ} (hm : 0 < m) (hj : j < m) (hi : i < m) + (H r X Y : ℝ) (hA : 0 ≤ connectorEntry H r X Y) {z : ℂ} + (hz : z.re ∈ Icc + ((retainedStep (connectorEntry H r X Y) m j i : ℝ) + 1 / 3) + ((retainedStep (connectorEntry H r X Y) m j i : ℝ) + 2 / 3)) : + normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z = + arrayCenter m X Y + columnPoint 0 (100 * m) m j + (side j * (-100 * m + (z.re - connectorEntry H r X Y - columnEntry 0 m j))) z.im := by + apply complete_normal_on_column hm (by omega) + have he := retained_middle_in_straight hA hi hz + have hc : columnEntry (connectorEntry H r X Y) m j = + connectorEntry H r X Y + columnEntry 0 m j := by unfold columnEntry; ring + rw [hc] at he + dsimp only [arrayEnd] + constructor <;> linarith [he.1, he.2] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineRetentionCount.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineRetentionCount.lean index e7edbf230cf9bebac5bce5b374a678b899c96e6a..1640ccb2da0c0594a43b4d7a946e79f2bb6c215d 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineRetentionCount.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineRetentionCount.lean @@ -1,38 +1,38 @@ -import OAI.Probability.SLE.Gauge.SerpentineParameters - -namespace OAI - -/-! All selected integer strips occur strictly before the rounded -completion, with exactly m² distinct selected steps. -/ - -noncomputable section -open Set - -namespace SLEExactGauge.Serpentine - -lemma retainedStep_pos (A : ℝ) (m j i : ℕ) : 0 < retainedStep A m j i := by - unfold retainedStep - omega - -lemma retainedStep_lt_completion {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} - (hm : 0 < m) (hj : j < m) (hi : i < m) : - retainedStep A m j i < completionStep A m := by - have hupper := (retainedStep_bounds (j := j) hA hi).2 - have hsep := columnEntry_separation (A := A) hm (show j < 2 * m by omega) - have hceil := Nat.le_ceil (columnEntry A m (2 * m) + 200 * (m : ℝ)) - change columnEntry A m (2 * m) + 200 * (m : ℝ) ≤ (completionStep A m : ℝ) at hceil - have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hr : (retainedStep A m j i : ℝ) < completionStep A m := by linarith - exact_mod_cast hr - -theorem retainedSteps_bounds {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : - ∀ k ∈ retainedSteps A m, 0 < k ∧ k < completionStep A m := by - intro k hk - obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk - exact ⟨retainedStep_pos A m j i, retainedStep_lt_completion hA hm j.isLt i.isLt⟩ - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineParameters + +namespace OAI + +/-! All selected integer strips occur strictly before the rounded +completion, with exactly m² distinct selected steps. -/ + +noncomputable section +open Set + +namespace SLEExactGauge.Serpentine + +lemma retainedStep_pos (A : ℝ) (m j i : ℕ) : 0 < retainedStep A m j i := by + unfold retainedStep + omega + +lemma retainedStep_lt_completion {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} + (hm : 0 < m) (hj : j < m) (hi : i < m) : + retainedStep A m j i < completionStep A m := by + have hupper := (retainedStep_bounds (j := j) hA hi).2 + have hsep := columnEntry_separation (A := A) hm (show j < 2 * m by omega) + have hceil := Nat.le_ceil (columnEntry A m (2 * m) + 200 * (m : ℝ)) + change columnEntry A m (2 * m) + 200 * (m : ℝ) ≤ (completionStep A m : ℝ) at hceil + have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hr : (retainedStep A m j i : ℝ) < completionStep A m := by linarith + exact_mod_cast hr + +theorem retainedSteps_bounds {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : + ∀ k ∈ retainedSteps A m, 0 < k ∧ k < completionStep A m := by + intro k hk + obtain ⟨⟨j, i⟩, _, rfl⟩ := Finset.mem_image.mp hk + exact ⟨retainedStep_pos A m j i, retainedStep_lt_completion hA hm j.isLt i.isLt⟩ + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineShadow.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineShadow.lean index f8cdcfdb74fbf0caea0f1f15b4c614a442bd9c8c..b63c7ad49d31ec59ae133d63784a5c9a7c7dbded 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineShadow.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineShadow.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Gauge.SerpentineLayout - -namespace OAI - -/-! The displaced array and its vertical attachments avoid an explicit -upward shadow from the marked point. This gives the free upward route -used to control the conformal radius of every completed prefix. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma center_horizontal_distance {s : ℝ} (hs : 0 ≤ s) (w : ℂ) : - |(center s w).re - w.re| = s / 4 := by - simp only [center, Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, abs_mul] - have hh : |displacementSign w| = 1 := by unfold displacementSign; split <;> norm_num - rw [hh, one_mul, abs_of_nonneg (div_nonneg hs (by norm_num))] - -lemma disk_horizontal_separation {s : ℝ} (hs : 0 < s) (w z : ℂ) - (hz : z ∈ closedBall (center s w) (s / 20)) : - s / 5 ≤ |z.re - w.re| := by - have ha := abs_sub_le (center s w).re z.re w.re - have hb : |(center s w).re - z.re| ≤ s / 20 := by - have hn := Complex.abs_re_le_norm (center s w - z) - have hd : ‖center s w - z‖ ≤ s / 20 := by - rw [← dist_eq_norm, dist_comm] - exact hz - have hn' : |(center s w).re - z.re| ≤ ‖center s w - z‖ := by - simpa only [Complex.sub_re] using hn - exact hn'.trans hd - rw [center_horizontal_distance hs.le w] at ha - linarith - -lemma column_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) - {m j : ℕ} (hm : 0 < m) (hj : j ≤ 2 * m) {y a : ℝ} - (hy : |y| ≤ halfHeight s) (ha : |a| ≤ 1) : - ¬ (|((columnPoint (center s w) (halfHeight s) m j y a).re - w.re)| < s / 100 ∧ - w.im - s / 100 < (columnPoint (center s w) (halfHeight s) m j y a).im) := by - have hh := disk_horizontal_separation hs w _ (column_inside_displaced_disk hs w hm hj hy ha) - intro h - linarith [h.1] - -lemma turn_inside_displaced_disk {s : ℝ} (hs : 0 < s) (w : ℂ) - {m j : ℕ} (hm : 0 < m) (hj : j + 1 ≤ 2 * m) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : - turnPoint (center s w) (halfHeight s) m j t a ∈ closedBall (center s w) (s / 20) := by - have hh : 0 < halfHeight s := by unfold halfHeight; positivity - have hb := turnPoint_bound hh hm hj (center s w) t ha - change dist _ _ ≤ s / 20 - dsimp only [halfHeight] at hb ⊢ - linarith - -lemma turn_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) - {m j : ℕ} (hm : 0 < m) (hj : j + 1 ≤ 2 * m) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : - ¬ (|(turnPoint (center s w) (halfHeight s) m j t a).re - w.re| < s / 100 ∧ - w.im - s / 100 < (turnPoint (center s w) (halfHeight s) m j t a).im) := by - have hh := disk_horizontal_separation hs w _ (turn_inside_displaced_disk hs w hm hj t ha) - intro h - linarith [h.1] - -/-- The low part of the connector lies below the shadow. -/ -lemma low_connector_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w z : ℂ) - (hw : s ≤ w.im) (hz : z.im ≤ s / 2) : - ¬ (|z.re - w.re| < s / 100 ∧ w.im - s / 100 < z.im) := by - intro h - linarith [h.2] - -/-- The vertical attachment over either outer column stays horizontally -separated, regardless of its height. -/ -lemma attachment_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w z : ℂ) - {x : ℝ} (hx : |x| ≤ halfHeight s) - (hz : |z.re - ((center s w).re + x)| ≤ s / 1000) : - ¬ (|z.re - w.re| < s / 100 ∧ w.im - s / 100 < z.im) := by - have h1 := abs_sub_le (center s w).re (((center s w).re + x)) w.re - have h2 := abs_sub_le (((center s w).re + x)) z.re w.re - have hxx : |(center s w).re - ((center s w).re + x)| = |x| := by - rw [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg] - rw [center_horizontal_distance hs.le w, hxx] at h1 - rw [abs_sub_comm (((center s w).re + x)) z.re] at h2 - dsimp only [halfHeight] at hx - intro h - linarith [h.1] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineLayout + +namespace OAI + +/-! The displaced array and its vertical attachments avoid an explicit +upward shadow from the marked point. This gives the free upward route +used to control the conformal radius of every completed prefix. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma center_horizontal_distance {s : ℝ} (hs : 0 ≤ s) (w : ℂ) : + |(center s w).re - w.re| = s / 4 := by + simp only [center, Complex.add_re, Complex.ofReal_re, add_sub_cancel_left, abs_mul] + have hh : |displacementSign w| = 1 := by unfold displacementSign; split <;> norm_num + rw [hh, one_mul, abs_of_nonneg (div_nonneg hs (by norm_num))] + +lemma disk_horizontal_separation {s : ℝ} (hs : 0 < s) (w z : ℂ) + (hz : z ∈ closedBall (center s w) (s / 20)) : + s / 5 ≤ |z.re - w.re| := by + have ha := abs_sub_le (center s w).re z.re w.re + have hb : |(center s w).re - z.re| ≤ s / 20 := by + have hn := Complex.abs_re_le_norm (center s w - z) + have hd : ‖center s w - z‖ ≤ s / 20 := by + rw [← dist_eq_norm, dist_comm] + exact hz + have hn' : |(center s w).re - z.re| ≤ ‖center s w - z‖ := by + simpa only [Complex.sub_re] using hn + exact hn'.trans hd + rw [center_horizontal_distance hs.le w] at ha + linarith + +lemma column_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) + {m j : ℕ} (hm : 0 < m) (hj : j ≤ 2 * m) {y a : ℝ} + (hy : |y| ≤ halfHeight s) (ha : |a| ≤ 1) : + ¬ (|((columnPoint (center s w) (halfHeight s) m j y a).re - w.re)| < s / 100 ∧ + w.im - s / 100 < (columnPoint (center s w) (halfHeight s) m j y a).im) := by + have hh := disk_horizontal_separation hs w _ (column_inside_displaced_disk hs w hm hj hy ha) + intro h + linarith [h.1] + +lemma turn_inside_displaced_disk {s : ℝ} (hs : 0 < s) (w : ℂ) + {m j : ℕ} (hm : 0 < m) (hj : j + 1 ≤ 2 * m) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : + turnPoint (center s w) (halfHeight s) m j t a ∈ closedBall (center s w) (s / 20) := by + have hh : 0 < halfHeight s := by unfold halfHeight; positivity + have hb := turnPoint_bound hh hm hj (center s w) t ha + change dist _ _ ≤ s / 20 + dsimp only [halfHeight] at hb ⊢ + linarith + +lemma turn_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w : ℂ) + {m j : ℕ} (hm : 0 < m) (hj : j + 1 ≤ 2 * m) (t : ℝ) {a : ℝ} (ha : |a| ≤ 1) : + ¬ (|(turnPoint (center s w) (halfHeight s) m j t a).re - w.re| < s / 100 ∧ + w.im - s / 100 < (turnPoint (center s w) (halfHeight s) m j t a).im) := by + have hh := disk_horizontal_separation hs w _ (turn_inside_displaced_disk hs w hm hj t ha) + intro h + linarith [h.1] + +/-- The low part of the connector lies below the shadow. -/ +lemma low_connector_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w z : ℂ) + (hw : s ≤ w.im) (hz : z.im ≤ s / 2) : + ¬ (|z.re - w.re| < s / 100 ∧ w.im - s / 100 < z.im) := by + intro h + linarith [h.2] + +/-- The vertical attachment over either outer column stays horizontally +separated, regardless of its height. -/ +lemma attachment_avoids_upward_shadow {s : ℝ} (hs : 0 < s) (w z : ℂ) + {x : ℝ} (hx : |x| ≤ halfHeight s) + (hz : |z.re - ((center s w).re + x)| ≤ s / 1000) : + ¬ (|z.re - w.re| < s / 100 ∧ w.im - s / 100 < z.im) := by + have h1 := abs_sub_le (center s w).re (((center s w).re + x)) w.re + have h2 := abs_sub_le (((center s w).re + x)) z.re w.re + have hxx : |(center s w).re - ((center s w).re + x)| = |x| := by + rw [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg] + rw [center_horizontal_distance hs.le w, hxx] at h1 + rw [abs_sub_comm (((center s w).re + x)) z.re] at h2 + dsimp only [halfHeight] at hx + intro h + linarith [h.1] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineStepCount.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineStepCount.lean index c8a2d2c8574c4eb6b05c43969e0b5a98a55d26c9..c4f8ad7d81aa7ad763292b028d1b39d255dc5ab0 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineStepCount.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineStepCount.lean @@ -1,121 +1,121 @@ -import OAI.Probability.SLE.Gauge.RibbonConcatenation - -namespace OAI - -/-! Integer unit-step alignment inside the actual straight passes of the -serpentine array. A full unit middle strip is retained in each selected -step; rounding the column entry times does not lose the quadratic count. -/ - -noncomputable section -open Set -open scoped NNReal - -namespace SLEExactGauge.Serpentine - -/-- Column entry times in ribbon-width units: straight length `200m`, -then a semicircle of radius 50. The connector contributes `A`. -/ -def columnEntry (A : ℝ) (m j : ℕ) : ℝ := - A + j * (200 * (m : ℝ) + 50 * Real.pi) - -def retainedStep (A : ℝ) (m j i : ℕ) : ℕ := ⌈columnEntry A m j⌉₊ + i + 1 - -lemma columnEntry_nonneg {A : ℝ} (hA : 0 ≤ A) (m j : ℕ) : - 0 ≤ columnEntry A m j := by - unfold columnEntry - positivity - -lemma retainedStep_bounds {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} - (hi : i < m) : - columnEntry A m j + 1 ≤ retainedStep A m j i ∧ - (retainedStep A m j i : ℝ) < columnEntry A m j + m + 1 := by - have hl := Nat.le_ceil (columnEntry A m j) - have hu := Nat.ceil_lt_add_one (columnEntry_nonneg hA m j) - have hir : (i : ℝ) + 1 ≤ m := by exact_mod_cast Nat.succ_le_of_lt hi - simp only [retainedStep, Nat.cast_add, Nat.cast_one] - constructor - · have hiz : (0 : ℝ) ≤ i := Nat.cast_nonneg i - linarith - · linarith - -/-- The complete closed middle interval of each chosen integer step lies -strictly in its corresponding straight pass. -/ -theorem retained_middle_in_straight {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} - (hi : i < m) : - Icc ((retainedStep A m j i : ℝ) + 1 / 3) - ((retainedStep A m j i : ℝ) + 2 / 3) ⊆ - Ioo (columnEntry A m j) (columnEntry A m j + 200 * m) := by - have hm : (1 : ℝ) ≤ m := by exact_mod_cast Nat.succ_le_of_lt (Nat.zero_le i |>.trans_lt hi) - obtain ⟨hl, hu⟩ := retainedStep_bounds hA hi - intro t ht - constructor <;> linarith [ht.1, ht.2] - -lemma columnEntry_separation {A : ℝ} {m j k : ℕ} (hm : 0 < m) (hjk : j < k) : - columnEntry A m j + 200 * m ≤ columnEntry A m k := by - have hjkr : (j : ℝ) + 1 ≤ k := by exact_mod_cast Nat.succ_le_of_lt hjk - have hperiod : 0 ≤ 200 * (m : ℝ) + 50 * Real.pi := by positivity - have hh := mul_le_mul_of_nonneg_right hjkr hperiod - unfold columnEntry - nlinarith [Real.pi_pos] - -lemma retainedStep_strict_column {A : ℝ} (hA : 0 ≤ A) {m j k i l : ℕ} - (hm : 0 < m) (hjk : j < k) (hi : i < m) (hl : l < m) : - retainedStep A m j i < retainedStep A m k l := by - obtain ⟨_, hui⟩ := retainedStep_bounds hA hi - obtain ⟨hll, _⟩ := retainedStep_bounds hA hl - have hsep := columnEntry_separation (A := A) hm hjk - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hh : (retainedStep A m j i : ℝ) < retainedStep A m k l := by linarith - exact_mod_cast hh - -theorem retainedStep_injective {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : - Function.Injective (fun ji : Fin m × Fin m => retainedStep A m ji.1 ji.2) := by - rintro ⟨j, i⟩ ⟨k, l⟩ he - have hjk : j = k := by - by_contra hn - rcases lt_or_gt_of_ne hn with hlt | hgt - · have hh := retainedStep_strict_column hA hm hlt i.isLt l.isLt - exact (ne_of_lt hh) he - · have hh := retainedStep_strict_column hA hm hgt l.isLt i.isLt - exact (ne_of_lt hh) he.symm - subst k - have hil : i = l := by - apply Fin.ext - change ⌈columnEntry A m (j : ℕ)⌉₊ + (i : ℕ) + 1 = - ⌈columnEntry A m (j : ℕ)⌉₊ + (l : ℕ) + 1 at he - omega - subst l - rfl - -def retainedSteps (A : ℝ) (m : ℕ) : Finset ℕ := - Finset.univ.image (fun ji : Fin m × Fin m => retainedStep A m ji.1 ji.2) - -theorem retainedSteps_card {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : - (retainedSteps A m).card = m ^ 2 := by - rw [retainedSteps, Finset.card_image_of_injective _ (retainedStep_injective hA hm)] - simp [pow_two] - -/-- Rounded completion of all `2m+1` passes, before extending vertically. -/ -def completionStep (A : ℝ) (m : ℕ) : ℕ := - ⌈columnEntry A m (2 * m) + 200 * (m : ℝ)⌉₊ - -theorem completionStep_quadratic_bound {A A₀ : ℝ} (hA : 0 ≤ A) (hA₀ : 0 ≤ A₀) - {m : ℕ} (hm : 0 < m) (hbound : A ≤ A₀ * m) : - (completionStep A m : ℝ) ≤ (A₀ + 601 + 100 * Real.pi) * (m : ℝ) ^ 2 := by - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - have hsq : (m : ℝ) ≤ (m : ℝ) ^ 2 := by nlinarith - have hAn := mul_le_mul_of_nonneg_left hsq hA₀ - have hπn := mul_le_mul_of_nonneg_left hsq (show 0 ≤ 100 * Real.pi by positivity) - have hl := Nat.ceil_lt_add_one - (show 0 ≤ columnEntry A m (2 * m) + 200 * (m : ℝ) by - exact add_nonneg (columnEntry_nonneg hA m (2 * m)) (by positivity)) - change (completionStep A m : ℝ) < _ at hl - unfold completionStep columnEntry at * - push_cast at hl ⊢ - nlinarith - - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonConcatenation + +namespace OAI + +/-! Integer unit-step alignment inside the actual straight passes of the +serpentine array. A full unit middle strip is retained in each selected +step; rounding the column entry times does not lose the quadratic count. -/ + +noncomputable section +open Set +open scoped NNReal + +namespace SLEExactGauge.Serpentine + +/-- Column entry times in ribbon-width units: straight length `200m`, +then a semicircle of radius 50. The connector contributes `A`. -/ +def columnEntry (A : ℝ) (m j : ℕ) : ℝ := + A + j * (200 * (m : ℝ) + 50 * Real.pi) + +def retainedStep (A : ℝ) (m j i : ℕ) : ℕ := ⌈columnEntry A m j⌉₊ + i + 1 + +lemma columnEntry_nonneg {A : ℝ} (hA : 0 ≤ A) (m j : ℕ) : + 0 ≤ columnEntry A m j := by + unfold columnEntry + positivity + +lemma retainedStep_bounds {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} + (hi : i < m) : + columnEntry A m j + 1 ≤ retainedStep A m j i ∧ + (retainedStep A m j i : ℝ) < columnEntry A m j + m + 1 := by + have hl := Nat.le_ceil (columnEntry A m j) + have hu := Nat.ceil_lt_add_one (columnEntry_nonneg hA m j) + have hir : (i : ℝ) + 1 ≤ m := by exact_mod_cast Nat.succ_le_of_lt hi + simp only [retainedStep, Nat.cast_add, Nat.cast_one] + constructor + · have hiz : (0 : ℝ) ≤ i := Nat.cast_nonneg i + linarith + · linarith + +/-- The complete closed middle interval of each chosen integer step lies +strictly in its corresponding straight pass. -/ +theorem retained_middle_in_straight {A : ℝ} (hA : 0 ≤ A) {m j i : ℕ} + (hi : i < m) : + Icc ((retainedStep A m j i : ℝ) + 1 / 3) + ((retainedStep A m j i : ℝ) + 2 / 3) ⊆ + Ioo (columnEntry A m j) (columnEntry A m j + 200 * m) := by + have hm : (1 : ℝ) ≤ m := by exact_mod_cast Nat.succ_le_of_lt (Nat.zero_le i |>.trans_lt hi) + obtain ⟨hl, hu⟩ := retainedStep_bounds hA hi + intro t ht + constructor <;> linarith [ht.1, ht.2] + +lemma columnEntry_separation {A : ℝ} {m j k : ℕ} (hm : 0 < m) (hjk : j < k) : + columnEntry A m j + 200 * m ≤ columnEntry A m k := by + have hjkr : (j : ℝ) + 1 ≤ k := by exact_mod_cast Nat.succ_le_of_lt hjk + have hperiod : 0 ≤ 200 * (m : ℝ) + 50 * Real.pi := by positivity + have hh := mul_le_mul_of_nonneg_right hjkr hperiod + unfold columnEntry + nlinarith [Real.pi_pos] + +lemma retainedStep_strict_column {A : ℝ} (hA : 0 ≤ A) {m j k i l : ℕ} + (hm : 0 < m) (hjk : j < k) (hi : i < m) (hl : l < m) : + retainedStep A m j i < retainedStep A m k l := by + obtain ⟨_, hui⟩ := retainedStep_bounds hA hi + obtain ⟨hll, _⟩ := retainedStep_bounds hA hl + have hsep := columnEntry_separation (A := A) hm hjk + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hh : (retainedStep A m j i : ℝ) < retainedStep A m k l := by linarith + exact_mod_cast hh + +theorem retainedStep_injective {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : + Function.Injective (fun ji : Fin m × Fin m => retainedStep A m ji.1 ji.2) := by + rintro ⟨j, i⟩ ⟨k, l⟩ he + have hjk : j = k := by + by_contra hn + rcases lt_or_gt_of_ne hn with hlt | hgt + · have hh := retainedStep_strict_column hA hm hlt i.isLt l.isLt + exact (ne_of_lt hh) he + · have hh := retainedStep_strict_column hA hm hgt l.isLt i.isLt + exact (ne_of_lt hh) he.symm + subst k + have hil : i = l := by + apply Fin.ext + change ⌈columnEntry A m (j : ℕ)⌉₊ + (i : ℕ) + 1 = + ⌈columnEntry A m (j : ℕ)⌉₊ + (l : ℕ) + 1 at he + omega + subst l + rfl + +def retainedSteps (A : ℝ) (m : ℕ) : Finset ℕ := + Finset.univ.image (fun ji : Fin m × Fin m => retainedStep A m ji.1 ji.2) + +theorem retainedSteps_card {A : ℝ} (hA : 0 ≤ A) {m : ℕ} (hm : 0 < m) : + (retainedSteps A m).card = m ^ 2 := by + rw [retainedSteps, Finset.card_image_of_injective _ (retainedStep_injective hA hm)] + simp [pow_two] + +/-- Rounded completion of all `2m+1` passes, before extending vertically. -/ +def completionStep (A : ℝ) (m : ℕ) : ℕ := + ⌈columnEntry A m (2 * m) + 200 * (m : ℝ)⌉₊ + +theorem completionStep_quadratic_bound {A A₀ : ℝ} (hA : 0 ≤ A) (hA₀ : 0 ≤ A₀) + {m : ℕ} (hm : 0 < m) (hbound : A ≤ A₀ * m) : + (completionStep A m : ℝ) ≤ (A₀ + 601 + 100 * Real.pi) * (m : ℝ) ^ 2 := by + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + have hsq : (m : ℝ) ≤ (m : ℝ) ^ 2 := by nlinarith + have hAn := mul_le_mul_of_nonneg_left hsq hA₀ + have hπn := mul_le_mul_of_nonneg_left hsq (show 0 ≤ 100 * Real.pi by positivity) + have hl := Nat.ceil_lt_add_one + (show 0 ≤ columnEntry A m (2 * m) + 200 * (m : ℝ) by + exact add_nonneg (columnEntry_nonneg hA m (2 * m)) (by positivity)) + change (completionStep A m : ℝ) < _ at hl + unfold completionStep columnEntry at * + push_cast at hl ⊢ + nlinarith + + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineTerminal.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineTerminal.lean index 6d7c7d959e1513ab977b3c733eae578ec70d2bd0..4f9359b1ac02b504ee7bf927ab3d11cdfe89d3e5 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineTerminal.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineTerminal.lean @@ -1,59 +1,59 @@ -import OAI.Probability.SLE.Gauge.SerpentineEmbedding - -namespace OAI - -noncomputable section -open Set - -namespace SLEExactGauge.Serpentine - -/-- A turn cannot meet any column except its two attached columns, -regardless of the height on that column. -/ -theorem turnPoint_column_ne_nonadjacent {h : ℝ} (hh : 0 < h) {m i k : ℕ} (hm : 0 < m) - (hk : k < i ∨ i + 2 ≤ k) (c : ℂ) (t y : ℝ) {a b : ℝ} - (ha : |a| ≤ 1) (hb : |b| ≤ 1) : - turnPoint c h m i t a ≠ columnPoint c h m k y b := by - intro he - have ht := abs_le.mp (turn_horizontal_bound hh hm c i t ha) - have hc := congrArg Complex.re he - rw [columnPoint_re] at hc - have hw := ribbonWidth_pos hh hm - have hb' : |side k * ribbonWidth h m * b| ≤ ribbonWidth h m := by - rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_pos hw] - exact mul_le_of_le_one_right hw.le hb - have hb'' := abs_le.mp hb' - simp only [columnOffset] at ht hc - rw [turnRadius_eq_width, spacing_eq_width] at ht - rw [spacing_eq_width] at hc - rcases hk with hk | hk - · have hki : (k : ℝ) + 1 ≤ i := by exact_mod_cast hk - nlinarith - · have hik : (i : ℝ) + 2 ≤ k := by exact_mod_cast hk - nlinarith - -lemma side_even (m : ℕ) : side (2 * m) = 1 := by - simp only [side, ite_eq_left (even_two_mul m)] - -/-- The upward terminal continuation is disjoint from every turn. -The preceding turn bends downwards; earlier turns are horizontally -separated from the terminal column. -/ -theorem turnPoint_terminal_column_ne {h : ℝ} (hh : 0 < h) {m i : ℕ} (hm : 0 < m) - (hi : i < 2 * m) (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) - (hy : h < y) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : - turnPoint c h m i t a ≠ columnPoint c h m (2 * m) y b := by - by_cases hil : i + 1 = 2 * m - · intro he - have hs : side i = -1 := by - have hh := side_succ i - rw [hil, side_even] at hh - linarith - have hh := turn_oriented_height hh hm c i ht ha - rw [hs, he, columnPoint_im] at hh - linarith - · exact turnPoint_column_ne_nonadjacent hh hm (Or.inr (by omega)) c t y ha hb - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineEmbedding + +namespace OAI + +noncomputable section +open Set + +namespace SLEExactGauge.Serpentine + +/-- A turn cannot meet any column except its two attached columns, +regardless of the height on that column. -/ +theorem turnPoint_column_ne_nonadjacent {h : ℝ} (hh : 0 < h) {m i k : ℕ} (hm : 0 < m) + (hk : k < i ∨ i + 2 ≤ k) (c : ℂ) (t y : ℝ) {a b : ℝ} + (ha : |a| ≤ 1) (hb : |b| ≤ 1) : + turnPoint c h m i t a ≠ columnPoint c h m k y b := by + intro he + have ht := abs_le.mp (turn_horizontal_bound hh hm c i t ha) + have hc := congrArg Complex.re he + rw [columnPoint_re] at hc + have hw := ribbonWidth_pos hh hm + have hb' : |side k * ribbonWidth h m * b| ≤ ribbonWidth h m := by + rw [abs_mul, abs_mul, side_abs, one_mul, abs_of_pos hw] + exact mul_le_of_le_one_right hw.le hb + have hb'' := abs_le.mp hb' + simp only [columnOffset] at ht hc + rw [turnRadius_eq_width, spacing_eq_width] at ht + rw [spacing_eq_width] at hc + rcases hk with hk | hk + · have hki : (k : ℝ) + 1 ≤ i := by exact_mod_cast hk + nlinarith + · have hik : (i : ℝ) + 2 ≤ k := by exact_mod_cast hk + nlinarith + +lemma side_even (m : ℕ) : side (2 * m) = 1 := by + simp only [side, ite_eq_left (even_two_mul m)] + +/-- The upward terminal continuation is disjoint from every turn. +The preceding turn bends downwards; earlier turns are horizontally +separated from the terminal column. -/ +theorem turnPoint_terminal_column_ne {h : ℝ} (hh : 0 < h) {m i : ℕ} (hm : 0 < m) + (hi : i < 2 * m) (c : ℂ) {t y a b : ℝ} (ht : t ∈ Icc 0 Real.pi) + (hy : h < y) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : + turnPoint c h m i t a ≠ columnPoint c h m (2 * m) y b := by + by_cases hil : i + 1 = 2 * m + · intro he + have hs : side i = -1 := by + have hh := side_succ i + rw [hil, side_even] at hh + linarith + have hh := turn_oriented_height hh hm c i ht ha + rw [hs, he, columnPoint_im] at hh + linarith + · exact turnPoint_column_ne_nonadjacent hh hm (Or.inr (by omega)) c t y ha hb + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineTerminalGuide.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineTerminalGuide.lean index b9c332fd0c0531f0500634a5137f914f088f2cf5..09bbe7ddfcc1ef0743a94a593276dce197307a4f 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineTerminalGuide.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineTerminalGuide.lean @@ -1,163 +1,163 @@ -import OAI.Probability.SLE.Gauge.SerpentineMarkedClearance - -namespace OAI - -/-! The actual last front is on the terminal vertical piece. Its access -guide reaches height two before the explicit marked hook is attached. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates - -def fitEntry (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) - -def fitCompletion (s : ℝ) (w : ℂ) (m : ℕ) : ℕ := completionStep (fitEntry s w m) m - -def fitTerminalX (s : ℝ) (w : ℂ) : ℝ := |w.re| + s / 4 + s / 200 - -def fitTerminalHeight (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - ribbonWidth (halfHeight s) m * - (fitY s w m + fitCompletion s w m - fitEntry s w m - columnEntry 0 m (2 * m)) - -def fitHighTime (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := - (2 - fitTerminalHeight s w m) / ribbonWidth (halfHeight s) m - -lemma fitCompletion_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - fitEntry s w m + columnEntry 0 m (2 * m) + 200 * m ≤ fitCompletion s w m ∧ - (fitCompletion s w m : ℝ) < fitEntry s w m + columnEntry 0 m (2 * m) + 200 * m + 1 := by - have hA : 0 ≤ fitEntry s w m := by - have h := fit_entry_lower hs w hw hm - exact le_trans (by unfold fitHeight; positivity) h - have hlo := Nat.le_ceil (columnEntry (fitEntry s w m) m (2 * m) + 200 * m) - have hhi := Nat.ceil_lt_add_one - (add_nonneg (columnEntry_nonneg hA m (2 * m)) (by positivity : (0 : ℝ) ≤ 200 * m)) - change _ ≤ (fitCompletion s w m : ℝ) at hlo - change (fitCompletion s w m : ℝ) < _ at hhi - have he : columnEntry (fitEntry s w m) m (2 * m) = - fitEntry s w m + columnEntry 0 m (2 * m) := by simp [columnEntry] - rw [he] at hlo hhi - exact ⟨hlo, hhi⟩ - -lemma fit_scaledY {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : - ribbonWidth (halfHeight s) m * fitY s w m = w.im - s / 200 := by - unfold fitY halfHeight - exact mul_div_cancel₀ _ (fit_width_pos hs hm).ne' - -lemma fitTerminalHeight_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) : - w.im + s / 200 ≤ fitTerminalHeight s w m ∧ - fitTerminalHeight s w m < w.im + s / 200 + ribbonWidth (halfHeight s) m := by - obtain ⟨hlo, hhi⟩ := fitCompletion_bounds hs w hw hm - have hℓ := fit_width_pos hs hm - have hl := mul_le_mul_of_nonneg_left hlo hℓ.le - have hh := mul_lt_mul_of_pos_left hhi hℓ - have hY := fit_scaledY hs w hm - have hmℓ := fit_width_mul hs hm - unfold fitTerminalHeight - constructor <;> nlinarith - -lemma physicalFitChart_terminal {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : (fitCompletion s w m : ℝ) ≤ z.re) : - physicalFitChart s w m z = - ((fitTerminalX s w - ribbonWidth (halfHeight s) m * z.im : ℝ) : ℂ) + - ((fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * - (z.re - fitCompletion s w m) : ℝ) : ℂ) * Complex.I := by - have hthreshold : fitEntry s w m + columnEntry 0 m (2 * m) + 1 ≤ z.re := by - have h := (fitCompletion_bounds hs w hw hm).1 - have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm - linarith - obtain ⟨hσ, hT⟩ := complete_linear_tail m (fitHeight m) (fitRadius m) - (fitX s w m) (fitY s w m) hthreshold - have hX := fit_scaledX hs w hm - have hmℓ := fit_width_mul hs hm - change ((ribbonWidth (halfHeight s) m : ℂ) * - (completeSpine m _ _ _ _ z.re + (z.im : ℂ) * Complex.I * completeTangent m _ _ _ _ z.re)) = _ - rw [hσ, hT] - apply Complex.ext <;> - simp only [Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - mul_zero, zero_mul, mul_one, sub_zero, add_zero, zero_add] - · unfold fitTerminalX - nlinarith - · unfold fitTerminalHeight fitEntry - ring - -lemma fitHighTime_large {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) (w : ℂ) - (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : - 1 < fitHighTime s w m := by - have hheight := (fitTerminalHeight_bounds hs w hw hm).2 - have hwi := (Complex.im_le_norm w).trans hw1 - have hwidth := fit_width_le hs hm - rw [fitHighTime, lt_div_iff₀ (fit_width_pos hs hm)] - nlinarith - -/-- Exact physical formula for the tapered terminal access guide. -/ -lemma physicalFitChart_terminal_guide {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} (ht : 0 ≤ t) : - physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t) = - ((fitTerminalX s w - ribbonWidth (halfHeight s) m * x * max (1 - 4 * t) 0 : ℝ) : ℂ) + - ((fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * t : ℝ) : ℂ) * Complex.I := by - rw [physicalFitChart_terminal hs w hw hm (by rw [guideCoordinate_re]; linarith), - guideCoordinate_re, guideCoordinate_im] - push_cast - ring - -lemma physicalFitChart_terminal_guide_endpoint {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) - (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) (x : ℝ) : - physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x (fitHighTime s w m)) = - (fitTerminalX s w : ℂ) + (2 : ℂ) * Complex.I := by - have hT := fitHighTime_large hs hs1 w hw hw1 hm - rw [physicalFitChart_terminal_guide hs w hw hm x (by linarith), - max_eq_right (by linarith : 1 - 4 * fitHighTime s w m ≤ 0), mul_zero, sub_zero] - have he : fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * fitHighTime s w m = 2 := by - unfold fitHighTime - rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] - ring - rw [he] - norm_num - -lemma physicalFitChart_terminal_guide_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} - (ht : t ∈ Icc (0 : ℝ) (fitHighTime s w m)) : - (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)).im ≤ 2 := by - rw [physicalFitChart_terminal_guide hs w hw hm x ht.1] - simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] - have h := mul_le_mul_of_nonneg_left ht.2 (fit_width_pos hs hm).le - unfold fitHighTime at h - rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] at h - linarith - -lemma physicalFitChart_terminal_guide_strict_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} - (ht : t ∈ Ico (0 : ℝ) (fitHighTime s w m)) : - (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)).im < 2 := by - rw [physicalFitChart_terminal_guide hs w hw hm x ht.1] - simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] - have h := mul_lt_mul_of_pos_left ht.2 (fit_width_pos hs hm) - unfold fitHighTime at h - rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] at h - linarith - -lemma physicalFitChart_terminal_guide_injective {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) - {m : ℕ} (hm : 0 < m) (x : ℝ) : - InjOn (fun t => physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)) (Ici 0) := by - intro t ht u hu he - have him := congrArg Complex.im he - dsimp only at him - rw [physicalFitChart_terminal_guide hs w hw hm x ht, - physicalFitChart_terminal_guide hs w hw hm x hu] at him - simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] at him - nlinarith [fit_width_pos hs hm] - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineMarkedClearance + +namespace OAI + +/-! The actual last front is on the terminal vertical piece. Its access +guide reaches height two before the explicit marked hook is attached. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates + +def fitEntry (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + connectorEntry (fitHeight m) (fitRadius m) (fitX s w m) (fitY s w m) + +def fitCompletion (s : ℝ) (w : ℂ) (m : ℕ) : ℕ := completionStep (fitEntry s w m) m + +def fitTerminalX (s : ℝ) (w : ℂ) : ℝ := |w.re| + s / 4 + s / 200 + +def fitTerminalHeight (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + ribbonWidth (halfHeight s) m * + (fitY s w m + fitCompletion s w m - fitEntry s w m - columnEntry 0 m (2 * m)) + +def fitHighTime (s : ℝ) (w : ℂ) (m : ℕ) : ℝ := + (2 - fitTerminalHeight s w m) / ribbonWidth (halfHeight s) m + +lemma fitCompletion_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + fitEntry s w m + columnEntry 0 m (2 * m) + 200 * m ≤ fitCompletion s w m ∧ + (fitCompletion s w m : ℝ) < fitEntry s w m + columnEntry 0 m (2 * m) + 200 * m + 1 := by + have hA : 0 ≤ fitEntry s w m := by + have h := fit_entry_lower hs w hw hm + exact le_trans (by unfold fitHeight; positivity) h + have hlo := Nat.le_ceil (columnEntry (fitEntry s w m) m (2 * m) + 200 * m) + have hhi := Nat.ceil_lt_add_one + (add_nonneg (columnEntry_nonneg hA m (2 * m)) (by positivity : (0 : ℝ) ≤ 200 * m)) + change _ ≤ (fitCompletion s w m : ℝ) at hlo + change (fitCompletion s w m : ℝ) < _ at hhi + have he : columnEntry (fitEntry s w m) m (2 * m) = + fitEntry s w m + columnEntry 0 m (2 * m) := by simp [columnEntry] + rw [he] at hlo hhi + exact ⟨hlo, hhi⟩ + +lemma fit_scaledY {s : ℝ} (hs : 0 < s) (w : ℂ) {m : ℕ} (hm : 0 < m) : + ribbonWidth (halfHeight s) m * fitY s w m = w.im - s / 200 := by + unfold fitY halfHeight + exact mul_div_cancel₀ _ (fit_width_pos hs hm).ne' + +lemma fitTerminalHeight_bounds {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) : + w.im + s / 200 ≤ fitTerminalHeight s w m ∧ + fitTerminalHeight s w m < w.im + s / 200 + ribbonWidth (halfHeight s) m := by + obtain ⟨hlo, hhi⟩ := fitCompletion_bounds hs w hw hm + have hℓ := fit_width_pos hs hm + have hl := mul_le_mul_of_nonneg_left hlo hℓ.le + have hh := mul_lt_mul_of_pos_left hhi hℓ + have hY := fit_scaledY hs w hm + have hmℓ := fit_width_mul hs hm + unfold fitTerminalHeight + constructor <;> nlinarith + +lemma physicalFitChart_terminal {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) {z : ℂ} (hz : (fitCompletion s w m : ℝ) ≤ z.re) : + physicalFitChart s w m z = + ((fitTerminalX s w - ribbonWidth (halfHeight s) m * z.im : ℝ) : ℂ) + + ((fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * + (z.re - fitCompletion s w m) : ℝ) : ℂ) * Complex.I := by + have hthreshold : fitEntry s w m + columnEntry 0 m (2 * m) + 1 ≤ z.re := by + have h := (fitCompletion_bounds hs w hw hm).1 + have hmr : (1 : ℝ) ≤ m := by exact_mod_cast hm + linarith + obtain ⟨hσ, hT⟩ := complete_linear_tail m (fitHeight m) (fitRadius m) + (fitX s w m) (fitY s w m) hthreshold + have hX := fit_scaledX hs w hm + have hmℓ := fit_width_mul hs hm + change ((ribbonWidth (halfHeight s) m : ℂ) * + (completeSpine m _ _ _ _ z.re + (z.im : ℂ) * Complex.I * completeTangent m _ _ _ _ z.re)) = _ + rw [hσ, hT] + apply Complex.ext <;> + simp only [Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + mul_zero, zero_mul, mul_one, sub_zero, add_zero, zero_add] + · unfold fitTerminalX + nlinarith + · unfold fitTerminalHeight fitEntry + ring + +lemma fitHighTime_large {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) (w : ℂ) + (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) : + 1 < fitHighTime s w m := by + have hheight := (fitTerminalHeight_bounds hs w hw hm).2 + have hwi := (Complex.im_le_norm w).trans hw1 + have hwidth := fit_width_le hs hm + rw [fitHighTime, lt_div_iff₀ (fit_width_pos hs hm)] + nlinarith + +/-- Exact physical formula for the tapered terminal access guide. -/ +lemma physicalFitChart_terminal_guide {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} (ht : 0 ≤ t) : + physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t) = + ((fitTerminalX s w - ribbonWidth (halfHeight s) m * x * max (1 - 4 * t) 0 : ℝ) : ℂ) + + ((fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * t : ℝ) : ℂ) * Complex.I := by + rw [physicalFitChart_terminal hs w hw hm (by rw [guideCoordinate_re]; linarith), + guideCoordinate_re, guideCoordinate_im] + push_cast + ring + +lemma physicalFitChart_terminal_guide_endpoint {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) + (w : ℂ) (hw : s ≤ w.im) (hw1 : ‖w‖ ≤ 1) {m : ℕ} (hm : 0 < m) (x : ℝ) : + physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x (fitHighTime s w m)) = + (fitTerminalX s w : ℂ) + (2 : ℂ) * Complex.I := by + have hT := fitHighTime_large hs hs1 w hw hw1 hm + rw [physicalFitChart_terminal_guide hs w hw hm x (by linarith), + max_eq_right (by linarith : 1 - 4 * fitHighTime s w m ≤ 0), mul_zero, sub_zero] + have he : fitTerminalHeight s w m + ribbonWidth (halfHeight s) m * fitHighTime s w m = 2 := by + unfold fitHighTime + rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] + ring + rw [he] + norm_num + +lemma physicalFitChart_terminal_guide_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} + (ht : t ∈ Icc (0 : ℝ) (fitHighTime s w m)) : + (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)).im ≤ 2 := by + rw [physicalFitChart_terminal_guide hs w hw hm x ht.1] + simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] + have h := mul_le_mul_of_nonneg_left ht.2 (fit_width_pos hs hm).le + unfold fitHighTime at h + rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] at h + linarith + +lemma physicalFitChart_terminal_guide_strict_height {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (x : ℝ) {t : ℝ} + (ht : t ∈ Ico (0 : ℝ) (fitHighTime s w m)) : + (physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)).im < 2 := by + rw [physicalFitChart_terminal_guide hs w hw hm x ht.1] + simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] + have h := mul_lt_mul_of_pos_left ht.2 (fit_width_pos hs hm) + unfold fitHighTime at h + rw [mul_div_cancel₀ _ (fit_width_pos hs hm).ne'] at h + linarith + +lemma physicalFitChart_terminal_guide_injective {s : ℝ} (hs : 0 < s) (w : ℂ) (hw : s ≤ w.im) + {m : ℕ} (hm : 0 < m) (x : ℝ) : + InjOn (fun t => physicalFitChart s w m (guideCoordinate (fitCompletion s w m) x t)) (Ici 0) := by + intro t ht u hu he + have him := congrArg Complex.im he + dsimp only at him + rw [physicalFitChart_terminal_guide hs w hw hm x ht, + physicalFitChart_terminal_guide hs w hw hm x hu] at him + simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] at him + nlinarith [fit_width_pos hs hm] + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/SerpentineUpperHalfPlane.lean b/lean/OAI/Probability/SLE/Gauge/SerpentineUpperHalfPlane.lean index c003883a173b461d57a1aff0ddc6dbfa1a40aec4..b46dd97b3cc4a55799b60fc4b435676729a8186c 100644 --- a/lean/OAI/Probability/SLE/Gauge/SerpentineUpperHalfPlane.lean +++ b/lean/OAI/Probability/SLE/Gauge/SerpentineUpperHalfPlane.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding - -namespace OAI - -/-! The complete normal ribbon starts on the real boundary and every -positive longitudinal coordinate lies in the upper half-plane. -/ - -noncomputable section -open Set Metric - -namespace SLEExactGauge.Serpentine - -lemma lowConnector_normal_im_pos {H r X : ℝ} (hH : 1 < H) (hr : 1 < r) (hX : 2 * r < X) - {z : ℂ} (hz : 0 < z.re) (ha : |z.im| ≤ 1) : - 0 < (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).im := by - have hr0 : 0 < r := by linarith - obtain ⟨hc1, hc2, hc3⟩ := connector_cuts (H := H) hr0 hX - by_cases h4 : z.re ≤ connectorFourth H r X - · by_cases h3 : z.re ≤ connectorThird H r X - · by_cases h2 : z.re ≤ connectorSecond H r - · by_cases h1 : z.re ≤ H - · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, - curveJoin_of_le h3, curveJoin_of_le h2, curveJoin_of_le h1, connectorInitial, - Complex.add_im, Complex.mul_im, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero] - exact hz - · have ht : z.re - H ∈ Icc (0 : ℝ) (quarterLength r) := by - constructor - · linarith - · dsimp only [connectorSecond] at h2 - linarith - have hb := (quarterNormal_bounds hr ht ha).2.2.1 - simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, - curveJoin_of_le h3, curveJoin_of_le h2, curveJoin_of_gt (lt_of_not_ge h1), connectorFirstTurn] - rw [add_assoc] - change 0 < ((H : ℂ) * Complex.I + quarterNormal r (z.re - H) z.im).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero] - linarith - · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, - curveJoin_of_le h3, curveJoin_of_gt (lt_of_not_ge h2), connectorHorizontal, - Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, mul_one, add_zero, zero_add] - linarith [(abs_le.mp ha).1] - · have ht : z.re - connectorThird H r X ∈ Icc (0 : ℝ) (quarterLength r) := by - constructor - · linarith - · dsimp only [connectorFourth] at h4 - linarith - have hb := (upperQuarterNormal_bounds hr ht ha).2.2.1 - simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, - curveJoin_of_gt (lt_of_not_ge h3), connectorLastTurn] - rw [add_assoc] - change 0 < ((((X - r : ℝ) : ℂ) + ((H + r : ℝ) : ℂ) * Complex.I) + - upperQuarterNormal r (z.re - connectorThird H r X) z.im).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] - linarith - · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_gt (lt_of_not_ge h4), - connectorTail, Complex.add_im, Complex.mul_im, Complex.mul_re, - Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, - mul_zero, mul_one, sub_zero, add_zero, zero_add] - linarith - -/-- The actual entire ribbon lies above the real boundary at every -positive longitudinal parameter. -/ -theorem complete_normal_upper {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} - (hH : 1 < H) (hr : 1 < r) (hX : 2 * r < X) (hY : 300 * (m : ℝ) < Y) - {z : ℂ} (hz : 0 < z.re) (ha : |z.im| ≤ 1) : - 0 < (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z).im := by - by_cases hA : z.re ≤ connectorEntry H r X Y - · rw [complete_normal_of_le m H r X Y hA] - exact lowConnector_normal_im_pos hH hr hX hz ha - · have hz' : 0 ≤ (z - (connectorEntry H r X Y : ℂ)).re ∧ - |(z - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by - simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] - exact ⟨by linarith, ha⟩ - have hb := array_normal_im_lower hm hz' - rw [complete_normal_of_gt m H r X Y (lt_of_not_ge hA)] - simp only [arrayCenter, Complex.add_im, Complex.ofReal_im, Complex.mul_im, - Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] - linarith - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.SerpentineCompleteEmbedding + +namespace OAI + +/-! The complete normal ribbon starts on the real boundary and every +positive longitudinal coordinate lies in the upper half-plane. -/ + +noncomputable section +open Set Metric + +namespace SLEExactGauge.Serpentine + +lemma lowConnector_normal_im_pos {H r X : ℝ} (hH : 1 < H) (hr : 1 < r) (hX : 2 * r < X) + {z : ℂ} (hz : 0 < z.re) (ha : |z.im| ≤ 1) : + 0 < (normalRibbon (lowConnector H r X) (lowConnectorTangent H r X) z).im := by + have hr0 : 0 < r := by linarith + obtain ⟨hc1, hc2, hc3⟩ := connector_cuts (H := H) hr0 hX + by_cases h4 : z.re ≤ connectorFourth H r X + · by_cases h3 : z.re ≤ connectorThird H r X + · by_cases h2 : z.re ≤ connectorSecond H r + · by_cases h1 : z.re ≤ H + · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, + curveJoin_of_le h3, curveJoin_of_le h2, curveJoin_of_le h1, connectorInitial, + Complex.add_im, Complex.mul_im, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero] + exact hz + · have ht : z.re - H ∈ Icc (0 : ℝ) (quarterLength r) := by + constructor + · linarith + · dsimp only [connectorSecond] at h2 + linarith + have hb := (quarterNormal_bounds hr ht ha).2.2.1 + simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, + curveJoin_of_le h3, curveJoin_of_le h2, curveJoin_of_gt (lt_of_not_ge h1), connectorFirstTurn] + rw [add_assoc] + change 0 < ((H : ℂ) * Complex.I + quarterNormal r (z.re - H) z.im).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero] + linarith + · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, + curveJoin_of_le h3, curveJoin_of_gt (lt_of_not_ge h2), connectorHorizontal, + Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, mul_one, add_zero, zero_add] + linarith [(abs_le.mp ha).1] + · have ht : z.re - connectorThird H r X ∈ Icc (0 : ℝ) (quarterLength r) := by + constructor + · linarith + · dsimp only [connectorFourth] at h4 + linarith + have hb := (upperQuarterNormal_bounds hr ht ha).2.2.1 + simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_le h4, + curveJoin_of_gt (lt_of_not_ge h3), connectorLastTurn] + rw [add_assoc] + change 0 < ((((X - r : ℝ) : ℂ) + ((H + r : ℝ) : ℂ) * Complex.I) + + upperQuarterNormal r (z.re - connectorThird H r X) z.im).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] + linarith + · simp only [normalRibbon, lowConnector, lowConnectorTangent, curveJoin_of_gt (lt_of_not_ge h4), + connectorTail, Complex.add_im, Complex.mul_im, Complex.mul_re, + Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, + mul_zero, mul_one, sub_zero, add_zero, zero_add] + linarith + +/-- The actual entire ribbon lies above the real boundary at every +positive longitudinal parameter. -/ +theorem complete_normal_upper {m : ℕ} (hm : 0 < m) {H r X Y : ℝ} + (hH : 1 < H) (hr : 1 < r) (hX : 2 * r < X) (hY : 300 * (m : ℝ) < Y) + {z : ℂ} (hz : 0 < z.re) (ha : |z.im| ≤ 1) : + 0 < (normalRibbon (completeSpine m H r X Y) (completeTangent m H r X Y) z).im := by + by_cases hA : z.re ≤ connectorEntry H r X Y + · rw [complete_normal_of_le m H r X Y hA] + exact lowConnector_normal_im_pos hH hr hX hz ha + · have hz' : 0 ≤ (z - (connectorEntry H r X Y : ℂ)).re ∧ + |(z - (connectorEntry H r X Y : ℂ)).im| ≤ 1 := by + simp only [Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im, sub_zero] + exact ⟨by linarith, ha⟩ + have hb := array_normal_im_lower hm hz' + rw [complete_normal_of_gt m H r X Y (lt_of_not_ge hA)] + simp only [arrayCenter, Complex.add_im, Complex.ofReal_im, Complex.mul_im, + Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_one, mul_zero, add_zero, zero_add] + linarith + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/ShrinkingCoverAtoms.lean b/lean/OAI/Probability/SLE/Gauge/ShrinkingCoverAtoms.lean index 6115c039e0e846f4b4fe0e1c6ceecb85773b4f33..a3024c0f93735ab8f44abec03b216b852a46f84b 100644 --- a/lean/OAI/Probability/SLE/Gauge/ShrinkingCoverAtoms.lean +++ b/lean/OAI/Probability/SLE/Gauge/ShrinkingCoverAtoms.lean @@ -1,79 +1,79 @@ -import Mathlib.MeasureTheory.Integral.Lebesgue.Add -import Mathlib.MeasureTheory.Measure.GiryMonad -import Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass -import Mathlib.Topology.Order.LiminfLimsup - -namespace OAI - -/-! The nonatomicity step of `mass:construction`: a squared atom is -bounded by every squared covering mass sum. Fatou therefore rules out -all atoms at once, without a countable union over points. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal Topology - -namespace SLEExactGauge - -variable {X Ω : Type*} [MeasurableSpace X] [MeasurableSpace Ω] - -/-- One atom is charged to one cell in any finite cover. -/ -lemma singleton_sq_le_cover_sum (μ : Measure X) (I : Finset ℕ) (C : ℕ → Set X) - {x : X} (hcover : ∃ i ∈ I, x ∈ C i) : - (μ {x})^2 ≤ ∑ i ∈ I, (μ (C i))^2 := by - obtain ⟨i,hi,hxi⟩ := hcover - have hsub : ({x} : Set X) ⊆ C i := singleton_subset_iff.mpr hxi - exact (pow_le_pow_left' (measure_mono hsub) 2).trans - (Finset.single_le_sum (f := fun j => (μ (C j))^2) (fun _ _ => zero_le) hi) - -/-- Vanishing expected upper bounds for squared atoms exclude every atom -on a set simultaneously almost surely. -/ -theorem ae_null_singletons_on_of_square_bounds (P : Measure Ω) (μ : Ω → Measure X) - (K : Set X) (A : ℕ → Ω → ℝ≥0∞) (hA : ∀ n, Measurable (A n)) - (hbound : ∀ ω x, x ∈ K → ∀ n, (μ ω {x})^2 ≤ A n ω) - (hlim : Tendsto (fun n => ∫⁻ ω, A n ω ∂P) atTop (𝓝 0)) : - ∀ᵐ ω ∂P, ∀ x ∈ K, μ ω {x}=0 := by - have hfatou : (∫⁻ ω, liminf (fun n => A n ω) atTop ∂P) ≤ 0 := by - simpa only [hlim.liminf_eq] using lintegral_liminf_le (μ := P) (u := atTop) hA - have hz : ∀ᵐ ω ∂P, liminf (fun n => A n ω) atTop=0 := - (lintegral_eq_zero_iff (Measurable.liminf hA)).mp (le_antisymm hfatou bot_le) - filter_upwards [hz] with ω hω - intro x hx - have hb : (μ ω {x})^2 ≤ liminf (fun n => A n ω) atTop := - le_liminf_of_le (by isBoundedDefault) (Eventually.of_forall (hbound ω x hx)) - rw [hω] at hb - exact eq_zero_of_pow_eq_zero (le_antisymm hb bot_le) - -/-- The finite-cover form used for compact exhaustion. Cover diameters -need not be named here: all geometric decay is captured in the expected -squared mass sum. -/ -theorem ae_null_singletons_on_of_cover_sums (P : Measure Ω) (μ : Ω → Measure X) - (hμ : Measurable μ) (K : Set X) (I : ℕ → Finset ℕ) (C : ℕ → ℕ → Set X) - (hC : ∀ n i, MeasurableSet (C n i)) - (hcover : ∀ n x, x ∈ K → ∃ i ∈ I n, x ∈ C n i) - (hlim : Tendsto (fun n => ∫⁻ ω, ∑ i ∈ I n, (μ ω (C n i))^2 ∂P) atTop (𝓝 0)) : - ∀ᵐ ω ∂P, ∀ x ∈ K, μ ω {x}=0 := by - apply ae_null_singletons_on_of_square_bounds P μ K - (fun n ω => ∑ i ∈ I n, (μ ω (C n i))^2) _ _ hlim - · intro n - exact Finset.measurable_sum (I n) (fun i _ => ((Measure.measurable_coe (hC n i)).comp hμ).pow_const 2) - · intro ω x hx n - exact singleton_sq_le_cover_sum (μ ω) (I n) (C n) (hcover n x hx) - -/-- A countable covering of the whole space upgrades the compact-by-compact -criterion to a nonatomic random measure. -/ -theorem ae_nullSingletonClass_of_exhaustion (P : Measure Ω) (μ : Ω → Measure X) - (K : ℕ → Set X) (hcover : ∀ x, ∃ j, x ∈ K j) - (hK : ∀ j, ∀ᵐ ω ∂P, ∀ x ∈ K j, μ ω {x}=0) : - ∀ᵐ ω ∂P, NullSingletonClass (μ ω) := by - filter_upwards [ae_all_iff.mpr hK] with ω hω - constructor - intro x - obtain ⟨j,hj⟩ := hcover x - exact hω j x hj - -end SLEExactGauge - -end - -end OAI +import Mathlib.MeasureTheory.Integral.Lebesgue.Add +import Mathlib.MeasureTheory.Measure.GiryMonad +import Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass +import Mathlib.Topology.Order.LiminfLimsup + +namespace OAI + +/-! The nonatomicity step of `mass:construction`: a squared atom is +bounded by every squared covering mass sum. Fatou therefore rules out +all atoms at once, without a countable union over points. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal Topology + +namespace SLEExactGauge + +variable {X Ω : Type*} [MeasurableSpace X] [MeasurableSpace Ω] + +/-- One atom is charged to one cell in any finite cover. -/ +lemma singleton_sq_le_cover_sum (μ : Measure X) (I : Finset ℕ) (C : ℕ → Set X) + {x : X} (hcover : ∃ i ∈ I, x ∈ C i) : + (μ {x})^2 ≤ ∑ i ∈ I, (μ (C i))^2 := by + obtain ⟨i,hi,hxi⟩ := hcover + have hsub : ({x} : Set X) ⊆ C i := singleton_subset_iff.mpr hxi + exact (pow_le_pow_left' (measure_mono hsub) 2).trans + (Finset.single_le_sum (f := fun j => (μ (C j))^2) (fun _ _ => zero_le) hi) + +/-- Vanishing expected upper bounds for squared atoms exclude every atom +on a set simultaneously almost surely. -/ +theorem ae_null_singletons_on_of_square_bounds (P : Measure Ω) (μ : Ω → Measure X) + (K : Set X) (A : ℕ → Ω → ℝ≥0∞) (hA : ∀ n, Measurable (A n)) + (hbound : ∀ ω x, x ∈ K → ∀ n, (μ ω {x})^2 ≤ A n ω) + (hlim : Tendsto (fun n => ∫⁻ ω, A n ω ∂P) atTop (𝓝 0)) : + ∀ᵐ ω ∂P, ∀ x ∈ K, μ ω {x}=0 := by + have hfatou : (∫⁻ ω, liminf (fun n => A n ω) atTop ∂P) ≤ 0 := by + simpa only [hlim.liminf_eq] using lintegral_liminf_le (μ := P) (u := atTop) hA + have hz : ∀ᵐ ω ∂P, liminf (fun n => A n ω) atTop=0 := + (lintegral_eq_zero_iff (Measurable.liminf hA)).mp (le_antisymm hfatou bot_le) + filter_upwards [hz] with ω hω + intro x hx + have hb : (μ ω {x})^2 ≤ liminf (fun n => A n ω) atTop := + le_liminf_of_le (by isBoundedDefault) (Eventually.of_forall (hbound ω x hx)) + rw [hω] at hb + exact eq_zero_of_pow_eq_zero (le_antisymm hb bot_le) + +/-- The finite-cover form used for compact exhaustion. Cover diameters +need not be named here: all geometric decay is captured in the expected +squared mass sum. -/ +theorem ae_null_singletons_on_of_cover_sums (P : Measure Ω) (μ : Ω → Measure X) + (hμ : Measurable μ) (K : Set X) (I : ℕ → Finset ℕ) (C : ℕ → ℕ → Set X) + (hC : ∀ n i, MeasurableSet (C n i)) + (hcover : ∀ n x, x ∈ K → ∃ i ∈ I n, x ∈ C n i) + (hlim : Tendsto (fun n => ∫⁻ ω, ∑ i ∈ I n, (μ ω (C n i))^2 ∂P) atTop (𝓝 0)) : + ∀ᵐ ω ∂P, ∀ x ∈ K, μ ω {x}=0 := by + apply ae_null_singletons_on_of_square_bounds P μ K + (fun n ω => ∑ i ∈ I n, (μ ω (C n i))^2) _ _ hlim + · intro n + exact Finset.measurable_sum (I n) (fun i _ => ((Measure.measurable_coe (hC n i)).comp hμ).pow_const 2) + · intro ω x hx n + exact singleton_sq_le_cover_sum (μ ω) (I n) (C n) (hcover n x hx) + +/-- A countable covering of the whole space upgrades the compact-by-compact +criterion to a nonatomic random measure. -/ +theorem ae_nullSingletonClass_of_exhaustion (P : Measure Ω) (μ : Ω → Measure X) + (K : ℕ → Set X) (hcover : ∀ x, ∃ j, x ∈ K j) + (hK : ∀ j, ∀ᵐ ω ∂P, ∀ x ∈ K j, μ ω {x}=0) : + ∀ᵐ ω ∂P, NullSingletonClass (μ ω) := by + filter_upwards [ae_all_iff.mpr hK] with ω hω + constructor + intro x + obtain ⟨j,hj⟩ := hcover x + exact hω j x hj + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StoppedSplicePassage.lean b/lean/OAI/Probability/SLE/Gauge/StoppedSplicePassage.lean index 11a6c37df3ec9881fdd8825d187d6a82321eba60..613eee9d8966f37b739c3dbd6485ee2aaf512c61 100644 --- a/lean/OAI/Probability/SLE/Gauge/StoppedSplicePassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/StoppedSplicePassage.lean @@ -1,128 +1,128 @@ -import OAI.Probability.SLE.Geometry.ActualStoppedTubeSupport -import OAI.Probability.SLE.Gauge.RibbonSurvival - -namespace OAI - -/-! The actual semicircle splice gives conditional passage after a narrow -first-height stop. The physical tube here is in the original parameter -plane, so compact continuation prefixes stay strictly inside the half-plane -and can subsequently be transported through a locally convergent family. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal unitInterval ENNReal - -namespace SLEExactGauge - -def parameterContinuationRegion (F : ℂ → ℂ) (a : ℂ) (r : ℝ) (O : Set ℂ) : Set ℂ := - upperHalfPlane ∩ F ⁻¹' (ball a (r / 2) ∪ O) - -theorem parameterContinuationRegion_isOpen - {F : ℂ → ℂ} (hF : ContinuousOn F upperHalfPlane) - (a : ℂ) (r : ℝ) {O : Set ℂ} (hO : IsOpen O) : - IsOpen (parameterContinuationRegion F a r O) := - hF.isOpen_inter_preimage isOpen_upperHalfPlane (isOpen_ball.union hO) - -theorem exists_physical_splice_route - (γ : ℝ≥0 → ℂ) (τ : ℝ≥0) {B r tB T : ℝ} (hB : 0 < B) (hr : 0 < r) - (hheight : (γ τ).im = B / 4) (hx : |(γ τ).re| < B / 16) - (hpast : γ '' Icc 0 τ ⊆ GuideSplice.closedInitialRectangle B) - (F : ℂ → ℂ) (a : ℂ) (O : Set ℂ) (β : ℝ → ℂ) - (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) : - ∃ p : Path (γ τ) (β T), Function.Injective p ∧ - (∀ t, t ≠ 0 → p t ∈ survivingDomain γ τ) ∧ - range p ⊆ parameterContinuationRegion F a r O := by - let x := (γ τ).re - let ζ : ℂ := (x : ℂ) + ((B / 4 : ℝ) : ℂ) * Complex.I - have htip : γ τ = ζ := by - apply Complex.ext <;> simp [ζ, x, hheight] - have hα : (GuideSplice.connectorTop B x).arg ∈ Ioo 0 Real.pi := by - apply GuideSplice.arg_mem_upper_interval - simpa [GuideSplice.connectorTop] using - Real.sqrt_pos.mpr (GuideSplice.circle_height_pos hB hx) - have hω : (β tB).arg ∈ Ioo 0 Real.pi := - GuideSplice.arg_mem_upper_interval (hβH tB ⟨htB, htBT.le⟩) - obtain ⟨p, hp, _, hpH, _, hpr⟩ := GuideSplice.exists_simple_splice_path hB hx - htB htBT hβ hi hβH hcross hafter - have hrest := GuideSplice.continuationRoute_rest_subset_tube hr hB hx hα hω htB - hβH hcross hafter hcircle hsector htail - have hsurv := continuationTube_subset_survivingDomain γ τ B hpast F a r O - have hpne (t : unitInterval) (ht : t ≠ 0) : p t ≠ ζ := by - intro he - apply ht - apply hp - simpa only [Path.source] using he - refine ⟨p.cast htip rfl, hp, ?_, ?_⟩ - · intro t ht - apply hsurv - apply hrest - exact ⟨hpr ▸ mem_range_self t, hpne t ht⟩ - · intro z hz - refine ⟨hpH hz, ?_⟩ - by_cases he : z = ζ - · subst z - have hζV : ζ ∈ GuideSplice.verticalConnector B x := by - simp only [ζ, GuideSplice.verticalConnector, mem_ofPred_eq, Complex.add_re, - Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.ofReal_im, - Complex.I_im, mul_zero, sub_self, add_zero, Complex.add_im, - Complex.mul_im, mul_one, zero_add] - exact ⟨trivial, le_rfl, (GuideSplice.circle_height_gt_quarter hB hx).le⟩ - exact Or.inl (ball_subset_ball (by linarith) - (GuideSplice.initial_splice_image_subset_quarter_ball hB hx hα hω - hcircle hsector (mem_image_of_mem F (Or.inl hζV)))) - · exact (hrest ⟨hpr ▸ hz, he⟩).1.2 - -theorem conditional_stopped_splice_positive - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (E : Set Ω) - (hgeometry : ∀ ω ∈ E, 0 < τ ω ∧ (γ ω (τ ω)).im = B / 4 ∧ - |(γ ω (τ ω)).re| < B / 16 ∧ - (∀ u : ℝ≥0, u < τ ω → (γ ω u).im < (γ ω (τ ω)).im) ∧ - γ ω '' Icc 0 (τ ω) ⊆ GuideSplice.closedInitialRectangle B) - (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : - ∀ᵐ ω ∂P, ω ∈ E → - 0 < P⁻[(stoppedPhysicalTube γ τ (parameterContinuationRegion F a r O) W).indicator - (fun _ => 1) | hτ.measurableSpace] ω := by - apply conditional_stopped_physical_tube_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw - hboundary hsupport τ hτ _ W (parameterContinuationRegion_isOpen hF a r hO) hW E - · intro ω hω - obtain ⟨hτpos, hh, _, hprior, _⟩ := hgeometry ω hω - exact ⟨hτpos, by rw [hh]; positivity, hprior⟩ - · intro ω hω - obtain ⟨_, hh, hx, _, hpast⟩ := hgeometry ω hω - obtain ⟨p, hp, hpD, hpO⟩ := exists_physical_splice_route (γ ω) (τ ω) hB hr hh hx hpast - F a O β htB htBT hβ hi hβH hcross hafter hcircle hsector htail - exact ⟨β T, p, hp, hpD, hpO, hend⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ActualStoppedTubeSupport +import OAI.Probability.SLE.Gauge.RibbonSurvival + +namespace OAI + +/-! The actual semicircle splice gives conditional passage after a narrow +first-height stop. The physical tube here is in the original parameter +plane, so compact continuation prefixes stay strictly inside the half-plane +and can subsequently be transported through a locally convergent family. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal unitInterval ENNReal + +namespace SLEExactGauge + +def parameterContinuationRegion (F : ℂ → ℂ) (a : ℂ) (r : ℝ) (O : Set ℂ) : Set ℂ := + upperHalfPlane ∩ F ⁻¹' (ball a (r / 2) ∪ O) + +theorem parameterContinuationRegion_isOpen + {F : ℂ → ℂ} (hF : ContinuousOn F upperHalfPlane) + (a : ℂ) (r : ℝ) {O : Set ℂ} (hO : IsOpen O) : + IsOpen (parameterContinuationRegion F a r O) := + hF.isOpen_inter_preimage isOpen_upperHalfPlane (isOpen_ball.union hO) + +theorem exists_physical_splice_route + (γ : ℝ≥0 → ℂ) (τ : ℝ≥0) {B r tB T : ℝ} (hB : 0 < B) (hr : 0 < r) + (hheight : (γ τ).im = B / 4) (hx : |(γ τ).re| < B / 16) + (hpast : γ '' Icc 0 τ ⊆ GuideSplice.closedInitialRectangle B) + (F : ℂ → ℂ) (a : ℂ) (O : Set ℂ) (β : ℝ → ℂ) + (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) : + ∃ p : Path (γ τ) (β T), Function.Injective p ∧ + (∀ t, t ≠ 0 → p t ∈ survivingDomain γ τ) ∧ + range p ⊆ parameterContinuationRegion F a r O := by + let x := (γ τ).re + let ζ : ℂ := (x : ℂ) + ((B / 4 : ℝ) : ℂ) * Complex.I + have htip : γ τ = ζ := by + apply Complex.ext <;> simp [ζ, x, hheight] + have hα : (GuideSplice.connectorTop B x).arg ∈ Ioo 0 Real.pi := by + apply GuideSplice.arg_mem_upper_interval + simpa [GuideSplice.connectorTop] using + Real.sqrt_pos.mpr (GuideSplice.circle_height_pos hB hx) + have hω : (β tB).arg ∈ Ioo 0 Real.pi := + GuideSplice.arg_mem_upper_interval (hβH tB ⟨htB, htBT.le⟩) + obtain ⟨p, hp, _, hpH, _, hpr⟩ := GuideSplice.exists_simple_splice_path hB hx + htB htBT hβ hi hβH hcross hafter + have hrest := GuideSplice.continuationRoute_rest_subset_tube hr hB hx hα hω htB + hβH hcross hafter hcircle hsector htail + have hsurv := continuationTube_subset_survivingDomain γ τ B hpast F a r O + have hpne (t : unitInterval) (ht : t ≠ 0) : p t ≠ ζ := by + intro he + apply ht + apply hp + simpa only [Path.source] using he + refine ⟨p.cast htip rfl, hp, ?_, ?_⟩ + · intro t ht + apply hsurv + apply hrest + exact ⟨hpr ▸ mem_range_self t, hpne t ht⟩ + · intro z hz + refine ⟨hpH hz, ?_⟩ + by_cases he : z = ζ + · subst z + have hζV : ζ ∈ GuideSplice.verticalConnector B x := by + simp only [ζ, GuideSplice.verticalConnector, mem_ofPred_eq, Complex.add_re, + Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.ofReal_im, + Complex.I_im, mul_zero, sub_self, add_zero, Complex.add_im, + Complex.mul_im, mul_one, zero_add] + exact ⟨trivial, le_rfl, (GuideSplice.circle_height_gt_quarter hB hx).le⟩ + exact Or.inl (ball_subset_ball (by linarith) + (GuideSplice.initial_splice_image_subset_quarter_ball hB hx hα hω + hcircle hsector (mem_image_of_mem F (Or.inl hζV)))) + · exact (hrest ⟨hpr ▸ hz, he⟩).1.2 + +theorem conditional_stopped_splice_positive + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (E : Set Ω) + (hgeometry : ∀ ω ∈ E, 0 < τ ω ∧ (γ ω (τ ω)).im = B / 4 ∧ + |(γ ω (τ ω)).re| < B / 16 ∧ + (∀ u : ℝ≥0, u < τ ω → (γ ω u).im < (γ ω (τ ω)).im) ∧ + γ ω '' Icc 0 (τ ω) ⊆ GuideSplice.closedInitialRectangle B) + (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : + ∀ᵐ ω ∂P, ω ∈ E → + 0 < P⁻[(stoppedPhysicalTube γ τ (parameterContinuationRegion F a r O) W).indicator + (fun _ => 1) | hτ.measurableSpace] ω := by + apply conditional_stopped_physical_tube_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw + hboundary hsupport τ hτ _ W (parameterContinuationRegion_isOpen hF a r hO) hW E + · intro ω hω + obtain ⟨hτpos, hh, _, hprior, _⟩ := hgeometry ω hω + exact ⟨hτpos, by rw [hh]; positivity, hprior⟩ + · intro ω hω + obtain ⟨_, hh, hx, _, hpast⟩ := hgeometry ω hω + obtain ⟨p, hp, hpD, hpO⟩ := exists_physical_splice_route (γ ω) (τ ω) hB hr hh hx hpast + F a O β htB htBT hβ hi hβH hcross hafter hcircle hsector htail + exact ⟨β T, p, hp, hpD, hpO, hend⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StoppedTubePassage.lean b/lean/OAI/Probability/SLE/Gauge/StoppedTubePassage.lean index a606b2e0078bd23cd4ca1dabc991cc6797f08c35..1b67da7717cfbf9b720472bfbc06b8e80dfc798e 100644 --- a/lean/OAI/Probability/SLE/Gauge/StoppedTubePassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/StoppedTubePassage.lean @@ -1,163 +1,163 @@ -import OAI.Probability.SLE.StoppedMeasures -import OAI.Probability.SLE.Geometry.GeometricTubeSupport -import OAI.Probability.SLE.Gauge.VaryingPassageEvents - -namespace OAI - -/-! Passage after a stopping time from countably many open tube certificates. -The stopped trace law gives independence for each fixed tube. A countable -cover of pasts then gives pointwise positive conditional passage probability; -no common support probability is assumed. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -/-- A fixed positive-law test, certified on a past event, forces the -conditional passage probability to be positive on that event. -/ -theorem conditional_passage_positive_on_certificate - {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] - (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Y : Ω → X) (hY : Measurable[mΩ] Y) - (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → - @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) - (A C : Set Ω) (hA : MeasurableSet[m] A) (hC : MeasurableSet[mΩ] C) - (B : Set X) (hB : MeasurableSet B) (hBpos : 0 < ν B) - (hcert : A ∩ Y ⁻¹' B ⊆ C) : - ∀ᵐ ω ∂P, ω ∈ A → 0 < P⁻[C.indicator (fun _ => 1) | m] ω := by - let : MeasurableSpace Ω := mΩ - let q := P⁻[C.indicator (fun _ => 1) | m] - let Z := A ∩ {ω | q ω = 0} - have hZ : MeasurableSet[m] Z := - hA.inter (measurableSet_eq_fun (measurable_condLExp m P _) measurable_const) - have hzero : P (Z ∩ C) = 0 := by - calc - P (Z ∩ C) = ∫⁻ ω in Z, C.indicator (fun _ => 1) ω ∂P := by - rw [setLIntegral_indicator hC, setLIntegral_const, one_mul] - congr 1 - exact inter_comm Z C - _ = ∫⁻ ω in Z, q ω ∂P := (setLIntegral_condLExp hm P _ hZ).symm - _ = 0 := by - apply (setLIntegral_eq_zero_iff (hm Z hZ) - ((measurable_condLExp m P _).mono hm le_rfl)).mpr - exact Eventually.of_forall (fun ω hω => hω.2) - have hproduct : P Z * ν B = 0 := by - apply le_antisymm ?_ bot_le - calc - P Z * ν B = ((P.restrict Z).map Y) B := by - rw [hlaw Z hZ, Measure.smul_apply, smul_eq_mul] - _ = P (Y ⁻¹' B ∩ Z) := by - rw [Measure.map_apply hY hB, Measure.restrict_apply (hY hB)] - _ ≤ P (Z ∩ C) := measure_mono (by - rintro ω ⟨hωB, hωZ⟩ - exact ⟨hωZ, hcert ⟨hωZ.1, hωB⟩⟩) - _ = 0 := hzero - have hZzero : P Z = 0 := - (mul_eq_zero.mp hproduct).resolve_right hBpos.ne' - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hZzero] with ω hω hωA - by_contra h - exact hω ⟨hωA, le_antisymm (not_lt.mp h) bot_le⟩ - -/-- Countably many past-measurable tube certificates suffice even when -their positive support probabilities depend on the certificate. -/ -theorem conditional_passage_positive_of_countable_certificates - {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] - (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Y : Ω → X) (hY : Measurable[mΩ] Y) - (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → - @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) - (E C : Set Ω) (hC : MeasurableSet[mΩ] C) - (A : ℕ → Set Ω) (hA : ∀ i, MeasurableSet[m] (A i)) - (B : ℕ → Set X) (hB : ∀ i, MeasurableSet (B i)) - (hBpos : ∀ i, 0 < ν (B i)) - (hcert : ∀ i, A i ∩ Y ⁻¹' B i ⊆ C) - (hcover : E ⊆ ⋃ i, A i) : - ∀ᵐ ω ∂P, ω ∈ E → 0 < P⁻[C.indicator (fun _ => 1) | m] ω := by - let : MeasurableSpace Ω := mΩ - have hall : ∀ᵐ ω ∂P, ∀ i, ω ∈ A i → - 0 < P⁻[C.indicator (fun _ => 1) | m] ω := - ae_all_iff.mpr (fun i => conditional_passage_positive_on_certificate - P ν m hm Y hY hlaw (A i) C (hA i) hC (B i) (hB i) (hBpos i) (hcert i)) - filter_upwards [hall] with ω hω hωE - obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hωE) - exact hω i hi - -/-- The actual Brownian stopped trace law and the published geometric -support input give conditional positivity for countably certified geometric -continuations. The same realization also has the actual conformal restart -identity, so the certificates can be checked in the original domain. -/ -theorem stopped_geometric_tube_passages - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (hpublished : PublishedGeometricTubeSupport P γ) : - ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧ - (∀ ω, Φ (sourceDriverPath B hc ω) = - sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ - ∀ (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime - (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))), - (∀ᵐ ω ∂P, ∀ t : ℝ≥0, - Φ (sourceRestartDriverPath B hc τ ω) t ∈ upperHalfPlane → - γ ω (τ ω + t) = F ω (τ ω) - (Φ (sourceRestartDriverPath B hc τ ω) t + - (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ - ∀ (E C : Set Ω), MeasurableSet[mΩ] C → - ∀ (A : ℕ → Set Ω), (∀ i, MeasurableSet[hτ.measurableSpace] (A i)) → - ∀ (z : ℕ → ℂ) (p : ∀ i, Path (0 : ℂ) (z i)), - (∀ i, Function.Injective (p i)) → - (∀ i t, t ≠ 0 → p i t ∈ upperHalfPlane) → - ∀ (N V : ℕ → Set ℂ), (∀ i, IsOpen (N i)) → (∀ i, IsOpen (V i)) → - (∀ i, range (p i) ⊆ N i) → (∀ i, z i ∈ V i) → - (∀ i, A i ∩ (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) ⁻¹' - pathTubeEvent (N i) (V i) ⊆ C) → - E ⊆ ⋃ i, A i → - ∀ᵐ ω ∂P, ω ∈ E → - 0 < P⁻[C.indicator (fun _ => 1) | hτ.measurableSpace] ω := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - obtain ⟨Φ, hΦm, hΦ, hlaw⟩ := - sourceStoppedTraceLaw κ hκ hκ8 B hB hm hc γ G F hw - refine ⟨Φ, hΦm, hΦ, ?_⟩ - intro τ hτ - refine ⟨(hlaw τ hτ).2, ?_⟩ - intro E C hC A hA z p hp hpH N V hN hV hpN hzV hcert hcover - let : MeasurableSpace Ω := mΩ - let gp := sourceTracePath γ (fun ω => (hw ω).1) - have hgm : Measurable gp := by - have he : gp = fun ω => Φ (sourceDriverPath B hc ω) := funext fun ω => (hΦ ω).symm - rw [he] - exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm) - have hrm : Measurable (sourceRestartDriverPath B hc τ) := - source_path_measurable _ (fun t => (measurable_pi_apply t).comp - (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) - have htests : ∀ i, 0 < P.map gp (pathTubeEvent (N i) (V i)) := by - intro i - rw [Measure.map_apply hgm (measurableSet_pathTubeEvent (hN i) (hV i))] - exact hpublished (z i) (p i) (hp i) (hpH i) (N i) (V i) - (hN i) (hV i) (hpN i) (hzV i) - exact conditional_passage_positive_of_countable_certificates P (P.map gp) - hτ.measurableSpace hτ.measurableSpace_le - (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) (hΦm.comp hrm) - ((hlaw τ hτ).1) E C hC A hA - (fun i => pathTubeEvent (N i) (V i)) - (fun i => measurableSet_pathTubeEvent (hN i) (hV i)) htests hcert hcover - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.StoppedMeasures +import OAI.Probability.SLE.Geometry.GeometricTubeSupport +import OAI.Probability.SLE.Gauge.VaryingPassageEvents + +namespace OAI + +/-! Passage after a stopping time from countably many open tube certificates. +The stopped trace law gives independence for each fixed tube. A countable +cover of pasts then gives pointwise positive conditional passage probability; +no common support probability is assumed. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +/-- A fixed positive-law test, certified on a past event, forces the +conditional passage probability to be positive on that event. -/ +theorem conditional_passage_positive_on_certificate + {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] + (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Y : Ω → X) (hY : Measurable[mΩ] Y) + (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → + @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) + (A C : Set Ω) (hA : MeasurableSet[m] A) (hC : MeasurableSet[mΩ] C) + (B : Set X) (hB : MeasurableSet B) (hBpos : 0 < ν B) + (hcert : A ∩ Y ⁻¹' B ⊆ C) : + ∀ᵐ ω ∂P, ω ∈ A → 0 < P⁻[C.indicator (fun _ => 1) | m] ω := by + let : MeasurableSpace Ω := mΩ + let q := P⁻[C.indicator (fun _ => 1) | m] + let Z := A ∩ {ω | q ω = 0} + have hZ : MeasurableSet[m] Z := + hA.inter (measurableSet_eq_fun (measurable_condLExp m P _) measurable_const) + have hzero : P (Z ∩ C) = 0 := by + calc + P (Z ∩ C) = ∫⁻ ω in Z, C.indicator (fun _ => 1) ω ∂P := by + rw [setLIntegral_indicator hC, setLIntegral_const, one_mul] + congr 1 + exact inter_comm Z C + _ = ∫⁻ ω in Z, q ω ∂P := (setLIntegral_condLExp hm P _ hZ).symm + _ = 0 := by + apply (setLIntegral_eq_zero_iff (hm Z hZ) + ((measurable_condLExp m P _).mono hm le_rfl)).mpr + exact Eventually.of_forall (fun ω hω => hω.2) + have hproduct : P Z * ν B = 0 := by + apply le_antisymm ?_ bot_le + calc + P Z * ν B = ((P.restrict Z).map Y) B := by + rw [hlaw Z hZ, Measure.smul_apply, smul_eq_mul] + _ = P (Y ⁻¹' B ∩ Z) := by + rw [Measure.map_apply hY hB, Measure.restrict_apply (hY hB)] + _ ≤ P (Z ∩ C) := measure_mono (by + rintro ω ⟨hωB, hωZ⟩ + exact ⟨hωZ, hcert ⟨hωZ.1, hωB⟩⟩) + _ = 0 := hzero + have hZzero : P Z = 0 := + (mul_eq_zero.mp hproduct).resolve_right hBpos.ne' + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hZzero] with ω hω hωA + by_contra h + exact hω ⟨hωA, le_antisymm (not_lt.mp h) bot_le⟩ + +/-- Countably many past-measurable tube certificates suffice even when +their positive support probabilities depend on the certificate. -/ +theorem conditional_passage_positive_of_countable_certificates + {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] + (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Y : Ω → X) (hY : Measurable[mΩ] Y) + (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → + @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) + (E C : Set Ω) (hC : MeasurableSet[mΩ] C) + (A : ℕ → Set Ω) (hA : ∀ i, MeasurableSet[m] (A i)) + (B : ℕ → Set X) (hB : ∀ i, MeasurableSet (B i)) + (hBpos : ∀ i, 0 < ν (B i)) + (hcert : ∀ i, A i ∩ Y ⁻¹' B i ⊆ C) + (hcover : E ⊆ ⋃ i, A i) : + ∀ᵐ ω ∂P, ω ∈ E → 0 < P⁻[C.indicator (fun _ => 1) | m] ω := by + let : MeasurableSpace Ω := mΩ + have hall : ∀ᵐ ω ∂P, ∀ i, ω ∈ A i → + 0 < P⁻[C.indicator (fun _ => 1) | m] ω := + ae_all_iff.mpr (fun i => conditional_passage_positive_on_certificate + P ν m hm Y hY hlaw (A i) C (hA i) hC (B i) (hB i) (hBpos i) (hcert i)) + filter_upwards [hall] with ω hω hωE + obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hωE) + exact hω i hi + +/-- The actual Brownian stopped trace law and the published geometric +support input give conditional positivity for countably certified geometric +continuations. The same realization also has the actual conformal restart +identity, so the certificates can be checked in the original domain. -/ +theorem stopped_geometric_tube_passages + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (hpublished : PublishedGeometricTubeSupport P γ) : + ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧ + (∀ ω, Φ (sourceDriverPath B hc ω) = + sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ + ∀ (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime + (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))), + (∀ᵐ ω ∂P, ∀ t : ℝ≥0, + Φ (sourceRestartDriverPath B hc τ ω) t ∈ upperHalfPlane → + γ ω (τ ω + t) = F ω (τ ω) + (Φ (sourceRestartDriverPath B hc τ ω) t + + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ + ∀ (E C : Set Ω), MeasurableSet[mΩ] C → + ∀ (A : ℕ → Set Ω), (∀ i, MeasurableSet[hτ.measurableSpace] (A i)) → + ∀ (z : ℕ → ℂ) (p : ∀ i, Path (0 : ℂ) (z i)), + (∀ i, Function.Injective (p i)) → + (∀ i t, t ≠ 0 → p i t ∈ upperHalfPlane) → + ∀ (N V : ℕ → Set ℂ), (∀ i, IsOpen (N i)) → (∀ i, IsOpen (V i)) → + (∀ i, range (p i) ⊆ N i) → (∀ i, z i ∈ V i) → + (∀ i, A i ∩ (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) ⁻¹' + pathTubeEvent (N i) (V i) ⊆ C) → + E ⊆ ⋃ i, A i → + ∀ᵐ ω ∂P, ω ∈ E → + 0 < P⁻[C.indicator (fun _ => 1) | hτ.measurableSpace] ω := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + obtain ⟨Φ, hΦm, hΦ, hlaw⟩ := + sourceStoppedTraceLaw κ hκ hκ8 B hB hm hc γ G F hw + refine ⟨Φ, hΦm, hΦ, ?_⟩ + intro τ hτ + refine ⟨(hlaw τ hτ).2, ?_⟩ + intro E C hC A hA z p hp hpH N V hN hV hpN hzV hcert hcover + let : MeasurableSpace Ω := mΩ + let gp := sourceTracePath γ (fun ω => (hw ω).1) + have hgm : Measurable gp := by + have he : gp = fun ω => Φ (sourceDriverPath B hc ω) := funext fun ω => (hΦ ω).symm + rw [he] + exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm) + have hrm : Measurable (sourceRestartDriverPath B hc τ) := + source_path_measurable _ (fun t => (measurable_pi_apply t).comp + (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) + have htests : ∀ i, 0 < P.map gp (pathTubeEvent (N i) (V i)) := by + intro i + rw [Measure.map_apply hgm (measurableSet_pathTubeEvent (hN i) (hV i))] + exact hpublished (z i) (p i) (hp i) (hpH i) (N i) (V i) + (hN i) (hV i) (hpN i) (hzV i) + exact conditional_passage_positive_of_countable_certificates P (P.map gp) + hτ.measurableSpace hτ.measurableSpace_le + (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) (hΦm.comp hrm) + ((hlaw τ hτ).1) E C hC A hA + (fun i => pathTubeEvent (N i) (V i)) + (fun i => measurableSet_pathTubeEvent (hN i) (hV i)) htests hcert hcover + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StrictPastCountable.lean b/lean/OAI/Probability/SLE/Gauge/StrictPastCountable.lean index 45cb7c5584da70c1df408201580d4efdc3203447..5f983936043e442693f56b674a2f0839a38ef3f2 100644 --- a/lean/OAI/Probability/SLE/Gauge/StrictPastCountable.lean +++ b/lean/OAI/Probability/SLE/Gauge/StrictPastCountable.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Gauge.StrictPastExpectation - -namespace OAI - -/-! Countable exhaustion of the strict past-mass identity. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - -include C hB hc h0 hκ hκ8 F hw - -lemma finite_strict_past_mass_aemeasurable (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) : - AEMeasurable (fun ω => C.rational.terminal ω (K ∩ ⋃ t∈I, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅)) P := by - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - let f (t : ℝ≥0) := (firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) - have hf (t) (ht : t∈I) : AEMeasurable (f t) P := - (C.strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw t K hK).indicator - ((ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht)) - have hm := I.aemeasurable_sum hf - apply hm.congr - filter_upwards [] with ω - simp only [Finset.sum_apply] - symm - simpa only [f,Set.indicator,mem_ofPred_eq] using firstPastSuccess_future_mass - (C.rational.terminal ω) K I E - (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) - (fun s t hst z hz => by - obtain ⟨r,hr,he⟩ := hz - exact ⟨r,⟨hr.1,hr.2.trans hst⟩,he⟩) ω - -lemma countable_strict_past_expectation (J : ℕ → Finset ℝ≥0) (hJ : Monotone J) - (E : ℝ≥0 → Set Ω) - (hE : ∀ t, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) : - (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ n, ⋃ t∈J n, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= - ⨆ n, ∑ t∈J n, ∫⁻ ω in firstPastSuccess (J n) E t, - halfPlaneGreenMeasure κ B γ G t ω K ∂P := by - let A (n : ℕ) (ω : Ω) := K ∩ ⋃ t∈J n, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅ - have hAm (ω) : Monotone (fun n => A n ω) := by - intro n m hnm z hz - obtain ⟨hzK,hz⟩ := hz - obtain ⟨t,ht,hzt⟩ := mem_iUnion₂.mp hz - exact ⟨hzK,mem_iUnion₂.mpr ⟨t,hJ hnm ht,hzt⟩⟩ - have hfm (n) : AEMeasurable (fun ω => C.rational.terminal ω (A n ω)) P := - C.finite_strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw (J n) E (fun t _ => hE t) K hK - calc - _ = ∫⁻ ω, ⨆ n, C.rational.terminal ω (A n ω) ∂P := by - apply lintegral_congr - intro ω - rw [inter_iUnion] - exact (hAm ω).measure_iUnion - _ = ⨆ n, ∫⁻ ω, C.rational.terminal ω (A n ω) ∂P := - lintegral_iSup' hfm (Eventually.of_forall (fun ω n m hnm => measure_mono (hAm ω hnm))) - _ = _ := iSup_congr (fun n => C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw - (J n) E (fun t _ => hE t) K hK) - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Gauge.StrictPastExpectation + +namespace OAI + +/-! Countable exhaustion of the strict past-mass identity. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + +include C hB hc h0 hκ hκ8 F hw + +lemma finite_strict_past_mass_aemeasurable (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) : + AEMeasurable (fun ω => C.rational.terminal ω (K ∩ ⋃ t∈I, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅)) P := by + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + let f (t : ℝ≥0) := (firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) + have hf (t) (ht : t∈I) : AEMeasurable (f t) P := + (C.strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw t K hK).indicator + ((ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht)) + have hm := I.aemeasurable_sum hf + apply hm.congr + filter_upwards [] with ω + simp only [Finset.sum_apply] + symm + simpa only [f,Set.indicator,mem_ofPred_eq] using firstPastSuccess_future_mass + (C.rational.terminal ω) K I E + (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) + (fun s t hst z hz => by + obtain ⟨r,hr,he⟩ := hz + exact ⟨r,⟨hr.1,hr.2.trans hst⟩,he⟩) ω + +lemma countable_strict_past_expectation (J : ℕ → Finset ℝ≥0) (hJ : Monotone J) + (E : ℝ≥0 → Set Ω) + (hE : ∀ t, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) : + (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ n, ⋃ t∈J n, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= + ⨆ n, ∑ t∈J n, ∫⁻ ω in firstPastSuccess (J n) E t, + halfPlaneGreenMeasure κ B γ G t ω K ∂P := by + let A (n : ℕ) (ω : Ω) := K ∩ ⋃ t∈J n, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅ + have hAm (ω) : Monotone (fun n => A n ω) := by + intro n m hnm z hz + obtain ⟨hzK,hz⟩ := hz + obtain ⟨t,ht,hzt⟩ := mem_iUnion₂.mp hz + exact ⟨hzK,mem_iUnion₂.mpr ⟨t,hJ hnm ht,hzt⟩⟩ + have hfm (n) : AEMeasurable (fun ω => C.rational.terminal ω (A n ω)) P := + C.finite_strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw (J n) E (fun t _ => hE t) K hK + calc + _ = ∫⁻ ω, ⨆ n, C.rational.terminal ω (A n ω) ∂P := by + apply lintegral_congr + intro ω + rw [inter_iUnion] + exact (hAm ω).measure_iUnion + _ = ⨆ n, ∫⁻ ω, C.rational.terminal ω (A n ω) ∂P := + lintegral_iSup' hfm (Eventually.of_forall (fun ω n m hnm => measure_mono (hAm ω hnm))) + _ = _ := iSup_congr (fun n => C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw + (J n) E (fun t _ => hE t) K hK) + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StrictPastExpectation.lean b/lean/OAI/Probability/SLE/Gauge/StrictPastExpectation.lean index 554277d93605eb0ef3e7d6f84a486ea46f38a862..aa235f9b74cee59afc42ec6f8b81360feb1d01a2 100644 --- a/lean/OAI/Probability/SLE/Gauge/StrictPastExpectation.lean +++ b/lean/OAI/Probability/SLE/Gauge/StrictPastExpectation.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Geometry.FirstSuccessfulPast - -namespace OAI - -/-! The finite past-mass identity before exhaustion over rational times. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - -include C hB hc h0 hκ hκ8 F hw - -lemma strict_past_mass_aemeasurable (t : ℝ≥0) (K : Set upperHalfPlane) (hK : IsCompact K) : - AEMeasurable (fun ω => C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) P := by - have hNm : Measurable (C.process t) := by - simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] - using C.atTime_measurable (fun _ => (t:ℝ≥0∞)) measurable_const - have hm : Measurable (fun ω => C.rational.terminal ω K-C.process t ω K) := - ((Measure.measurable_coe hK.measurableSet).comp C.rational.terminal_measurable).sub - ((Measure.measurable_coe hK.measurableSet).comp hNm) - apply hm.aemeasurable.congr - have hp : ∀ᵐ ω ∂P, C.process t ω=(C.rational.terminal ω).restrict - {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t} := by - simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] - using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw t - filter_upwards [hp] with ω hω - have hsum : C.process t ω K+C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})=C.rational.terminal ω K := by - rw [hω,Measure.restrict_apply hK.measurableSet] - exact measure_inter_add_sdiff K - (((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet) - let := (C.radon t ω).2 - rw [← hsum,ENNReal.add_sub_cancel_left hK.measure_lt_top.ne] - -lemma finite_strict_past_expectation (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) : - (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= - ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, halfPlaneGreenMeasure κ B γ G t ω K ∂P := by - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - let f (t : ℝ≥0) := (firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) - have hfirst (t) (ht : t∈I) : MeasurableSet (firstPastSuccess I E t) := - (ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht) - have hf (t) (ht : t∈I) : AEMeasurable (f t) P := - (C.strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw t K hK).indicator (hfirst t ht) - calc - _ = ∫⁻ ω, ∑ t∈I, f t ω ∂P := by - apply lintegral_congr - intro ω - simpa only [f,Set.indicator,mem_ofPred_eq] using firstPastSuccess_future_mass - (C.rational.terminal ω) K I E - (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) - (fun s t hst z hz => by - obtain ⟨r,hr,he⟩ := hz - exact ⟨r,⟨hr.1,hr.2.trans hst⟩,he⟩) ω - _ = ∑ t∈I, ∫⁻ ω, f t ω ∂P := lintegral_finsetSum' I hf - _ = _ := by - apply Finset.sum_congr rfl - intro t ht - rw [show f t=(firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) from rfl, - lintegral_indicator (hfirst t ht)] - exact C.strict_past_mass_kernel hB hc h0 hκ hκ8 F hw t _ - (firstPastSuccess_measurable ℱ I E hE ht) K hK - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.FirstSuccessfulPast + +namespace OAI + +/-! The finite past-mass identity before exhaustion over rational times. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + +include C hB hc h0 hκ hκ8 F hw + +lemma strict_past_mass_aemeasurable (t : ℝ≥0) (K : Set upperHalfPlane) (hK : IsCompact K) : + AEMeasurable (fun ω => C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) P := by + have hNm : Measurable (C.process t) := by + simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] + using C.atTime_measurable (fun _ => (t:ℝ≥0∞)) measurable_const + have hm : Measurable (fun ω => C.rational.terminal ω K-C.process t ω K) := + ((Measure.measurable_coe hK.measurableSet).comp C.rational.terminal_measurable).sub + ((Measure.measurable_coe hK.measurableSet).comp hNm) + apply hm.aemeasurable.congr + have hp : ∀ᵐ ω ∂P, C.process t ω=(C.rational.terminal ω).restrict + {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t} := by + simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] + using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw t + filter_upwards [hp] with ω hω + have hsum : C.process t ω K+C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})=C.rational.terminal ω K := by + rw [hω,Measure.restrict_apply hK.measurableSet] + exact measure_inter_add_sdiff K + (((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet) + let := (C.radon t ω).2 + rw [← hsum,ENNReal.add_sub_cancel_left hK.measure_lt_top.ne] + +lemma finite_strict_past_expectation (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) : + (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= + ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, halfPlaneGreenMeasure κ B γ G t ω K ∂P := by + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + let f (t : ℝ≥0) := (firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) + have hfirst (t) (ht : t∈I) : MeasurableSet (firstPastSuccess I E t) := + (ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht) + have hf (t) (ht : t∈I) : AEMeasurable (f t) P := + (C.strict_past_mass_aemeasurable hB hc h0 hκ hκ8 F hw t K hK).indicator (hfirst t ht) + calc + _ = ∫⁻ ω, ∑ t∈I, f t ω ∂P := by + apply lintegral_congr + intro ω + simpa only [f,Set.indicator,mem_ofPred_eq] using firstPastSuccess_future_mass + (C.rational.terminal ω) K I E + (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) + (fun s t hst z hz => by + obtain ⟨r,hr,he⟩ := hz + exact ⟨r,⟨hr.1,hr.2.trans hst⟩,he⟩) ω + _ = ∑ t∈I, ∫⁻ ω, f t ω ∂P := lintegral_finsetSum' I hf + _ = _ := by + apply Finset.sum_congr rfl + intro t ht + rw [show f t=(firstPastSuccess I E t).indicator (fun ω => C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})) from rfl, + lintegral_indicator (hfirst t ht)] + exact C.strict_past_mass_kernel hB hc h0 hκ hκ8 F hw t _ + (firstPastSuccess_measurable ℱ I E hE ht) K hK + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StrictPastSpatial.lean b/lean/OAI/Probability/SLE/Gauge/StrictPastSpatial.lean index 280e67581f6c66e8a159e1937c0f75d1e0d8caa1..83aa72d4366c44527e3dd955db88aa620d16753c 100644 --- a/lean/OAI/Probability/SLE/Gauge/StrictPastSpatial.lean +++ b/lean/OAI/Probability/SLE/Gauge/StrictPastSpatial.lean @@ -1,67 +1,67 @@ -import OAI.Probability.SLE.Gauge.StrictPastExpectation -import OAI.Probability.SLE.Geometry.GreenCompactDecay - -namespace OAI - -/-! Spatial Green-integral versions of the strict past-mass identities. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - -include C hB hc h0 hκ hκ8 F hw - -lemma strict_past_mass_spatial (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) - (K : Set upperHalfPlane) (hK : IsCompact K) : - (∫⁻ ω in E, C.rational.terminal ω - (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P)= - ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, ∫⁻ ω in E, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - rw [C.strict_past_mass_kernel hB hc h0 hκ hκ8 F hw t E hE K hK] - rw [← Measure.bind_apply hK.measurableSet - (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t).aemeasurable] - exact halfPlaneGreenMeasure_mean_apply (P.restrict E) hBm hc κ γ G F hw t K hK.measurableSet - -lemma finite_strict_past_spatial (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) : - (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= - ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, - ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - rw [C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw I E hE K hK] - have hswap (t : ℝ≥0) : - (∫⁻ ω in firstPastSuccess I E t, halfPlaneGreenMeasure κ B γ G t ω K ∂P)= - ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by - rw [← Measure.bind_apply hK.measurableSet - (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t).aemeasurable] - exact halfPlaneGreenMeasure_mean_apply (P.restrict _) hBm hc κ γ G F hw t K hK.measurableSet - simp_rw [hswap] - symm - apply lintegral_finsetSum - intro t _ - exact (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.lintegral_prod_left' - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Gauge.StrictPastExpectation +import OAI.Probability.SLE.Geometry.GreenCompactDecay + +namespace OAI + +/-! Spatial Green-integral versions of the strict past-mass identities. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + +include C hB hc h0 hκ hκ8 F hw + +lemma strict_past_mass_spatial (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) + (K : Set upperHalfPlane) (hK : IsCompact K) : + (∫⁻ ω in E, C.rational.terminal ω + (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P)= + ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, ∫⁻ ω in E, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + rw [C.strict_past_mass_kernel hB hc h0 hκ hκ8 F hw t E hE K hK] + rw [← Measure.bind_apply hK.measurableSet + (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t).aemeasurable] + exact halfPlaneGreenMeasure_mean_apply (P.restrict E) hBm hc κ γ G F hw t K hK.measurableSet + +lemma finite_strict_past_spatial (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) : + (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)= + ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, + ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + rw [C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw I E hE K hK] + have hswap (t : ℝ≥0) : + (∫⁻ ω in firstPastSuccess I E t, halfPlaneGreenMeasure κ B γ G t ω K ∂P)= + ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by + rw [← Measure.bind_apply hK.measurableSet + (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t).aemeasurable] + exact halfPlaneGreenMeasure_mean_apply (P.restrict _) hBm hc κ γ G F hw t K hK.measurableSet + simp_rw [hswap] + symm + apply lintegral_finsetSum + intro t _ + exact (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.lintegral_prod_left' + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/StrictPastTilted.lean b/lean/OAI/Probability/SLE/Gauge/StrictPastTilted.lean index 3a64b04bced85df048c4d34bb195d75a01c9e4be..86e80b69b26090efd9510b66ae59b171a2e0be2b 100644 --- a/lean/OAI/Probability/SLE/Gauge/StrictPastTilted.lean +++ b/lean/OAI/Probability/SLE/Gauge/StrictPastTilted.lean @@ -1,109 +1,109 @@ -import OAI.Probability.SLE.Geometry.FinitePastKernel -import OAI.Probability.SLE.Gauge.StrictPastSpatial - -namespace OAI - -/-! The strict past-mass formula expressed through finite centered laws. -Every spatial integrand is identified before applying monotone convergence. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - -include hB hBm hc h0 hκ hκ8 γ G F hw - -lemma finiteCenteredLaw_tests_spatial_tendsto - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) : - Tendsto (fun n => ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) * - finiteCenteredLaw P κ B γ G z (u n) - (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) atTop - (𝓝 (∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume)) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - have hlevel (n : ℕ) : - (∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) - (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) = - ∑ t∈I, ∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume := by - calc - _ = ∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume := by - apply setLIntegral_congr_fun hK - intro z hz - exact finiteCenteredLaw_tests hB hBm hc h0 hκ hκ8 γ G F hw z (hKH hz) - (u n) (hu n) I E hE - _ = _ := by - apply lintegral_finsetSum - intro t _ - exact (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).ennreal_ofReal.lintegral_prod_left' - have hlast : - (∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume) = - ∑ t∈I, ∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by - apply lintegral_finsetSum - intro t _ - exact (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.lintegral_prod_left' - rw [hlast] - simp_rw [hlevel] - apply tendsto_finsetSum - intro t _ - have hh := currentGreen_spatial_mean_tendsto P hBm hc κ γ G F hw u hu hmono hlim - t K (fun _ => 1) measurable_const (firstPastSuccess I E t) - have hs (n : ℕ) : - (∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume) = - ∫⁻ ω in firstPastSuccess I E t, ∫⁻ z in K, - ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂volume ∂P := - (lintegral_lintegral_swap - (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).ennreal_ofReal.aemeasurable).symm - have hs' : - (∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume) = - ∫⁻ ω in firstPastSuccess I E t, ∫⁻ z in K, - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume ∂P := - (lintegral_lintegral_swap - (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.aemeasurable).symm - simpa only [one_mul, hs, hs'] using hh - -namespace CanonicalMass - -lemma finite_strict_past_tilted_limit - (C : CanonicalMass P κ B hBm γ G) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) : - Tendsto (fun n => ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, - ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) - (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) atTop - (𝓝 (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)) := by - rw [C.finite_strict_past_spatial hB hc h0 hκ hκ8 F hw I E hE K hK] - apply finiteCenteredLaw_tests_spatial_tendsto hB hBm hc h0 hκ hκ8 γ G F hw u hu hmono hlim I E hE - _ (hK.image continuous_subtype_val).measurableSet - rintro _ ⟨z,_,rfl⟩ - exact z.property - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.FinitePastKernel +import OAI.Probability.SLE.Gauge.StrictPastSpatial + +namespace OAI + +/-! The strict past-mass formula expressed through finite centered laws. +Every spatial integrand is identified before applying monotone convergence. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + +include hB hBm hc h0 hκ hκ8 γ G F hw + +lemma finiteCenteredLaw_tests_spatial_tendsto + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) : + Tendsto (fun n => ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) * + finiteCenteredLaw P κ B γ G z (u n) + (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) atTop + (𝓝 (∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume)) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + have hlevel (n : ℕ) : + (∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) + (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) = + ∑ t∈I, ∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume := by + calc + _ = ∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume := by + apply setLIntegral_congr_fun hK + intro z hz + exact finiteCenteredLaw_tests hB hBm hc h0 hκ hκ8 γ G F hw z (hKH hz) + (u n) (hu n) I E hE + _ = _ := by + apply lintegral_finsetSum + intro t _ + exact (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).ennreal_ofReal.lintegral_prod_left' + have hlast : + (∫⁻ z in K, ∑ t∈I, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume) = + ∑ t∈I, ∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume := by + apply lintegral_finsetSum + intro t _ + exact (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.lintegral_prod_left' + rw [hlast] + simp_rw [hlevel] + apply tendsto_finsetSum + intro t _ + have hh := currentGreen_spatial_mean_tendsto P hBm hc κ γ G F hw u hu hmono hlim + t K (fun _ => 1) measurable_const (firstPastSuccess I E t) + have hs (n : ℕ) : + (∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂P ∂volume) = + ∫⁻ ω in firstPastSuccess I E t, ∫⁻ z in K, + ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂volume ∂P := + (lintegral_lintegral_swap + (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).ennreal_ofReal.aemeasurable).symm + have hs' : + (∫⁻ z in K, ∫⁻ ω in firstPastSuccess I E t, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P ∂volume) = + ∫⁻ ω in firstPastSuccess I E t, ∫⁻ z in K, + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume ∂P := + (lintegral_lintegral_swap + (actual_greenDensity_joint_measurable hBm hc κ γ G F hw t).ennreal_ofReal.aemeasurable).symm + simpa only [one_mul, hs, hs'] using hh + +namespace CanonicalMass + +lemma finite_strict_past_tilted_limit + (C : CanonicalMass P κ B hBm γ G) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) : + Tendsto (fun n => ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, + ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) + (finiteStrictPastEvent I E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) atTop + (𝓝 (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ t∈I, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P)) := by + rw [C.finite_strict_past_spatial hB hc h0 hκ hκ8 F hw I E hE K hK] + apply finiteCenteredLaw_tests_spatial_tendsto hB hBm hc h0 hκ hκ8 γ G F hw u hu hmono hlim I E hE + _ (hK.image continuous_subtype_val).measurableSet + rintro _ ⟨z,_,rfl⟩ + exact z.property + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/UniformActualRibbonPassage.lean b/lean/OAI/Probability/SLE/Gauge/UniformActualRibbonPassage.lean index eacee98da87c4ba61273622d37457104cdf1ec49..a0df2cd30f7405bef3401f88a86753f6762596ed 100644 --- a/lean/OAI/Probability/SLE/Gauge/UniformActualRibbonPassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/UniformActualRibbonPassage.lean @@ -1,191 +1,191 @@ -import OAI.Probability.SLE.Gauge.UniformRecenteredPassage -import OAI.Probability.SLE.Geometry.ActualWaypointAngle -import OAI.Probability.SLE.Gauge.RibbonStepPatch - -namespace OAI - -/-! Uniform one-step passage for actual ribbon histories. The two-sided -waypoint angle and the compactness hypotheses are derived from the fixed -patch bounds, the actual first-front past, and the published analytic and -probabilistic inputs. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates BoundaryScaleLaw Target - -/-- Data at an actual attained ribbon front, before any restart. -/ -structure RibbonHistory (C L : ℝ≥0) (ε : ℝ) where - U : ℝ≥0 → ℝ - trace : ℝ≥0 → ℂ - forward : ℝ≥0 → ℂ → ℂ - inverse : ℝ≥0 → ℂ → ℂ - driver_continuous : Continuous U - witness : IsCapacityTwoWitness U trace forward inverse - chart : ℂ → ℂ - chart_continuous : ContinuousOn chart closedRibbon - chart_injective : InjOn chart closedRibbon - chart_upper : ∀ z ∈ closedRibbon, 0 < z.re → chart z ∈ upperHalfPlane - chart_escape : Tendsto (fun t : ℝ => ‖chart (t : ℂ)‖) atTop atTop - front : ℝ - offset : ℝ - front_ge_one : 1 ≤ front - offset_bound : |offset| ≤ ε / 4 - patch_upper : LipschitzOnWith C chart (ribbonStepPatch front) - patch_lower : ∀ u ∈ ribbonStepPatch front, ∀ v ∈ ribbonStepPatch front, - dist u v ≤ L * dist (chart u) (chart v) - time : ℝ≥0 - time_pos : 0 < time - coordinates : ℝ≥0 → ℂ - coordinates_continuous : ContinuousOn coordinates (Icc 0 time) - trace_coordinates : EqOn trace (chart ∘ coordinates) (Icc 0 time) - initial_front : (coordinates 0).re = 0 - attained : coordinates time = (front : ℂ) + (offset : ℂ) * Complex.I - coordinates_ribbon : MapsTo coordinates (Icc 0 time) closedRibbon - past_core : ∀ t ∈ Icc 0 time, (coordinates t).re ≤ front ∧ |(coordinates t).im| ≤ ε - strict_past : ∀ t : ℝ≥0, t < time → (coordinates t).re < front - -def RibbonHistory.guide {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (t : ℝ) : ℂ := - d.chart (guideCoordinate d.front d.offset t) - -def RibbonHistory.recenteredChart {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (z : ℂ) : ℂ := - d.chart ((d.front : ℂ) + z) - d.trace d.time - -theorem uniform_actual_ribbon_passage - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) - (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (Gbase ω) (Fbase ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ Fb : RibbonHistory C L ε → C(closedHalfPlane, ℂ), - (∀ d (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb d z = - d.inverse d.time ((d.U d.time : ℂ) + z) - d.trace d.time) ∧ - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ d : RibbonHistory C L ε, ∀ v : ℝ, 0 < v → v < v₀ → - ENNReal.ofReal q ≤ P {ω | - finiteBoundaryPath (Fb d) (sourceTracePath γ (fun ω => (hwbase ω).1) ω) ∈ - frontMassEvent (longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) - (physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) - (d.recenteredChart 1) (ε / L) κ v c - (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by - have hLr : 0 < (L : ℝ) := hL - have hx (d : RibbonHistory C L ε) : |d.offset| ≤ 1 / 4 := by - linarith [d.offset_bound] - have hguideL (d : RibbonHistory C L ε) : - LipschitzOnWith C d.chart (guidePatch d.front) := - d.patch_upper.mono (guidePatch_subset_ribbonStepPatch _) - have hguideLower (d : RibbonHistory C L ε) : - ∀ u ∈ guidePatch d.front, ∀ v ∈ guidePatch d.front, - dist u v ≤ L * dist (d.chart u) (d.chart v) := by - intro u hu v hv - exact d.patch_lower u (guidePatch_subset_ribbonStepPatch _ hu) - v (guidePatch_subset_ribbonStepPatch _ hv) - have hgeom (d : RibbonHistory C L ε) := ribbon_access_guide_geometry hplanar - d.chart d.chart_continuous d.chart_injective d.chart_upper d.chart_escape - d.trace d.coordinates d.time d.coordinates_continuous d.initial_front - d.coordinates_ribbon d.trace_coordinates d.front d.offset - (by linarith [d.front_ge_one]) (hx d) d.attained d.strict_past C L hLr - (hguideL d) (hguideLower d) - have hpast (d : RibbonHistory C L ε) : - d.trace '' Icc 0 d.time ⊆ d.chart '' {z ∈ closedRibbon | z.re ≤ d.front} := by - rintro z ⟨t, ht, rfl⟩ - exact ⟨d.coordinates t, ⟨d.coordinates_ribbon ht, (d.past_core t ht).1⟩, - (d.trace_coordinates ht).symm⟩ - have hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ (d : RibbonHistory C L ε) t, t ∈ Icc δ 2 → - ball (d.guide t) r ⊆ survivingDomain d.trace d.time := by - intro δ hδ _ - refine ⟨min δ 1 / (4 * L), div_pos (lt_min hδ zero_lt_one) (by positivity), ?_⟩ - intro d t ht - apply (ball_subset_ball ?_).trans - (physical_guide_clearance_of_patch hplanar d.chart d.chart_continuous d.chart_injective - (by linarith [d.front_ge_one]) (hx d) hLr d.chart_upper d.trace d.time (hpast d) - (futureRibbon_unbounded_of_centerline_escape d.chart d.front d.chart_escape) - (hguideLower d) t ⟨hδ.trans_le ht.1, ht.2⟩) - exact div_le_div_of_nonneg_right (min_le_min_right 1 ht.1) (by positivity) - obtain ⟨s, hs, hangle⟩ := uniform_actual_ribbon_waypoint_angle hplanar hstraight hexit - hwiener W hW C L hLr - have hang (d : RibbonHistory C L ε) : - s ≤ (d.forward d.time (d.guide (1 / 2)) - (d.U d.time : ℂ)).im / - ‖d.forward d.time (d.guide (1 / 2)) - (d.U d.time : ℂ)‖ := by - have hh := hangle d.U d.trace d.forward d.inverse d.witness d.driver_continuous - d.chart d.chart_continuous d.chart_injective d.chart_upper d.chart_escape - d.front ε d.offset d.front_ge_one hε hεsmall (by linarith [d.offset_bound]) - (d.patch_upper.mono (crossingPatch_subset_ribbonStepPatch _)) - (fun u hu v hv => d.patch_lower u (crossingPatch_subset_ribbonStepPatch _ hu) - v (crossingPatch_subset_ribbonStepPatch _ hv)) - d.coordinates d.time d.time_pos d.coordinates_continuous d.trace_coordinates - d.initial_front d.attained d.coordinates_ribbon d.past_core d.strict_past - simpa only [RibbonHistory.guide, guideCoordinate_waypoint] using hh - have hQU (d : RibbonHistory C L ε) : - LipschitzOnWith C d.recenteredChart ribbonStepCompact := by - apply lipschitzOnWith_iff_dist_le_mul.mpr - intro u hu v hv - simpa only [RibbonHistory.recenteredChart, dist_sub_right, dist_add_left] using - d.patch_upper.dist_le_mul _ (translate_ribbonStepCompact hu) _ - (translate_ribbonStepCompact hv) - have hQL (d : RibbonHistory C L ε) : - ∀ u ∈ ribbonStepCompact, ∀ v ∈ ribbonStepCompact, - dist u v ≤ L * dist (d.recenteredChart u) (d.recenteredChart v) := by - intro u hu v hv - simpa only [RibbonHistory.recenteredChart, dist_sub_right, dist_add_left] using - d.patch_lower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hv) - have hQ0 (d : RibbonHistory C L ε) : ‖d.recenteredChart 0‖ ≤ C := by - have htip : d.chart ((d.front : ℂ) + (d.offset : ℂ) * Complex.I) = d.trace d.time := by - rw [← d.attained] - exact (d.trace_coordinates ⟨bot_le, le_rfl⟩).symm - have hmem : (d.front : ℂ) + (d.offset : ℂ) * Complex.I ∈ ribbonStepPatch d.front := by - change d.front - 1 ≤ ((d.front : ℂ) + (d.offset : ℂ) * Complex.I).re ∧ - ((d.front : ℂ) + (d.offset : ℂ) * Complex.I).re ≤ d.front + 3 ∧ - |((d.front : ℂ) + (d.offset : ℂ) * Complex.I).im| ≤ 1 - simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, - Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero, - Complex.add_im, Complex.mul_im, zero_add] - exact ⟨by linarith, by linarith, (hx d).trans (by norm_num)⟩ - have hzero : (d.front : ℂ) ∈ ribbonStepPatch d.front := by - change d.front - 1 ≤ d.front ∧ d.front ≤ d.front + 3 ∧ |(0 : ℝ)| ≤ 1 - constructor - · linarith - · constructor <;> norm_num - have hd := d.patch_upper.dist_le_mul _ hzero _ hmem - have hdist : dist (d.front : ℂ) ((d.front : ℂ) + (d.offset : ℂ) * Complex.I) = - |d.offset| := by - rw [dist_self_add_right, norm_mul, Complex.norm_I, mul_one, Complex.norm_real, - Real.norm_eq_abs] - rw [hdist, htip, dist_eq_norm] at hd - simpa only [RibbonHistory.recenteredChart, add_zero] using - hd.trans (mul_le_of_le_one_right C.coe_nonneg ((hx d).trans (by norm_num))) - apply uniform_recentered_passage_from_geometry P κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase - hwbase hcompact hboundary hsupport hplanar - RibbonHistory.U RibbonHistory.trace RibbonHistory.forward RibbonHistory.inverse - RibbonHistory.driver_continuous RibbonHistory.witness RibbonHistory.time RibbonHistory.time_pos - (fun d => (hgeom d).2.2.2.2.1) RibbonHistory.guide (2 * C) s (1 / L) - (by positivity) hs (one_div_pos.mpr hLr) - (fun d => (hgeom d).2.2.2.2.2.1) (fun d => (hgeom d).2.2.1) - (fun d => (hgeom d).2.2.2.2.2.2.1) (fun d => (hgeom d).2.2.2.1) - hclear RibbonHistory.recenteredChart ribbonStepCompact ribbonStepCompact_isCompact - 0 ribbonStepCompact_zero C L C hC hL hQU hQL hQ0 RibbonHistory.offset ε hε - RibbonHistory.offset_bound (ribbonStepCompact_core hεsmall) - (fun d t _ => by simp only [RibbonHistory.recenteredChart, RibbonHistory.guide, - guideCoordinate_translate]) hang (ribbonStepCompact_passage hεsmall C (ε / L)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.UniformRecenteredPassage +import OAI.Probability.SLE.Geometry.ActualWaypointAngle +import OAI.Probability.SLE.Gauge.RibbonStepPatch + +namespace OAI + +/-! Uniform one-step passage for actual ribbon histories. The two-sided +waypoint angle and the compactness hypotheses are derived from the fixed +patch bounds, the actual first-front past, and the published analytic and +probabilistic inputs. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates BoundaryScaleLaw Target + +/-- Data at an actual attained ribbon front, before any restart. -/ +structure RibbonHistory (C L : ℝ≥0) (ε : ℝ) where + U : ℝ≥0 → ℝ + trace : ℝ≥0 → ℂ + forward : ℝ≥0 → ℂ → ℂ + inverse : ℝ≥0 → ℂ → ℂ + driver_continuous : Continuous U + witness : IsCapacityTwoWitness U trace forward inverse + chart : ℂ → ℂ + chart_continuous : ContinuousOn chart closedRibbon + chart_injective : InjOn chart closedRibbon + chart_upper : ∀ z ∈ closedRibbon, 0 < z.re → chart z ∈ upperHalfPlane + chart_escape : Tendsto (fun t : ℝ => ‖chart (t : ℂ)‖) atTop atTop + front : ℝ + offset : ℝ + front_ge_one : 1 ≤ front + offset_bound : |offset| ≤ ε / 4 + patch_upper : LipschitzOnWith C chart (ribbonStepPatch front) + patch_lower : ∀ u ∈ ribbonStepPatch front, ∀ v ∈ ribbonStepPatch front, + dist u v ≤ L * dist (chart u) (chart v) + time : ℝ≥0 + time_pos : 0 < time + coordinates : ℝ≥0 → ℂ + coordinates_continuous : ContinuousOn coordinates (Icc 0 time) + trace_coordinates : EqOn trace (chart ∘ coordinates) (Icc 0 time) + initial_front : (coordinates 0).re = 0 + attained : coordinates time = (front : ℂ) + (offset : ℂ) * Complex.I + coordinates_ribbon : MapsTo coordinates (Icc 0 time) closedRibbon + past_core : ∀ t ∈ Icc 0 time, (coordinates t).re ≤ front ∧ |(coordinates t).im| ≤ ε + strict_past : ∀ t : ℝ≥0, t < time → (coordinates t).re < front + +def RibbonHistory.guide {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (t : ℝ) : ℂ := + d.chart (guideCoordinate d.front d.offset t) + +def RibbonHistory.recenteredChart {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (z : ℂ) : ℂ := + d.chart ((d.front : ℂ) + z) - d.trace d.time + +theorem uniform_actual_ribbon_passage + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) + (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (Gbase ω) (Fbase ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ Fb : RibbonHistory C L ε → C(closedHalfPlane, ℂ), + (∀ d (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb d z = + d.inverse d.time ((d.U d.time : ℂ) + z) - d.trace d.time) ∧ + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ d : RibbonHistory C L ε, ∀ v : ℝ, 0 < v → v < v₀ → + ENNReal.ofReal q ≤ P {ω | + finiteBoundaryPath (Fb d) (sourceTracePath γ (fun ω => (hwbase ω).1) ω) ∈ + frontMassEvent (longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) + (physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) + (d.recenteredChart 1) (ε / L) κ v c + (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by + have hLr : 0 < (L : ℝ) := hL + have hx (d : RibbonHistory C L ε) : |d.offset| ≤ 1 / 4 := by + linarith [d.offset_bound] + have hguideL (d : RibbonHistory C L ε) : + LipschitzOnWith C d.chart (guidePatch d.front) := + d.patch_upper.mono (guidePatch_subset_ribbonStepPatch _) + have hguideLower (d : RibbonHistory C L ε) : + ∀ u ∈ guidePatch d.front, ∀ v ∈ guidePatch d.front, + dist u v ≤ L * dist (d.chart u) (d.chart v) := by + intro u hu v hv + exact d.patch_lower u (guidePatch_subset_ribbonStepPatch _ hu) + v (guidePatch_subset_ribbonStepPatch _ hv) + have hgeom (d : RibbonHistory C L ε) := ribbon_access_guide_geometry hplanar + d.chart d.chart_continuous d.chart_injective d.chart_upper d.chart_escape + d.trace d.coordinates d.time d.coordinates_continuous d.initial_front + d.coordinates_ribbon d.trace_coordinates d.front d.offset + (by linarith [d.front_ge_one]) (hx d) d.attained d.strict_past C L hLr + (hguideL d) (hguideLower d) + have hpast (d : RibbonHistory C L ε) : + d.trace '' Icc 0 d.time ⊆ d.chart '' {z ∈ closedRibbon | z.re ≤ d.front} := by + rintro z ⟨t, ht, rfl⟩ + exact ⟨d.coordinates t, ⟨d.coordinates_ribbon ht, (d.past_core t ht).1⟩, + (d.trace_coordinates ht).symm⟩ + have hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ (d : RibbonHistory C L ε) t, t ∈ Icc δ 2 → + ball (d.guide t) r ⊆ survivingDomain d.trace d.time := by + intro δ hδ _ + refine ⟨min δ 1 / (4 * L), div_pos (lt_min hδ zero_lt_one) (by positivity), ?_⟩ + intro d t ht + apply (ball_subset_ball ?_).trans + (physical_guide_clearance_of_patch hplanar d.chart d.chart_continuous d.chart_injective + (by linarith [d.front_ge_one]) (hx d) hLr d.chart_upper d.trace d.time (hpast d) + (futureRibbon_unbounded_of_centerline_escape d.chart d.front d.chart_escape) + (hguideLower d) t ⟨hδ.trans_le ht.1, ht.2⟩) + exact div_le_div_of_nonneg_right (min_le_min_right 1 ht.1) (by positivity) + obtain ⟨s, hs, hangle⟩ := uniform_actual_ribbon_waypoint_angle hplanar hstraight hexit + hwiener W hW C L hLr + have hang (d : RibbonHistory C L ε) : + s ≤ (d.forward d.time (d.guide (1 / 2)) - (d.U d.time : ℂ)).im / + ‖d.forward d.time (d.guide (1 / 2)) - (d.U d.time : ℂ)‖ := by + have hh := hangle d.U d.trace d.forward d.inverse d.witness d.driver_continuous + d.chart d.chart_continuous d.chart_injective d.chart_upper d.chart_escape + d.front ε d.offset d.front_ge_one hε hεsmall (by linarith [d.offset_bound]) + (d.patch_upper.mono (crossingPatch_subset_ribbonStepPatch _)) + (fun u hu v hv => d.patch_lower u (crossingPatch_subset_ribbonStepPatch _ hu) + v (crossingPatch_subset_ribbonStepPatch _ hv)) + d.coordinates d.time d.time_pos d.coordinates_continuous d.trace_coordinates + d.initial_front d.attained d.coordinates_ribbon d.past_core d.strict_past + simpa only [RibbonHistory.guide, guideCoordinate_waypoint] using hh + have hQU (d : RibbonHistory C L ε) : + LipschitzOnWith C d.recenteredChart ribbonStepCompact := by + apply lipschitzOnWith_iff_dist_le_mul.mpr + intro u hu v hv + simpa only [RibbonHistory.recenteredChart, dist_sub_right, dist_add_left] using + d.patch_upper.dist_le_mul _ (translate_ribbonStepCompact hu) _ + (translate_ribbonStepCompact hv) + have hQL (d : RibbonHistory C L ε) : + ∀ u ∈ ribbonStepCompact, ∀ v ∈ ribbonStepCompact, + dist u v ≤ L * dist (d.recenteredChart u) (d.recenteredChart v) := by + intro u hu v hv + simpa only [RibbonHistory.recenteredChart, dist_sub_right, dist_add_left] using + d.patch_lower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hv) + have hQ0 (d : RibbonHistory C L ε) : ‖d.recenteredChart 0‖ ≤ C := by + have htip : d.chart ((d.front : ℂ) + (d.offset : ℂ) * Complex.I) = d.trace d.time := by + rw [← d.attained] + exact (d.trace_coordinates ⟨bot_le, le_rfl⟩).symm + have hmem : (d.front : ℂ) + (d.offset : ℂ) * Complex.I ∈ ribbonStepPatch d.front := by + change d.front - 1 ≤ ((d.front : ℂ) + (d.offset : ℂ) * Complex.I).re ∧ + ((d.front : ℂ) + (d.offset : ℂ) * Complex.I).re ≤ d.front + 3 ∧ + |((d.front : ℂ) + (d.offset : ℂ) * Complex.I).im| ≤ 1 + simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, + Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero, + Complex.add_im, Complex.mul_im, zero_add] + exact ⟨by linarith, by linarith, (hx d).trans (by norm_num)⟩ + have hzero : (d.front : ℂ) ∈ ribbonStepPatch d.front := by + change d.front - 1 ≤ d.front ∧ d.front ≤ d.front + 3 ∧ |(0 : ℝ)| ≤ 1 + constructor + · linarith + · constructor <;> norm_num + have hd := d.patch_upper.dist_le_mul _ hzero _ hmem + have hdist : dist (d.front : ℂ) ((d.front : ℂ) + (d.offset : ℂ) * Complex.I) = + |d.offset| := by + rw [dist_self_add_right, norm_mul, Complex.norm_I, mul_one, Complex.norm_real, + Real.norm_eq_abs] + rw [hdist, htip, dist_eq_norm] at hd + simpa only [RibbonHistory.recenteredChart, add_zero] using + hd.trans (mul_le_of_le_one_right C.coe_nonneg ((hx d).trans (by norm_num))) + apply uniform_recentered_passage_from_geometry P κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase + hwbase hcompact hboundary hsupport hplanar + RibbonHistory.U RibbonHistory.trace RibbonHistory.forward RibbonHistory.inverse + RibbonHistory.driver_continuous RibbonHistory.witness RibbonHistory.time RibbonHistory.time_pos + (fun d => (hgeom d).2.2.2.2.1) RibbonHistory.guide (2 * C) s (1 / L) + (by positivity) hs (one_div_pos.mpr hLr) + (fun d => (hgeom d).2.2.2.2.2.1) (fun d => (hgeom d).2.2.1) + (fun d => (hgeom d).2.2.2.2.2.2.1) (fun d => (hgeom d).2.2.2.1) + hclear RibbonHistory.recenteredChart ribbonStepCompact ribbonStepCompact_isCompact + 0 ribbonStepCompact_zero C L C hC hL hQU hQL hQ0 RibbonHistory.offset ε hε + RibbonHistory.offset_bound (ribbonStepCompact_core hεsmall) + (fun d t _ => by simp only [RibbonHistory.recenteredChart, RibbonHistory.guide, + guideCoordinate_translate]) hang (ribbonStepCompact_passage hεsmall C (ε / L)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/UniformFrontPassage.lean b/lean/OAI/Probability/SLE/Gauge/UniformFrontPassage.lean index c89730d3507b679c5d14d468433505685e8c9857..9b8f2dfe252807ac5ebf0eb753529c9223241259 100644 --- a/lean/OAI/Probability/SLE/Gauge/UniformFrontPassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/UniformFrontPassage.lean @@ -1,171 +1,171 @@ -import OAI.Probability.SLE.Geometry.FrontSuccessStopped -import OAI.Probability.SLE.Gauge.UniformPassageThreshold - -namespace OAI - -/-! The uniform quantifier step for actual front passages. A compactness -argument need only produce an eventual positive success probability along -one strictly increasing subsequence of each vanishing-mesh sequence. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -/-- The diagonal obstruction to a uniform passage estimate persists along -all subsequences, so compactness need only control one of them. -/ -theorem uniform_passage_threshold_of_subsequences - {A : Type*} (p : A → ℝ → ℝ → ℝ≥0∞) - (hmono : ∀ a v, Antitone (p a v)) - (hsequence : ∀ (a : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → - Tendsto v atTop (𝓝 0) → - ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ c : ℝ, 0 < c ∧ ∃ q : ℝ≥0∞, 0 < q ∧ - ∀ᶠ n in atTop, q ≤ p (a (φ n)) (v (φ n)) c) : - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ a : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ p a v c := by - classical - by_contra h - push Not at h - let ε : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) - have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity - have hfail (n : ℕ) : ∃ a : A, ∃ v : ℝ, - 0 < v ∧ v < ε n ∧ p a v (ε n) < ENNReal.ofReal (ε n) := by - obtain ⟨a, v, hv, hvs, hp⟩ := h (ε n) (ε n) (ε n) (hε n) (hε n) (hε n) - exact ⟨a, v, hv, hvs, hp⟩ - choose a v hv hvs hbad using hfail - have hεlim : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat - have hvlim : Tendsto v atTop (𝓝 0) := - squeeze_zero (fun n => (hv n).le) (fun n => (hvs n).le) hεlim - obtain ⟨φ, hφ, c, hc, q, hq, hlower⟩ := hsequence a v hv hvlim - have hεsub : Tendsto (fun n => ε (φ n)) atTop (𝓝 0) := hεlim.comp hφ.tendsto_atTop - have hsmall : ∀ᶠ n in atTop, ε (φ n) ≤ c := - (hεsub.eventually (gt_mem_nhds hc)).mono fun _ hn => hn.le - have hupper : ∀ᶠ n in atTop, q ≤ ENNReal.ofReal (ε (φ n)) := by - filter_upwards [hlower, hsmall] with n hn hs - exact (hn.trans (hmono (a (φ n)) (v (φ n)) hs)).trans (hbad (φ n)).le - have hlim : Tendsto (fun n => ENNReal.ofReal (ε (φ n))) atTop (𝓝 0) := by - simpa only [Function.comp_def, ENNReal.ofReal_zero] using - (ENNReal.continuous_ofReal.tendsto 0).comp hεsub - exact (not_le_of_gt hq) (ge_of_tendsto hlim hupper) - -namespace FirstFrontPassage - -/-- Before an attained first hit, a continuous coordinate that started -below the front remains strictly below it. -/ -theorem lt_before_firstFrontTime - (x : ℝ≥0 → ℝ) (q : ℝ) (τ : ℝ≥0) - (hτ : firstFrontTime x q = (τ : WithTop ℝ≥0)) - (hc : ContinuousOn x (Icc 0 τ)) (hstart : x 0 < q) - (t : ℝ≥0) (ht : t < τ) : x t < q := by - by_contra hn - obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le) - (hc.mono (Icc_subset_Icc_right ht.le)) ⟨hstart.le, le_of_not_gt hn⟩ - have hh := hittingAfter_le_of_mem (u := fun s (_ : Unit) => x s - q) - (ω := ()) hu.1 (show x u - q ∈ ({0} : Set ℝ) by simp only [hue, sub_self, mem_singleton_iff]) - change firstFrontTime x q ≤ (u : WithTop ℝ≥0) at hh - rw [hτ, WithTop.coe_le_coe] at hh - exact (not_le_of_gt ht) (hh.trans hu.2) - -end FirstFrontPassage - -namespace PassageSuccess -open FirstFrontPassage - -/-- The open-envelope success event gives exactly the strict pre-front -containment used by the manuscript. The front point itself is specified -separately by its attained coordinate and terminal offset. -/ -theorem frontSuccess_source_prefix - {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p : ℕ → ℂ → ℝ) - (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (a : ℕ → ℂ) (ε : ℝ) (n : ℕ) - (ω : Ω) (hω : ω ∈ FrontSuccess η p σ V a ε n) - (hc : ContinuousOn (fun t => p n (η n ω t)) - (Icc 0 (frontTime (fun t => p n (η n ω t)) 1))) - (hstart : p n (η n ω 0) < 1) : - MapsTo (η n ω) (Ico 0 (frontTime (fun t => p n (η n ω t)) 1)) - (V n ∩ {z | p n z < 1}) ∧ - p n (η n ω (frontTime (fun t => p n (η n ω t)) 1)) = 1 ∧ - ‖η n ω (frontTime (fun t => p n (η n ω t)) 1) - a n‖ < ε / 4 := by - obtain ⟨htime, hhit, _, hpath, hterminal⟩ := hω - refine ⟨?_, hhit, hterminal⟩ - intro t ht - exact ⟨hpath ⟨ht.1, ht.2.le⟩, - lt_before_firstFrontTime _ 1 _ htime hc hstart t ht.2⟩ - -/-- Probability of the concrete full-prefix front success and its physical -sausage mass. The index records an admissible physical configuration. -/ -def frontPassageProbability - {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) - (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) - (d : A) (v c : ℝ) : ℝ≥0∞ := - P {ω | ω ∈ FrontSuccess (fun _ => η d) (fun _ => p d) (σ d) - (fun _ => V d) (fun _ => a d) (ε d) 0 ∧ ENNReal.ofReal c ≤ - segmentSausageCharge κ (η d ω) (σ d ω) - (frontTime (fun t => p d (η d ω t)) 1) v (R d)} - -/-- Increasing the required mass only decreases the concrete passage -probability; no probability or measurability assumption is needed here. -/ -theorem frontPassageProbability_antitone - {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) - (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) - (d : A) (v : ℝ) : Antitone (frontPassageProbability P η p σ V a ε κ R d v) := by - intro c₁ c₂ hc - apply measure_mono - intro ω hω - exact ⟨hω.1, (ENNReal.ofReal_le_ofReal hc).trans hω.2⟩ - -/-- A later retained starting point gives a subset of the full passage -success-and-mass event. This removes the auxiliary first-height stop from -the final one-step conclusion. -/ -theorem frontPassageProbability_mono_start - {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ₁ σ₂ : A → Ω → ℝ≥0) - (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) - (d : A) (v c : ℝ) (hσ : ∀ ω, σ₁ d ω ≤ σ₂ d ω) : - frontPassageProbability P η p σ₂ V a ε κ R d v c ≤ - frontPassageProbability P η p σ₁ V a ε κ R d v c := by - apply measure_mono - rintro ω ⟨hfront, hmass⟩ - obtain ⟨hfinite, hhit, hbefore, hpath, hterminal⟩ := hfront - refine ⟨⟨hfinite, hhit, (hσ ω).trans_lt hbefore, hpath, hterminal⟩, ?_⟩ - exact hmass.trans (segmentSausageCharge_mono κ (η d ω) (hσ ω) le_rfl v Subset.rfl) - -theorem frontPassageProbability_le_from_zero - {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) - (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) - (d : A) (v c : ℝ) : - frontPassageProbability P η p σ V a ε κ R d v c ≤ - frontPassageProbability P η p (fun _ _ => 0) V a ε κ R d v c := - frontPassageProbability_mono_start P η p (fun _ _ => 0) σ V a ε κ R d v c (fun _ => bot_le) - -/-- Once actual conformal compactness and the sequence passage theorem -supply the subsequential estimate, the full uniform passage constants -follow for the original concrete success-and-mass event. -/ -theorem uniform_frontPassage_probability_of_subsequences - {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) - (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) - (hsequence : ∀ (d : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → - Tendsto v atTop (𝓝 0) → - ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ - ∀ᶠ n in atTop, ENNReal.ofReal q ≤ - frontPassageProbability P η p σ V a ε κ R (d (φ n)) (v (φ n)) c) : - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ d : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ - frontPassageProbability P η p σ V a ε κ R d v c := by - apply uniform_passage_threshold_of_subsequences - (frontPassageProbability P η p σ V a ε κ R) - (frontPassageProbability_antitone P η p σ V a ε κ R) - intro d v hv hlim - obtain ⟨φ, hφ, c, q, hc, hq, hlower⟩ := hsequence d v hv hlim - exact ⟨φ, hφ, c, hc, ENNReal.ofReal q, ENNReal.ofReal_pos.mpr hq, hlower⟩ - -end PassageSuccess -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FrontSuccessStopped +import OAI.Probability.SLE.Gauge.UniformPassageThreshold + +namespace OAI + +/-! The uniform quantifier step for actual front passages. A compactness +argument need only produce an eventual positive success probability along +one strictly increasing subsequence of each vanishing-mesh sequence. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +/-- The diagonal obstruction to a uniform passage estimate persists along +all subsequences, so compactness need only control one of them. -/ +theorem uniform_passage_threshold_of_subsequences + {A : Type*} (p : A → ℝ → ℝ → ℝ≥0∞) + (hmono : ∀ a v, Antitone (p a v)) + (hsequence : ∀ (a : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → + Tendsto v atTop (𝓝 0) → + ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ c : ℝ, 0 < c ∧ ∃ q : ℝ≥0∞, 0 < q ∧ + ∀ᶠ n in atTop, q ≤ p (a (φ n)) (v (φ n)) c) : + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ a : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ p a v c := by + classical + by_contra h + push Not at h + let ε : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) + have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity + have hfail (n : ℕ) : ∃ a : A, ∃ v : ℝ, + 0 < v ∧ v < ε n ∧ p a v (ε n) < ENNReal.ofReal (ε n) := by + obtain ⟨a, v, hv, hvs, hp⟩ := h (ε n) (ε n) (ε n) (hε n) (hε n) (hε n) + exact ⟨a, v, hv, hvs, hp⟩ + choose a v hv hvs hbad using hfail + have hεlim : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat + have hvlim : Tendsto v atTop (𝓝 0) := + squeeze_zero (fun n => (hv n).le) (fun n => (hvs n).le) hεlim + obtain ⟨φ, hφ, c, hc, q, hq, hlower⟩ := hsequence a v hv hvlim + have hεsub : Tendsto (fun n => ε (φ n)) atTop (𝓝 0) := hεlim.comp hφ.tendsto_atTop + have hsmall : ∀ᶠ n in atTop, ε (φ n) ≤ c := + (hεsub.eventually (gt_mem_nhds hc)).mono fun _ hn => hn.le + have hupper : ∀ᶠ n in atTop, q ≤ ENNReal.ofReal (ε (φ n)) := by + filter_upwards [hlower, hsmall] with n hn hs + exact (hn.trans (hmono (a (φ n)) (v (φ n)) hs)).trans (hbad (φ n)).le + have hlim : Tendsto (fun n => ENNReal.ofReal (ε (φ n))) atTop (𝓝 0) := by + simpa only [Function.comp_def, ENNReal.ofReal_zero] using + (ENNReal.continuous_ofReal.tendsto 0).comp hεsub + exact (not_le_of_gt hq) (ge_of_tendsto hlim hupper) + +namespace FirstFrontPassage + +/-- Before an attained first hit, a continuous coordinate that started +below the front remains strictly below it. -/ +theorem lt_before_firstFrontTime + (x : ℝ≥0 → ℝ) (q : ℝ) (τ : ℝ≥0) + (hτ : firstFrontTime x q = (τ : WithTop ℝ≥0)) + (hc : ContinuousOn x (Icc 0 τ)) (hstart : x 0 < q) + (t : ℝ≥0) (ht : t < τ) : x t < q := by + by_contra hn + obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le) + (hc.mono (Icc_subset_Icc_right ht.le)) ⟨hstart.le, le_of_not_gt hn⟩ + have hh := hittingAfter_le_of_mem (u := fun s (_ : Unit) => x s - q) + (ω := ()) hu.1 (show x u - q ∈ ({0} : Set ℝ) by simp only [hue, sub_self, mem_singleton_iff]) + change firstFrontTime x q ≤ (u : WithTop ℝ≥0) at hh + rw [hτ, WithTop.coe_le_coe] at hh + exact (not_le_of_gt ht) (hh.trans hu.2) + +end FirstFrontPassage + +namespace PassageSuccess +open FirstFrontPassage + +/-- The open-envelope success event gives exactly the strict pre-front +containment used by the manuscript. The front point itself is specified +separately by its attained coordinate and terminal offset. -/ +theorem frontSuccess_source_prefix + {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p : ℕ → ℂ → ℝ) + (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (a : ℕ → ℂ) (ε : ℝ) (n : ℕ) + (ω : Ω) (hω : ω ∈ FrontSuccess η p σ V a ε n) + (hc : ContinuousOn (fun t => p n (η n ω t)) + (Icc 0 (frontTime (fun t => p n (η n ω t)) 1))) + (hstart : p n (η n ω 0) < 1) : + MapsTo (η n ω) (Ico 0 (frontTime (fun t => p n (η n ω t)) 1)) + (V n ∩ {z | p n z < 1}) ∧ + p n (η n ω (frontTime (fun t => p n (η n ω t)) 1)) = 1 ∧ + ‖η n ω (frontTime (fun t => p n (η n ω t)) 1) - a n‖ < ε / 4 := by + obtain ⟨htime, hhit, _, hpath, hterminal⟩ := hω + refine ⟨?_, hhit, hterminal⟩ + intro t ht + exact ⟨hpath ⟨ht.1, ht.2.le⟩, + lt_before_firstFrontTime _ 1 _ htime hc hstart t ht.2⟩ + +/-- Probability of the concrete full-prefix front success and its physical +sausage mass. The index records an admissible physical configuration. -/ +def frontPassageProbability + {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) + (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) + (d : A) (v c : ℝ) : ℝ≥0∞ := + P {ω | ω ∈ FrontSuccess (fun _ => η d) (fun _ => p d) (σ d) + (fun _ => V d) (fun _ => a d) (ε d) 0 ∧ ENNReal.ofReal c ≤ + segmentSausageCharge κ (η d ω) (σ d ω) + (frontTime (fun t => p d (η d ω t)) 1) v (R d)} + +/-- Increasing the required mass only decreases the concrete passage +probability; no probability or measurability assumption is needed here. -/ +theorem frontPassageProbability_antitone + {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) + (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) + (d : A) (v : ℝ) : Antitone (frontPassageProbability P η p σ V a ε κ R d v) := by + intro c₁ c₂ hc + apply measure_mono + intro ω hω + exact ⟨hω.1, (ENNReal.ofReal_le_ofReal hc).trans hω.2⟩ + +/-- A later retained starting point gives a subset of the full passage +success-and-mass event. This removes the auxiliary first-height stop from +the final one-step conclusion. -/ +theorem frontPassageProbability_mono_start + {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ₁ σ₂ : A → Ω → ℝ≥0) + (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) + (d : A) (v c : ℝ) (hσ : ∀ ω, σ₁ d ω ≤ σ₂ d ω) : + frontPassageProbability P η p σ₂ V a ε κ R d v c ≤ + frontPassageProbability P η p σ₁ V a ε κ R d v c := by + apply measure_mono + rintro ω ⟨hfront, hmass⟩ + obtain ⟨hfinite, hhit, hbefore, hpath, hterminal⟩ := hfront + refine ⟨⟨hfinite, hhit, (hσ ω).trans_lt hbefore, hpath, hterminal⟩, ?_⟩ + exact hmass.trans (segmentSausageCharge_mono κ (η d ω) (hσ ω) le_rfl v Subset.rfl) + +theorem frontPassageProbability_le_from_zero + {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) + (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) + (d : A) (v c : ℝ) : + frontPassageProbability P η p σ V a ε κ R d v c ≤ + frontPassageProbability P η p (fun _ _ => 0) V a ε κ R d v c := + frontPassageProbability_mono_start P η p (fun _ _ => 0) σ V a ε κ R d v c (fun _ => bot_le) + +/-- Once actual conformal compactness and the sequence passage theorem +supply the subsequential estimate, the full uniform passage constants +follow for the original concrete success-and-mass event. -/ +theorem uniform_frontPassage_probability_of_subsequences + {A Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (η : A → Ω → ℝ≥0 → ℂ) (p : A → ℂ → ℝ) (σ : A → Ω → ℝ≥0) + (V : A → Set ℂ) (a : A → ℂ) (ε : A → ℝ) (κ : ℝ) (R : A → Set ℂ) + (hsequence : ∀ (d : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → + Tendsto v atTop (𝓝 0) → + ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ + ∀ᶠ n in atTop, ENNReal.ofReal q ≤ + frontPassageProbability P η p σ V a ε κ R (d (φ n)) (v (φ n)) c) : + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ d : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ + frontPassageProbability P η p σ V a ε κ R d v c := by + apply uniform_passage_threshold_of_subsequences + (frontPassageProbability P η p σ V a ε κ R) + (frontPassageProbability_antitone P η p σ V a ε κ R) + intro d v hv hlim + obtain ⟨φ, hφ, c, q, hc, hq, hlower⟩ := hsequence d v hv hlim + exact ⟨φ, hφ, c, hc, ENNReal.ofReal q, ENNReal.ofReal_pos.mpr hq, hlower⟩ + +end PassageSuccess +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/UniformPassageThreshold.lean b/lean/OAI/Probability/SLE/Gauge/UniformPassageThreshold.lean index 03749af7bd3fa0d897d90ccdc9adfa0ee9990230..aeca6a5e0b155ae48b2409ab0796f974981ef5b6 100644 --- a/lean/OAI/Probability/SLE/Gauge/UniformPassageThreshold.lean +++ b/lean/OAI/Probability/SLE/Gauge/UniformPassageThreshold.lean @@ -1,52 +1,52 @@ -import Mathlib - -namespace OAI - -/-! The diagonal quantifier step in the uniform ribbon passage proof. -Uniform probability, mass threshold and mesh cutoff follow once every -vanishing-mesh sequence admits one positive eventual lower bound. -/ - -noncomputable section -open Set Filter -open scoped Topology ENNReal - -namespace SLEExactGauge - -theorem uniform_passage_threshold_of_sequences - {A : Type*} (p : A → ℝ → ℝ → ℝ≥0∞) - (hmono : ∀ a v, Antitone (p a v)) - (hsequence : ∀ (a : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → - Tendsto v atTop (𝓝 0) → - ∃ c : ℝ, 0 < c ∧ ∃ q : ℝ≥0∞, 0 < q ∧ - ∀ᶠ n in atTop, q ≤ p (a n) (v n) c) : - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ a : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ p a v c := by - classical - by_contra h - push Not at h - let ε : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) - have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity - have hfail (n : ℕ) : ∃ a : A, ∃ v : ℝ, - 0 < v ∧ v < ε n ∧ p a v (ε n) < ENNReal.ofReal (ε n) := by - obtain ⟨a, v, hv, hvs, hp⟩ := h (ε n) (ε n) (ε n) (hε n) (hε n) (hε n) - exact ⟨a, v, hv, hvs, hp⟩ - choose a v hv hvs hbad using hfail - have hεlim : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat - have hvlim : Tendsto v atTop (𝓝 0) := - squeeze_zero (fun n => (hv n).le) (fun n => (hvs n).le) hεlim - obtain ⟨c, hc, q, hq, hlower⟩ := hsequence a v hv hvlim - have hsmall : ∀ᶠ n in atTop, ε n ≤ c := - (hεlim.eventually (gt_mem_nhds hc)).mono fun _ h => h.le - have hupper : ∀ᶠ n in atTop, q ≤ ENNReal.ofReal (ε n) := by - filter_upwards [hlower, hsmall] with n hn hs - exact (hn.trans (hmono (a n) (v n) hs)).trans (hbad n).le - have hlim : Tendsto (fun n => ENNReal.ofReal (ε n)) atTop (𝓝 0) := by - simpa only [Function.comp_def, ENNReal.ofReal_zero] using - (ENNReal.continuous_ofReal.tendsto 0).comp hεlim - exact (not_le_of_gt hq) (ge_of_tendsto hlim hupper) - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! The diagonal quantifier step in the uniform ribbon passage proof. +Uniform probability, mass threshold and mesh cutoff follow once every +vanishing-mesh sequence admits one positive eventual lower bound. -/ + +noncomputable section +open Set Filter +open scoped Topology ENNReal + +namespace SLEExactGauge + +theorem uniform_passage_threshold_of_sequences + {A : Type*} (p : A → ℝ → ℝ → ℝ≥0∞) + (hmono : ∀ a v, Antitone (p a v)) + (hsequence : ∀ (a : ℕ → A) (v : ℕ → ℝ), (∀ n, 0 < v n) → + Tendsto v atTop (𝓝 0) → + ∃ c : ℝ, 0 < c ∧ ∃ q : ℝ≥0∞, 0 < q ∧ + ∀ᶠ n in atTop, q ≤ p (a n) (v n) c) : + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ a : A, ∀ v : ℝ, 0 < v → v < v₀ → ENNReal.ofReal q ≤ p a v c := by + classical + by_contra h + push Not at h + let ε : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) + have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity + have hfail (n : ℕ) : ∃ a : A, ∃ v : ℝ, + 0 < v ∧ v < ε n ∧ p a v (ε n) < ENNReal.ofReal (ε n) := by + obtain ⟨a, v, hv, hvs, hp⟩ := h (ε n) (ε n) (ε n) (hε n) (hε n) (hε n) + exact ⟨a, v, hv, hvs, hp⟩ + choose a v hv hvs hbad using hfail + have hεlim : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat + have hvlim : Tendsto v atTop (𝓝 0) := + squeeze_zero (fun n => (hv n).le) (fun n => (hvs n).le) hεlim + obtain ⟨c, hc, q, hq, hlower⟩ := hsequence a v hv hvlim + have hsmall : ∀ᶠ n in atTop, ε n ≤ c := + (hεlim.eventually (gt_mem_nhds hc)).mono fun _ h => h.le + have hupper : ∀ᶠ n in atTop, q ≤ ENNReal.ofReal (ε n) := by + filter_upwards [hlower, hsmall] with n hn hs + exact (hn.trans (hmono (a n) (v n) hs)).trans (hbad n).le + have hlim : Tendsto (fun n => ENNReal.ofReal (ε n)) atTop (𝓝 0) := by + simpa only [Function.comp_def, ENNReal.ofReal_zero] using + (ENNReal.continuous_ofReal.tendsto 0).comp hεlim + exact (not_le_of_gt hq) (ge_of_tendsto hlim hupper) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/UniformRecenteredPassage.lean b/lean/OAI/Probability/SLE/Gauge/UniformRecenteredPassage.lean index 9600087c54f3ac3da9b62c181eda27b1200ea8f7..874889930b5204fbd744e6635cd506c4bdd6cccc 100644 --- a/lean/OAI/Probability/SLE/Gauge/UniformRecenteredPassage.lean +++ b/lean/OAI/Probability/SLE/Gauge/UniformRecenteredPassage.lean @@ -1,157 +1,157 @@ -import OAI.Probability.SLE.Gauge.RecenteredRibbonPassage -import OAI.Probability.SLE.Geometry.BoundaryScaleLaw - -namespace OAI - -/-! Uniform passage constants for actual recentered ribbon configurations. -The finite chart realization and normalization law are constructed, and -subsequence compactness supplies the diagonal estimate. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates BoundaryScaleLaw - -local instance uniformRecenteredPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance uniformRecenteredPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem uniform_recentered_passage_from_geometry - {A Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) - (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (Gbase ω) (Fbase ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (U : A → ℝ≥0 → ℝ) (γₙ : A → ℝ≥0 → ℂ) - (G F₀ : A → ℝ≥0 → ℂ → ℂ) - (hU : ∀ n, Continuous (U n)) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) - (T : A → ℝ≥0) (hT : ∀ n, 0 < T n) - (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) - (αₙ : A → ℝ → ℂ) (Cg : ℝ≥0) (s c : ℝ) - (hCg : 0 < (Cg : ℝ)) (hs : 0 < s) (hcpos : 0 < c) - (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) - (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) - (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → - c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) - (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) - (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) - (Qₙ : A → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) - (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C Lq : ℝ≥0) (Rq : ℝ) - (hC : 0 < C) (hLq : 0 < Lq) - (hQupper : ∀ n, LipschitzOnWith C (Qₙ n) K) - (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) - (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) - (xₙ : A → ℝ) (ε : ℝ) (hε : 0 < ε) - (hxₙ : ∀ n, |xₙ n| ≤ ε / 4) - (hcore : {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ interior K) - (hshape : ∀ n, EqOn (fun t => αₙ n t - γₙ n (T n)) - (fun t => Qₙ n (guideCoordinate 0 (xₙ n) t)) (Icc 0 2)) - (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / - ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) - (hregion : K ∩ passageCoordinateRegion C ε (ε / Lq) ⊆ interior K) : - ∃ Fb : A → C(closedHalfPlane, ℂ), - (∀ d (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb d z = - F₀ d (T d) ((U d (T d) : ℂ) + z) - γₙ d (T d)) ∧ - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ d : A, ∀ v : ℝ, 0 < v → v < v₀ → - ENNReal.ofReal q ≤ P {ω | - finiteBoundaryPath (Fb d) (sourceTracePath γ (fun ω => (hwbase ω).1) ω) ∈ - BoundaryScaleLaw.frontMassEvent (longitudinalCoordinate (Qₙ d) K 0) - (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) (Qₙ d 1) - (ε / Lq) κ v c (physicalMiddleRegion (Qₙ d) K 0 ε)} := by - choose Fb₀ hFb₀ _ using fun d => normalized_finite_capacity_inverse_boundary_extension - hboundary (hU d) (hw d) (T d) 1 zero_lt_one - let Fb : A → C(closedHalfPlane, ℂ) := fun d => Fb₀ d - ContinuousMap.const _ (γₙ d (T d)) - have hext d (z : closedHalfPlane) (hz : z.1 ∈ upperHalfPlane) : Fb d z = - F₀ d (T d) ((U d (T d) : ℂ) + z) - γₙ d (T d) := by - change Fb₀ d z - γₙ d (T d) = _ - rw [hFb₀ d z hz] - simp only [Complex.ofReal_one, one_mul] - let gp := sourceTracePath γ (fun ω => (hwbase ω).1) - let prob := fun d v c => P {ω | finiteBoundaryPath (Fb d) (gp ω) ∈ - frontMassEvent (longitudinalCoordinate (Qₙ d) K 0) - (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) (Qₙ d 1) - (ε / Lq) κ v c (physicalMiddleRegion (Qₙ d) K 0 ε)} - have hmono : ∀ d v, Antitone (prob d v) := by - intro d v c₁ c₂ hc' - apply measure_mono - intro ω hω - exact ⟨hω.1, (ENNReal.ofReal_le_ofReal hc').trans hω.2⟩ - have hgp : Measurable gp := by - apply source_path_measurable - intro t - exact (brownian_capacity_trace_adapted B₀ hm hc κ γ Gbase Fbase hwbase t).mono - ((Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)).le t) le_rfl - have hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (gp ω) := - Eventually.of_forall (fun ω t => (hwbase ω).2.2.1 t) - have hVopen d : IsOpen (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) := by - have hi := lower_lipschitz_injOn (Qₙ d) K Lq (hQlower d) - have he : K ∩ passageCoordinateRegion C ε (ε / Lq) = - interior K ∩ passageCoordinateRegion C ε (ε / Lq) := - Subset.antisymm (fun _ hz => ⟨hregion hz, hz.2⟩) - (fun _ hz => ⟨interior_subset hz.1, hz.2⟩) - unfold physicalPassageRegion - rw [he] - exact isOpen_image_coordinate_subset hplanar (Qₙ d) K _ (hQupper d).continuousOn hi - (isOpen_interior.inter (passageCoordinateRegion_isOpen _ _ _)) - (inter_subset_left.trans interior_subset) - have hVK d : physicalPassageRegion (Qₙ d) K C ε (ε / Lq) ⊆ Qₙ d '' K := - image_mono inter_subset_left - have hpc d : ContinuousOn (longitudinalCoordinate (Qₙ d) K 0) (Qₙ d '' K) := - longitudinalCoordinate_continuousOn (Qₙ d) K Lq 0 (hQlower d) - have hpK d : IsClosed (Qₙ d '' K) := (hK.image_of_continuousOn (hQupper d).continuousOn).isClosed - obtain ⟨q, c₁, v₀, hq, hc₁, hv₀, hprob⟩ := - uniform_passage_threshold_of_subsequences prob hmono (by - intro d v hv hlim - obtain ⟨φ, b, Fbn, hφ, hb, hFbn, c, q, hcpos', hqpos, hseq⟩ := - recentered_ribbon_subsequence_passage P κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase hwbase - hcompact hboundary hsupport hplanar - (fun n => U (d n)) (fun n => γₙ (d n)) (fun n => G (d n)) (fun n => F₀ (d n)) - (fun n => hU (d n)) (fun n => hw (d n)) (fun n => T (d n)) (fun n => hT (d n)) - (fun n => hfresh (d n)) (fun n => αₙ (d n)) Cg s c hCg hs hcpos - (fun n => hαL (d n)) (fun n => hα0 (d n)) (fun n => hchord (d n)) - (fun n => hαD (d n)) - (fun δ hδ hδ2 => by - obtain ⟨r, hr, hh⟩ := hclear δ hδ hδ2 - exact ⟨r, hr, fun n => hh (d n)⟩) - (fun n => Qₙ (d n)) K hK x₀ hx₀ C Lq Rq hC hLq - (fun n => hQupper (d n)) (fun n => hQlower (d n)) (fun n => hQtip (d n)) - (fun n => xₙ (d n)) ε hε (fun n => hxₙ (d n)) hcore - (fun n => hshape (d n)) (fun n => hangle (d n)) v hv hlim - refine ⟨φ, hφ, c, hcpos', ENNReal.ofReal q, ENNReal.ofReal_pos.mpr hqpos, ?_⟩ - filter_upwards [hseq] with n hn - let ell : ℝ≥0 := ⟨b n, (hb n).le⟩ - have hell : 0 < ell := by exact_mod_cast hb n - have hFbeq : Fbn n = (Fb (d (φ n))).comp (scaleClosed ell) := by - apply normalized_boundary_eq - (fun z => F₀ (d (φ n)) (T (d (φ n))) z - γₙ (d (φ n)) (T (d (φ n)))) - (U (d (φ n)) (T (d (φ n)))) ell hell _ _ (hext (d (φ n))) - intro z hz - exact hFbn n z hz - have hscale := ScaleTraceLaw.scaled_trace_path_law κ hκ hκ8 B₀ hB₀ hm hc - γ Gbase Fbase hwbase ell hell - have he := frontMassEvent_boundary_scale_probability P gp hgp hclosed ell hell - (Fb (d (φ n))) hscale (longitudinalCoordinate (Qₙ (d (φ n))) K 0) - (physicalPassageRegion (Qₙ (d (φ n))) K C ε (ε / Lq)) (Qₙ (d (φ n)) 1) - (ε / Lq) κ (v (φ n)) c (physicalMiddleRegion (Qₙ (d (φ n))) K 0 ε) - (Qₙ (d (φ n)) '' K) (hpK _) (hpc _) (hVopen _) (hVK _) - rw [← hFbeq] at he - dsimp only [prob] - rw [← he] - exact hn) - exact ⟨Fb, hext, q, c₁, v₀, hq, hc₁, hv₀, hprob⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RecenteredRibbonPassage +import OAI.Probability.SLE.Geometry.BoundaryScaleLaw + +namespace OAI + +/-! Uniform passage constants for actual recentered ribbon configurations. +The finite chart realization and normalization law are constructed, and +subsequence compactness supplies the diagonal estimate. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates BoundaryScaleLaw + +local instance uniformRecenteredPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance uniformRecenteredPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem uniform_recentered_passage_from_geometry + {A Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (Gbase Fbase : Ω → ℝ≥0 → ℂ → ℂ) + (hwbase : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (Gbase ω) (Fbase ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (U : A → ℝ≥0 → ℝ) (γₙ : A → ℝ≥0 → ℂ) + (G F₀ : A → ℝ≥0 → ℂ → ℂ) + (hU : ∀ n, Continuous (U n)) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) + (T : A → ℝ≥0) (hT : ∀ n, 0 < T n) + (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) + (αₙ : A → ℝ → ℂ) (Cg : ℝ≥0) (s c : ℝ) + (hCg : 0 < (Cg : ℝ)) (hs : 0 < s) (hcpos : 0 < c) + (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) + (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) + (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → + c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) + (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) + (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) + (Qₙ : A → ℂ → ℂ) (K : Set ℂ) (hK : IsCompact K) + (x₀ : ℂ) (hx₀ : x₀ ∈ K) (C Lq : ℝ≥0) (Rq : ℝ) + (hC : 0 < C) (hLq : 0 < Lq) + (hQupper : ∀ n, LipschitzOnWith C (Qₙ n) K) + (hQlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y)) + (hQtip : ∀ n, ‖Qₙ n x₀‖ ≤ Rq) + (xₙ : A → ℝ) (ε : ℝ) (hε : 0 < ε) + (hxₙ : ∀ n, |xₙ n| ≤ ε / 4) + (hcore : {z : ℂ | 0 ≤ z.re ∧ z.re ≤ 2 ∧ |z.im| ≤ ε / 4} ⊆ interior K) + (hshape : ∀ n, EqOn (fun t => αₙ n t - γₙ n (T n)) + (fun t => Qₙ n (guideCoordinate 0 (xₙ n) t)) (Icc 0 2)) + (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / + ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) + (hregion : K ∩ passageCoordinateRegion C ε (ε / Lq) ⊆ interior K) : + ∃ Fb : A → C(closedHalfPlane, ℂ), + (∀ d (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb d z = + F₀ d (T d) ((U d (T d) : ℂ) + z) - γₙ d (T d)) ∧ + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ d : A, ∀ v : ℝ, 0 < v → v < v₀ → + ENNReal.ofReal q ≤ P {ω | + finiteBoundaryPath (Fb d) (sourceTracePath γ (fun ω => (hwbase ω).1) ω) ∈ + BoundaryScaleLaw.frontMassEvent (longitudinalCoordinate (Qₙ d) K 0) + (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) (Qₙ d 1) + (ε / Lq) κ v c (physicalMiddleRegion (Qₙ d) K 0 ε)} := by + choose Fb₀ hFb₀ _ using fun d => normalized_finite_capacity_inverse_boundary_extension + hboundary (hU d) (hw d) (T d) 1 zero_lt_one + let Fb : A → C(closedHalfPlane, ℂ) := fun d => Fb₀ d - ContinuousMap.const _ (γₙ d (T d)) + have hext d (z : closedHalfPlane) (hz : z.1 ∈ upperHalfPlane) : Fb d z = + F₀ d (T d) ((U d (T d) : ℂ) + z) - γₙ d (T d) := by + change Fb₀ d z - γₙ d (T d) = _ + rw [hFb₀ d z hz] + simp only [Complex.ofReal_one, one_mul] + let gp := sourceTracePath γ (fun ω => (hwbase ω).1) + let prob := fun d v c => P {ω | finiteBoundaryPath (Fb d) (gp ω) ∈ + frontMassEvent (longitudinalCoordinate (Qₙ d) K 0) + (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) (Qₙ d 1) + (ε / Lq) κ v c (physicalMiddleRegion (Qₙ d) K 0 ε)} + have hmono : ∀ d v, Antitone (prob d v) := by + intro d v c₁ c₂ hc' + apply measure_mono + intro ω hω + exact ⟨hω.1, (ENNReal.ofReal_le_ofReal hc').trans hω.2⟩ + have hgp : Measurable gp := by + apply source_path_measurable + intro t + exact (brownian_capacity_trace_adapted B₀ hm hc κ γ Gbase Fbase hwbase t).mono + ((Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)).le t) le_rfl + have hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (gp ω) := + Eventually.of_forall (fun ω t => (hwbase ω).2.2.1 t) + have hVopen d : IsOpen (physicalPassageRegion (Qₙ d) K C ε (ε / Lq)) := by + have hi := lower_lipschitz_injOn (Qₙ d) K Lq (hQlower d) + have he : K ∩ passageCoordinateRegion C ε (ε / Lq) = + interior K ∩ passageCoordinateRegion C ε (ε / Lq) := + Subset.antisymm (fun _ hz => ⟨hregion hz, hz.2⟩) + (fun _ hz => ⟨interior_subset hz.1, hz.2⟩) + unfold physicalPassageRegion + rw [he] + exact isOpen_image_coordinate_subset hplanar (Qₙ d) K _ (hQupper d).continuousOn hi + (isOpen_interior.inter (passageCoordinateRegion_isOpen _ _ _)) + (inter_subset_left.trans interior_subset) + have hVK d : physicalPassageRegion (Qₙ d) K C ε (ε / Lq) ⊆ Qₙ d '' K := + image_mono inter_subset_left + have hpc d : ContinuousOn (longitudinalCoordinate (Qₙ d) K 0) (Qₙ d '' K) := + longitudinalCoordinate_continuousOn (Qₙ d) K Lq 0 (hQlower d) + have hpK d : IsClosed (Qₙ d '' K) := (hK.image_of_continuousOn (hQupper d).continuousOn).isClosed + obtain ⟨q, c₁, v₀, hq, hc₁, hv₀, hprob⟩ := + uniform_passage_threshold_of_subsequences prob hmono (by + intro d v hv hlim + obtain ⟨φ, b, Fbn, hφ, hb, hFbn, c, q, hcpos', hqpos, hseq⟩ := + recentered_ribbon_subsequence_passage P κ hκ hκ8 B₀ hB₀ hm hc γ Gbase Fbase hwbase + hcompact hboundary hsupport hplanar + (fun n => U (d n)) (fun n => γₙ (d n)) (fun n => G (d n)) (fun n => F₀ (d n)) + (fun n => hU (d n)) (fun n => hw (d n)) (fun n => T (d n)) (fun n => hT (d n)) + (fun n => hfresh (d n)) (fun n => αₙ (d n)) Cg s c hCg hs hcpos + (fun n => hαL (d n)) (fun n => hα0 (d n)) (fun n => hchord (d n)) + (fun n => hαD (d n)) + (fun δ hδ hδ2 => by + obtain ⟨r, hr, hh⟩ := hclear δ hδ hδ2 + exact ⟨r, hr, fun n => hh (d n)⟩) + (fun n => Qₙ (d n)) K hK x₀ hx₀ C Lq Rq hC hLq + (fun n => hQupper (d n)) (fun n => hQlower (d n)) (fun n => hQtip (d n)) + (fun n => xₙ (d n)) ε hε (fun n => hxₙ (d n)) hcore + (fun n => hshape (d n)) (fun n => hangle (d n)) v hv hlim + refine ⟨φ, hφ, c, hcpos', ENNReal.ofReal q, ENNReal.ofReal_pos.mpr hqpos, ?_⟩ + filter_upwards [hseq] with n hn + let ell : ℝ≥0 := ⟨b n, (hb n).le⟩ + have hell : 0 < ell := by exact_mod_cast hb n + have hFbeq : Fbn n = (Fb (d (φ n))).comp (scaleClosed ell) := by + apply normalized_boundary_eq + (fun z => F₀ (d (φ n)) (T (d (φ n))) z - γₙ (d (φ n)) (T (d (φ n)))) + (U (d (φ n)) (T (d (φ n)))) ell hell _ _ (hext (d (φ n))) + intro z hz + exact hFbn n z hz + have hscale := ScaleTraceLaw.scaled_trace_path_law κ hκ hκ8 B₀ hB₀ hm hc + γ Gbase Fbase hwbase ell hell + have he := frontMassEvent_boundary_scale_probability P gp hgp hclosed ell hell + (Fb (d (φ n))) hscale (longitudinalCoordinate (Qₙ (d (φ n))) K 0) + (physicalPassageRegion (Qₙ (d (φ n))) K C ε (ε / Lq)) (Qₙ (d (φ n)) 1) + (ε / Lq) κ (v (φ n)) c (physicalMiddleRegion (Qₙ (d (φ n))) K 0 ε) + (Qₙ (d (φ n)) '' K) (hpK _) (hpc _) (hVopen _) (hVK _) + rw [← hFbeq] at he + dsimp only [prob] + rw [← he] + exact hn) + exact ⟨Fb, hext, q, c₁, v₀, hq, hc₁, hv₀, hprob⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/UniformRibbonStep.lean b/lean/OAI/Probability/SLE/Gauge/UniformRibbonStep.lean index af83e8c96e2c3a68f7829bb9cc81299d0f52771e..e9dc98bc7de0994956503ba516ef1693852c2532 100644 --- a/lean/OAI/Probability/SLE/Gauge/UniformRibbonStep.lean +++ b/lean/OAI/Probability/SLE/Gauge/UniformRibbonStep.lean @@ -1,152 +1,152 @@ -import OAI.Probability.SLE.Gauge.RibbonStepTranslation - -namespace OAI - -/-! Uniform one-step probability in the original physical ribbon, for any -actual continuous boundary realization of the stopped inverse map. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonCoordinates BoundaryScaleLaw RibbonPassageScaling Target - -local instance uniformStepPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance uniformStepPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -def translateFiniteBoundary (a : ℂ) (f : C(closedHalfPlane, ℂ)) : - C(closedHalfPlane, ℂ) := ⟨fun z => a + f z, continuous_const.add f.continuous⟩ - -lemma finiteBoundaryPath_translate (a : ℂ) (f : C(closedHalfPlane, ℂ)) - (p : C(ℝ≥0, ℂ)) (t : ℝ≥0) : - finiteBoundaryPath (translateFiniteBoundary a f) p t = - affine a 1 (finiteBoundaryPath f p t) := by - simp only [finiteBoundaryPath, translateFiniteBoundary, ContinuousMap.comp_apply, - ContinuousMap.coe_mk, affine, Complex.ofReal_one, one_mul] - -def RibbonHistory.stepEvent {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (κ v c : ℝ) : Set C(ℝ≥0, ℂ) := - frontMassEvent (longitudinalCoordinate d.chart closedRibbon d.front) - (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) - (d.chart ((d.front : ℂ) + 1)) (ε / L) κ v c - (physicalMiddleRegion d.chart closedRibbon d.front ε) - -lemma RibbonHistory.finiteBoundary_step_probability - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) - (f : C(closedHalfPlane, ℂ)) (p : Ω → C(ℝ≥0, ℂ)) - (κ v c : ℝ) (hv : 0 < v) : - P {ω | finiteBoundaryPath (translateFiniteBoundary (d.trace d.time) f) (p ω) ∈ - d.stepEvent κ v c} = - P {ω | finiteBoundaryPath f (p ω) ∈ - frontMassEvent (longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) - (physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) - (d.recenteredChart 1) (ε / L) κ v c - (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by - have hη (ω : Ω) : (finiteBoundaryPath - (translateFiniteBoundary (d.trace d.time) f) (p ω) : ℝ≥0 → ℂ) = - fun t => affine (d.trace d.time) 1 (finiteBoundaryPath f (p ω) t) := by - funext t - exact finiteBoundaryPath_translate _ _ _ _ - have hh := d.frontMass_probability_translate P hε - (fun ω => (finiteBoundaryPath f (p ω) : ℝ≥0 → ℂ)) κ v c hv - simpa only [RibbonHistory.stepEvent, frontMassEvent, hη, finiteBoundaryPath_translate, - PassageSuccess.FrontSuccess, mem_ofPred_eq] using hh - -lemma RibbonHistory.step_lower {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) : - ∀ u ∈ ribbonStepCompact, ∀ v ∈ ribbonStepCompact, - dist u v ≤ L * dist (d.stepChart u) (d.stepChart v) := by - intro u hu v hv - simpa only [RibbonHistory.stepChart, dist_add_left] using - d.patch_lower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hv) - -lemma RibbonHistory.stepChart_continuousOn {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : ContinuousOn d.stepChart ribbonStepCompact := - d.chart_continuous.comp (continuous_const.add continuous_id).continuousOn d.stepCompact_maps - -lemma RibbonHistory.global_progress_eq_step {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) : - EqOn (longitudinalCoordinate d.chart closedRibbon d.front) - (longitudinalCoordinate d.stepChart ribbonStepCompact 0) - (d.stepChart '' ribbonStepCompact) := by - have hi := lower_lipschitz_injOn d.stepChart ribbonStepCompact L d.step_lower - rintro z ⟨w, hw, rfl⟩ - simp only [longitudinalCoordinate, RibbonHistory.stepChart, - coordinateInverse_left d.chart_injective (d.stepCompact_maps hw), Complex.add_re, - Complex.ofReal_re, add_sub_cancel_left, sub_zero] - exact congrArg Complex.re (coordinateInverse_left hi hw).symm - -lemma RibbonHistory.stepEvent_measurable {C L : ℝ≥0} {ε : ℝ} - (d : RibbonHistory C L ε) (hplanar : PublishedPlanarDiskOpenness) - (hε : ε ≤ 1 / 8) (κ v c : ℝ) : MeasurableSet (d.stepEvent κ v c) := by - have hK := ribbonStepCompact_isCompact.image_of_continuousOn d.stepChart_continuousOn - have hp : ContinuousOn (longitudinalCoordinate d.chart closedRibbon d.front) - (d.stepChart '' ribbonStepCompact) := - (longitudinalCoordinate_continuousOn d.stepChart ribbonStepCompact L 0 d.step_lower).congr - (fun z hz => d.global_progress_eq_step hz) - have hV : IsOpen (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) := by - have he : ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) = - interior ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) := by - apply Subset.antisymm - · exact fun z hz => ⟨ribbonStepCompact_passage hε C (ε / L) hz, hz.2⟩ - · exact inter_subset_inter_left _ interior_subset - unfold physicalPassageRegion - rw [he] - exact isOpen_image_coordinate_subset hplanar d.stepChart ribbonStepCompact _ - d.stepChart_continuousOn (lower_lipschitz_injOn _ _ L d.step_lower) - (isOpen_interior.inter (passageCoordinateRegion_isOpen C ε (ε / L))) - (inter_subset_left.trans interior_subset) - exact measurableSet_frontMassEvent _ _ _ _ κ v c _ _ hK.isClosed hp hV - (image_mono inter_subset_left) - -/-- This is the uniform one-step estimate for an actual stopped history. -The boundary chart can be the already past-measurable chart used in its -restart law; no measurable selection of compactness outputs is required. -/ -theorem uniform_actual_ribbon_step - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hcont : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) - (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : - ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ - ∀ (d : RibbonHistory C L ε) (Fb : C(closedHalfPlane, ℂ)), - (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → - Fb z = d.inverse d.time ((d.U d.time : ℂ) + z)) → - ∀ v : ℝ, 0 < v → v < v₀ → - ENNReal.ofReal q ≤ P {ω | - finiteBoundaryPath Fb (sourceTracePath γ (fun ω => (hw ω).1) ω) ∈ - d.stepEvent κ v c} := by - obtain ⟨Fc, hFc, q, c, v₀, hq, hc, hv₀, hprob⟩ := - uniform_actual_ribbon_passage P κ hκ hκ8 B hB hm hcont γ G F hw - hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW C L hC hL - ε hε hεsmall - refine ⟨q, c, v₀, hq, hc, hv₀, ?_⟩ - intro d Fb hFb v hv hvcut - have he : Fb = translateFiniteBoundary (d.trace d.time) (Fc d) := by - apply finite_boundary_ext - intro z hz - change Fb z = d.trace d.time + Fc d z - rw [hFb z hz, hFc d z hz] - abel - rw [he, d.finiteBoundary_step_probability P hεsmall (Fc d) - (sourceTracePath γ (fun ω => (hw ω).1)) κ v c hv] - exact hprob d v hv hvcut - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonStepTranslation + +namespace OAI + +/-! Uniform one-step probability in the original physical ribbon, for any +actual continuous boundary realization of the stopped inverse map. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonCoordinates BoundaryScaleLaw RibbonPassageScaling Target + +local instance uniformStepPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance uniformStepPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +def translateFiniteBoundary (a : ℂ) (f : C(closedHalfPlane, ℂ)) : + C(closedHalfPlane, ℂ) := ⟨fun z => a + f z, continuous_const.add f.continuous⟩ + +lemma finiteBoundaryPath_translate (a : ℂ) (f : C(closedHalfPlane, ℂ)) + (p : C(ℝ≥0, ℂ)) (t : ℝ≥0) : + finiteBoundaryPath (translateFiniteBoundary a f) p t = + affine a 1 (finiteBoundaryPath f p t) := by + simp only [finiteBoundaryPath, translateFiniteBoundary, ContinuousMap.comp_apply, + ContinuousMap.coe_mk, affine, Complex.ofReal_one, one_mul] + +def RibbonHistory.stepEvent {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (κ v c : ℝ) : Set C(ℝ≥0, ℂ) := + frontMassEvent (longitudinalCoordinate d.chart closedRibbon d.front) + (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) + (d.chart ((d.front : ℂ) + 1)) (ε / L) κ v c + (physicalMiddleRegion d.chart closedRibbon d.front ε) + +lemma RibbonHistory.finiteBoundary_step_probability + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (hε : ε ≤ 1 / 8) + (f : C(closedHalfPlane, ℂ)) (p : Ω → C(ℝ≥0, ℂ)) + (κ v c : ℝ) (hv : 0 < v) : + P {ω | finiteBoundaryPath (translateFiniteBoundary (d.trace d.time) f) (p ω) ∈ + d.stepEvent κ v c} = + P {ω | finiteBoundaryPath f (p ω) ∈ + frontMassEvent (longitudinalCoordinate d.recenteredChart ribbonStepCompact 0) + (physicalPassageRegion d.recenteredChart ribbonStepCompact C ε (ε / L)) + (d.recenteredChart 1) (ε / L) κ v c + (physicalMiddleRegion d.recenteredChart ribbonStepCompact 0 ε)} := by + have hη (ω : Ω) : (finiteBoundaryPath + (translateFiniteBoundary (d.trace d.time) f) (p ω) : ℝ≥0 → ℂ) = + fun t => affine (d.trace d.time) 1 (finiteBoundaryPath f (p ω) t) := by + funext t + exact finiteBoundaryPath_translate _ _ _ _ + have hh := d.frontMass_probability_translate P hε + (fun ω => (finiteBoundaryPath f (p ω) : ℝ≥0 → ℂ)) κ v c hv + simpa only [RibbonHistory.stepEvent, frontMassEvent, hη, finiteBoundaryPath_translate, + PassageSuccess.FrontSuccess, mem_ofPred_eq] using hh + +lemma RibbonHistory.step_lower {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) : + ∀ u ∈ ribbonStepCompact, ∀ v ∈ ribbonStepCompact, + dist u v ≤ L * dist (d.stepChart u) (d.stepChart v) := by + intro u hu v hv + simpa only [RibbonHistory.stepChart, dist_add_left] using + d.patch_lower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hv) + +lemma RibbonHistory.stepChart_continuousOn {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : ContinuousOn d.stepChart ribbonStepCompact := + d.chart_continuous.comp (continuous_const.add continuous_id).continuousOn d.stepCompact_maps + +lemma RibbonHistory.global_progress_eq_step {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) : + EqOn (longitudinalCoordinate d.chart closedRibbon d.front) + (longitudinalCoordinate d.stepChart ribbonStepCompact 0) + (d.stepChart '' ribbonStepCompact) := by + have hi := lower_lipschitz_injOn d.stepChart ribbonStepCompact L d.step_lower + rintro z ⟨w, hw, rfl⟩ + simp only [longitudinalCoordinate, RibbonHistory.stepChart, + coordinateInverse_left d.chart_injective (d.stepCompact_maps hw), Complex.add_re, + Complex.ofReal_re, add_sub_cancel_left, sub_zero] + exact congrArg Complex.re (coordinateInverse_left hi hw).symm + +lemma RibbonHistory.stepEvent_measurable {C L : ℝ≥0} {ε : ℝ} + (d : RibbonHistory C L ε) (hplanar : PublishedPlanarDiskOpenness) + (hε : ε ≤ 1 / 8) (κ v c : ℝ) : MeasurableSet (d.stepEvent κ v c) := by + have hK := ribbonStepCompact_isCompact.image_of_continuousOn d.stepChart_continuousOn + have hp : ContinuousOn (longitudinalCoordinate d.chart closedRibbon d.front) + (d.stepChart '' ribbonStepCompact) := + (longitudinalCoordinate_continuousOn d.stepChart ribbonStepCompact L 0 d.step_lower).congr + (fun z hz => d.global_progress_eq_step hz) + have hV : IsOpen (physicalPassageRegion d.stepChart ribbonStepCompact C ε (ε / L)) := by + have he : ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) = + interior ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) := by + apply Subset.antisymm + · exact fun z hz => ⟨ribbonStepCompact_passage hε C (ε / L) hz, hz.2⟩ + · exact inter_subset_inter_left _ interior_subset + unfold physicalPassageRegion + rw [he] + exact isOpen_image_coordinate_subset hplanar d.stepChart ribbonStepCompact _ + d.stepChart_continuousOn (lower_lipschitz_injOn _ _ L d.step_lower) + (isOpen_interior.inter (passageCoordinateRegion_isOpen C ε (ε / L))) + (inter_subset_left.trans interior_subset) + exact measurableSet_frontMassEvent _ _ _ _ κ v c _ _ hK.isClosed hp hV + (image_mono inter_subset_left) + +/-- This is the uniform one-step estimate for an actual stopped history. +The boundary chart can be the already past-measurable chart used in its +restart law; no measurable selection of compactness outputs is required. -/ +theorem uniform_actual_ribbon_step + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hcont : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L) + (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) : + ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧ + ∀ (d : RibbonHistory C L ε) (Fb : C(closedHalfPlane, ℂ)), + (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → + Fb z = d.inverse d.time ((d.U d.time : ℂ) + z)) → + ∀ v : ℝ, 0 < v → v < v₀ → + ENNReal.ofReal q ≤ P {ω | + finiteBoundaryPath Fb (sourceTracePath γ (fun ω => (hw ω).1) ω) ∈ + d.stepEvent κ v c} := by + obtain ⟨Fc, hFc, q, c, v₀, hq, hc, hv₀, hprob⟩ := + uniform_actual_ribbon_passage P κ hκ hκ8 B hB hm hcont γ G F hw + hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW C L hC hL + ε hε hεsmall + refine ⟨q, c, v₀, hq, hc, hv₀, ?_⟩ + intro d Fb hFb v hv hvcut + have he : Fb = translateFiniteBoundary (d.trace d.time) (Fc d) := by + apply finite_boundary_ext + intro z hz + change Fb z = d.trace d.time + Fc d z + rw [hFb z hz, hFc d z hz] + abel + rw [he, d.finiteBoundary_step_probability P hεsmall (Fc d) + (sourceTracePath γ (fun ω => (hw ω).1)) κ v c hv] + exact hprob d v hv hvcut + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/VaryingPassageEvents.lean b/lean/OAI/Probability/SLE/Gauge/VaryingPassageEvents.lean index 181a9e40ab2fe1542d08efacaa3125a900d1c674..b869bb5e5819cf2555c8a6890b824f2974261c18 100644 --- a/lean/OAI/Probability/SLE/Gauge/VaryingPassageEvents.lean +++ b/lean/OAI/Probability/SLE/Gauge/VaryingPassageEvents.lean @@ -1,103 +1,103 @@ -import Mathlib - -namespace OAI - -/-! -The `rc:varying-good-events` step of the ribbon argument. Positivity of a -conditional passage probability on one event gives a fixed lower probability -after intersecting with varying early-containment events. No uniform -conditional passage bound is assumed. --/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -theorem varying_passage_events_lower_bound - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (E C : Set Ω) (G : ℕ → Set Ω) - (hE : MeasurableSet[m] E) (hC : MeasurableSet[mΩ] C) - (hG : ∀ n, MeasurableSet[m] (G n)) (hEpos : 0 < P E) - (hpassage : ∀ᵐ ω ∂P, ω ∈ E → 0 < P⁻[C.indicator (fun _ => 1) | m] ω) - (hmissing : ∀ n, P (E \ G n) ≤ P E / 2) : - ∃ q₂ : ℝ≥0∞, 0 < q₂ ∧ ∀ n, q₂ ≤ P (E ∩ G n ∩ C) := by - let : MeasurableSpace Ω := mΩ - let q : Ω → ℝ≥0∞ := P⁻[C.indicator (fun _ => 1) | m] - have hqm : Measurable[m] q := measurable_condLExp m P _ - have hqmΩ : Measurable q := hqm.mono hm le_rfl - let b : ℕ → ℝ≥0∞ := fun n => (1 / 2 : ℝ≥0∞) ^ n - let S : ℕ → Set Ω := fun n => E ∩ {ω | q ω < b n} - have hSm (n : ℕ) : MeasurableSet (S n) := - (hm E hE).inter (measurableSet_lt hqmΩ measurable_const) - have hSanti : Antitone S := by - intro n k hnk ω hω - exact ⟨hω.1, hω.2.trans_le - (pow_le_pow_right_of_le_one' (by norm_num : (1 / 2 : ℝ≥0∞) ≤ 1) hnk)⟩ - have hblim : Tendsto b atTop (𝓝 0) := - ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) - have hinter : P (⋂ n, S n) = 0 := by - apply measure_eq_zero_iff_ae_notMem.mpr - filter_upwards [hpassage] with ω hω - intro hmem - have hEω : ω ∈ E := (mem_iInter.mp hmem 0).1 - have hle : q ω ≤ 0 := ge_of_tendsto hblim - (Eventually.of_forall fun n => ((mem_iInter.mp hmem n).2).le) - exact (not_le_of_gt (hω hEω)) hle - have hsmalllim : Tendsto (fun n => P (S n)) atTop (𝓝 0) := by - simpa only [Function.comp_def, hinter] using - tendsto_measure_iInter_atTop (fun n => (hSm n).nullMeasurableSet) - hSanti ⟨0, measure_ne_top P _⟩ - have hquarter : 0 < P E / 4 := ENNReal.div_pos hEpos.ne' (by norm_num) - obtain ⟨N, hN⟩ := - (hsmalllim.eventually (gt_mem_nhds hquarter)).exists - let q₁ : ℝ≥0∞ := b N - have hq₁ : 0 < q₁ := by - exact ENNReal.pow_pos (ENNReal.div_pos one_ne_zero (by norm_num)) N - refine ⟨q₁ * (P E / 4), ENNReal.mul_pos hq₁.ne' hquarter.ne', ?_⟩ - intro n - let A : Set Ω := E ∩ G n - let B : Set Ω := A ∩ {ω | q₁ ≤ q ω} - have hAm : MeasurableSet[m] A := hE.inter (hG n) - have hBm : MeasurableSet[m] B := - hAm.inter (measurableSet_le measurable_const hqm) - have hcover : E ⊆ B ∪ (E \ G n) ∪ S N := by - intro ω hω - by_cases hωG : ω ∈ G n - · by_cases hωq : q₁ ≤ q ω - · exact Or.inl (Or.inl ⟨⟨hω, hωG⟩, hωq⟩) - · exact Or.inr ⟨hω, lt_of_not_ge hωq⟩ - · exact Or.inl (Or.inr ⟨hω, hωG⟩) - have hcovermeasure : P E ≤ P B + P (E \ G n) + P (S N) := - (measure_mono hcover).trans - ((measure_union_le _ _).trans (add_le_add (measure_union_le _ _) le_rfl)) - have hBmass : P E / 4 ≤ P B := by - have hcR := ENNReal.toReal_mono (by finiteness) hcovermeasure - rw [ENNReal.toReal_add (by finiteness) (by finiteness), - ENNReal.toReal_add (by finiteness) (by finiteness)] at hcR - have hmR := ENNReal.toReal_mono (by finiteness) (hmissing n) - have hsR := ENNReal.toReal_mono (by finiteness) hN.le - norm_num only [ENNReal.toReal_div, ENNReal.toReal_ofNat] at hmR hsR - apply (ENNReal.toReal_le_toReal (by finiteness) (by finiteness)).mp - norm_num only [ENNReal.toReal_div, ENNReal.toReal_ofNat] - linarith - calc - q₁ * (P E / 4) ≤ q₁ * P B := mul_le_mul' le_rfl hBmass - _ = ∫⁻ _ in B, q₁ ∂P := (setLIntegral_const _ _).symm - _ ≤ ∫⁻ ω in B, q ω ∂P := - setLIntegral_mono' (hm B hBm) (fun _ hω => hω.2) - _ ≤ ∫⁻ ω in A, q ω ∂P := lintegral_mono_set inter_subset_left - _ = ∫⁻ ω in A, C.indicator (fun _ => 1) ω ∂P := - setLIntegral_condLExp hm P _ hAm - _ = P (E ∩ G n ∩ C) := by - rw [setLIntegral_indicator hC, setLIntegral_const, one_mul] - congr 1 - exact inter_comm C A - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! +The `rc:varying-good-events` step of the ribbon argument. Positivity of a +conditional passage probability on one event gives a fixed lower probability +after intersecting with varying early-containment events. No uniform +conditional passage bound is assumed. +-/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +theorem varying_passage_events_lower_bound + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (E C : Set Ω) (G : ℕ → Set Ω) + (hE : MeasurableSet[m] E) (hC : MeasurableSet[mΩ] C) + (hG : ∀ n, MeasurableSet[m] (G n)) (hEpos : 0 < P E) + (hpassage : ∀ᵐ ω ∂P, ω ∈ E → 0 < P⁻[C.indicator (fun _ => 1) | m] ω) + (hmissing : ∀ n, P (E \ G n) ≤ P E / 2) : + ∃ q₂ : ℝ≥0∞, 0 < q₂ ∧ ∀ n, q₂ ≤ P (E ∩ G n ∩ C) := by + let : MeasurableSpace Ω := mΩ + let q : Ω → ℝ≥0∞ := P⁻[C.indicator (fun _ => 1) | m] + have hqm : Measurable[m] q := measurable_condLExp m P _ + have hqmΩ : Measurable q := hqm.mono hm le_rfl + let b : ℕ → ℝ≥0∞ := fun n => (1 / 2 : ℝ≥0∞) ^ n + let S : ℕ → Set Ω := fun n => E ∩ {ω | q ω < b n} + have hSm (n : ℕ) : MeasurableSet (S n) := + (hm E hE).inter (measurableSet_lt hqmΩ measurable_const) + have hSanti : Antitone S := by + intro n k hnk ω hω + exact ⟨hω.1, hω.2.trans_le + (pow_le_pow_right_of_le_one' (by norm_num : (1 / 2 : ℝ≥0∞) ≤ 1) hnk)⟩ + have hblim : Tendsto b atTop (𝓝 0) := + ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) + have hinter : P (⋂ n, S n) = 0 := by + apply measure_eq_zero_iff_ae_notMem.mpr + filter_upwards [hpassage] with ω hω + intro hmem + have hEω : ω ∈ E := (mem_iInter.mp hmem 0).1 + have hle : q ω ≤ 0 := ge_of_tendsto hblim + (Eventually.of_forall fun n => ((mem_iInter.mp hmem n).2).le) + exact (not_le_of_gt (hω hEω)) hle + have hsmalllim : Tendsto (fun n => P (S n)) atTop (𝓝 0) := by + simpa only [Function.comp_def, hinter] using + tendsto_measure_iInter_atTop (fun n => (hSm n).nullMeasurableSet) + hSanti ⟨0, measure_ne_top P _⟩ + have hquarter : 0 < P E / 4 := ENNReal.div_pos hEpos.ne' (by norm_num) + obtain ⟨N, hN⟩ := + (hsmalllim.eventually (gt_mem_nhds hquarter)).exists + let q₁ : ℝ≥0∞ := b N + have hq₁ : 0 < q₁ := by + exact ENNReal.pow_pos (ENNReal.div_pos one_ne_zero (by norm_num)) N + refine ⟨q₁ * (P E / 4), ENNReal.mul_pos hq₁.ne' hquarter.ne', ?_⟩ + intro n + let A : Set Ω := E ∩ G n + let B : Set Ω := A ∩ {ω | q₁ ≤ q ω} + have hAm : MeasurableSet[m] A := hE.inter (hG n) + have hBm : MeasurableSet[m] B := + hAm.inter (measurableSet_le measurable_const hqm) + have hcover : E ⊆ B ∪ (E \ G n) ∪ S N := by + intro ω hω + by_cases hωG : ω ∈ G n + · by_cases hωq : q₁ ≤ q ω + · exact Or.inl (Or.inl ⟨⟨hω, hωG⟩, hωq⟩) + · exact Or.inr ⟨hω, lt_of_not_ge hωq⟩ + · exact Or.inl (Or.inr ⟨hω, hωG⟩) + have hcovermeasure : P E ≤ P B + P (E \ G n) + P (S N) := + (measure_mono hcover).trans + ((measure_union_le _ _).trans (add_le_add (measure_union_le _ _) le_rfl)) + have hBmass : P E / 4 ≤ P B := by + have hcR := ENNReal.toReal_mono (by finiteness) hcovermeasure + rw [ENNReal.toReal_add (by finiteness) (by finiteness), + ENNReal.toReal_add (by finiteness) (by finiteness)] at hcR + have hmR := ENNReal.toReal_mono (by finiteness) (hmissing n) + have hsR := ENNReal.toReal_mono (by finiteness) hN.le + norm_num only [ENNReal.toReal_div, ENNReal.toReal_ofNat] at hmR hsR + apply (ENNReal.toReal_le_toReal (by finiteness) (by finiteness)).mp + norm_num only [ENNReal.toReal_div, ENNReal.toReal_ofNat] + linarith + calc + q₁ * (P E / 4) ≤ q₁ * P B := mul_le_mul' le_rfl hBmass + _ = ∫⁻ _ in B, q₁ ∂P := (setLIntegral_const _ _).symm + _ ≤ ∫⁻ ω in B, q ω ∂P := + setLIntegral_mono' (hm B hBm) (fun _ hω => hω.2) + _ ≤ ∫⁻ ω in A, q ω ∂P := lintegral_mono_set inter_subset_left + _ = ∫⁻ ω in A, C.indicator (fun _ => 1) ω ∂P := + setLIntegral_condLExp hm P _ hAm + _ = P (E ∩ G n ∩ C) := by + rw [setLIntegral_indicator hC, setLIntegral_const, one_mul] + congr 1 + exact inter_comm C A + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/WaypointBrownianPassages.lean b/lean/OAI/Probability/SLE/Gauge/WaypointBrownianPassages.lean index 3c0233bc2176f894c40f14e5ec3989ed82aac509..0526c67443b099cd29cedf9c593a5a913d2df8d4 100644 --- a/lean/OAI/Probability/SLE/Gauge/WaypointBrownianPassages.lean +++ b/lean/OAI/Probability/SLE/Gauge/WaypointBrownianPassages.lean @@ -1,158 +1,158 @@ -import OAI.Probability.SLE.Geometry.WaypointRouteProbability - -namespace OAI - -/-! Uniform Brownian waypoint tube events stay initially in the surviving -domain, then avoid the complete access guide and hit the stopped past. -/ - -noncomputable section -open Set Metric MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RibbonCoordinates -open Target - -theorem early_waypoint_coordinate_ball_safe - (k side : ℝ) (hside : |side| = 1) {ε t : ℝ} - (hk : 1 ≤ k) (hεsmall : ε ≤ 1 / 8) (ht : t ∈ Icc (0 : ℝ) 2) : - ball (waypointRoute k side t) (1 / 32) ⊆ - futureRibbon k ∪ sideCorridor ε 1 ∪ sideCorridor (-1) (-ε) := by - intro z hz - have hn : ‖z - waypointRoute k side t‖ < 1 / 32 := by - simpa only [mem_ball, dist_eq_norm] using hz - have hre := (Complex.abs_re_le_norm (z - waypointRoute k side t)).trans_lt hn - have him := (Complex.abs_im_le_norm (z - waypointRoute k side t)).trans_lt hn - rw [Complex.sub_re, waypointRoute_re] at hre - rw [Complex.sub_im] at him - have hr := abs_lt.mp hre - have hi := abs_lt.mp him - have hrouteim := abs_le.mp (waypointRoute_im_bound k side hside t) - have hzim : |z.im| < 1 := abs_lt.mpr ⟨by linarith, by linarith⟩ - by_cases ht1 : t ≤ 1 - · left; left - rw [unitClip_of_nonpos (by linarith : t - 1 ≤ 0)] at hr - exact ⟨by linarith, hzim⟩ - · have hclip0 := unitClip_nonneg (t - 1) - have hclip1 := unitClip_le_one (t - 1) - have hzr : 0 < z.re := by linarith - have hrouteabs : |(waypointRoute k side t).im| = 1 / 2 := by - rw [waypointRoute_im, unitClip_of_one_le (by linarith : 1 ≤ t), - unitClip_of_nonpos (by linarith [ht.2] : t - 2 ≤ 0), abs_mul, hside] - norm_num - have hh : |(waypointRoute k side t).im| ≤ - |(waypointRoute k side t).im - z.im| + |z.im| := by - simpa only [sub_add_cancel] using abs_add_le - ((waypointRoute k side t).im - z.im) z.im - rw [abs_sub_comm] at hh - have hgap : ε < |z.im| := by linarith - by_cases hz0 : 0 ≤ z.im - · left; right - exact ⟨hzr, by simpa only [abs_of_nonneg hz0] using hgap, (abs_lt.mp hzim).2⟩ - · right - rw [abs_of_neg (lt_of_not_ge hz0)] at hgap - exact ⟨hzr, (abs_lt.mp hzim).1, by linarith⟩ - -theorem physical_early_waypoint_tube_survives - (hpublished : PublishedPlanarDiskOpenness) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) - (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) - (K : ℝ≥0) (hK : 0 < (K : ℝ)) (γ : ℝ≥0 → ℂ) (T : ℝ≥0) - {k ε side : ℝ} (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) - (hside : |side| = 1) - (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε}) - (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) - (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, - dist u v ≤ K * dist (P u) (P v)) - (ζ : ℝ → ℂ) - (hnear : ∀ t ∈ Icc (0 : ℝ) 2, - dist (ζ t) (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K) : - MapsTo ζ (Icc 0 2) (survivingDomain γ T) := by - have hk0 : 0 ≤ k := by linarith - have hfuture := futureRibbon_survives P hP hi hk0 hH γ T - (by rintro z ⟨u, hu, rfl⟩; obtain ⟨v, hv, he⟩ := hpast ⟨u, hu, rfl⟩; - exact ⟨v, ⟨hv.1, hv.2.1⟩, he⟩) hunbounded - have hupper := sideCorridor_survives P hP hi hH γ T hk0 - (show ε < 1 by linarith) (show -1 ≤ ε by linarith) le_rfl - (fun x hx => hx.1.trans_le (le_abs_self x)) hpast hunbounded - have hlowerSide := sideCorridor_survives P hP hi hH γ T hk0 - (show -1 < -ε by linarith) le_rfl (show -ε ≤ 1 by linarith) - (fun x hx => (show ε < -x by linarith [hx.2]).trans_le (neg_le_abs x)) hpast hunbounded - intro t ht - have hpatch := waypointRoute_ball_subset_patch k side hside t - have hsub := crossingPatch_subset_closedRibbon hk - have hball := ball_subset_closedBall.trans (hpatch.trans hsub) - obtain ⟨z, hz, he⟩ := ball_subset_image_ball P (waypointRoute k side t) - (by norm_num : (0 : ℝ) < 1 / 32) hK (hP.mono (hpatch.trans hsub)) - (hpublished P _ _ (by norm_num) (hP.mono hball) (hi.mono hball)) - (fun z hz => hlower z (hpatch hz) _ (hpatch (mem_closedBall_self (by norm_num)))) - (hnear t ht) - rw [← he] - rcases early_waypoint_coordinate_ball_safe k side hside hk hεsmall ht hz with - (hf | hu) | hl - · exact hfuture (mem_image_of_mem P hf) - · exact hupper (mem_image_of_mem P hu) - · exact hlowerSide (mem_image_of_mem P hl) - -/-- One positive probability gives the complete route geometry: the early -part survives, the later part avoids the full guide, and the final leg -hits the actual stopped past. No harmonic-measure labels are assumed. -/ -theorem uniform_brownian_waypoint_passage_geometry - (hopen : PublishedPlanarDiskOpenness) (hsupport : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) (C K : ℝ≥0) (hK : 0 < (K : ℝ)) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ P : ℂ → ℂ, - ContinuousOn P closedRibbon → InjOn P closedRibbon → - (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → - ∀ k ε x side : ℝ, 1 ≤ k → 0 < ε → ε ≤ 1 / 8 → |x| ≤ 1 / 16 → - (side = -1 ∨ side = 1) → LipschitzOnWith C P (crossingPatch k) → - (∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, - dist u v ≤ K * dist (P u) (P v)) → - ∀ (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0), 0 < T → - ContinuousOn xp (Icc 0 T) → EqOn γ (P ∘ xp) (Icc 0 T) → - (xp 0).re = 0 → (xp T).re = k → - MapsTo xp (Icc 0 T) closedRibbon → - (∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε) → - (¬ Bornology.IsBounded (P '' futureRibbon k)) → - q ≤ (W.prod W) {b | - let ζ := physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b - MapsTo ζ (Icc 0 2) (survivingDomain γ T) ∧ - (∀ t ∈ Icc (1 / 4 : ℝ) 3, ∀ u : ℝ, 0 ≤ u → ζ t ≠ P (guideCoordinate k x u)) ∧ - ∃ s ∈ Icc 0 T, ∃ t ∈ Icc (0 : ℝ) 3, γ s = ζ t} := by - obtain ⟨q, hq, hprob⟩ := uniform_physical_waypoint_tube_probability hsupport W hW C - (div_pos (by norm_num : (0 : ℝ) < 1 / 32) hK) - refine ⟨q, hq, ?_⟩ - intro P hP hi hH k ε x side hk hε hεsmall hx hside hL hlower γ xp T hT - hxp hγ hxp0 hxpT hcoords hcore hunbounded - have habs : |side| = 1 := by rcases hside with rfl | rfl <;> norm_num - have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε} := by - rintro z ⟨s, hs, rfl⟩ - exact ⟨xp s, ⟨hcoords hs, hcore s hs⟩, (hγ hs).symm⟩ - refine (hprob P k side habs hL).trans (measure_mono ?_) - intro b hb - let ζ := physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b - refine ⟨physical_early_waypoint_tube_survives hopen P hP hi hH K hK γ T hk hε hεsmall - habs hpast hunbounded hlower ζ (fun t ht => hb t ⟨ht.1, ht.2.trans (by norm_num)⟩), ?_, ?_⟩ - · intro t ht u hu he - have hh := physical_waypointRoute_avoids_complete_guide hopen P hP hi K hK hk hx habs - hlower ht.1 hu - apply hh - rw [← he] - exact hb t ⟨by linarith [ht.1], ht.2⟩ - · let γr : ℝ → ℂ := fun s => γ s.toNNReal - let xpr : ℝ → ℂ := fun s => xp s.toNNReal - have hmap : MapsTo Real.toNNReal (Icc (0 : ℝ) T) (Icc 0 T) := by - intro s hs - exact ⟨bot_le, (Real.toNNReal_le_iff_le_coe).mpr hs.2⟩ - have hcont : ContinuousOn xpr (Icc (0 : ℝ) T) := - hxp.comp (by fun_prop) hmap - obtain ⟨s, hs, t, ht, he⟩ := physical_waypoint_tube_hits_past hopen P hP hi K hK hk - hε hεsmall (by exact_mod_cast hT : (0 : ℝ) < T) hside hlower γr xpr ζ hcont - (fun s hs => hγ (hmap hs)) (by simpa [xpr] using hxp0) - (by simpa [xpr] using hxpT) (fun s hs => (hcore _ (hmap hs)).2) - (physicalUnitBrownianPath_continuous _ b).continuousOn hb - exact ⟨s.toNNReal, hmap hs, t, ht, he⟩ - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Geometry.WaypointRouteProbability + +namespace OAI + +/-! Uniform Brownian waypoint tube events stay initially in the surviving +domain, then avoid the complete access guide and hit the stopped past. -/ + +noncomputable section +open Set Metric MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RibbonCoordinates +open Target + +theorem early_waypoint_coordinate_ball_safe + (k side : ℝ) (hside : |side| = 1) {ε t : ℝ} + (hk : 1 ≤ k) (hεsmall : ε ≤ 1 / 8) (ht : t ∈ Icc (0 : ℝ) 2) : + ball (waypointRoute k side t) (1 / 32) ⊆ + futureRibbon k ∪ sideCorridor ε 1 ∪ sideCorridor (-1) (-ε) := by + intro z hz + have hn : ‖z - waypointRoute k side t‖ < 1 / 32 := by + simpa only [mem_ball, dist_eq_norm] using hz + have hre := (Complex.abs_re_le_norm (z - waypointRoute k side t)).trans_lt hn + have him := (Complex.abs_im_le_norm (z - waypointRoute k side t)).trans_lt hn + rw [Complex.sub_re, waypointRoute_re] at hre + rw [Complex.sub_im] at him + have hr := abs_lt.mp hre + have hi := abs_lt.mp him + have hrouteim := abs_le.mp (waypointRoute_im_bound k side hside t) + have hzim : |z.im| < 1 := abs_lt.mpr ⟨by linarith, by linarith⟩ + by_cases ht1 : t ≤ 1 + · left; left + rw [unitClip_of_nonpos (by linarith : t - 1 ≤ 0)] at hr + exact ⟨by linarith, hzim⟩ + · have hclip0 := unitClip_nonneg (t - 1) + have hclip1 := unitClip_le_one (t - 1) + have hzr : 0 < z.re := by linarith + have hrouteabs : |(waypointRoute k side t).im| = 1 / 2 := by + rw [waypointRoute_im, unitClip_of_one_le (by linarith : 1 ≤ t), + unitClip_of_nonpos (by linarith [ht.2] : t - 2 ≤ 0), abs_mul, hside] + norm_num + have hh : |(waypointRoute k side t).im| ≤ + |(waypointRoute k side t).im - z.im| + |z.im| := by + simpa only [sub_add_cancel] using abs_add_le + ((waypointRoute k side t).im - z.im) z.im + rw [abs_sub_comm] at hh + have hgap : ε < |z.im| := by linarith + by_cases hz0 : 0 ≤ z.im + · left; right + exact ⟨hzr, by simpa only [abs_of_nonneg hz0] using hgap, (abs_lt.mp hzim).2⟩ + · right + rw [abs_of_neg (lt_of_not_ge hz0)] at hgap + exact ⟨hzr, (abs_lt.mp hzim).1, by linarith⟩ + +theorem physical_early_waypoint_tube_survives + (hpublished : PublishedPlanarDiskOpenness) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon) + (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) + (K : ℝ≥0) (hK : 0 < (K : ℝ)) (γ : ℝ≥0 → ℂ) (T : ℝ≥0) + {k ε side : ℝ} (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) + (hside : |side| = 1) + (hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε}) + (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k)) + (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, + dist u v ≤ K * dist (P u) (P v)) + (ζ : ℝ → ℂ) + (hnear : ∀ t ∈ Icc (0 : ℝ) 2, + dist (ζ t) (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K) : + MapsTo ζ (Icc 0 2) (survivingDomain γ T) := by + have hk0 : 0 ≤ k := by linarith + have hfuture := futureRibbon_survives P hP hi hk0 hH γ T + (by rintro z ⟨u, hu, rfl⟩; obtain ⟨v, hv, he⟩ := hpast ⟨u, hu, rfl⟩; + exact ⟨v, ⟨hv.1, hv.2.1⟩, he⟩) hunbounded + have hupper := sideCorridor_survives P hP hi hH γ T hk0 + (show ε < 1 by linarith) (show -1 ≤ ε by linarith) le_rfl + (fun x hx => hx.1.trans_le (le_abs_self x)) hpast hunbounded + have hlowerSide := sideCorridor_survives P hP hi hH γ T hk0 + (show -1 < -ε by linarith) le_rfl (show -ε ≤ 1 by linarith) + (fun x hx => (show ε < -x by linarith [hx.2]).trans_le (neg_le_abs x)) hpast hunbounded + intro t ht + have hpatch := waypointRoute_ball_subset_patch k side hside t + have hsub := crossingPatch_subset_closedRibbon hk + have hball := ball_subset_closedBall.trans (hpatch.trans hsub) + obtain ⟨z, hz, he⟩ := ball_subset_image_ball P (waypointRoute k side t) + (by norm_num : (0 : ℝ) < 1 / 32) hK (hP.mono (hpatch.trans hsub)) + (hpublished P _ _ (by norm_num) (hP.mono hball) (hi.mono hball)) + (fun z hz => hlower z (hpatch hz) _ (hpatch (mem_closedBall_self (by norm_num)))) + (hnear t ht) + rw [← he] + rcases early_waypoint_coordinate_ball_safe k side hside hk hεsmall ht hz with + (hf | hu) | hl + · exact hfuture (mem_image_of_mem P hf) + · exact hupper (mem_image_of_mem P hu) + · exact hlowerSide (mem_image_of_mem P hl) + +/-- One positive probability gives the complete route geometry: the early +part survives, the later part avoids the full guide, and the final leg +hits the actual stopped past. No harmonic-measure labels are assumed. -/ +theorem uniform_brownian_waypoint_passage_geometry + (hopen : PublishedPlanarDiskOpenness) (hsupport : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) (C K : ℝ≥0) (hK : 0 < (K : ℝ)) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ P : ℂ → ℂ, + ContinuousOn P closedRibbon → InjOn P closedRibbon → + (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → + ∀ k ε x side : ℝ, 1 ≤ k → 0 < ε → ε ≤ 1 / 8 → |x| ≤ 1 / 16 → + (side = -1 ∨ side = 1) → LipschitzOnWith C P (crossingPatch k) → + (∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, + dist u v ≤ K * dist (P u) (P v)) → + ∀ (γ xp : ℝ≥0 → ℂ) (T : ℝ≥0), 0 < T → + ContinuousOn xp (Icc 0 T) → EqOn γ (P ∘ xp) (Icc 0 T) → + (xp 0).re = 0 → (xp T).re = k → + MapsTo xp (Icc 0 T) closedRibbon → + (∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε) → + (¬ Bornology.IsBounded (P '' futureRibbon k)) → + q ≤ (W.prod W) {b | + let ζ := physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b + MapsTo ζ (Icc 0 2) (survivingDomain γ T) ∧ + (∀ t ∈ Icc (1 / 4 : ℝ) 3, ∀ u : ℝ, 0 ≤ u → ζ t ≠ P (guideCoordinate k x u)) ∧ + ∃ s ∈ Icc 0 T, ∃ t ∈ Icc (0 : ℝ) 3, γ s = ζ t} := by + obtain ⟨q, hq, hprob⟩ := uniform_physical_waypoint_tube_probability hsupport W hW C + (div_pos (by norm_num : (0 : ℝ) < 1 / 32) hK) + refine ⟨q, hq, ?_⟩ + intro P hP hi hH k ε x side hk hε hεsmall hx hside hL hlower γ xp T hT + hxp hγ hxp0 hxpT hcoords hcore hunbounded + have habs : |side| = 1 := by rcases hside with rfl | rfl <;> norm_num + have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k ∧ |z.im| ≤ ε} := by + rintro z ⟨s, hs, rfl⟩ + exact ⟨xp s, ⟨hcoords hs, hcore s hs⟩, (hγ hs).symm⟩ + refine (hprob P k side habs hL).trans (measure_mono ?_) + intro b hb + let ζ := physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b + refine ⟨physical_early_waypoint_tube_survives hopen P hP hi hH K hK γ T hk hε hεsmall + habs hpast hunbounded hlower ζ (fun t ht => hb t ⟨ht.1, ht.2.trans (by norm_num)⟩), ?_, ?_⟩ + · intro t ht u hu he + have hh := physical_waypointRoute_avoids_complete_guide hopen P hP hi K hK hk hx habs + hlower ht.1 hu + apply hh + rw [← he] + exact hb t ⟨by linarith [ht.1], ht.2⟩ + · let γr : ℝ → ℂ := fun s => γ s.toNNReal + let xpr : ℝ → ℂ := fun s => xp s.toNNReal + have hmap : MapsTo Real.toNNReal (Icc (0 : ℝ) T) (Icc 0 T) := by + intro s hs + exact ⟨bot_le, (Real.toNNReal_le_iff_le_coe).mpr hs.2⟩ + have hcont : ContinuousOn xpr (Icc (0 : ℝ) T) := + hxp.comp (by fun_prop) hmap + obtain ⟨s, hs, t, ht, he⟩ := physical_waypoint_tube_hits_past hopen P hP hi K hK hk + hε hεsmall (by exact_mod_cast hT : (0 : ℝ) < T) hside hlower γr xpr ζ hcont + (fun s hs => hγ (hmap hs)) (by simpa [xpr] using hxp0) + (by simpa [xpr] using hxpT) (fun s hs => (hcore _ (hmap hs)).2) + (physicalUnitBrownianPath_continuous _ b).continuousOn hb + exact ⟨s.toNNReal, hmap hs, t, ht, he⟩ + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/WeightedMomentIntegration.lean b/lean/OAI/Probability/SLE/Gauge/WeightedMomentIntegration.lean index 7e18ca93ae7b9e9c2d3e05d3977c4be9f9cf990f..a7b0c14231ddfc35b4d2df2cf0dac8d098d954b0 100644 --- a/lean/OAI/Probability/SLE/Gauge/WeightedMomentIntegration.lean +++ b/lean/OAI/Probability/SLE/Gauge/WeightedMomentIntegration.lean @@ -1,32 +1,32 @@ -import OAI.Probability.SLE.Multipoint.MultipointMomentIntegration - -namespace OAI - -noncomputable section -open MeasureTheory -open scoped ENNReal - -namespace SLEExactGauge.Multipoint - -lemma lintegral_weighted_mass_pow_eq - {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ] - (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) - (f : X → ℝ≥0∞) (hf : Measurable f) (m : ℕ) : - (∫⁻ ω, (∫⁻ z, f z * W ω z ∂μ)^m ∂P) = - ∫⁻ z : Fin m → X, (∏ i, f (z i)) * (∫⁻ ω, ∏ i, W ω (z i) ∂P) - ∂Measure.pi (fun _ : Fin m => μ) := by - rw [lintegral_mass_pow_eq P μ (fun ω z => f z * W ω z) - ((hf.comp measurable_snd).mul hW) m] - apply lintegral_congr - intro z - have hg : Measurable (fun ω => ∏ i, W ω (z i)) := - Finset.measurable_prod _ (fun i _ => hW.comp (measurable_id.prodMk measurable_const)) - simp_rw [Finset.prod_mul_distrib] - exact lintegral_const_mul _ hg - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Multipoint.MultipointMomentIntegration + +namespace OAI + +noncomputable section +open MeasureTheory +open scoped ENNReal + +namespace SLEExactGauge.Multipoint + +lemma lintegral_weighted_mass_pow_eq + {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ] + (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) + (f : X → ℝ≥0∞) (hf : Measurable f) (m : ℕ) : + (∫⁻ ω, (∫⁻ z, f z * W ω z ∂μ)^m ∂P) = + ∫⁻ z : Fin m → X, (∏ i, f (z i)) * (∫⁻ ω, ∏ i, W ω (z i) ∂P) + ∂Measure.pi (fun _ : Fin m => μ) := by + rw [lintegral_mass_pow_eq P μ (fun ω z => f z * W ω z) + ((hf.comp measurable_snd).mul hW) m] + apply lintegral_congr + intro z + have hg : Measurable (fun ω => ∏ i, W ω (z i)) := + Finset.measurable_prod _ (fun i _ => hW.comp (measurable_id.prodMk measurable_const)) + simp_rw [Finset.prod_mul_distrib] + exact lintegral_const_mul _ hg + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/WholeSausageScaling.lean b/lean/OAI/Probability/SLE/Gauge/WholeSausageScaling.lean index 7ef04c00f8ff4bf94515a5a83b873a1193675126..fd6bacbd2071ac4f9d5ff7f8924ed548c48fd8c4 100644 --- a/lean/OAI/Probability/SLE/Gauge/WholeSausageScaling.lean +++ b/lean/OAI/Probability/SLE/Gauge/WholeSausageScaling.lean @@ -1,79 +1,79 @@ -import OAI.Probability.SLE.Gauge.RibbonPassageScaling -import OAI.Probability.SLE.Loewner.ScaleTraceLaw - -namespace OAI - -/-! Exact whole-sausage scaling at the one common physical mesh. The -capacity time change does not change the trace's range. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open RibbonPassageScaling ScaleTraceLaw - -lemma wholeSausageSet_surjective_clock (γ : ℝ≥0 → ℂ) (e : ℝ≥0 → ℝ≥0) - (he : Function.Surjective e) (v : ℝ) : - wholeSausageSet (γ ∘ e) v = wholeSausageSet γ v := by - ext z - constructor - · rintro ⟨t, ht⟩ - exact ⟨e t, ht⟩ - · rintro ⟨t, ht⟩ - obtain ⟨u, rfl⟩ := he t - exact ⟨u, ht⟩ - -lemma wholeSausageSet_affine (a : ℂ) {ell : ℝ} (hell : 0 < ell) - (γ : ℝ≥0 → ℂ) (v : ℝ) : - wholeSausageSet (fun t => affine a ell (γ t)) (ell * v) = - affine a ell '' wholeSausageSet γ v := by - ext z - obtain ⟨w, rfl⟩ := (affine_bijective a hell).2 z - constructor - · rintro ⟨t, ht⟩ - rw [affine_dist a hell.le] at ht - exact ⟨w, ⟨t, (mul_lt_mul_iff_right₀ hell).mp ht⟩, rfl⟩ - · rintro ⟨w', ⟨t, ht⟩, he⟩ - have hw : w' = w := (affine_bijective a hell).1 he - subst w' - exact ⟨t, by rw [affine_dist a hell.le]; exact mul_lt_mul_of_pos_left ht hell⟩ - -theorem wholeSausageCharge_affine (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) - (γ : ℝ≥0 → ℂ) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : - wholeSausageCharge κ (fun t => affine a ell (γ t)) (ell * v) (affine a ell '' R) = - ENNReal.ofReal (ell ^ dimension κ) * wholeSausageCharge κ γ v R := by - have he : (ell * v) ^ (-codimension κ) * ell ^ 2 = - ell ^ dimension κ * v ^ (-codimension κ) := by - rw [Real.mul_rpow hell.le hv.le] - calc - _ = (ell ^ (-codimension κ) * ell ^ (2 : ℝ)) * v ^ (-codimension κ) := by - rw [Real.rpow_two] - ring - _ = ell ^ (-codimension κ + 2) * v ^ (-codimension κ) := by rw [Real.rpow_add hell] - _ = _ := by congr 2; unfold codimension; ring - simp only [wholeSausageCharge, wholeSausageSet_affine a hell] - rw [← image_inter (affine_bijective a hell).1, affine_volume a hell.le] - rw [← mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg (mul_pos hell hv).le _), he, - ENNReal.ofReal_mul (Real.rpow_nonneg hell.le _), mul_assoc] - -theorem wholeSausageCharge_scalePath (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - (p : C(ℝ≥0, ℂ)) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : - wholeSausageCharge κ (scalePath c p) ((c : ℝ) * v) (affine 0 c '' R) = - ENNReal.ofReal ((c : ℝ) ^ dimension κ) * wholeSausageCharge κ p v R := by - have he : Function.Surjective (fun t : ℝ≥0 => t / c ^ 2) := by - intro t - exact ⟨t * c ^ 2, mul_div_cancel_right₀ t (pow_ne_zero 2 hc.ne')⟩ - have hp : (scalePath c p : ℝ≥0 → ℂ) = - (fun t => affine 0 c (p t)) ∘ (fun t => t / c ^ 2) := by - funext t - change (c : ℂ) * p (t / c ^ 2) = 0 + (c : ℂ) * p (t / c ^ 2) - exact (zero_add _).symm - simp only [wholeSausageCharge, hp, wholeSausageSet_surjective_clock _ _ he] - exact wholeSausageCharge_affine κ 0 (show 0 < (c : ℝ) from hc) p hv R - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonPassageScaling +import OAI.Probability.SLE.Loewner.ScaleTraceLaw + +namespace OAI + +/-! Exact whole-sausage scaling at the one common physical mesh. The +capacity time change does not change the trace's range. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open RibbonPassageScaling ScaleTraceLaw + +lemma wholeSausageSet_surjective_clock (γ : ℝ≥0 → ℂ) (e : ℝ≥0 → ℝ≥0) + (he : Function.Surjective e) (v : ℝ) : + wholeSausageSet (γ ∘ e) v = wholeSausageSet γ v := by + ext z + constructor + · rintro ⟨t, ht⟩ + exact ⟨e t, ht⟩ + · rintro ⟨t, ht⟩ + obtain ⟨u, rfl⟩ := he t + exact ⟨u, ht⟩ + +lemma wholeSausageSet_affine (a : ℂ) {ell : ℝ} (hell : 0 < ell) + (γ : ℝ≥0 → ℂ) (v : ℝ) : + wholeSausageSet (fun t => affine a ell (γ t)) (ell * v) = + affine a ell '' wholeSausageSet γ v := by + ext z + obtain ⟨w, rfl⟩ := (affine_bijective a hell).2 z + constructor + · rintro ⟨t, ht⟩ + rw [affine_dist a hell.le] at ht + exact ⟨w, ⟨t, (mul_lt_mul_iff_right₀ hell).mp ht⟩, rfl⟩ + · rintro ⟨w', ⟨t, ht⟩, he⟩ + have hw : w' = w := (affine_bijective a hell).1 he + subst w' + exact ⟨t, by rw [affine_dist a hell.le]; exact mul_lt_mul_of_pos_left ht hell⟩ + +theorem wholeSausageCharge_affine (κ : ℝ) (a : ℂ) {ell : ℝ} (hell : 0 < ell) + (γ : ℝ≥0 → ℂ) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : + wholeSausageCharge κ (fun t => affine a ell (γ t)) (ell * v) (affine a ell '' R) = + ENNReal.ofReal (ell ^ dimension κ) * wholeSausageCharge κ γ v R := by + have he : (ell * v) ^ (-codimension κ) * ell ^ 2 = + ell ^ dimension κ * v ^ (-codimension κ) := by + rw [Real.mul_rpow hell.le hv.le] + calc + _ = (ell ^ (-codimension κ) * ell ^ (2 : ℝ)) * v ^ (-codimension κ) := by + rw [Real.rpow_two] + ring + _ = ell ^ (-codimension κ + 2) * v ^ (-codimension κ) := by rw [Real.rpow_add hell] + _ = _ := by congr 2; unfold codimension; ring + simp only [wholeSausageCharge, wholeSausageSet_affine a hell] + rw [← image_inter (affine_bijective a hell).1, affine_volume a hell.le] + rw [← mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg (mul_pos hell hv).le _), he, + ENNReal.ofReal_mul (Real.rpow_nonneg hell.le _), mul_assoc] + +theorem wholeSausageCharge_scalePath (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + (p : C(ℝ≥0, ℂ)) {v : ℝ} (hv : 0 < v) (R : Set ℂ) : + wholeSausageCharge κ (scalePath c p) ((c : ℝ) * v) (affine 0 c '' R) = + ENNReal.ofReal ((c : ℝ) ^ dimension κ) * wholeSausageCharge κ p v R := by + have he : Function.Surjective (fun t : ℝ≥0 => t / c ^ 2) := by + intro t + exact ⟨t * c ^ 2, mul_div_cancel_right₀ t (pow_ne_zero 2 hc.ne')⟩ + have hp : (scalePath c p : ℝ≥0 → ℂ) = + (fun t => affine 0 c (p t)) ∘ (fun t => t / c ^ 2) := by + funext t + change (c : ℂ) * p (t / c ^ 2) = 0 + (c : ℂ) * p (t / c ^ 2) + exact (zero_add _).symm + simp only [wholeSausageCharge, hp, wholeSausageSet_surjective_clock _ _ he] + exact wholeSausageCharge_affine κ 0 (show 0 < (c : ℝ) from hc) p hv R + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Gauge/WindowDenseTests.lean b/lean/OAI/Probability/SLE/Gauge/WindowDenseTests.lean index 65561c2cceade37d899532ed865001ac62308969..485a2e20f021a626f905e7b5ea1545ba0dcb1691 100644 --- a/lean/OAI/Probability/SLE/Gauge/WindowDenseTests.lean +++ b/lean/OAI/Probability/SLE/Gauge/WindowDenseTests.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests - -namespace OAI - -/-! Adapted tests need conditional estimates only inside the retained -scale window. The discarded early scales need no dense estimate. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem cover_window_failure_bound_on_window - {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (F : Filtration ℕ mΩ) (success : ℕ → Set Ω) - (n : ℕ) (hn : 1 ≤ n) - (hmeas : ∀ j ∈ coverTestWindow n, MeasurableSet[F (j + 1)] (success j)) - (hstep : ∀ j ∈ coverTestWindow n, ∀ B : Set Ω, MeasurableSet[F j] B → - ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ success j)) : - μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤ - ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by - classical - let S : ℕ → Set Ω := fun j => if j ∈ coverTestWindow n then success j else univ - have hS (j : ℕ) : MeasurableSet[F (j + 1)] (S j) := by - dsimp only [S] - split_ifs with hj - · exact hmeas j hj - · exact MeasurableSet.univ - have hp : ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) ≤ 1 := by - apply ENNReal.ofReal_le_one.mpr - have hnr : (1 : ℝ) ≤ n := by exact_mod_cast hn - have hs : 1 ≤ Real.sqrt (n : ℝ) := by - simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hnr - exact (div_le_one (zero_lt_one.trans_le hs)).mpr hs - have hstep' (j : ℕ) (B : Set Ω) (hB : MeasurableSet[F j] B) : - ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ S j) := by - dsimp only [S] - split_ifs with hj - · exact hstep j hj B hB - · simpa only [inter_univ, one_mul] using mul_le_mul' hp (le_refl (μ B)) - have he : (⋂ j ∈ coverTestWindow n, (S j)ᶜ) = - ⋂ j ∈ coverTestWindow n, (success j)ᶜ := by - apply iInter_congr - intro j - apply iInter_congr - intro hj - simp only [S, ite_eq_left hj] - simpa only [he] using cover_window_failure_bound μ F S hS n hn hstep' - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests + +namespace OAI + +/-! Adapted tests need conditional estimates only inside the retained +scale window. The discarded early scales need no dense estimate. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem cover_window_failure_bound_on_window + {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (F : Filtration ℕ mΩ) (success : ℕ → Set Ω) + (n : ℕ) (hn : 1 ≤ n) + (hmeas : ∀ j ∈ coverTestWindow n, MeasurableSet[F (j + 1)] (success j)) + (hstep : ∀ j ∈ coverTestWindow n, ∀ B : Set Ω, MeasurableSet[F j] B → + ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ success j)) : + μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤ + ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by + classical + let S : ℕ → Set Ω := fun j => if j ∈ coverTestWindow n then success j else univ + have hS (j : ℕ) : MeasurableSet[F (j + 1)] (S j) := by + dsimp only [S] + split_ifs with hj + · exact hmeas j hj + · exact MeasurableSet.univ + have hp : ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) ≤ 1 := by + apply ENNReal.ofReal_le_one.mpr + have hnr : (1 : ℝ) ≤ n := by exact_mod_cast hn + have hs : 1 ≤ Real.sqrt (n : ℝ) := by + simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hnr + exact (div_le_one (zero_lt_one.trans_le hs)).mpr hs + have hstep' (j : ℕ) (B : Set Ω) (hB : MeasurableSet[F j] B) : + ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ S j) := by + dsimp only [S] + split_ifs with hj + · exact hstep j hj B hB + · simpa only [inter_univ, one_mul] using mul_le_mul' hp (le_refl (μ B)) + have he : (⋂ j ∈ coverTestWindow n, (S j)ᶜ) = + ⋂ j ∈ coverTestWindow n, (success j)ᶜ := by + apply iInter_congr + intro j + apply iInter_congr + intro hj + simp only [S, ite_eq_left hj] + simpa only [he] using cover_window_failure_bound μ F S hS n hn hstep' + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/AccessGuideCollapse.lean b/lean/OAI/Probability/SLE/Geometry/AccessGuideCollapse.lean index a68a6c9db97ae6e82985c59d9c4f587a0f49e347..9a2cc4ca7565cb05e1c616011f58d43befceb55b 100644 --- a/lean/OAI/Probability/SLE/Geometry/AccessGuideCollapse.lean +++ b/lean/OAI/Probability/SLE/Geometry/AccessGuideCollapse.lean @@ -1,111 +1,111 @@ -import OAI.Probability.SLE.Loewner.SlitSupport -import OAI.Probability.SLE.StripProbabilities - -namespace OAI - -/-! Uniform collapse of access guides from the already proved conformal -localization estimate and pointwise slit support. An interior separated set -prevents arbitrarily small physical images of a unit-modulus access guide. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -theorem uniform_access_image_radius_lower_bound - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (T : ℝ≥0) (R : ℝ) (Q : Set ℂ) (hQ : Q ⊆ upperHalfPlane) - (hvisit : 0 < P {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q}) - (r₀ : ℝ) (hr₀ : 0 < r₀) : - ∃ rmin : ℝ, 0 < rmin ∧ - ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ), - DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane → - ContinuousOn Fbar {z : ℂ | 0 ≤ z.im} → - (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) → - ∀ (a : ℂ) (r : ℝ), 0 < r → - (∀ z ∈ Q, r₀ ≤ ‖F z - a‖) → - ∀ (η : ℝ → ℂ) (S : ℝ), 0 < S → ContinuousOn η (Icc 0 S) → - η 0 = 0 → ‖η S‖ = 1 → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) → - (∀ t ∈ Icc 0 S, ∃ w : ℂ, - Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w - a‖ ≤ r) → rmin ≤ r := by - obtain ⟨b, C, c, L₀, hb, _, hc, hL₀, hloc⟩ := - sourceUniformConformalLocalization κ hκ hκ8 - let E : Set Ω := {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q} - have ht : Tendsto (fun L : ℝ => ENNReal.ofReal (C * L ^ (-c))) atTop (𝓝 0) := by - simpa only [Function.comp_def, mul_zero, ENNReal.ofReal_zero] using - ENNReal.continuous_ofReal.continuousAt.tendsto.comp - ((tendsto_rpow_neg_atTop hc).const_mul C) - have hlarge := (eventually_ge_atTop L₀).and - (((tendsto_rpow_atTop hb).eventually_gt_atTop R).and - (ht.eventually (gt_mem_nhds hvisit))) - obtain ⟨L, hL, hLR, hLprob⟩ := hlarge.exists - have hLp : 0 < L := (zero_lt_one.trans hL₀).trans_le hL - refine ⟨r₀ / L, div_pos hr₀ hLp, ?_⟩ - intro F Fbar hF hinj hbar hext a r hr hsep η S hS hη hη0 hηS hupper himage - by_contra! hsmall - have hLr : L * r < r₀ := by - have hh := (lt_div_iff₀ hLp).mp hsmall - simpa only [mul_comm] using hh - have hbound := hloc Ω inferInstance P inferInstance γ hSLE F Fbar hF hinj - hbar hext a r hr η S hS hη hη0 hηS hupper himage L hL - have hsub : E ⊆ {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) (L ^ b) t ∧ - Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a (L * r)} := by - rintro ω ⟨hω, hωQ⟩ - refine ⟨T, ?_, ?_⟩ - · intro s hs - exact (hω s ⟨zero_le, hs.le⟩).trans hLR - · rintro ⟨w, hw, he⟩ - have heq : w = F (γ ω T) := - OnePoint.coe_injective (he.trans (hext _ (hQ hωQ))) - subst w - have hh : ‖F (γ ω T) - a‖ < L * r := by - simpa only [mem_ball, dist_eq_norm] using hw - exact (not_lt_of_ge (hsep _ hωQ)) (hh.trans hLr) - exact (not_lt_of_ge ((measure_mono hsub).trans hbound)) hLprob - -/-- The separated interior set and published pointwise support rule out the -bad sequence in the access-guide compactness argument. All map and guide -conditions refer to the actual complex maps, rather than to an abstract state. -/ -theorem no_shrinking_access_guides_of_published_support - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hpublished : PublishedSimpleTraceSupport P γ) - (T : ℝ≥0) (hT : 0 < T) (β : ℝ≥0 → ℂ) (U : ℝ≥0 → ℝ) - (hU : Continuous U) (hβ : IsCapacityTwoTrace U β) - (hinjβ : InjOn β (Icc 0 T)) (hβH : ∀ t ∈ Ioc 0 T, β t ∈ upperHalfPlane) - (R : ℝ) (hβR : β '' Icc 0 T ⊆ ball 0 R) - (Q : Set ℂ) (hQopen : IsOpen Q) (hQ : Q ⊆ upperHalfPlane) (hβQ : β T ∈ Q) - (F : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) - (hF : ∀ n, DifferentiableOn ℂ (F n) upperHalfPlane) - (hinj : ∀ n, InjOn (F n) upperHalfPlane) - (hbar : ∀ n, ContinuousOn (Fbar n) {z : ℂ | 0 ≤ z.im}) - (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (F n z : OnePoint ℂ)) - (a : ℕ → ℂ) (r : ℕ → ℝ) (hr : ∀ n, 0 < r n) - (hlim : Tendsto r atTop (𝓝 0)) (r₀ : ℝ) (hr₀ : 0 < r₀) - (hsep : ∀ᶠ n in atTop, ∀ z ∈ Q, r₀ ≤ ‖F n z - a n‖) - (η : ℕ → ℝ → ℂ) (S : ℕ → ℝ) (hS : ∀ n, 0 < S n) - (hη : ∀ n, ContinuousOn (η n) (Icc 0 (S n))) - (hη0 : ∀ n, η n 0 = 0) (hηS : ∀ n, ‖η n (S n)‖ = 1) - (hupper : ∀ n t, t ∈ Icc 0 (S n) → 0 ≤ (η n t).im) - (himage : ∀ n t, t ∈ Icc 0 (S n) → ∃ w : ℂ, - Fbar n (η n t) = (w : OnePoint ℂ) ∧ ‖w - a n‖ ≤ r n) : False := by - have hp := slit_tube_support_of_published_support P γ hpublished T hT β U hU hβ - hinjβ hβH (ball 0 R) Q isOpen_ball hQopen hβR hβQ - have hvisit : 0 < P {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q} := by - simpa only [mem_ball, dist_zero_right] using hp - obtain ⟨rmin, hrmin, hbound⟩ := uniform_access_image_radius_lower_bound - P κ hκ hκ8 γ hSLE T R Q hQ hvisit r₀ hr₀ - obtain ⟨n, hn, hsepn⟩ := ((hlim.eventually (gt_mem_nhds hrmin)).and hsep).exists - exact (not_lt_of_ge (hbound (F n) (Fbar n) (hF n) (hinj n) (hbar n) - (hext n) (a n) (r n) (hr n) hsepn (η n) (S n) (hS n) (hη n) - (hη0 n) (hηS n) (hupper n) (himage n))) hn - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitSupport +import OAI.Probability.SLE.StripProbabilities + +namespace OAI + +/-! Uniform collapse of access guides from the already proved conformal +localization estimate and pointwise slit support. An interior separated set +prevents arbitrarily small physical images of a unit-modulus access guide. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +theorem uniform_access_image_radius_lower_bound + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (T : ℝ≥0) (R : ℝ) (Q : Set ℂ) (hQ : Q ⊆ upperHalfPlane) + (hvisit : 0 < P {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q}) + (r₀ : ℝ) (hr₀ : 0 < r₀) : + ∃ rmin : ℝ, 0 < rmin ∧ + ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ), + DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane → + ContinuousOn Fbar {z : ℂ | 0 ≤ z.im} → + (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) → + ∀ (a : ℂ) (r : ℝ), 0 < r → + (∀ z ∈ Q, r₀ ≤ ‖F z - a‖) → + ∀ (η : ℝ → ℂ) (S : ℝ), 0 < S → ContinuousOn η (Icc 0 S) → + η 0 = 0 → ‖η S‖ = 1 → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) → + (∀ t ∈ Icc 0 S, ∃ w : ℂ, + Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w - a‖ ≤ r) → rmin ≤ r := by + obtain ⟨b, C, c, L₀, hb, _, hc, hL₀, hloc⟩ := + sourceUniformConformalLocalization κ hκ hκ8 + let E : Set Ω := {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q} + have ht : Tendsto (fun L : ℝ => ENNReal.ofReal (C * L ^ (-c))) atTop (𝓝 0) := by + simpa only [Function.comp_def, mul_zero, ENNReal.ofReal_zero] using + ENNReal.continuous_ofReal.continuousAt.tendsto.comp + ((tendsto_rpow_neg_atTop hc).const_mul C) + have hlarge := (eventually_ge_atTop L₀).and + (((tendsto_rpow_atTop hb).eventually_gt_atTop R).and + (ht.eventually (gt_mem_nhds hvisit))) + obtain ⟨L, hL, hLR, hLprob⟩ := hlarge.exists + have hLp : 0 < L := (zero_lt_one.trans hL₀).trans_le hL + refine ⟨r₀ / L, div_pos hr₀ hLp, ?_⟩ + intro F Fbar hF hinj hbar hext a r hr hsep η S hS hη hη0 hηS hupper himage + by_contra! hsmall + have hLr : L * r < r₀ := by + have hh := (lt_div_iff₀ hLp).mp hsmall + simpa only [mul_comm] using hh + have hbound := hloc Ω inferInstance P inferInstance γ hSLE F Fbar hF hinj + hbar hext a r hr η S hS hη hη0 hηS hupper himage L hL + have hsub : E ⊆ {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) (L ^ b) t ∧ + Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a (L * r)} := by + rintro ω ⟨hω, hωQ⟩ + refine ⟨T, ?_, ?_⟩ + · intro s hs + exact (hω s ⟨zero_le, hs.le⟩).trans hLR + · rintro ⟨w, hw, he⟩ + have heq : w = F (γ ω T) := + OnePoint.coe_injective (he.trans (hext _ (hQ hωQ))) + subst w + have hh : ‖F (γ ω T) - a‖ < L * r := by + simpa only [mem_ball, dist_eq_norm] using hw + exact (not_lt_of_ge (hsep _ hωQ)) (hh.trans hLr) + exact (not_lt_of_ge ((measure_mono hsub).trans hbound)) hLprob + +/-- The separated interior set and published pointwise support rule out the +bad sequence in the access-guide compactness argument. All map and guide +conditions refer to the actual complex maps, rather than to an abstract state. -/ +theorem no_shrinking_access_guides_of_published_support + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hpublished : PublishedSimpleTraceSupport P γ) + (T : ℝ≥0) (hT : 0 < T) (β : ℝ≥0 → ℂ) (U : ℝ≥0 → ℝ) + (hU : Continuous U) (hβ : IsCapacityTwoTrace U β) + (hinjβ : InjOn β (Icc 0 T)) (hβH : ∀ t ∈ Ioc 0 T, β t ∈ upperHalfPlane) + (R : ℝ) (hβR : β '' Icc 0 T ⊆ ball 0 R) + (Q : Set ℂ) (hQopen : IsOpen Q) (hQ : Q ⊆ upperHalfPlane) (hβQ : β T ∈ Q) + (F : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) + (hF : ∀ n, DifferentiableOn ℂ (F n) upperHalfPlane) + (hinj : ∀ n, InjOn (F n) upperHalfPlane) + (hbar : ∀ n, ContinuousOn (Fbar n) {z : ℂ | 0 ≤ z.im}) + (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (F n z : OnePoint ℂ)) + (a : ℕ → ℂ) (r : ℕ → ℝ) (hr : ∀ n, 0 < r n) + (hlim : Tendsto r atTop (𝓝 0)) (r₀ : ℝ) (hr₀ : 0 < r₀) + (hsep : ∀ᶠ n in atTop, ∀ z ∈ Q, r₀ ≤ ‖F n z - a n‖) + (η : ℕ → ℝ → ℂ) (S : ℕ → ℝ) (hS : ∀ n, 0 < S n) + (hη : ∀ n, ContinuousOn (η n) (Icc 0 (S n))) + (hη0 : ∀ n, η n 0 = 0) (hηS : ∀ n, ‖η n (S n)‖ = 1) + (hupper : ∀ n t, t ∈ Icc 0 (S n) → 0 ≤ (η n t).im) + (himage : ∀ n t, t ∈ Icc 0 (S n) → ∃ w : ℂ, + Fbar n (η n t) = (w : OnePoint ℂ) ∧ ‖w - a n‖ ≤ r n) : False := by + have hp := slit_tube_support_of_published_support P γ hpublished T hT β U hU hβ + hinjβ hβH (ball 0 R) Q isOpen_ball hQopen hβR hβQ + have hvisit : 0 < P {ω | (∀ t ∈ Icc 0 T, ‖γ ω t‖ < R) ∧ γ ω T ∈ Q} := by + simpa only [mem_ball, dist_zero_right] using hp + obtain ⟨rmin, hrmin, hbound⟩ := uniform_access_image_radius_lower_bound + P κ hκ hκ8 γ hSLE T R Q hQ hvisit r₀ hr₀ + obtain ⟨n, hn, hsepn⟩ := ((hlim.eventually (gt_mem_nhds hrmin)).and hsep).exists + exact (not_lt_of_ge (hbound (F n) (Fbar n) (hF n) (hinj n) (hbar n) + (hext n) (a n) (r n) (hr n) hsepn (η n) (S n) (hS n) (hη n) + (hη0 n) (hηS n) (hupper n) (himage n))) hn + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualCompactAvoidance.lean b/lean/OAI/Probability/SLE/Geometry/ActualCompactAvoidance.lean index d7839c75568e0cb697a83d7ebedacb6d05aa727b..b030afb11ea4d8f0d61b8808697af983d3948afe 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualCompactAvoidance.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualCompactAvoidance.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Loewner.LoewnerCompactDriverNeighborhood -import OAI.Probability.SLE.Stochastic.BrownianFiniteTube -import OAI.Probability.SLE.Geometry.ActualFirstHeightSupport - -namespace OAI - -/-! Positive probability of preserving a compact surviving set of a -prescribed continuous Loewner driver. This controls compact interior -obstacles in the geometric-support construction. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem regular_scaled_driver_tube_positive - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) - (T : ℝ≥0) (hT : 0 < T) {δ : ℝ} (hδ : 0 < δ) : - 0 < (P.comap (Subtype.val : a.samples → Ω)) - {ω | ∀ t ∈ Icc 0 T, |Real.sqrt κ*a.B t ω-V t| < δ} := by - have hroot : 0 < Real.sqrt κ := Real.sqrt_pos.mpr hκ - let f : C(ℝ≥0,ℝ) := ⟨fun t => V t / Real.sqrt κ, - V.continuous.div_const _⟩ - have hf0 : f 0=0 := by simp [f,hV0] - have hp := brownian_finite_uniform_tube_positive a.B a.brownian a.measurable_B - a.continuous_B a.zero_B T hT f hf0 (div_pos hδ hroot) - apply hp.trans_le (measure_mono ?_) - intro ω hω t ht - have hh := hω t ht - change |a.B t ω-V t / Real.sqrt κ| < δ/Real.sqrt κ at hh - have he : Real.sqrt κ*a.B t ω-V t = - Real.sqrt κ*(a.B t ω-V t/Real.sqrt κ) := by field_simp - rw [he,abs_mul,abs_of_pos hroot] - simpa only [mul_comm] using (lt_div_iff₀ hroot).mp hh - -theorem regular_compact_survival_positive - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) - {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) (hT : 0 < T) - {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : - 0 < (P.comap (Subtype.val : a.samples → Ω)) - {ω | K ⊆ survivingDomain (γ ω) T} := by - obtain ⟨δ,hδ,hkeep⟩ := driver_neighborhood_preserves_compact hv T hK hKD - have hp := regular_scaled_driver_tube_positive a hκ V hV0 T hT hδ - apply hp.trans_le (measure_mono ?_) - intro ω hω - apply hkeep (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) - exact fun t ht => (hω t ht).le - -theorem ordinary_compact_survival_positive - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) - (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) - {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) (hT : 0 < T) - {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : - 0 < P {ω | K ⊆ survivingDomain (γ ω) T} := by - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - have hp := regular_compact_survival_positive a hκ V hV0 hv T hT hK hKD - apply hp.trans_le - have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) - a.measurePreserving_val.measurable.aemeasurable - {ω | K ⊆ survivingDomain (γ ω) T} - rw [a.measurePreserving_val.map_eq] at hh - convert! hh using 1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerCompactDriverNeighborhood +import OAI.Probability.SLE.Stochastic.BrownianFiniteTube +import OAI.Probability.SLE.Geometry.ActualFirstHeightSupport + +namespace OAI + +/-! Positive probability of preserving a compact surviving set of a +prescribed continuous Loewner driver. This controls compact interior +obstacles in the geometric-support construction. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem regular_scaled_driver_tube_positive + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) + (T : ℝ≥0) (hT : 0 < T) {δ : ℝ} (hδ : 0 < δ) : + 0 < (P.comap (Subtype.val : a.samples → Ω)) + {ω | ∀ t ∈ Icc 0 T, |Real.sqrt κ*a.B t ω-V t| < δ} := by + have hroot : 0 < Real.sqrt κ := Real.sqrt_pos.mpr hκ + let f : C(ℝ≥0,ℝ) := ⟨fun t => V t / Real.sqrt κ, + V.continuous.div_const _⟩ + have hf0 : f 0=0 := by simp [f,hV0] + have hp := brownian_finite_uniform_tube_positive a.B a.brownian a.measurable_B + a.continuous_B a.zero_B T hT f hf0 (div_pos hδ hroot) + apply hp.trans_le (measure_mono ?_) + intro ω hω t ht + have hh := hω t ht + change |a.B t ω-V t / Real.sqrt κ| < δ/Real.sqrt κ at hh + have he : Real.sqrt κ*a.B t ω-V t = + Real.sqrt κ*(a.B t ω-V t/Real.sqrt κ) := by field_simp + rw [he,abs_mul,abs_of_pos hroot] + simpa only [mul_comm] using (lt_div_iff₀ hroot).mp hh + +theorem regular_compact_survival_positive + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) + {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) (hT : 0 < T) + {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : + 0 < (P.comap (Subtype.val : a.samples → Ω)) + {ω | K ⊆ survivingDomain (γ ω) T} := by + obtain ⟨δ,hδ,hkeep⟩ := driver_neighborhood_preserves_compact hv T hK hKD + have hp := regular_scaled_driver_tube_positive a hκ V hV0 T hT hδ + apply hp.trans_le (measure_mono ?_) + intro ω hω + apply hkeep (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) + exact fun t ht => (hω t ht).le + +theorem ordinary_compact_survival_positive + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) + (V : C(ℝ≥0,ℝ)) (hV0 : V 0=0) + {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) (hT : 0 < T) + {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : + 0 < P {ω | K ⊆ survivingDomain (γ ω) T} := by + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + have hp := regular_compact_survival_positive a hκ V hV0 hv T hT hK hKD + apply hp.trans_le + have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) + a.measurePreserving_val.measurable.aemeasurable + {ω | K ⊆ survivingDomain (γ ω) T} + rw [a.measurePreserving_val.map_eq] at hh + convert! hh using 1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualConformalExitLaw.lean b/lean/OAI/Probability/SLE/Geometry/ActualConformalExitLaw.lean index 96aece0426a483874e09fb69154138e60fa9e872..aa5c5112fed5e552b6265dae0ef49a524afa1c14 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualConformalExitLaw.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualConformalExitLaw.lean @@ -1,43 +1,43 @@ -import OAI.Probability.SLE.Geometry.UnitWindowPlanarExit - -namespace OAI - -/-! The published finite-window conformal exit input, proved from the -original planar Wiener law and bounded harmonic localization. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Target RibbonCoordinates - -lemma unitWindowExitArc_subset_planar (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ) (positive : Bool) : - unitWindowExitArc D g w (if positive then 1 else -1) ⊆ - planarHalfPlaneExitArc D g w 1 positive := by - rintro b ⟨τ,hτ,x,hx,hl⟩ - obtain ⟨hS,hS1,hp,he⟩ := unitWindowFirstExit_planar hτ - refine ⟨unitWindowStop τ,hS,hS1,hp,he,x,?_,?_⟩ - · cases positive <;> simp only [Bool.false_eq_true,↓reduceIte] at hx ⊢ <;> linarith - · exact (unitWindow_map_limit_iff w b g hτ.1 (x:ℂ)).mp hl - -theorem actual_halfPlaneConformalExitLaw : PublishedHalfPlaneConformalExitLaw := by - intro W hW D g f hD _hDH hg hf hgH hfD hfg _hgf w hw - refine ⟨?_,?_,?_⟩ - · have hl := halfPlane_finite_exit_limit W hW hD hg hf hgH hfD hfg w hw 1 - filter_upwards [hl] with b hb - intro τ hτ - obtain ⟨hS,hS1,hp,he⟩ := unitWindowFirstExit_planar hτ - obtain ⟨x,hx,hm⟩ := hb (unitWindowStop τ) hS hS1 hp he - exact ⟨x,hx,(unitWindow_map_limit_iff w b g hτ.1 (x:ℂ)).mpr hm⟩ - · have hh := (measure_mono (unitWindowExitArc_subset_planar D g w false)).trans - (planarHalfPlaneExitArc_bound W hW hD w hw hg hgH 1 false) - simpa only [Bool.false_eq_true,↓reduceIte,halfPlaneExitWeight] using hh - · have hh := (measure_mono (unitWindowExitArc_subset_planar D g w true)).trans - (planarHalfPlaneExitArc_bound W hW hD w hw hg hgH 1 true) - simpa only [↓reduceIte,halfPlaneExitWeight] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.UnitWindowPlanarExit + +namespace OAI + +/-! The published finite-window conformal exit input, proved from the +original planar Wiener law and bounded harmonic localization. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target RibbonCoordinates + +lemma unitWindowExitArc_subset_planar (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ) (positive : Bool) : + unitWindowExitArc D g w (if positive then 1 else -1) ⊆ + planarHalfPlaneExitArc D g w 1 positive := by + rintro b ⟨τ,hτ,x,hx,hl⟩ + obtain ⟨hS,hS1,hp,he⟩ := unitWindowFirstExit_planar hτ + refine ⟨unitWindowStop τ,hS,hS1,hp,he,x,?_,?_⟩ + · cases positive <;> simp only [Bool.false_eq_true,↓reduceIte] at hx ⊢ <;> linarith + · exact (unitWindow_map_limit_iff w b g hτ.1 (x:ℂ)).mp hl + +theorem actual_halfPlaneConformalExitLaw : PublishedHalfPlaneConformalExitLaw := by + intro W hW D g f hD _hDH hg hf hgH hfD hfg _hgf w hw + refine ⟨?_,?_,?_⟩ + · have hl := halfPlane_finite_exit_limit W hW hD hg hf hgH hfD hfg w hw 1 + filter_upwards [hl] with b hb + intro τ hτ + obtain ⟨hS,hS1,hp,he⟩ := unitWindowFirstExit_planar hτ + obtain ⟨x,hx,hm⟩ := hb (unitWindowStop τ) hS hS1 hp he + exact ⟨x,hx,(unitWindow_map_limit_iff w b g hτ.1 (x:ℂ)).mpr hm⟩ + · have hh := (measure_mono (unitWindowExitArc_subset_planar D g w false)).trans + (planarHalfPlaneExitArc_bound W hW hD w hw hg hgH 1 false) + simpa only [Bool.false_eq_true,↓reduceIte,halfPlaneExitWeight] using hh + · have hh := (measure_mono (unitWindowExitArc_subset_planar D g w true)).trans + (planarHalfPlaneExitArc_bound W hW hD w hw hg hgH 1 true) + simpa only [↓reduceIte,halfPlaneExitWeight] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualCrosscutStraightening.lean b/lean/OAI/Probability/SLE/Geometry/ActualCrosscutStraightening.lean index 645bd0ad6e1435edb50945293db3a86ae2267899..dc8045014b300a794613430f0c12595ed0234b89 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualCrosscutStraightening.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualCrosscutStraightening.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening -import OAI.Probability.SLE.Geometry.DiskCrosscutHomeomorph -import OAI.Probability.SLE.Geometry.HalfPlaneDiskTransport -import OAI.Probability.SLE.Geometry.ClosedHalfPlaneExtension -import OAI.Probability.SLE.Geometry.CrosscutParameterInverse - -namespace OAI - -/-! The relative half-plane crosscut theorem from the specified-boundary -Schönflies extension. -/ -noncomputable section -open Set Complex Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem actual_halfPlaneCrosscutStraightening : PublishedHalfPlaneCrosscutStraightening := by - intro β hc hi h0 hH hinfty - have hβ (t : ℝ≥0) : 0 ≤ (β t).im := by - by_cases ht : t=0 - · simp [ht,h0] - · exact (hH t (pos_iff_ne_zero.mpr ht)).le - obtain ⟨hpc,hpi,hpint⟩ := compactifiedCrosscut_properties β hc hi h0 hH hinfty - obtain ⟨e,hep,hef⟩ := disk_crosscut_homeomorph (compactifiedCrosscut β) hpc hpi - (compactifiedCrosscut_zero β h0) (compactifiedCrosscut_one β) hpint - have he1 : e ⟨1,by norm_num⟩=⟨1,by norm_num⟩ := - Subtype.ext (hef ⟨1,by norm_num⟩ (by norm_num)) - obtain ⟨F,hF,hFr,hFH⟩ := closedHalfPlane_extension (halfPlaneDiskTransport e he1) - (halfPlaneDiskTransport_boundary e he1 hef) - refine ⟨F,hFr,hFH,?_⟩ - intro t - rw [hF (β t) (hβ t),halfPlaneDiskTransport_apply] - have hs := rayDiskParameter_mem t - have hpt := compactifiedCrosscut_rayDiskParameter β t - have hball : halfPlaneCayley (β t) ∈ closedBall (0:ℂ) 1 := by - simpa only [mem_closedBall,dist_zero_right] using halfPlaneCayley_norm_le (hβ t) - have hps : compactifiedCrosscut β (rayDiskParameter t) ∈ closedBall (0:ℂ) 1 := - hpt.symm ▸ hball - have heq := hep (rayDiskParameter t) ⟨hs.1,hs.2.le⟩ hps - have heq' : (e ⟨halfPlaneCayley (β t),hball⟩:ℂ)=diskDiameter (rayDiskParameter t) := by - simpa only [hpt] using heq - rw [heq',diskDiameter_rayDiskParameter,halfPlaneCayleyInv_cayley] - simp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening +import OAI.Probability.SLE.Geometry.DiskCrosscutHomeomorph +import OAI.Probability.SLE.Geometry.HalfPlaneDiskTransport +import OAI.Probability.SLE.Geometry.ClosedHalfPlaneExtension +import OAI.Probability.SLE.Geometry.CrosscutParameterInverse + +namespace OAI + +/-! The relative half-plane crosscut theorem from the specified-boundary +Schönflies extension. -/ +noncomputable section +open Set Complex Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem actual_halfPlaneCrosscutStraightening : PublishedHalfPlaneCrosscutStraightening := by + intro β hc hi h0 hH hinfty + have hβ (t : ℝ≥0) : 0 ≤ (β t).im := by + by_cases ht : t=0 + · simp [ht,h0] + · exact (hH t (pos_iff_ne_zero.mpr ht)).le + obtain ⟨hpc,hpi,hpint⟩ := compactifiedCrosscut_properties β hc hi h0 hH hinfty + obtain ⟨e,hep,hef⟩ := disk_crosscut_homeomorph (compactifiedCrosscut β) hpc hpi + (compactifiedCrosscut_zero β h0) (compactifiedCrosscut_one β) hpint + have he1 : e ⟨1,by norm_num⟩=⟨1,by norm_num⟩ := + Subtype.ext (hef ⟨1,by norm_num⟩ (by norm_num)) + obtain ⟨F,hF,hFr,hFH⟩ := closedHalfPlane_extension (halfPlaneDiskTransport e he1) + (halfPlaneDiskTransport_boundary e he1 hef) + refine ⟨F,hFr,hFH,?_⟩ + intro t + rw [hF (β t) (hβ t),halfPlaneDiskTransport_apply] + have hs := rayDiskParameter_mem t + have hpt := compactifiedCrosscut_rayDiskParameter β t + have hball : halfPlaneCayley (β t) ∈ closedBall (0:ℂ) 1 := by + simpa only [mem_closedBall,dist_zero_right] using halfPlaneCayley_norm_le (hβ t) + have hps : compactifiedCrosscut β (rayDiskParameter t) ∈ closedBall (0:ℂ) 1 := + hpt.symm ▸ hball + have heq := hep (rayDiskParameter t) ⟨hs.1,hs.2.le⟩ hps + have heq' : (e ⟨halfPlaneCayley (β t),hball⟩:ℂ)=diskDiameter (rayDiskParameter t) := by + simpa only [hpt] using heq + rw [heq',diskDiameter_rayDiskParameter,halfPlaneCayleyInv_cayley] + simp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualExplicitMain.lean b/lean/OAI/Probability/SLE/Geometry/ActualExplicitMain.lean index cb2654518cfd4c4e41062636a0d3d6f94f4e7035..f9ed21eed0569368774c15e3b25682f3e133c7a4 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualExplicitMain.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualExplicitMain.lean @@ -1,101 +1,101 @@ -import OAI.Probability.SLE.Gauge.ActualDenseCoverProbability -import OAI.Probability.SLE.Gauge.DenseCoverExpectation -import OAI.Probability.SLE.Geometry.SpatialExhaustion - -namespace OAI - -/-! The explicit-gauge main theorem for a regular Brownian-driven Loewner -presentation, conditional on the precise published inputs used in the -geometric and stochastic construction. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Target - -local instance actualMainDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance actualMainDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -lemma compact_upperHalfPlane_height_lower_bound {K : Set ℂ} - (hK : IsCompact K) (hH : K ⊆ upperHalfPlane) : - ∃ η : ℝ, 0 < η ∧ ∀ z ∈ K, η ≤ z.im := by - by_cases hne : K.Nonempty - · obtain ⟨z, hz, hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn - exact ⟨z.im, hH hz, hmin⟩ - · refine ⟨1, zero_lt_one, ?_⟩ - intro z hz - exact False.elim (hne ⟨z, hz⟩) - -theorem regular_driver_gauge_mainConclusion - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hγm : ∀ t, Measurable (fun ω => γ ω t)) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) : - MainConclusion P h γ := by - have hSLE : IsOrdinaryChordalSLE κ γ P := - ⟨hγm, B, hB, Eventually.of_forall (fun ω => ⟨G ω, F ω, hw ω⟩)⟩ - obtain ⟨T, b, c, hT, hb, hcpos, v, hv, _, hprob⟩ := - uniform_actual_dense_cover_probability P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw - htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW - obtain ⟨C, _, hbound⟩ := expected_hausdorff_of_dense_candidates P κ hκ hκ8 γ hSLE - (fun ω => (hw ω).1) hT hb hcpos h hh hf - apply mainConclusion_of_uniform_compact_interior_bounds P κ hκ hκ8 γ hSLE h hh hf - intro R hR - refine ⟨(C : ℝ≥0∞) * wholeSausageChargeBound κ (R + 3), ?_, ?_⟩ - · exact ENNReal.mul_lt_top ENNReal.coe_lt_top (wholeSausageChargeBound_lt_top κ _ hκ hκ8) - · intro t K hK hKR - obtain ⟨η, hη, hheight⟩ := compact_upperHalfPlane_height_lower_bound hK - (fun z hz => (hKR hz).1) - obtain ⟨D, hD, hmiss⟩ := hprob η hη - have hwhole := hbound R hR.le K (fun z hz => (hKR hz).2) v hv D hD.le - (by simpa only [denseMissedProbabilityBound] using hmiss R K (fun z hz => (hKR hz).2) hheight) - refine (lintegral_mono fun ω => measure_mono ?_).trans hwhole - intro z hz - exact ⟨hz.1, image_subset_range (γ ω) (Icc 0 t) hz.2⟩ - -/-- The regular-driver form of the explicit exact-gauge result. The same -conclusion includes finite expected measure in every bounded disk. -/ -theorem regular_driver_explicit_mainConclusion - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hγm : ∀ t, Measurable (fun ω => γ ω t)) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) : - MainConclusion P (explicitGauge κ) γ := - regular_driver_gauge_mainConclusion P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw - htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW - (explicitGauge κ) (explicitGauge_isGauge hκ) (explicitGauge_small_formula κ) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ActualDenseCoverProbability +import OAI.Probability.SLE.Gauge.DenseCoverExpectation +import OAI.Probability.SLE.Geometry.SpatialExhaustion + +namespace OAI + +/-! The explicit-gauge main theorem for a regular Brownian-driven Loewner +presentation, conditional on the precise published inputs used in the +geometric and stochastic construction. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target + +local instance actualMainDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance actualMainDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +lemma compact_upperHalfPlane_height_lower_bound {K : Set ℂ} + (hK : IsCompact K) (hH : K ⊆ upperHalfPlane) : + ∃ η : ℝ, 0 < η ∧ ∀ z ∈ K, η ≤ z.im := by + by_cases hne : K.Nonempty + · obtain ⟨z, hz, hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn + exact ⟨z.im, hH hz, hmin⟩ + · refine ⟨1, zero_lt_one, ?_⟩ + intro z hz + exact False.elim (hne ⟨z, hz⟩) + +theorem regular_driver_gauge_mainConclusion + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hγm : ∀ t, Measurable (fun ω => γ ω t)) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) : + MainConclusion P h γ := by + have hSLE : IsOrdinaryChordalSLE κ γ P := + ⟨hγm, B, hB, Eventually.of_forall (fun ω => ⟨G ω, F ω, hw ω⟩)⟩ + obtain ⟨T, b, c, hT, hb, hcpos, v, hv, _, hprob⟩ := + uniform_actual_dense_cover_probability P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw + htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW + obtain ⟨C, _, hbound⟩ := expected_hausdorff_of_dense_candidates P κ hκ hκ8 γ hSLE + (fun ω => (hw ω).1) hT hb hcpos h hh hf + apply mainConclusion_of_uniform_compact_interior_bounds P κ hκ hκ8 γ hSLE h hh hf + intro R hR + refine ⟨(C : ℝ≥0∞) * wholeSausageChargeBound κ (R + 3), ?_, ?_⟩ + · exact ENNReal.mul_lt_top ENNReal.coe_lt_top (wholeSausageChargeBound_lt_top κ _ hκ hκ8) + · intro t K hK hKR + obtain ⟨η, hη, hheight⟩ := compact_upperHalfPlane_height_lower_bound hK + (fun z hz => (hKR hz).1) + obtain ⟨D, hD, hmiss⟩ := hprob η hη + have hwhole := hbound R hR.le K (fun z hz => (hKR hz).2) v hv D hD.le + (by simpa only [denseMissedProbabilityBound] using hmiss R K (fun z hz => (hKR hz).2) hheight) + refine (lintegral_mono fun ω => measure_mono ?_).trans hwhole + intro z hz + exact ⟨hz.1, image_subset_range (γ ω) (Icc 0 t) hz.2⟩ + +/-- The regular-driver form of the explicit exact-gauge result. The same +conclusion includes finite expected measure in every bounded disk. -/ +theorem regular_driver_explicit_mainConclusion + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hγm : ∀ t, Measurable (fun ω => γ ω t)) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) : + MainConclusion P (explicitGauge κ) γ := + regular_driver_gauge_mainConclusion P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw + htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW + (explicitGauge κ) (explicitGauge_isGauge hκ) (explicitGauge_small_formula κ) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualExplicitTransfer.lean b/lean/OAI/Probability/SLE/Geometry/ActualExplicitTransfer.lean index 3fe224dc6809fdaae954edee49d20003e1f3ae1d..1d16f749df5ba38e0dae4f7fac3ed07dd28c14fd 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualExplicitTransfer.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualExplicitTransfer.lean @@ -1,164 +1,164 @@ -import OAI.Probability.SLE.Geometry.ActualExplicitMain - -namespace OAI - -/-! Passage from a regular presentation to an arbitrary ordinary SLE. -The subtype restriction preserves the published geometric support input, -and whole-trace expectations transfer through its measure-preserving map. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Target - -local instance explicitTransferDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance explicitTransferDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -/-- The same continuous Brownian path, retaining its pointwise zero start. -/ -def sourceBasedDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) - (ω : Ω) : Driver where - toContinuousMap := sourceDriverPath B hc ω - map_zero' := h0 ω - -lemma sourceBasedDriverPath_measurable - {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ) - (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) : - Measurable (sourceBasedDriverPath B hc h0) := by - apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp - exact source_path_measurable _ hm - -/-- An ordinary regular presentation supplies the based Wiener law needed -by the two-dimensional Brownian support input. -/ -theorem sourceBasedDriverLaw_isWiener - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) : - IsWiener (P.map (sourceBasedDriverPath B hc h0)) := by - have hbp := sourceBasedDriverPath_measurable B hm hc h0 - refine ⟨⟨fun I => ?_⟩, Eventually.of_forall (fun b => map_continuous b)⟩ - have hI : Measurable (fun b : Driver => I.restrict (fun t => b t)) := - Measurable.of_eval (fun i => (continuous_eval_const (i : ℝ≥0)).measurable) - refine ⟨hI.aemeasurable, ?_⟩ - rw [Measure.map_map hI hbp] - exact (hB.hasLaw I).map_eq - -theorem RegularSLEPresentation.geometric_support - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hsupport : PublishedGeometricTubeSupport P γ) : - PublishedGeometricTubeSupport (P.comap (Subtype.val : a.samples → Ω)) - (fun ω => γ ω) := by - intro z p hp hpH N V hN hV hpN hzV - let E : Set Ω := {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) N ∧ γ ω T ∈ V} - have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := - MeasurableEmbedding.subtype_coe a.measurable_samples - have he : (P.comap (Subtype.val : a.samples → Ω)) - ((Subtype.val : a.samples → Ω) ⁻¹' E) = P E := by - rw [← hemb.map_apply, a.measurePreserving_val.map_eq] - change 0 < (P.comap (Subtype.val : a.samples → Ω)) - ((Subtype.val : a.samples → Ω) ⁻¹' E) - rw [he] - exact hsupport z p hp hpH N V hN hV hpN hzV - -theorem ordinary_explicit_main_of_regular_presentation - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (a : RegularSLEPresentation P κ γ) - (htrace : PublishedTraceGeneration - ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) - κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) : - MainConclusion P (explicitGauge κ) γ := by - let Q := P.comap (Subtype.val : a.samples → Ω) - let : IsProbabilityMeasure Q := a.probability - have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) := - measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B - have hmain := regular_driver_explicit_mainConclusion Q κ hκ hκ8 a.B a.brownian hBm - a.continuous_B a.zero_B (fun ω => γ ω) a.G a.F - (fun t => (hSLE.1 t).comp measurable_subtype_coe) a.witness htrace - hcompact hboundary (a.geometric_support hsupport) hplanar hstraight hexit hwiener W hW - apply (mainConclusion_iff_finiteExpectedWholeTrace P κ hκ hκ8 γ hSLE - (explicitGauge κ) (explicitGauge_isGauge hκ) (explicitGauge_small_formula κ)).mpr - intro R hR - have hb := finiteExpectedWholeTrace_of_mainConclusion Q (explicitGauge κ) - (fun ω => γ ω) hmain R hR - have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := - MeasurableEmbedding.subtype_coe a.measurable_samples - have he := a.measurePreserving_val.lintegral_comp_emb hemb - (fun ω => hausdorffGauge (explicitGauge κ) (wholeTrace (γ ω) ∩ closedBall 0 R)) - exact he.symm.trans_lt hb - -/-- A fixed choice is used only to state the precise canonical-driver law -on which the Rohde--Schramm trace-generation input is needed. -/ -def ordinaryRegularPresentation - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) : - RegularSLEPresentation P κ γ := Classical.choice (exists_regular_sle_presentation hSLE) - -def ordinaryRegularDriverLaw - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) : Measure C(ℝ≥0, ℝ) := - let a := ordinaryRegularPresentation hSLE - (P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B) - -/-- The explicit exact-gauge main conclusion for any ordinary chordal SLE, -conditional on the same eight published inputs as the regular theorem. -/ -theorem ordinary_explicit_mainConclusion - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) : - MainConclusion P (explicitGauge κ) γ := - ordinary_explicit_main_of_regular_presentation P κ hκ hκ8 γ hSLE - (ordinaryRegularPresentation hSLE) htrace hcompact hboundary hsupport hplanar hstraight - hexit hwiener W hW - -/-- The final ordinary-SLE theorem has no auxiliary Wiener-law assumption: -that law is constructed from the chosen regular Brownian presentation. -/ -theorem ordinary_explicit_mainConclusion_of_published_inputs - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hwiener : PublishedPlanarWienerUnitSupport) : - MainConclusion P (explicitGauge κ) γ := by - let a := ordinaryRegularPresentation hSLE - let W : WienerLaw := (P.comap (Subtype.val : a.samples → Ω)).map - (sourceBasedDriverPath a.B a.continuous_B a.zero_B) - have hW : IsWiener W := sourceBasedDriverLaw_isWiener a.B a.brownian - a.measurable_B a.continuous_B a.zero_B - exact ordinary_explicit_mainConclusion P κ hκ hκ8 γ hSLE htrace hcompact hboundary - hsupport hplanar hstraight hexit hwiener W hW - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ActualExplicitMain + +namespace OAI + +/-! Passage from a regular presentation to an arbitrary ordinary SLE. +The subtype restriction preserves the published geometric support input, +and whole-trace expectations transfer through its measure-preserving map. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target + +local instance explicitTransferDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance explicitTransferDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +/-- The same continuous Brownian path, retaining its pointwise zero start. -/ +def sourceBasedDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) + (ω : Ω) : Driver where + toContinuousMap := sourceDriverPath B hc ω + map_zero' := h0 ω + +lemma sourceBasedDriverPath_measurable + {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ) + (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) : + Measurable (sourceBasedDriverPath B hc h0) := by + apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp + exact source_path_measurable _ hm + +/-- An ordinary regular presentation supplies the based Wiener law needed +by the two-dimensional Brownian support input. -/ +theorem sourceBasedDriverLaw_isWiener + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) : + IsWiener (P.map (sourceBasedDriverPath B hc h0)) := by + have hbp := sourceBasedDriverPath_measurable B hm hc h0 + refine ⟨⟨fun I => ?_⟩, Eventually.of_forall (fun b => map_continuous b)⟩ + have hI : Measurable (fun b : Driver => I.restrict (fun t => b t)) := + Measurable.of_eval (fun i => (continuous_eval_const (i : ℝ≥0)).measurable) + refine ⟨hI.aemeasurable, ?_⟩ + rw [Measure.map_map hI hbp] + exact (hB.hasLaw I).map_eq + +theorem RegularSLEPresentation.geometric_support + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hsupport : PublishedGeometricTubeSupport P γ) : + PublishedGeometricTubeSupport (P.comap (Subtype.val : a.samples → Ω)) + (fun ω => γ ω) := by + intro z p hp hpH N V hN hV hpN hzV + let E : Set Ω := {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) N ∧ γ ω T ∈ V} + have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := + MeasurableEmbedding.subtype_coe a.measurable_samples + have he : (P.comap (Subtype.val : a.samples → Ω)) + ((Subtype.val : a.samples → Ω) ⁻¹' E) = P E := by + rw [← hemb.map_apply, a.measurePreserving_val.map_eq] + change 0 < (P.comap (Subtype.val : a.samples → Ω)) + ((Subtype.val : a.samples → Ω) ⁻¹' E) + rw [he] + exact hsupport z p hp hpH N V hN hV hpN hzV + +theorem ordinary_explicit_main_of_regular_presentation + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (a : RegularSLEPresentation P κ γ) + (htrace : PublishedTraceGeneration + ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B)) + κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) : + MainConclusion P (explicitGauge κ) γ := by + let Q := P.comap (Subtype.val : a.samples → Ω) + let : IsProbabilityMeasure Q := a.probability + have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) := + measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B + have hmain := regular_driver_explicit_mainConclusion Q κ hκ hκ8 a.B a.brownian hBm + a.continuous_B a.zero_B (fun ω => γ ω) a.G a.F + (fun t => (hSLE.1 t).comp measurable_subtype_coe) a.witness htrace + hcompact hboundary (a.geometric_support hsupport) hplanar hstraight hexit hwiener W hW + apply (mainConclusion_iff_finiteExpectedWholeTrace P κ hκ hκ8 γ hSLE + (explicitGauge κ) (explicitGauge_isGauge hκ) (explicitGauge_small_formula κ)).mpr + intro R hR + have hb := finiteExpectedWholeTrace_of_mainConclusion Q (explicitGauge κ) + (fun ω => γ ω) hmain R hR + have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) := + MeasurableEmbedding.subtype_coe a.measurable_samples + have he := a.measurePreserving_val.lintegral_comp_emb hemb + (fun ω => hausdorffGauge (explicitGauge κ) (wholeTrace (γ ω) ∩ closedBall 0 R)) + exact he.symm.trans_lt hb + +/-- A fixed choice is used only to state the precise canonical-driver law +on which the Rohde--Schramm trace-generation input is needed. -/ +def ordinaryRegularPresentation + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) : + RegularSLEPresentation P κ γ := Classical.choice (exists_regular_sle_presentation hSLE) + +def ordinaryRegularDriverLaw + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) : Measure C(ℝ≥0, ℝ) := + let a := ordinaryRegularPresentation hSLE + (P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B) + +/-- The explicit exact-gauge main conclusion for any ordinary chordal SLE, +conditional on the same eight published inputs as the regular theorem. -/ +theorem ordinary_explicit_mainConclusion + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) : + MainConclusion P (explicitGauge κ) γ := + ordinary_explicit_main_of_regular_presentation P κ hκ hκ8 γ hSLE + (ordinaryRegularPresentation hSLE) htrace hcompact hboundary hsupport hplanar hstraight + hexit hwiener W hW + +/-- The final ordinary-SLE theorem has no auxiliary Wiener-law assumption: +that law is constructed from the chosen regular Brownian presentation. -/ +theorem ordinary_explicit_mainConclusion_of_published_inputs + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hwiener : PublishedPlanarWienerUnitSupport) : + MainConclusion P (explicitGauge κ) γ := by + let a := ordinaryRegularPresentation hSLE + let W : WienerLaw := (P.comap (Subtype.val : a.samples → Ω)).map + (sourceBasedDriverPath a.B a.continuous_B a.zero_B) + have hW : IsWiener W := sourceBasedDriverLaw_isWiener a.B a.brownian + a.measurable_B a.continuous_B a.zero_B + exact ordinary_explicit_mainConclusion P κ hκ hκ8 γ hSLE htrace hcompact hboundary + hsupport hplanar hstraight hexit hwiener W hW + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualFiniteBoundaryExtension.lean b/lean/OAI/Probability/SLE/Geometry/ActualFiniteBoundaryExtension.lean index 32d662ecba84838bd299ea79a9594304796b557c..e02de50d7c90cd1e491e7ef50b16e6d28b8f3933 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualFiniteBoundaryExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualFiniteBoundaryExtension.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Loewner.InverseBoundaryShrink -import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension -import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension - -namespace OAI - -/-! The finite boundary extension required by the SLE application, -proved from the capacity witness. Small inverse half-disks are Cauchy; -completeness gives the boundary limits and the existing tip limit fixes -the extension at the driver. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.finite_real_boundary_limit - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x : ℝ) : - ∃ z : ℂ, Tendsto (F t) (𝓝[upperHalfPlane] (x:ℂ)) (𝓝 z) := by - have hx : (x:ℂ)∈closure upperHalfPlane := - closedHalfPlane_subset_closure_upperHalfPlane (by simp) - have : NeBot (𝓝[upperHalfPlane] (x:ℂ)) := mem_closure_iff_nhdsWithin_neBot.mp hx - apply cauchy_map_iff_exists_tendsto.mp - apply Metric.cauchy_iff.mpr - refine ⟨inferInstance,fun ε hε => ?_⟩ - obtain ⟨r,hr,z,hsmall⟩ := hw.small_boundary_halfdisk t x (ε/3) (by positivity) - refine ⟨F t '' (upperHalfPlane∩ball (x:ℂ) r),?_,?_⟩ - · change (F t) ⁻¹' (F t '' (upperHalfPlane∩ball (x:ℂ) r))∈𝓝[upperHalfPlane] (x:ℂ) - filter_upwards [self_mem_nhdsWithin, - mem_nhdsWithin_of_mem_nhds (ball_mem_nhds (x:ℂ) hr)] with w hwH hwr - exact ⟨w,⟨hwH,hwr⟩,rfl⟩ - · rintro a ⟨u,hu,rfl⟩ b ⟨v,hv,rfl⟩ - have hsu := hsmall u hu.1 hu.2 - have hsv := hsmall v hv.1 hv.2 - have htri := dist_triangle (F t u) z (F t v) - rw [dist_comm z (F t v)] at htri - linarith - -/-- Continuous finite boundary extension of the inverse Loewner map. -/ -theorem actual_finiteLoewnerBoundaryExtension : PublishedFiniteLoewnerBoundaryExtension := by - intro U γ G F _hU hw t - exact hw.boundary_extension_of_real_limits t (hw.finite_real_boundary_limit t) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseBoundaryShrink +import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension +import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension + +namespace OAI + +/-! The finite boundary extension required by the SLE application, +proved from the capacity witness. Small inverse half-disks are Cauchy; +completeness gives the boundary limits and the existing tip limit fixes +the extension at the driver. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.finite_real_boundary_limit + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x : ℝ) : + ∃ z : ℂ, Tendsto (F t) (𝓝[upperHalfPlane] (x:ℂ)) (𝓝 z) := by + have hx : (x:ℂ)∈closure upperHalfPlane := + closedHalfPlane_subset_closure_upperHalfPlane (by simp) + have : NeBot (𝓝[upperHalfPlane] (x:ℂ)) := mem_closure_iff_nhdsWithin_neBot.mp hx + apply cauchy_map_iff_exists_tendsto.mp + apply Metric.cauchy_iff.mpr + refine ⟨inferInstance,fun ε hε => ?_⟩ + obtain ⟨r,hr,z,hsmall⟩ := hw.small_boundary_halfdisk t x (ε/3) (by positivity) + refine ⟨F t '' (upperHalfPlane∩ball (x:ℂ) r),?_,?_⟩ + · change (F t) ⁻¹' (F t '' (upperHalfPlane∩ball (x:ℂ) r))∈𝓝[upperHalfPlane] (x:ℂ) + filter_upwards [self_mem_nhdsWithin, + mem_nhdsWithin_of_mem_nhds (ball_mem_nhds (x:ℂ) hr)] with w hwH hwr + exact ⟨w,⟨hwH,hwr⟩,rfl⟩ + · rintro a ⟨u,hu,rfl⟩ b ⟨v,hv,rfl⟩ + have hsu := hsmall u hu.1 hu.2 + have hsv := hsmall v hv.1 hv.2 + have htri := dist_triangle (F t u) z (F t v) + rw [dist_comm z (F t v)] at htri + linarith + +/-- Continuous finite boundary extension of the inverse Loewner map. -/ +theorem actual_finiteLoewnerBoundaryExtension : PublishedFiniteLoewnerBoundaryExtension := by + intro U γ G F _hU hw t + exact hw.boundary_extension_of_real_limits t (hw.finite_real_boundary_limit t) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualFirstHeightSupport.lean b/lean/OAI/Probability/SLE/Geometry/ActualFirstHeightSupport.lean index 8dbe0bfe3e1f361684fc5d2eaf9e00cb3728f5aa..77259241ed9cd9da9aec6a3ab8221f9e5b706eb6 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualFirstHeightSupport.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualFirstHeightSupport.lean @@ -1,114 +1,114 @@ -import OAI.Probability.SLE.Stochastic.BrownianFiniteSmallBall -import OAI.Probability.SLE.Loewner.SmallDriverRise -import OAI.Probability.SLE.Geometry.FirstHeightSupport - -namespace OAI - -/-! The narrow first-height support event is proved directly from the -Brownian driver and the deterministic Loewner equation. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open FirstHeightSupport - -theorem regular_strip_rise_positive - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) {A H : ℝ} (hA : 0 < A) (hH : 0 ≤ H) : - 0 < (P.comap (Subtype.val : a.samples → Ω)) - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ - H < (γ ω T).im} := by - let T : ℝ≥0 := ⟨(H+1)^2,sq_nonneg _⟩ - have hT : 0 < T := by change 0 < (H+1)^2; positivity - have hHT : H < 2*Real.sqrt T := by - change H < 2*Real.sqrt ((H+1)^2) - rw [Real.sqrt_sq (by linarith : 0 ≤ H+1)] - linarith - obtain ⟨δ,hδ,hrise⟩ := exists_small_driver_rise T H hH hHT - let ε := min δ (A/2) - have hε : 0 < ε := lt_min hδ (by positivity) - have hroot : 0 < Real.sqrt κ := Real.sqrt_pos.mpr hκ - have hp := brownian_finite_smallBall_positive a.B a.brownian a.measurable_B - a.continuous_B a.zero_B T hT (div_pos hε hroot) - apply hp.trans_le (measure_mono ?_) - intro ω hω - have hu (t : ℝ≥0) (ht : t ∈ Icc 0 T) : |Real.sqrt κ*a.B t ω| < ε := by - rw [abs_mul,abs_of_pos hroot] - have hh := (lt_div_iff₀ hroot).mp (hω t ht) - nlinarith - obtain ⟨τ,hτ,hhigh⟩ := hrise (fun t => Real.sqrt κ*a.B t ω) (γ ω) - (a.G ω) (a.F ω) (a.witness ω) (fun t ht => (hu t ht).le.trans (min_le_left _ _)) - have hτ0 : 0 < τ := by - by_contra hn - have he : τ=0 := le_antisymm (le_of_not_gt hn) zero_le - rw [he,(a.witness ω).2.1,Complex.zero_im] at hhigh - linarith - refine ⟨τ,hτ0,?_,hhigh⟩ - intro t ht - have hb := (a.witness ω).trace_abs_re_le t (M := ε) - (fun s hs => (hu s ⟨zero_le,hs.2.trans (ht.2.trans hτ.2)⟩).le) - change |(γ ω t).re| < A - exact hb.trans_lt (lt_of_le_of_lt (min_le_right _ _) (by linarith)) - - -theorem regular_narrowRise_positive - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) {B : ℝ} (hB : 0 < B) : - 0 < (P.comap (Subtype.val : a.samples → Ω)) - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ - γ ω T ∈ aboveQuarterHeight B} := - regular_strip_rise_positive a hκ (by positivity : 0 < B/16) (by positivity : 0 ≤ B/4) - -theorem ordinary_strip_rise_positive - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) {A H : ℝ} (hA : 0 < A) (hH : 0 ≤ H) : - 0 < P {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ - H < (γ ω T).im} := by - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - have hp := regular_strip_rise_positive a hκ hA hH - apply hp.trans_le - have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) - a.measurePreserving_val.measurable.aemeasurable - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ - H < (γ ω T).im} - rw [a.measurePreserving_val.map_eq] at hh - convert! hh using 1 - -theorem ordinary_narrowRise_positive - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) {B : ℝ} (hB : 0 < B) : - 0 < P {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ - γ ω T ∈ aboveQuarterHeight B} := by - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - have hp := regular_narrowRise_positive a hκ hB - apply hp.trans_le - have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) - a.measurePreserving_val.measurable.aemeasurable - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ - γ ω T ∈ aboveQuarterHeight B} - rw [a.measurePreserving_val.map_eq] at hh - convert! hh using 1 - -theorem ordinary_narrowFirstHeight_positive - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) {B : ℝ} (hB : 0 < B) : - 0 < P {ω | NarrowFirstHeight B (γ ω)} := by - obtain ⟨_,_,htrace⟩ := hSLE.2 - apply (ordinary_narrowRise_positive P κ hκ γ hSLE hB).trans_le - apply measure_mono_ae - filter_upwards [htrace] with ω hω - intro hrise - obtain ⟨G,F,hw⟩ := hω - exact narrowRiseEvent_gives_first_height hB ⟨γ ω,hw.1⟩ hw.2.1 hw.2.2.1 hrise - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.BrownianFiniteSmallBall +import OAI.Probability.SLE.Loewner.SmallDriverRise +import OAI.Probability.SLE.Geometry.FirstHeightSupport + +namespace OAI + +/-! The narrow first-height support event is proved directly from the +Brownian driver and the deterministic Loewner equation. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open FirstHeightSupport + +theorem regular_strip_rise_positive + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) {A H : ℝ} (hA : 0 < A) (hH : 0 ≤ H) : + 0 < (P.comap (Subtype.val : a.samples → Ω)) + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ + H < (γ ω T).im} := by + let T : ℝ≥0 := ⟨(H+1)^2,sq_nonneg _⟩ + have hT : 0 < T := by change 0 < (H+1)^2; positivity + have hHT : H < 2*Real.sqrt T := by + change H < 2*Real.sqrt ((H+1)^2) + rw [Real.sqrt_sq (by linarith : 0 ≤ H+1)] + linarith + obtain ⟨δ,hδ,hrise⟩ := exists_small_driver_rise T H hH hHT + let ε := min δ (A/2) + have hε : 0 < ε := lt_min hδ (by positivity) + have hroot : 0 < Real.sqrt κ := Real.sqrt_pos.mpr hκ + have hp := brownian_finite_smallBall_positive a.B a.brownian a.measurable_B + a.continuous_B a.zero_B T hT (div_pos hε hroot) + apply hp.trans_le (measure_mono ?_) + intro ω hω + have hu (t : ℝ≥0) (ht : t ∈ Icc 0 T) : |Real.sqrt κ*a.B t ω| < ε := by + rw [abs_mul,abs_of_pos hroot] + have hh := (lt_div_iff₀ hroot).mp (hω t ht) + nlinarith + obtain ⟨τ,hτ,hhigh⟩ := hrise (fun t => Real.sqrt κ*a.B t ω) (γ ω) + (a.G ω) (a.F ω) (a.witness ω) (fun t ht => (hu t ht).le.trans (min_le_left _ _)) + have hτ0 : 0 < τ := by + by_contra hn + have he : τ=0 := le_antisymm (le_of_not_gt hn) zero_le + rw [he,(a.witness ω).2.1,Complex.zero_im] at hhigh + linarith + refine ⟨τ,hτ0,?_,hhigh⟩ + intro t ht + have hb := (a.witness ω).trace_abs_re_le t (M := ε) + (fun s hs => (hu s ⟨zero_le,hs.2.trans (ht.2.trans hτ.2)⟩).le) + change |(γ ω t).re| < A + exact hb.trans_lt (lt_of_le_of_lt (min_le_right _ _) (by linarith)) + + +theorem regular_narrowRise_positive + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) {B : ℝ} (hB : 0 < B) : + 0 < (P.comap (Subtype.val : a.samples → Ω)) + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ + γ ω T ∈ aboveQuarterHeight B} := + regular_strip_rise_positive a hκ (by positivity : 0 < B/16) (by positivity : 0 ≤ B/4) + +theorem ordinary_strip_rise_positive + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) {A H : ℝ} (hA : 0 < A) (hH : 0 ≤ H) : + 0 < P {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ + H < (γ ω T).im} := by + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + have hp := regular_strip_rise_positive a hκ hA hH + apply hp.trans_le + have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) + a.measurePreserving_val.measurable.aemeasurable + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) {z : ℂ | |z.re| < A} ∧ + H < (γ ω T).im} + rw [a.measurePreserving_val.map_eq] at hh + convert! hh using 1 + +theorem ordinary_narrowRise_positive + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) {B : ℝ} (hB : 0 < B) : + 0 < P {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ + γ ω T ∈ aboveQuarterHeight B} := by + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + have hp := regular_narrowRise_positive a hκ hB + apply hp.trans_le + have hh := Measure.le_map_apply (μ := P.comap (Subtype.val : a.samples → Ω)) + a.measurePreserving_val.measurable.aemeasurable + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (γ ω) (Icc 0 T) (narrowStrip B) ∧ + γ ω T ∈ aboveQuarterHeight B} + rw [a.measurePreserving_val.map_eq] at hh + convert! hh using 1 + +theorem ordinary_narrowFirstHeight_positive + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) {B : ℝ} (hB : 0 < B) : + 0 < P {ω | NarrowFirstHeight B (γ ω)} := by + obtain ⟨_,_,htrace⟩ := hSLE.2 + apply (ordinary_narrowRise_positive P κ hκ γ hSLE hB).trans_le + apply measure_mono_ae + filter_upwards [htrace] with ω hω + intro hrise + obtain ⟨G,F,hw⟩ := hω + exact narrowRiseEvent_gives_first_height hB ⟨γ ω,hw.1⟩ hw.2.1 hw.2.2.1 hrise + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualGeometricTubeSupport.lean b/lean/OAI/Probability/SLE/Geometry/ActualGeometricTubeSupport.lean index a8befd792ed0bdf38e10055b1763d025fa3e9098..05f9236f949463ff5f3f5659fb36bed9501b1232 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualGeometricTubeSupport.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualGeometricTubeSupport.lean @@ -1,58 +1,58 @@ -import OAI.Probability.SLE.Loewner.SlitRawCapacity -import OAI.Probability.SLE.Loewner.SlitRealCharacteristics -import OAI.Probability.SLE.Geometry.GeometricTubeSupport - -namespace OAI - -/-! The required open-tube support follows from the actual capacity -driver of a prescribed simple arc, finite Brownian driver tubes, and -deterministic trace stability. No geometric support input is assumed. -/ -noncomputable section -open Set Filter Function Complex _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem ordinary_geometricTubeSupport - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0<κ) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) : PublishedGeometricTubeSupport P γ := by - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - intro z p hi hH N V hN hV hpN hzV - let c := p.toContinuousMap - have hc0 : c 0=0 := p.source - have hcH : ∀ t : unitInterval, t≠0 → c t∈upperHalfPlane := hH - have hcl : (1:unitInterval)∈closure (Ioo (0:unitInterval) 1) := by - rw [closure_Ioo (by norm_num : (0:unitInterval)≠1)] - simp - have hnear : c ⁻¹' V∈𝓝 (1:unitInterval) := c.continuous.continuousAt.preimage_mem_nhds - (hV.mem_nhds (by change p 1∈V; rw [p.target]; exact hzV)) - obtain ⟨T,hTV,hT0,hT⟩ := mem_closure_iff_nhds.mp hcl (c ⁻¹' V) hnear - let A := slitCapacityHorizon c hi hc0 hcH T - let b := slitCapacityDriver c hi hc0 hcH hT - let β := fun t : ℝ≥0 => Target.trace 1 b t - have hA : 0 Target.trace 1 (slitCapacityDriver c hi hc0 hcH hT) t) '' Icc 0 A ⊆ N - rw [slitCapacity_raw_initial_range c hi hc0 hcH hT le_rfl, - slitCapacityCurve_initial_range c hi hc0 hcH T le_rfl, - slitCapacityTime_after c hi hc0 hcH T le_rfl] - exact (initialSubarc_range_subset c T).trans hpN - have hfree := (a.slitCapacity_raw_geometry hκ c hi hc0 hcH hT0 hT).2.2.2.2 A hA - exact ordinary_regular_slit_tube_positive P κ hκ γ hSLE b.toContinuousMap (map_zero b) - (a.slit_raw_capacity_witness hκ c hi hc0 hcH hT0 hT) A hA hfree - (slit_regularRealCharacteristics c hi hc0 hcH hT) - (by change 0<(β A).im; rw [hβA]; exact hcH T (ne_of_gt hT0)) - N V hN hV hsub (by change β A∈V; rw [hβA]; exact hTV) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitRawCapacity +import OAI.Probability.SLE.Loewner.SlitRealCharacteristics +import OAI.Probability.SLE.Geometry.GeometricTubeSupport + +namespace OAI + +/-! The required open-tube support follows from the actual capacity +driver of a prescribed simple arc, finite Brownian driver tubes, and +deterministic trace stability. No geometric support input is assumed. -/ +noncomputable section +open Set Filter Function Complex _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem ordinary_geometricTubeSupport + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0<κ) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) : PublishedGeometricTubeSupport P γ := by + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + intro z p hi hH N V hN hV hpN hzV + let c := p.toContinuousMap + have hc0 : c 0=0 := p.source + have hcH : ∀ t : unitInterval, t≠0 → c t∈upperHalfPlane := hH + have hcl : (1:unitInterval)∈closure (Ioo (0:unitInterval) 1) := by + rw [closure_Ioo (by norm_num : (0:unitInterval)≠1)] + simp + have hnear : c ⁻¹' V∈𝓝 (1:unitInterval) := c.continuous.continuousAt.preimage_mem_nhds + (hV.mem_nhds (by change p 1∈V; rw [p.target]; exact hzV)) + obtain ⟨T,hTV,hT0,hT⟩ := mem_closure_iff_nhds.mp hcl (c ⁻¹' V) hnear + let A := slitCapacityHorizon c hi hc0 hcH T + let b := slitCapacityDriver c hi hc0 hcH hT + let β := fun t : ℝ≥0 => Target.trace 1 b t + have hA : 0 Target.trace 1 (slitCapacityDriver c hi hc0 hcH hT) t) '' Icc 0 A ⊆ N + rw [slitCapacity_raw_initial_range c hi hc0 hcH hT le_rfl, + slitCapacityCurve_initial_range c hi hc0 hcH T le_rfl, + slitCapacityTime_after c hi hc0 hcH T le_rfl] + exact (initialSubarc_range_subset c T).trans hpN + have hfree := (a.slitCapacity_raw_geometry hκ c hi hc0 hcH hT0 hT).2.2.2.2 A hA + exact ordinary_regular_slit_tube_positive P κ hκ γ hSLE b.toContinuousMap (map_zero b) + (a.slit_raw_capacity_witness hκ c hi hc0 hcH hT0 hT) A hA hfree + (slit_regularRealCharacteristics c hi hc0 hcH hT) + (by change 0<(β A).im; rw [hβA]; exact hcH T (ne_of_gt hT0)) + N V hN hV hsub (by change β A∈V; rw [hβA]; exact hTV) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualGuideCompactness.lean b/lean/OAI/Probability/SLE/Geometry/ActualGuideCompactness.lean index 2138968b1ca7a91b7a23cc73261a3df318d494ec..02d1b1a0005f612160c9ed83967d3febcc9e2f4a 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualGuideCompactness.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualGuideCompactness.lean @@ -1,207 +1,207 @@ -import OAI.Probability.SLE.Geometry.NormalizedAccessGuide -import OAI.Probability.SLE.Geometry.NormalizedLoewnerCharts -import OAI.Probability.SLE.Geometry.GuideCompactness - -namespace OAI - -/-! Joint compactness for stopped capacity charts. Freshness, physical -clearance and a waypoint angle bound give parameter-guide compactness -and nondegenerate limits of the normalized analytic charts. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem actual_guide_joint_compactness - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedLoewnerBoundaryExtension) - (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) - (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) - (hU : ∀ n, Continuous (U n)) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) - (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) - (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) - (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) - (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) - (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) - (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) - (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → - c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) - (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) - (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) - (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / - ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : - ∃ (b : ℕ → ℝ) (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), - (∀ n, 0 < b n) ∧ StrictMono φ ∧ - DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ - (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ - TendstoLocallyUniformlyOn - (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) - F atTop upperHalfPlane ∧ - TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ - β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ - MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) := by - have hguide n := normalized_actual_access_guide hboundary (hw n) (hU n) (T n) (hT n) - (hfresh n) (αₙ n) (by norm_num : (0 : ℝ) < 2) - (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2) (hαL n).continuousOn - (PhysicalGuideCompactness.injective_of_lower_chord_bound hc (hchord n)) (hα0 n) (hαD n) - choose b hb Fb βₙ hbar hext hβc hβi hβ0 hβH hbarα hβnorm hβeq using hguide - let Fn := fun n => normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) - let g := fun n => normalizedForward (G n (T n)) (U n (T n)) (b n) - let D := fun n => survivingDomain (γₙ n) (T n) - have hcharts n := normalized_capacity_charts (hw n) (T n) (b n) (hb n) - have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hcharts n).1 - have hiFn n : InjOn (Fn n) upperHalfPlane := (hcharts n).2.1 - have hg n : DifferentiableOn ℂ (g n) (D n) := (hcharts n).2.2.1 - have hig n : InjOn (g n) (D n) := (hcharts n).2.2.2.1 - have hFD n : MapsTo (Fn n) upperHalfPlane (D n) := (hcharts n).2.2.2.2.1 - have hgH n : MapsTo (g n) (D n) upperHalfPlane := (hcharts n).2.2.2.2.2.1 - have hgf n : ∀ z ∈ upperHalfPlane, g n (Fn n z) = z := (hcharts n).2.2.2.2.2.2.1 - have hrelation n t (ht : t ∈ Ioc (0 : ℝ) 2) : Fn n (βₙ n t) = αₙ n t := by - apply OnePoint.coe_injective - exact (hext n _ (hβH n ht)).symm.trans (hbarα n t ⟨ht.1.le, ht.2⟩) - have hβHclosed n t (ht : t ∈ Icc (0 : ℝ) 2) : 0 ≤ (βₙ n t).im := by - by_cases ht0 : t = 0 - · simp only [ht0, hβ0, Complex.zero_im, le_refl] - · exact (show 0 < (βₙ n t).im from hβH n ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), ht.2⟩).le - have hway n : ‖βₙ n (1 / 2)‖ = 1 ∧ s ≤ (βₙ n (1 / 2)).im := by - refine ⟨hβnorm n, ?_⟩ - have hnorm := hβnorm n - rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), norm_div, - Complex.norm_real, Real.norm_eq_abs, abs_of_pos (hb n)] at hnorm - have he : b n = ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖ := - ((div_eq_one_iff_eq (hb n).ne').mp hnorm).symm - rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), Complex.div_ofReal_im, he] - exact hangle n - have homit n : γₙ n (T n) ∉ Fn n '' upperHalfPlane := by - rintro ⟨z, hz, he⟩ - have hh := hFD n hz - rw [he] at hh - exact hh.1.2 ⟨T n, ⟨bot_le, le_rfl⟩, rfl⟩ - obtain ⟨ρ, hρ, hballs⟩ := hclear (1 / 2) (by norm_num) (by norm_num) - have hwayclear n : ball (αₙ n (1 / 2)) ρ ⊆ D n := - hballs n _ (by norm_num) - obtain ⟨φ, F, α, β, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, _, _, - _, _, _, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := - GuideCompactness.exists_joint_guide_limit_of_clearance P κ hκ hκ8 γ hSLE hcompact - Fn g D Fb αₙ βₙ (fun n => γₙ n (T n)) C - (by norm_num) (by norm_num) (by norm_num) hs hρ hc hC - (fun n => isOpen_survivingDomain (hw n).1 _) hFn hiFn hg hig hFD hgH hgf - hbar hext hαL hα0 haBound hchord hβc hβ0 hβHclosed hway hwayclear homit - hrelation hbarα (fun n t ht => hβeq n t ht) hclear - exact ⟨b, φ, F, α, β, hb, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, - hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ - -/-- The same actual compactness extraction, retaining the chosen normalized -boundary charts and parameter-guide family. Uniform convergence of that -family supplies the fixed positive guide-time convergence needed by early -trace localization. -/ -theorem actual_guide_joint_compactness_with_family - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedLoewnerBoundaryExtension) - (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) - (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) - (hU : ∀ n, Continuous (U n)) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) - (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) - (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) - (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) - (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) - (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) - (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) - (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → - c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) - (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) - (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ - ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) - (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / - ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : - ∃ (b : ℕ → ℝ) (Fb : ℕ → ℂ → OnePoint ℂ) (βₙ : ℕ → ℝ → ℂ) - (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), - (∀ n, 0 < b n) ∧ - (∀ n, ContinuousOn (Fb n) {z : ℂ | 0 ≤ z.im}) ∧ - (∀ n z, z ∈ upperHalfPlane → Fb n z = - (normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) z : OnePoint ℂ)) ∧ - (∀ n, ContinuousOn (βₙ n) (Icc 0 2)) ∧ - (∀ n, βₙ n 0 = 0) ∧ - (∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im) ∧ - (∀ n t, t ∈ Ioc (0 : ℝ) 2 → βₙ n t = - normalizedForward (G n (T n)) (U n (T n)) (b n) (αₙ n t)) ∧ - (∀ n t, t ∈ Icc (0 : ℝ) 2 → Fb n (βₙ n t) = (αₙ n t : OnePoint ℂ)) ∧ - StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ - (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ - TendstoLocallyUniformlyOn - (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) - F atTop upperHalfPlane ∧ - TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ - TendstoUniformlyOn (fun n => βₙ (φ n)) β atTop (Icc 0 2) ∧ - Tendsto (fun n => γₙ (φ n) (T (φ n))) atTop (𝓝 (α 0)) ∧ - LipschitzOnWith C α (Icc 0 2) ∧ ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ - β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ - MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) := by - have hguide n := normalized_actual_access_guide hboundary (hw n) (hU n) (T n) (hT n) - (hfresh n) (αₙ n) (by norm_num : (0 : ℝ) < 2) - (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2) (hαL n).continuousOn - (PhysicalGuideCompactness.injective_of_lower_chord_bound hc (hchord n)) (hα0 n) (hαD n) - choose b hb Fb βₙ hbar hext hβc hβi hβ0 hβH hbarα hβnorm hβeq using hguide - let Fn := fun n => normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) - let g := fun n => normalizedForward (G n (T n)) (U n (T n)) (b n) - let D := fun n => survivingDomain (γₙ n) (T n) - have hcharts n := normalized_capacity_charts (hw n) (T n) (b n) (hb n) - have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hcharts n).1 - have hiFn n : InjOn (Fn n) upperHalfPlane := (hcharts n).2.1 - have hg n : DifferentiableOn ℂ (g n) (D n) := (hcharts n).2.2.1 - have hig n : InjOn (g n) (D n) := (hcharts n).2.2.2.1 - have hFD n : MapsTo (Fn n) upperHalfPlane (D n) := (hcharts n).2.2.2.2.1 - have hgH n : MapsTo (g n) (D n) upperHalfPlane := (hcharts n).2.2.2.2.2.1 - have hgf n : ∀ z ∈ upperHalfPlane, g n (Fn n z) = z := (hcharts n).2.2.2.2.2.2.1 - have hrelation n t (ht : t ∈ Ioc (0 : ℝ) 2) : Fn n (βₙ n t) = αₙ n t := by - apply OnePoint.coe_injective - exact (hext n _ (hβH n ht)).symm.trans (hbarα n t ⟨ht.1.le, ht.2⟩) - have hβHclosed n t (ht : t ∈ Icc (0 : ℝ) 2) : 0 ≤ (βₙ n t).im := by - by_cases ht0 : t = 0 - · simp only [ht0, hβ0, Complex.zero_im, le_refl] - · exact (show 0 < (βₙ n t).im from hβH n ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), ht.2⟩).le - have hway n : ‖βₙ n (1 / 2)‖ = 1 ∧ s ≤ (βₙ n (1 / 2)).im := by - refine ⟨hβnorm n, ?_⟩ - have hnorm := hβnorm n - rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), norm_div, - Complex.norm_real, Real.norm_eq_abs, abs_of_pos (hb n)] at hnorm - have he : b n = ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖ := - ((div_eq_one_iff_eq (hb n).ne').mp hnorm).symm - rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), Complex.div_ofReal_im, he] - exact hangle n - have homit n : γₙ n (T n) ∉ Fn n '' upperHalfPlane := by - rintro ⟨z, hz, he⟩ - have hh := hFD n hz - rw [he] at hh - exact hh.1.2 ⟨T n, ⟨bot_le, le_rfl⟩, rfl⟩ - obtain ⟨ρ, hρ, hballs⟩ := hclear (1 / 2) (by norm_num) (by norm_num) - have hwayclear n : ball (αₙ n (1 / 2)) ρ ⊆ D n := - hballs n _ (by norm_num) - obtain ⟨φ, F, α, β, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, - hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := - GuideCompactness.exists_joint_guide_limit_of_clearance P κ hκ hκ8 γ hSLE hcompact - Fn g D Fb αₙ βₙ (fun n => γₙ n (T n)) C - (by norm_num) (by norm_num) (by norm_num) hs hρ hc hC - (fun n => isOpen_survivingDomain (hw n).1 _) hFn hiFn hg hig hFD hgH hgf - hbar hext hαL hα0 haBound hchord hβc hβ0 hβHclosed hway hwayclear homit - hrelation hbarα (fun n t ht => hβeq n t ht) hclear - exact ⟨b, Fb, βₙ, φ, F, α, β, hb, hbar, hext, hβc, hβ0, hβHclosed, hβeq, - hbarα, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, - hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.NormalizedAccessGuide +import OAI.Probability.SLE.Geometry.NormalizedLoewnerCharts +import OAI.Probability.SLE.Geometry.GuideCompactness + +namespace OAI + +/-! Joint compactness for stopped capacity charts. Freshness, physical +clearance and a waypoint angle bound give parameter-guide compactness +and nondegenerate limits of the normalized analytic charts. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem actual_guide_joint_compactness + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedLoewnerBoundaryExtension) + (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) + (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) + (hU : ∀ n, Continuous (U n)) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) + (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) + (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) + (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) + (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) + (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) + (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) + (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → + c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) + (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) + (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) + (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / + ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : + ∃ (b : ℕ → ℝ) (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), + (∀ n, 0 < b n) ∧ StrictMono φ ∧ + DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ + (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ + TendstoLocallyUniformlyOn + (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) + F atTop upperHalfPlane ∧ + TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ + β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ + MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) := by + have hguide n := normalized_actual_access_guide hboundary (hw n) (hU n) (T n) (hT n) + (hfresh n) (αₙ n) (by norm_num : (0 : ℝ) < 2) + (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2) (hαL n).continuousOn + (PhysicalGuideCompactness.injective_of_lower_chord_bound hc (hchord n)) (hα0 n) (hαD n) + choose b hb Fb βₙ hbar hext hβc hβi hβ0 hβH hbarα hβnorm hβeq using hguide + let Fn := fun n => normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) + let g := fun n => normalizedForward (G n (T n)) (U n (T n)) (b n) + let D := fun n => survivingDomain (γₙ n) (T n) + have hcharts n := normalized_capacity_charts (hw n) (T n) (b n) (hb n) + have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hcharts n).1 + have hiFn n : InjOn (Fn n) upperHalfPlane := (hcharts n).2.1 + have hg n : DifferentiableOn ℂ (g n) (D n) := (hcharts n).2.2.1 + have hig n : InjOn (g n) (D n) := (hcharts n).2.2.2.1 + have hFD n : MapsTo (Fn n) upperHalfPlane (D n) := (hcharts n).2.2.2.2.1 + have hgH n : MapsTo (g n) (D n) upperHalfPlane := (hcharts n).2.2.2.2.2.1 + have hgf n : ∀ z ∈ upperHalfPlane, g n (Fn n z) = z := (hcharts n).2.2.2.2.2.2.1 + have hrelation n t (ht : t ∈ Ioc (0 : ℝ) 2) : Fn n (βₙ n t) = αₙ n t := by + apply OnePoint.coe_injective + exact (hext n _ (hβH n ht)).symm.trans (hbarα n t ⟨ht.1.le, ht.2⟩) + have hβHclosed n t (ht : t ∈ Icc (0 : ℝ) 2) : 0 ≤ (βₙ n t).im := by + by_cases ht0 : t = 0 + · simp only [ht0, hβ0, Complex.zero_im, le_refl] + · exact (show 0 < (βₙ n t).im from hβH n ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), ht.2⟩).le + have hway n : ‖βₙ n (1 / 2)‖ = 1 ∧ s ≤ (βₙ n (1 / 2)).im := by + refine ⟨hβnorm n, ?_⟩ + have hnorm := hβnorm n + rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), norm_div, + Complex.norm_real, Real.norm_eq_abs, abs_of_pos (hb n)] at hnorm + have he : b n = ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖ := + ((div_eq_one_iff_eq (hb n).ne').mp hnorm).symm + rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), Complex.div_ofReal_im, he] + exact hangle n + have homit n : γₙ n (T n) ∉ Fn n '' upperHalfPlane := by + rintro ⟨z, hz, he⟩ + have hh := hFD n hz + rw [he] at hh + exact hh.1.2 ⟨T n, ⟨bot_le, le_rfl⟩, rfl⟩ + obtain ⟨ρ, hρ, hballs⟩ := hclear (1 / 2) (by norm_num) (by norm_num) + have hwayclear n : ball (αₙ n (1 / 2)) ρ ⊆ D n := + hballs n _ (by norm_num) + obtain ⟨φ, F, α, β, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, _, _, + _, _, _, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := + GuideCompactness.exists_joint_guide_limit_of_clearance P κ hκ hκ8 γ hSLE hcompact + Fn g D Fb αₙ βₙ (fun n => γₙ n (T n)) C + (by norm_num) (by norm_num) (by norm_num) hs hρ hc hC + (fun n => isOpen_survivingDomain (hw n).1 _) hFn hiFn hg hig hFD hgH hgf + hbar hext hαL hα0 haBound hchord hβc hβ0 hβHclosed hway hwayclear homit + hrelation hbarα (fun n t ht => hβeq n t ht) hclear + exact ⟨b, φ, F, α, β, hb, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, + hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ + +/-- The same actual compactness extraction, retaining the chosen normalized +boundary charts and parameter-guide family. Uniform convergence of that +family supplies the fixed positive guide-time convergence needed by early +trace localization. -/ +theorem actual_guide_joint_compactness_with_family + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedLoewnerBoundaryExtension) + (U : ℕ → ℝ≥0 → ℝ) (γₙ : ℕ → ℝ≥0 → ℂ) + (G F₀ : ℕ → ℝ≥0 → ℂ → ℂ) + (hU : ∀ n, Continuous (U n)) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γₙ n) (G n) (F₀ n)) + (T : ℕ → ℝ≥0) (hT : ∀ n, 0 < T n) + (hfresh : ∀ n s, s < T n → γₙ n (T n) ∈ survivingDomain (γₙ n) s) + (αₙ : ℕ → ℝ → ℂ) (C : ℝ≥0) (s c A : ℝ) + (hC : 0 < (C : ℝ)) (hs : 0 < s) (hc : 0 < c) + (hαL : ∀ n, LipschitzOnWith C (αₙ n) (Icc 0 2)) + (hα0 : ∀ n, αₙ n 0 = γₙ n (T n)) (haBound : ∀ n, ‖γₙ n (T n)‖ ≤ A) + (hchord : ∀ n u, u ∈ Icc (0 : ℝ) 2 → ∀ v, v ∈ Icc (0 : ℝ) 2 → + c * |u - v| ≤ dist (αₙ n u) (αₙ n v)) + (hαD : ∀ n, MapsTo (αₙ n) (Ioc 0 2) (survivingDomain (γₙ n) (T n))) + (hclear : ∀ δ : ℝ, 0 < δ → δ ≤ 2 → ∃ r : ℝ, 0 < r ∧ + ∀ n t, t ∈ Icc δ 2 → ball (αₙ n t) r ⊆ survivingDomain (γₙ n) (T n)) + (hangle : ∀ n, s ≤ (G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)).im / + ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖) : + ∃ (b : ℕ → ℝ) (Fb : ℕ → ℂ → OnePoint ℂ) (βₙ : ℕ → ℝ → ℂ) + (φ : ℕ → ℕ) (F : ℂ → ℂ) (α β : ℝ → ℂ), + (∀ n, 0 < b n) ∧ + (∀ n, ContinuousOn (Fb n) {z : ℂ | 0 ≤ z.im}) ∧ + (∀ n z, z ∈ upperHalfPlane → Fb n z = + (normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) z : OnePoint ℂ)) ∧ + (∀ n, ContinuousOn (βₙ n) (Icc 0 2)) ∧ + (∀ n, βₙ n 0 = 0) ∧ + (∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im) ∧ + (∀ n t, t ∈ Ioc (0 : ℝ) 2 → βₙ n t = + normalizedForward (G n (T n)) (U n (T n)) (b n) (αₙ n t)) ∧ + (∀ n t, t ∈ Icc (0 : ℝ) 2 → Fb n (βₙ n t) = (αₙ n t : OnePoint ℂ)) ∧ + StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ InjOn F upperHalfPlane ∧ + (∀ z ∈ upperHalfPlane, deriv F z ≠ 0) ∧ α 0 ∉ F '' upperHalfPlane ∧ + TendstoLocallyUniformlyOn + (fun n => normalizedInverse (F₀ (φ n) (T (φ n))) (U (φ n) (T (φ n))) (b (φ n))) + F atTop upperHalfPlane ∧ + TendstoUniformlyOn (fun n => αₙ (φ n)) α atTop (Icc 0 2) ∧ + TendstoUniformlyOn (fun n => βₙ (φ n)) β atTop (Icc 0 2) ∧ + Tendsto (fun n => γₙ (φ n) (T (φ n))) atTop (𝓝 (α 0)) ∧ + LipschitzOnWith C α (Icc 0 2) ∧ ContinuousOn α (Icc 0 2) ∧ InjOn α (Icc 0 2) ∧ + β 0 = 0 ∧ ContinuousOn β (Icc 0 2) ∧ InjOn β (Icc 0 2) ∧ + MapsTo β (Ioc 0 2) upperHalfPlane ∧ EqOn (F ∘ β) α (Ioc 0 2) := by + have hguide n := normalized_actual_access_guide hboundary (hw n) (hU n) (T n) (hT n) + (hfresh n) (αₙ n) (by norm_num : (0 : ℝ) < 2) + (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2) (hαL n).continuousOn + (PhysicalGuideCompactness.injective_of_lower_chord_bound hc (hchord n)) (hα0 n) (hαD n) + choose b hb Fb βₙ hbar hext hβc hβi hβ0 hβH hbarα hβnorm hβeq using hguide + let Fn := fun n => normalizedInverse (F₀ n (T n)) (U n (T n)) (b n) + let g := fun n => normalizedForward (G n (T n)) (U n (T n)) (b n) + let D := fun n => survivingDomain (γₙ n) (T n) + have hcharts n := normalized_capacity_charts (hw n) (T n) (b n) (hb n) + have hFn n : DifferentiableOn ℂ (Fn n) upperHalfPlane := (hcharts n).1 + have hiFn n : InjOn (Fn n) upperHalfPlane := (hcharts n).2.1 + have hg n : DifferentiableOn ℂ (g n) (D n) := (hcharts n).2.2.1 + have hig n : InjOn (g n) (D n) := (hcharts n).2.2.2.1 + have hFD n : MapsTo (Fn n) upperHalfPlane (D n) := (hcharts n).2.2.2.2.1 + have hgH n : MapsTo (g n) (D n) upperHalfPlane := (hcharts n).2.2.2.2.2.1 + have hgf n : ∀ z ∈ upperHalfPlane, g n (Fn n z) = z := (hcharts n).2.2.2.2.2.2.1 + have hrelation n t (ht : t ∈ Ioc (0 : ℝ) 2) : Fn n (βₙ n t) = αₙ n t := by + apply OnePoint.coe_injective + exact (hext n _ (hβH n ht)).symm.trans (hbarα n t ⟨ht.1.le, ht.2⟩) + have hβHclosed n t (ht : t ∈ Icc (0 : ℝ) 2) : 0 ≤ (βₙ n t).im := by + by_cases ht0 : t = 0 + · simp only [ht0, hβ0, Complex.zero_im, le_refl] + · exact (show 0 < (βₙ n t).im from hβH n ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), ht.2⟩).le + have hway n : ‖βₙ n (1 / 2)‖ = 1 ∧ s ≤ (βₙ n (1 / 2)).im := by + refine ⟨hβnorm n, ?_⟩ + have hnorm := hβnorm n + rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), norm_div, + Complex.norm_real, Real.norm_eq_abs, abs_of_pos (hb n)] at hnorm + have he : b n = ‖G n (T n) (αₙ n (1 / 2)) - (U n (T n) : ℂ)‖ := + ((div_eq_one_iff_eq (hb n).ne').mp hnorm).symm + rw [hβeq n _ (by norm_num : (1 / 2 : ℝ) ∈ Ioc 0 2), Complex.div_ofReal_im, he] + exact hangle n + have homit n : γₙ n (T n) ∉ Fn n '' upperHalfPlane := by + rintro ⟨z, hz, he⟩ + have hh := hFD n hz + rw [he] at hh + exact hh.1.2 ⟨T n, ⟨bot_le, le_rfl⟩, rfl⟩ + obtain ⟨ρ, hρ, hballs⟩ := hclear (1 / 2) (by norm_num) (by norm_num) + have hwayclear n : ball (αₙ n (1 / 2)) ρ ⊆ D n := + hballs n _ (by norm_num) + obtain ⟨φ, F, α, β, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, + hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ := + GuideCompactness.exists_joint_guide_limit_of_clearance P κ hκ hκ8 γ hSLE hcompact + Fn g D Fb αₙ βₙ (fun n => γₙ n (T n)) C + (by norm_num) (by norm_num) (by norm_num) hs hρ hc hC + (fun n => isOpen_survivingDomain (hw n).1 _) hFn hiFn hg hig hFD hgH hgf + hbar hext hαL hα0 haBound hchord hβc hβ0 hβHclosed hway hwayclear homit + hrelation hbarα (fun n t ht => hβeq n t ht) hclear + exact ⟨b, Fb, βₙ, φ, F, α, β, hb, hbar, hext, hβc, hβ0, hβHclosed, hβeq, + hbarα, hφ, hF, hiF, hFne, homitlim, hFconv, hαconv, hβconv, hatip, + hαLlim, hαclim, hαilim, hβlim0, hβlimc, hβlimi, hβlimH, hlimrel⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualMarkedCell.lean b/lean/OAI/Probability/SLE/Geometry/ActualMarkedCell.lean index 1f3982155b3978a1dded5baefd228fc775b0b5e1..c92f49f447abc7103212795246a4dff15e35f13c 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualMarkedCell.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualMarkedCell.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Mass.CellMassComparison - -namespace OAI - -/-! The actual centered argument and modulus supply the marked disk -coordinates in the physical cell comparison. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Multipoint - -lemma IsCapacityTwoWitness.live_marked_polar - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ t) : - let q := G t z-(U t:ℂ) - 0 < ‖q‖₊ ∧ q.arg ∈ Ioo (0:ℝ) Real.pi ∧ - G t z=(U t:ℂ)+(‖q‖₊:ℂ)*Complex.exp (q.arg*Complex.I) := by - let q := G t z-(U t:ℂ) - have hq : 0 < q.im := by - simpa only [q,Complex.sub_im,Complex.ofReal_im,sub_zero] using hw.imag_positive t z hz - have hn : q ≠ 0 := by intro h; simp only [h,Complex.zero_im,lt_self_iff_false] at hq - refine ⟨by exact_mod_cast norm_pos_iff.mpr hn,?_,?_⟩ - · refine ⟨lt_of_le_of_ne (Complex.arg_nonneg_iff.mpr hq.le) ?_, - Complex.arg_lt_pi_iff.mpr (Or.inr hq.ne')⟩ - intro h - exact hq.ne' (Complex.arg_eq_zero_iff.mp h.symm).2 - · change G t z=(U t:ℂ)+(‖q‖:ℂ)*Complex.exp (q.arg*Complex.I) - rw [Complex.norm_mul_exp_arg_mul_I] - dsimp only [q] - ring - -lemma IsCapacityTwoWitness.cell_mass_comparison_at_marked_point - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) - (κ : ℝ) (hκ : 0 < κ) (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) - (r : ℝ) (hr : 0 < r) (hR : fullRadius G t z=(2048*Real.pi)*r) - (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : - ∃ (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (a : ℝ≥0) (ha : 0 < a), - θ=(G t z-(U t:ℂ)).arg ∧ a=‖G t z-(U t:ℂ)‖₊ ∧ - let ν := weightedImageMass (fun w : upperHalfPlane => F t ((w:ℂ)+(U t:ℂ))) - (fun w : upperHalfPlane => ENNReal.ofReal - (‖deriv (F t) ((w:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) - let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) μ) - ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*L ≤ - ν (closedBall z ((4096*Real.pi)*r)) ∧ - ∀ A : Set ℂ, A ⊆ closedBall z (2*r) → - ν A ≤ ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*L := by - obtain ⟨ha,hθ,he⟩ := hw.live_marked_polar t z hz - exact ⟨_,hθ,_,ha,rfl,rfl,hw.cell_mass_comparison hgrowth κ hκ _ hθ t z hz r hr hR _ ha he μ⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.CellMassComparison + +namespace OAI + +/-! The actual centered argument and modulus supply the marked disk +coordinates in the physical cell comparison. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Multipoint + +lemma IsCapacityTwoWitness.live_marked_polar + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ t) : + let q := G t z-(U t:ℂ) + 0 < ‖q‖₊ ∧ q.arg ∈ Ioo (0:ℝ) Real.pi ∧ + G t z=(U t:ℂ)+(‖q‖₊:ℂ)*Complex.exp (q.arg*Complex.I) := by + let q := G t z-(U t:ℂ) + have hq : 0 < q.im := by + simpa only [q,Complex.sub_im,Complex.ofReal_im,sub_zero] using hw.imag_positive t z hz + have hn : q ≠ 0 := by intro h; simp only [h,Complex.zero_im,lt_self_iff_false] at hq + refine ⟨by exact_mod_cast norm_pos_iff.mpr hn,?_,?_⟩ + · refine ⟨lt_of_le_of_ne (Complex.arg_nonneg_iff.mpr hq.le) ?_, + Complex.arg_lt_pi_iff.mpr (Or.inr hq.ne')⟩ + intro h + exact hq.ne' (Complex.arg_eq_zero_iff.mp h.symm).2 + · change G t z=(U t:ℂ)+(‖q‖:ℂ)*Complex.exp (q.arg*Complex.I) + rw [Complex.norm_mul_exp_arg_mul_I] + dsimp only [q] + ring + +lemma IsCapacityTwoWitness.cell_mass_comparison_at_marked_point + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) + (κ : ℝ) (hκ : 0 < κ) (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) + (r : ℝ) (hr : 0 < r) (hR : fullRadius G t z=(2048*Real.pi)*r) + (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : + ∃ (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (a : ℝ≥0) (ha : 0 < a), + θ=(G t z-(U t:ℂ)).arg ∧ a=‖G t z-(U t:ℂ)‖₊ ∧ + let ν := weightedImageMass (fun w : upperHalfPlane => F t ((w:ℂ)+(U t:ℂ))) + (fun w : upperHalfPlane => ENNReal.ofReal + (‖deriv (F t) ((w:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) + let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) μ) + ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*L ≤ + ν (closedBall z ((4096*Real.pi)*r)) ∧ + ∀ A : Set ℂ, A ⊆ closedBall z (2*r) → + ν A ≤ ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*L := by + obtain ⟨ha,hθ,he⟩ := hw.live_marked_polar t z hz + exact ⟨_,hθ,_,ha,rfl,rfl,hw.cell_mass_comparison hgrowth κ hκ _ hθ t z hz r hr hR _ ha he μ⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualMarkedStraightening.lean b/lean/OAI/Probability/SLE/Geometry/ActualMarkedStraightening.lean index afd4365a916773b9defbaef8327842985eb4751d..009c129d200158f1a593883b30ecb5fb1d5166df 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualMarkedStraightening.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualMarkedStraightening.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Geometry.CompleteAccessGuide - -namespace OAI - -/-! Straighten the actual complete marked access guide. The local chart's -zero set is identified with the explicit physical guide, including the -exclusion of every nonlocal guide piece. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -/-- An actual fresh-tip guide and its local marked-point chart give a -straightening with an exact physical zero set. -/ -theorem actual_marked_access_straightening - (hpublished : PublishedHalfPlaneCrosscutStraightening) - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) - (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) - (α : ℝ≥0 → ℂ) (hα : Continuous α) (hiα : Function.Injective α) - (hα0 : α 0 = γ T) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) - (hescape : Tendsto α atTop (cocompact ℂ)) - (Q : ℂ ≃ₜ ℂ) (r : ℝ) - (hQD : MapsTo Q (ball 0 r) (survivingDomain γ T)) - (hlocus : ∀ z ∈ ball (0 : ℂ) r, - (∃ t : ℝ≥0, 0 < t ∧ α t = Q z) ↔ z.im = 0) : - ∃ h : ℂ ≃ₜ ℂ, (∀ x : ℝ, h (x : ℂ) = (x : ℂ)) ∧ - MapsTo h upperHalfPlane upperHalfPlane ∧ - (∀ z ∈ survivingDomain γ T, - (h (G T z - (U T : ℂ))).re = 0 ↔ ∃ t : ℝ≥0, 0 < t ∧ α t = z) ∧ - ContinuousOn (fun z => h (G T (Q z) - (U T : ℂ))) (ball 0 r) ∧ - InjOn (fun z => h (G T (Q z) - (U T : ℂ))) (ball 0 r) ∧ - ∀ z ∈ ball (0 : ℂ) r, (h (G T (Q z) - (U T : ℂ))).re = 0 ↔ z.im = 0 := by - obtain ⟨β, hc, hi, hz, hH, hproper, heq⟩ := - actual_complete_access_guide hw hU T hT hfresh α hα hiα hα0 hαD hescape - obtain ⟨h, hreal, hhalf, hstraight⟩ := hpublished β hc hi hz hH hproper - have hzero (z : ℂ) (hzD : z ∈ survivingDomain γ T) : - (h (G T z - (U T : ℂ))).re = 0 ↔ ∃ t : ℝ≥0, 0 < t ∧ α t = z := by - have hzH : G T z - (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] - using hw.imag_positive T z hzD - rw [CrosscutSides.straightened_real_zero_iff β h hhalf hstraight hzH] - constructor - · rintro ⟨t, ht, he⟩ - rw [heq t ht] at he - have hg : G T (α t) = G T z := sub_left_inj.mp he - refine ⟨t, ht, ?_⟩ - calc - α t = F T (G T (α t)) := (hw.2.2.2.2.2.2.2.1 T _ (hαD t ht)).symm - _ = F T (G T z) := congrArg (F T) hg - _ = z := hw.2.2.2.2.2.2.2.1 T _ hzD - · rintro ⟨t, ht, he⟩ - exact ⟨t, ht, by rw [heq t ht, he]⟩ - refine ⟨h, hreal, hhalf, hzero, ?_, ?_, ?_⟩ - · exact h.continuous.comp_continuousOn - (((hw.2.2.2.1 T).continuousOn.comp Q.continuous.continuousOn hQD).sub continuousOn_const) - · intro z hz' w hw' he - have hg : G T (Q z) = G T (Q w) := sub_left_inj.mp (h.injective he) - apply Q.injective - calc - Q z = F T (G T (Q z)) := (hw.2.2.2.2.2.2.2.1 T _ (hQD hz')).symm - _ = F T (G T (Q w)) := congrArg (F T) hg - _ = Q w := hw.2.2.2.2.2.2.2.1 T _ (hQD hw') - · intro z hz' - exact (hzero (Q z) (hQD hz')).trans (hlocus z hz') - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompleteAccessGuide + +namespace OAI + +/-! Straighten the actual complete marked access guide. The local chart's +zero set is identified with the explicit physical guide, including the +exclusion of every nonlocal guide piece. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +/-- An actual fresh-tip guide and its local marked-point chart give a +straightening with an exact physical zero set. -/ +theorem actual_marked_access_straightening + (hpublished : PublishedHalfPlaneCrosscutStraightening) + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) + (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) + (α : ℝ≥0 → ℂ) (hα : Continuous α) (hiα : Function.Injective α) + (hα0 : α 0 = γ T) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) + (hescape : Tendsto α atTop (cocompact ℂ)) + (Q : ℂ ≃ₜ ℂ) (r : ℝ) + (hQD : MapsTo Q (ball 0 r) (survivingDomain γ T)) + (hlocus : ∀ z ∈ ball (0 : ℂ) r, + (∃ t : ℝ≥0, 0 < t ∧ α t = Q z) ↔ z.im = 0) : + ∃ h : ℂ ≃ₜ ℂ, (∀ x : ℝ, h (x : ℂ) = (x : ℂ)) ∧ + MapsTo h upperHalfPlane upperHalfPlane ∧ + (∀ z ∈ survivingDomain γ T, + (h (G T z - (U T : ℂ))).re = 0 ↔ ∃ t : ℝ≥0, 0 < t ∧ α t = z) ∧ + ContinuousOn (fun z => h (G T (Q z) - (U T : ℂ))) (ball 0 r) ∧ + InjOn (fun z => h (G T (Q z) - (U T : ℂ))) (ball 0 r) ∧ + ∀ z ∈ ball (0 : ℂ) r, (h (G T (Q z) - (U T : ℂ))).re = 0 ↔ z.im = 0 := by + obtain ⟨β, hc, hi, hz, hH, hproper, heq⟩ := + actual_complete_access_guide hw hU T hT hfresh α hα hiα hα0 hαD hescape + obtain ⟨h, hreal, hhalf, hstraight⟩ := hpublished β hc hi hz hH hproper + have hzero (z : ℂ) (hzD : z ∈ survivingDomain γ T) : + (h (G T z - (U T : ℂ))).re = 0 ↔ ∃ t : ℝ≥0, 0 < t ∧ α t = z := by + have hzH : G T z - (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] + using hw.imag_positive T z hzD + rw [CrosscutSides.straightened_real_zero_iff β h hhalf hstraight hzH] + constructor + · rintro ⟨t, ht, he⟩ + rw [heq t ht] at he + have hg : G T (α t) = G T z := sub_left_inj.mp he + refine ⟨t, ht, ?_⟩ + calc + α t = F T (G T (α t)) := (hw.2.2.2.2.2.2.2.1 T _ (hαD t ht)).symm + _ = F T (G T z) := congrArg (F T) hg + _ = z := hw.2.2.2.2.2.2.2.1 T _ hzD + · rintro ⟨t, ht, he⟩ + exact ⟨t, ht, by rw [heq t ht, he]⟩ + refine ⟨h, hreal, hhalf, hzero, ?_, ?_, ?_⟩ + · exact h.continuous.comp_continuousOn + (((hw.2.2.2.1 T).continuousOn.comp Q.continuous.continuousOn hQD).sub continuousOn_const) + · intro z hz' w hw' he + have hg : G T (Q z) = G T (Q w) := sub_left_inj.mp (h.injective he) + apply Q.injective + calc + Q z = F T (G T (Q z)) := (hw.2.2.2.2.2.2.2.1 T _ (hQD hz')).symm + _ = F T (G T (Q w)) := congrArg (F T) hg + _ = Q w := hw.2.2.2.2.2.2.2.1 T _ (hQD hw') + · intro z hz' + exact (hzero (Q z) (hQD hz')).trans (hlocus z hz') + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualStoppedTubeSupport.lean b/lean/OAI/Probability/SLE/Geometry/ActualStoppedTubeSupport.lean index 2fbd68411aef1504f4c9919f57e1416ac24f7108..84e065c45b8e511bd7f8ec7abb426c022d01a009 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualStoppedTubeSupport.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualStoppedTubeSupport.lean @@ -1,132 +1,132 @@ -import OAI.Probability.SLE.Geometry.MeasurableBoundaryRestart -import OAI.Probability.SLE.Loewner.FreshTipMapOut - -namespace OAI - -/-! Physical open-tube support after an actual finite first-height stop. -Simple continuation routes are mapped out by the actual Loewner chart; -their positive fresh-law probabilities transfer through the measurable -boundary restart, including all parameter-boundary visits. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -def stoppedPhysicalTube {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (τ : Ω → ℝ≥0) - (O W : Set ℂ) : Set Ω := - {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (fun t => γ ω (τ ω + t)) (Icc 0 T) O ∧ - γ ω (τ ω + T) ∈ W} - -theorem measurableSet_stoppedPhysicalTube - {Ω : Type*} [MeasurableSpace Ω] - (γ : Ω → ℝ≥0 → ℂ) (hγ : ∀ ω, Continuous (γ ω)) - (hγm : ∀ t, Measurable (fun ω => γ ω t)) - (τ : Ω → ℝ≥0) (hτ : Measurable τ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) : - MeasurableSet (stoppedPhysicalTube γ τ O W) := by - let p (ω : Ω) : C(ℝ≥0, ℂ) := - ⟨fun t => γ ω (τ ω + t), (hγ ω).comp (continuous_const.add continuous_id)⟩ - have hj : Measurable (fun x : ℝ≥0 × Ω => γ x.2 x.1) := - measurable_uncurry_of_continuous_of_measurable hγ hγm - have hp : Measurable p := by - apply ContinuousMap.measurable_iff_eval.mpr - intro t - exact hj.comp ((hτ.add_const t).prodMk measurable_id) - exact (measurableSet_pathTubeEvent hO hW).preimage hp - -theorem conditional_stopped_physical_tube_positive - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) (E : Set Ω) - (hfresh : ∀ ω ∈ E, 0 < τ ω ∧ 0 < (γ ω (τ ω)).im ∧ - ∀ u : ℝ≥0, u < τ ω → (γ ω u).im < (γ ω (τ ω)).im) - (hroutes : ∀ ω ∈ E, ∃ w : ℂ, ∃ p : Path (γ ω (τ ω)) w, - Function.Injective p ∧ - (∀ t, t ≠ 0 → p t ∈ survivingDomain (γ ω) (τ ω)) ∧ - range p ⊆ O ∧ w ∈ W) : - ∀ᵐ ω ∂P, ω ∈ E → - 0 < P⁻[(stoppedPhysicalTube γ τ O W).indicator (fun _ => 1) | - hτ.measurableSpace] ω := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - obtain ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, hFb0, hlaw, hfull⟩ := - source_measurable_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ - let gp := sourceTracePath γ (fun ω => (hw ω).1) - let Y := fun ω => Φ (sourceRestartDriverPath B hc τ ω) - have hgm : Measurable gp := by - have he : gp = Φ ∘ sourceDriverPath B hc := funext fun ω => (hΦ ω).symm - rw [he] - exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm) - have hYm : Measurable Y := hΦm.comp (source_path_measurable _ (fun t => - (measurable_pi_apply t).comp (brownianRestart_measurable hm hc - (stopping_time_measurable_nnreal hτ)))) - let O' : Set (OnePoint ℂ) := ((↑) : ℂ → OnePoint ℂ) '' O - let W' : Set (OnePoint ℂ) := ((↑) : ℂ → OnePoint ℂ) '' W - have hO' : IsOpen O' := OnePoint.isOpenEmbedding_coe.isOpenMap _ hO - have hW' : IsOpen W' := OnePoint.isOpenEmbedding_coe.isOpenMap _ hW - have hgeometry : ∀ ω ∈ E, ∃ z : ℂ, ∃ p : Path (0 : ℂ) z, - Function.Injective p ∧ - ∃ hpH : ∀ t, t ≠ 0 → p t ∈ upperHalfPlane, - (∀ t, Fb ω ⟨p t, by - by_cases ht : t = 0 - · simp [ht, closedHalfPlane] - · exact (show 0 < (p t).im from hpH t ht).le⟩ ∈ O') ∧ - Fb ω ⟨z, by simpa using - (show 0 < (p 1).im from hpH 1 (by norm_num)).le⟩ ∈ W' := by - intro ω hω - obtain ⟨hτpos, hheight, hfirst⟩ := hfresh ω hω - obtain ⟨w, p, hp, hpD, hpO, hwW⟩ := hroutes ω hω - obtain ⟨q, hq, hqH, hqback⟩ := (hw ω).map_out_access_at_first_height - (continuous_const.mul (hc ω)) (τ ω) hτpos hheight hfirst w p hp hpD - refine ⟨_, q, hq, hqH, ?_, ?_⟩ - · intro t - by_cases ht : t = 0 - · subst t - simpa only [Path.source, hFb0] using - (mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) (hpO (mem_range_self 0))) - · rw [hFb ω _ (hqH t ht), hqback t ht] - exact mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) (hpO (mem_range_self t)) - · rw [hFb ω _ (by simpa only [Path.target] using hqH 1 (by norm_num))] - have he := hqback 1 (by norm_num) - simp only [Path.target] at he - rw [he] - exact mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) hwW - have hpositive := conditional_random_boundary_tube_positive P - hτ.measurableSpace hτ.measurableSpace_le Y gp hYm hgm hlaw - (fun ω => (hw ω).2.2.1) hsupport Fb hFbm O' W' hO' hW' E hgeometry - let C : Set Ω := {ω | (Fb ω).comp (liftClosedPath (Y ω)) ∈ pathTubeEventIn O' W'} - have hsubset : ∀ᵐ ω ∂P, ω ∈ C → ω ∈ stoppedPhysicalTube γ τ O W := by - filter_upwards [hfull] with ω hω hC - obtain ⟨T, hT, htrace, hend⟩ := hC - have hmap (t : ℝ≥0) {S : Set ℂ} - (ht : Fb ω (liftClosedPath (Y ω) t) ∈ ((↑) : ℂ → OnePoint ℂ) '' S) : - γ ω (τ ω + t) ∈ S := by - obtain ⟨z, hz, he⟩ := ht - have he' : z = γ ω (τ ω + t) := OnePoint.coe_injective (he.trans (hω t).symm) - exact he' ▸ hz - exact ⟨T, hT, fun t ht => hmap t (htrace ht), hmap T hend⟩ - have hmono : C.indicator (fun _ => (1 : ℝ≥0∞)) ≤ᵐ[P] - (stoppedPhysicalTube γ τ O W).indicator (fun _ => 1) := by - filter_upwards [hsubset] with ω hω - by_cases hC : ω ∈ C - · simp only [indicator_of_mem hC, indicator_of_mem (hω hC), le_refl] - · simp only [indicator_of_notMem hC, zero_le] - filter_upwards [hpositive, condLExp_mono (mΩ := hτ.measurableSpace) hmono] with ω hp hm hω - exact (hp hω).trans_le hm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.MeasurableBoundaryRestart +import OAI.Probability.SLE.Loewner.FreshTipMapOut + +namespace OAI + +/-! Physical open-tube support after an actual finite first-height stop. +Simple continuation routes are mapped out by the actual Loewner chart; +their positive fresh-law probabilities transfer through the measurable +boundary restart, including all parameter-boundary visits. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +def stoppedPhysicalTube {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (τ : Ω → ℝ≥0) + (O W : Set ℂ) : Set Ω := + {ω | ∃ T : ℝ≥0, 0 < T ∧ MapsTo (fun t => γ ω (τ ω + t)) (Icc 0 T) O ∧ + γ ω (τ ω + T) ∈ W} + +theorem measurableSet_stoppedPhysicalTube + {Ω : Type*} [MeasurableSpace Ω] + (γ : Ω → ℝ≥0 → ℂ) (hγ : ∀ ω, Continuous (γ ω)) + (hγm : ∀ t, Measurable (fun ω => γ ω t)) + (τ : Ω → ℝ≥0) (hτ : Measurable τ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) : + MeasurableSet (stoppedPhysicalTube γ τ O W) := by + let p (ω : Ω) : C(ℝ≥0, ℂ) := + ⟨fun t => γ ω (τ ω + t), (hγ ω).comp (continuous_const.add continuous_id)⟩ + have hj : Measurable (fun x : ℝ≥0 × Ω => γ x.2 x.1) := + measurable_uncurry_of_continuous_of_measurable hγ hγm + have hp : Measurable p := by + apply ContinuousMap.measurable_iff_eval.mpr + intro t + exact hj.comp ((hτ.add_const t).prodMk measurable_id) + exact (measurableSet_pathTubeEvent hO hW).preimage hp + +theorem conditional_stopped_physical_tube_positive + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) (E : Set Ω) + (hfresh : ∀ ω ∈ E, 0 < τ ω ∧ 0 < (γ ω (τ ω)).im ∧ + ∀ u : ℝ≥0, u < τ ω → (γ ω u).im < (γ ω (τ ω)).im) + (hroutes : ∀ ω ∈ E, ∃ w : ℂ, ∃ p : Path (γ ω (τ ω)) w, + Function.Injective p ∧ + (∀ t, t ≠ 0 → p t ∈ survivingDomain (γ ω) (τ ω)) ∧ + range p ⊆ O ∧ w ∈ W) : + ∀ᵐ ω ∂P, ω ∈ E → + 0 < P⁻[(stoppedPhysicalTube γ τ O W).indicator (fun _ => 1) | + hτ.measurableSpace] ω := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + obtain ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, hFb0, hlaw, hfull⟩ := + source_measurable_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ + let gp := sourceTracePath γ (fun ω => (hw ω).1) + let Y := fun ω => Φ (sourceRestartDriverPath B hc τ ω) + have hgm : Measurable gp := by + have he : gp = Φ ∘ sourceDriverPath B hc := funext fun ω => (hΦ ω).symm + rw [he] + exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm) + have hYm : Measurable Y := hΦm.comp (source_path_measurable _ (fun t => + (measurable_pi_apply t).comp (brownianRestart_measurable hm hc + (stopping_time_measurable_nnreal hτ)))) + let O' : Set (OnePoint ℂ) := ((↑) : ℂ → OnePoint ℂ) '' O + let W' : Set (OnePoint ℂ) := ((↑) : ℂ → OnePoint ℂ) '' W + have hO' : IsOpen O' := OnePoint.isOpenEmbedding_coe.isOpenMap _ hO + have hW' : IsOpen W' := OnePoint.isOpenEmbedding_coe.isOpenMap _ hW + have hgeometry : ∀ ω ∈ E, ∃ z : ℂ, ∃ p : Path (0 : ℂ) z, + Function.Injective p ∧ + ∃ hpH : ∀ t, t ≠ 0 → p t ∈ upperHalfPlane, + (∀ t, Fb ω ⟨p t, by + by_cases ht : t = 0 + · simp [ht, closedHalfPlane] + · exact (show 0 < (p t).im from hpH t ht).le⟩ ∈ O') ∧ + Fb ω ⟨z, by simpa using + (show 0 < (p 1).im from hpH 1 (by norm_num)).le⟩ ∈ W' := by + intro ω hω + obtain ⟨hτpos, hheight, hfirst⟩ := hfresh ω hω + obtain ⟨w, p, hp, hpD, hpO, hwW⟩ := hroutes ω hω + obtain ⟨q, hq, hqH, hqback⟩ := (hw ω).map_out_access_at_first_height + (continuous_const.mul (hc ω)) (τ ω) hτpos hheight hfirst w p hp hpD + refine ⟨_, q, hq, hqH, ?_, ?_⟩ + · intro t + by_cases ht : t = 0 + · subst t + simpa only [Path.source, hFb0] using + (mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) (hpO (mem_range_self 0))) + · rw [hFb ω _ (hqH t ht), hqback t ht] + exact mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) (hpO (mem_range_self t)) + · rw [hFb ω _ (by simpa only [Path.target] using hqH 1 (by norm_num))] + have he := hqback 1 (by norm_num) + simp only [Path.target] at he + rw [he] + exact mem_image_of_mem ((↑) : ℂ → OnePoint ℂ) hwW + have hpositive := conditional_random_boundary_tube_positive P + hτ.measurableSpace hτ.measurableSpace_le Y gp hYm hgm hlaw + (fun ω => (hw ω).2.2.1) hsupport Fb hFbm O' W' hO' hW' E hgeometry + let C : Set Ω := {ω | (Fb ω).comp (liftClosedPath (Y ω)) ∈ pathTubeEventIn O' W'} + have hsubset : ∀ᵐ ω ∂P, ω ∈ C → ω ∈ stoppedPhysicalTube γ τ O W := by + filter_upwards [hfull] with ω hω hC + obtain ⟨T, hT, htrace, hend⟩ := hC + have hmap (t : ℝ≥0) {S : Set ℂ} + (ht : Fb ω (liftClosedPath (Y ω) t) ∈ ((↑) : ℂ → OnePoint ℂ) '' S) : + γ ω (τ ω + t) ∈ S := by + obtain ⟨z, hz, he⟩ := ht + have he' : z = γ ω (τ ω + t) := OnePoint.coe_injective (he.trans (hω t).symm) + exact he' ▸ hz + exact ⟨T, hT, fun t ht => hmap t (htrace ht), hmap T hend⟩ + have hmono : C.indicator (fun _ => (1 : ℝ≥0∞)) ≤ᵐ[P] + (stoppedPhysicalTube γ τ O W).indicator (fun _ => 1) := by + filter_upwards [hsubset] with ω hω + by_cases hC : ω ∈ C + · simp only [indicator_of_mem hC, indicator_of_mem (hω hC), le_refl] + · simp only [indicator_of_notMem hC, zero_le] + filter_upwards [hpositive, condLExp_mono (mΩ := hτ.measurableSpace) hmono] with ω hp hm hω + exact (hp hω).trans_le hm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualWaypointAngle.lean b/lean/OAI/Probability/SLE/Geometry/ActualWaypointAngle.lean index f81aff96de6524cfe40c42d22f69cf50ad7ff858..f111e96a2e610f91deee670c0612972abbaae9a8 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualWaypointAngle.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualWaypointAngle.lean @@ -1,95 +1,95 @@ -import OAI.Probability.SLE.Geometry.ActualWaypointStraightening -import OAI.Probability.SLE.Mass.WaypointHarmonicMeasure -import OAI.Probability.SLE.Geometry.NormalizedLoewnerCharts - -namespace OAI - -/-! The actual uniform ribbon waypoint angle. Uniform Brownian route -support, crosscut topology, and conformal Brownian exit bounds are the -published inputs. The positive ribbon angle is derived here. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RibbonCoordinates -open Target CrosscutSides - -theorem uniform_actual_ribbon_waypoint_angle - (hopen : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) - (hsupport : PublishedPlanarWienerUnitSupport) - (W : WienerLaw) (hW : IsWiener W) - (C K : ℝ≥0) (hK : 0 < (K : ℝ)) : - ∃ s : ℝ, 0 < s ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → - ∀ (P : ℂ → ℂ), ContinuousOn P closedRibbon → InjOn P closedRibbon → - (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → - Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop → - ∀ (k ε x : ℝ), 1 ≤ k → 0 < ε → ε ≤ 1 / 8 → |x| ≤ 1 / 16 → - LipschitzOnWith C P (crossingPatch k) → - (∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, - dist u v ≤ K * dist (P u) (P v)) → - ∀ (xp : ℝ≥0 → ℂ) (T : ℝ≥0), 0 < T → ContinuousOn xp (Icc 0 T) → - EqOn γ (P ∘ xp) (Icc 0 T) → (xp 0).re = 0 → - xp T = (k : ℂ) + (x : ℂ) * Complex.I → - MapsTo xp (Icc 0 T) closedRibbon → - (∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε) → - (∀ t : ℝ≥0, t < T → (xp t).re < k) → - s ≤ (G T (P ((k + 1 / 2 : ℝ) : ℂ)) - (U T : ℂ)).im / - ‖G T (P ((k + 1 / 2 : ℝ) : ℂ)) - (U T : ℂ)‖ := by - let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure - obtain ⟨q, hq, hprob⟩ := uniform_physical_waypoint_tube_probability - hsupport W hW C (div_pos (by norm_num : (0 : ℝ) < 1 / 32) hK) - have hqfin : q ≠ ⊤ := by - have hb := hprob (fun _ => (0 : ℂ)) 1 1 (by norm_num) - ((LipschitzWith.const (0 : ℂ)).weaken (show (0 : ℝ≥0) ≤ C from bot_le)).lipschitzOnWith - exact ne_top_of_le_ne_top (measure_ne_top _ _) hb - have hqreal : 0 < q.toReal := ENNReal.toReal_pos hq.ne' hqfin - refine ⟨2 * q.toReal, by positivity, ?_⟩ - intro U γ G F hw hU P hP hiP hH hescape k ε x hk hε hεsmall hx hupper hlower - xp T hT hxp hγ hxp0 htip hcoords hcore hfront - have htipre : (xp T).re = k := by simp [htip] - have htipim : (xp T).im = x := by simp [htip] - have hxquarter : |x| ≤ 1 / 4 := hx.trans (by norm_num) - have hfresh := ribbon_front_tip_survives_prior P hP hiP hH hescape γ xp T hxp hxp0 - hcoords hγ (fun t ht => by rw [htipre]; exact hfront t ht) - (by rw [htipim]; exact hx.trans_lt (by norm_num)) - have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by - rintro z ⟨t, ht, rfl⟩ - exact ⟨xp t, ⟨hcoords ht, (hcore t ht).1⟩, (hγ ht).symm⟩ - have hunbounded := futureRibbon_unbounded_of_centerline_escape P k hescape - have hfuture : P '' futureRibbon k ⊆ survivingDomain γ T := - futureRibbon_survives P hP hiP (by linarith) hH γ T hpast hunbounded - have hphysicalTip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T := by - rw [← htip] - exact (hγ ⟨bot_le, le_rfl⟩).symm - obtain ⟨h, hreal, _, hcut, hloc, hloci, hlocz⟩ := - actual_ribbon_waypoint_straightening hstraight hw hU T hT hfresh P hP hiP - (by linarith) hxquarter hphysicalTip hfuture hescape - let g := fun z => G T z - (U T : ℂ) - let f := fun z => F T ((U T : ℂ) + z) - have hcharts := normalized_capacity_charts hw T 1 (by norm_num) - have hgeq : normalizedForward (G T) (U T) 1 = g := by - funext z - simp [normalizedForward, g] - have hfeq : normalizedInverse (F T) (U T) 1 = f := by - funext z - simp [normalizedInverse, f] - rw [hgeq, hfeq] at hcharts - obtain ⟨hf, _, hg, _, hfD, hgH, hgf, hfg⟩ := hcharts - obtain ⟨hl, hr⟩ := ribbon_tube_harmonic_bounds hopen hexit W hW K hK P hP hiP hH - hk hε hεsmall hx hlower γ xp hw.1 T hT hxp hγ hxp0 htipre hcoords hcore hunbounded - g f hg hf hgH hfD hfg hgf h hreal hcut hloc hloci hlocz q - (fun side hs => hprob P k side (by rcases hs with rfl | rfl <;> norm_num) hupper) - have hwD : P ((k + 1 / 2 : ℝ) : ℂ) ∈ survivingDomain γ T := by - apply hfuture - exact ⟨((k + 1 / 2 : ℝ) : ℂ), ⟨by simp, by simp⟩, rfl⟩ - exact (positive_angle_of_two_exit_probability_bounds hq (hgH hwD) hl hr).2 - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Geometry.ActualWaypointStraightening +import OAI.Probability.SLE.Mass.WaypointHarmonicMeasure +import OAI.Probability.SLE.Geometry.NormalizedLoewnerCharts + +namespace OAI + +/-! The actual uniform ribbon waypoint angle. Uniform Brownian route +support, crosscut topology, and conformal Brownian exit bounds are the +published inputs. The positive ribbon angle is derived here. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RibbonCoordinates +open Target CrosscutSides + +theorem uniform_actual_ribbon_waypoint_angle + (hopen : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) + (hsupport : PublishedPlanarWienerUnitSupport) + (W : WienerLaw) (hW : IsWiener W) + (C K : ℝ≥0) (hK : 0 < (K : ℝ)) : + ∃ s : ℝ, 0 < s ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → Continuous U → + ∀ (P : ℂ → ℂ), ContinuousOn P closedRibbon → InjOn P closedRibbon → + (∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane) → + Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop → + ∀ (k ε x : ℝ), 1 ≤ k → 0 < ε → ε ≤ 1 / 8 → |x| ≤ 1 / 16 → + LipschitzOnWith C P (crossingPatch k) → + (∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k, + dist u v ≤ K * dist (P u) (P v)) → + ∀ (xp : ℝ≥0 → ℂ) (T : ℝ≥0), 0 < T → ContinuousOn xp (Icc 0 T) → + EqOn γ (P ∘ xp) (Icc 0 T) → (xp 0).re = 0 → + xp T = (k : ℂ) + (x : ℂ) * Complex.I → + MapsTo xp (Icc 0 T) closedRibbon → + (∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε) → + (∀ t : ℝ≥0, t < T → (xp t).re < k) → + s ≤ (G T (P ((k + 1 / 2 : ℝ) : ℂ)) - (U T : ℂ)).im / + ‖G T (P ((k + 1 / 2 : ℝ) : ℂ)) - (U T : ℂ)‖ := by + let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure + obtain ⟨q, hq, hprob⟩ := uniform_physical_waypoint_tube_probability + hsupport W hW C (div_pos (by norm_num : (0 : ℝ) < 1 / 32) hK) + have hqfin : q ≠ ⊤ := by + have hb := hprob (fun _ => (0 : ℂ)) 1 1 (by norm_num) + ((LipschitzWith.const (0 : ℂ)).weaken (show (0 : ℝ≥0) ≤ C from bot_le)).lipschitzOnWith + exact ne_top_of_le_ne_top (measure_ne_top _ _) hb + have hqreal : 0 < q.toReal := ENNReal.toReal_pos hq.ne' hqfin + refine ⟨2 * q.toReal, by positivity, ?_⟩ + intro U γ G F hw hU P hP hiP hH hescape k ε x hk hε hεsmall hx hupper hlower + xp T hT hxp hγ hxp0 htip hcoords hcore hfront + have htipre : (xp T).re = k := by simp [htip] + have htipim : (xp T).im = x := by simp [htip] + have hxquarter : |x| ≤ 1 / 4 := hx.trans (by norm_num) + have hfresh := ribbon_front_tip_survives_prior P hP hiP hH hescape γ xp T hxp hxp0 + hcoords hγ (fun t ht => by rw [htipre]; exact hfront t ht) + (by rw [htipim]; exact hx.trans_lt (by norm_num)) + have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by + rintro z ⟨t, ht, rfl⟩ + exact ⟨xp t, ⟨hcoords ht, (hcore t ht).1⟩, (hγ ht).symm⟩ + have hunbounded := futureRibbon_unbounded_of_centerline_escape P k hescape + have hfuture : P '' futureRibbon k ⊆ survivingDomain γ T := + futureRibbon_survives P hP hiP (by linarith) hH γ T hpast hunbounded + have hphysicalTip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T := by + rw [← htip] + exact (hγ ⟨bot_le, le_rfl⟩).symm + obtain ⟨h, hreal, _, hcut, hloc, hloci, hlocz⟩ := + actual_ribbon_waypoint_straightening hstraight hw hU T hT hfresh P hP hiP + (by linarith) hxquarter hphysicalTip hfuture hescape + let g := fun z => G T z - (U T : ℂ) + let f := fun z => F T ((U T : ℂ) + z) + have hcharts := normalized_capacity_charts hw T 1 (by norm_num) + have hgeq : normalizedForward (G T) (U T) 1 = g := by + funext z + simp [normalizedForward, g] + have hfeq : normalizedInverse (F T) (U T) 1 = f := by + funext z + simp [normalizedInverse, f] + rw [hgeq, hfeq] at hcharts + obtain ⟨hf, _, hg, _, hfD, hgH, hgf, hfg⟩ := hcharts + obtain ⟨hl, hr⟩ := ribbon_tube_harmonic_bounds hopen hexit W hW K hK P hP hiP hH + hk hε hεsmall hx hlower γ xp hw.1 T hT hxp hγ hxp0 htipre hcoords hcore hunbounded + g f hg hf hgH hfD hfg hgf h hreal hcut hloc hloci hlocz q + (fun side hs => hprob P k side (by rcases hs with rfl | rfl <;> norm_num) hupper) + have hwD : P ((k + 1 / 2 : ℝ) : ℂ) ∈ survivingDomain γ T := by + apply hfuture + exact ⟨((k + 1 / 2 : ℝ) : ℂ), ⟨by simp, by simp⟩, rfl⟩ + exact (positive_angle_of_two_exit_probability_bounds hq (hgH hwD) hl hr).2 + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualWaypointStraightening.lean b/lean/OAI/Probability/SLE/Geometry/ActualWaypointStraightening.lean index dd5784b46c517a0b761855f09b2c17fb9d6d991e..5f3a977ee2dd1d2bd8d10e21af2d82b048b58d16 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualWaypointStraightening.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualWaypointStraightening.lean @@ -1,253 +1,253 @@ -import OAI.Probability.SLE.Geometry.CompleteAccessGuide -import OAI.Probability.SLE.Geometry.WaypointCrosscutSides - -namespace OAI - -/-! The local straightening hypotheses at the actual ribbon waypoint. -The complete access guide and the inverse identities exclude nonlocal -pieces of the guide from the waypoint disk. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.RibbonCoordinates - -theorem waypoint_ball_coordinates {k : ℝ} {z : ℂ} - (hz : z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) : - k + 1 / 4 < z.re ∧ z.re < k + 3 / 4 ∧ |z.im| < 1 / 4 := by - have hd : ‖z - ((k + 1 / 2 : ℝ) : ℂ)‖ < 1 / 4 := by - simpa only [mem_ball, dist_eq_norm] using hz - have hr := (Complex.abs_re_le_norm (z - ((k + 1 / 2 : ℝ) : ℂ))).trans_lt hd - have hi := (Complex.abs_im_le_norm (z - ((k + 1 / 2 : ℝ) : ℂ))).trans_lt hd - simp only [Complex.sub_re, Complex.ofReal_re] at hr - simp only [Complex.sub_im, Complex.ofReal_im, sub_zero] at hi - exact ⟨by linarith [(abs_lt.mp hr).1], by linarith [(abs_lt.mp hr).2], hi⟩ - -theorem guideCoordinate_mem_future {k x : ℝ} (hx : |x| ≤ 1 / 4) - {t : ℝ} (ht : 0 < t) : guideCoordinate k x t ∈ futureRibbon k := by - exact ⟨by simp only [guideCoordinate_re]; linarith, - (guideCoordinate_im_bound ht.le).trans_lt (hx.trans_lt (by norm_num))⟩ - -/-- The complete straightening detects the whole physical access guide, -including points far outside the local coordinate disk. -/ -theorem actual_straightened_zero_lies_on_guide - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) - (P : ℂ → ℂ) {k x : ℝ} (hx : |x| ≤ 1 / 4) - (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) - (β : ℝ≥0 → ℂ) - (hβ : ∀ t : ℝ≥0, 0 < t → - β t = G T (P (guideCoordinate k x t)) - (U T : ℂ)) - (h : ℂ ≃ₜ ℂ) (hH : MapsTo h upperHalfPlane upperHalfPlane) - (hstraight : ∀ t : ℝ≥0, h (β t) = (t : ℂ) * Complex.I) - {z : ℂ} (hz : z ∈ survivingDomain γ T) - (hzero : (h (G T z - (U T : ℂ))).re = 0) : - ∃ t : ℝ, 0 < t ∧ P (guideCoordinate k x t) = z := by - have hzH : G T z - (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, - sub_zero] using hw.imag_positive T z hz - obtain ⟨t, ht, he⟩ := - (CrosscutSides.straightened_real_zero_iff β h hH hstraight hzH).mp hzero - have hD := hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) - have hg : G T (P (guideCoordinate k x t)) = G T z := by - rw [hβ t ht] at he - exact sub_left_inj.mp he - refine ⟨t, by exact_mod_cast ht, ?_⟩ - calc - P (guideCoordinate k x t) = F T (G T (P (guideCoordinate k x t))) := - (hw.2.2.2.2.2.2.2.1 T _ hD).symm - _ = F T (G T z) := congrArg (F T) hg - _ = z := hw.2.2.2.2.2.2.2.1 T z hz - -theorem actual_waypoint_chart - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) - (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) - (β : ℝ≥0 → ℂ) - (hβ : ∀ t : ℝ≥0, 0 < t → - β t = G T (P (guideCoordinate k x t)) - (U T : ℂ)) - (h : ℂ ≃ₜ ℂ) (hH : MapsTo h upperHalfPlane upperHalfPlane) - (hstraight : ∀ t : ℝ≥0, h (β t) = (t : ℂ) * Complex.I) : - let f := fun z => h (G T (P z) - (U T : ℂ)) - ContinuousOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ - InjOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ - ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4), (f z).re = 0 ↔ z.im = 0 := by - dsimp only - let B := ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4) - have hBF : B ⊆ futureRibbon k := by - intro z hz - obtain ⟨hr, _, him⟩ := waypoint_ball_coordinates hz - exact ⟨by linarith, him.trans (by norm_num)⟩ - have hBC := hBF.trans (futureRibbon_subset_closedRibbon hk) - have hBD : MapsTo P B (survivingDomain γ T) := by - intro z hz - exact hfuture (mem_image_of_mem P (hBF hz)) - have hclosed (t : ℝ≥0) (ht : 0 < t) : - guideCoordinate k x t ∈ closedRibbon := - futureRibbon_subset_closedRibbon hk (guideCoordinate_mem_future hx (by exact_mod_cast ht)) - have hD (t : ℝ≥0) (ht : 0 < t) : - P (guideCoordinate k x t) ∈ survivingDomain γ T := - hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) - have hgInj : InjOn (G T) (survivingDomain γ T) := by - intro z hz w hw' he - calc - z = F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hz).symm - _ = F T (G T w) := congrArg (F T) he - _ = w := hw.2.2.2.2.2.2.2.1 T w hw' - refine ⟨h.continuous.continuousOn.comp - (((hw.2.2.2.1 T).continuousOn.comp (hP.mono hBC) hBD).sub continuousOn_const) - (fun _ _ => mem_univ _), ?_, ?_⟩ - · intro z hz w hw' he - exact hiP (hBC hz) (hBC hw') - (hgInj (hBD hz) (hBD hw') (sub_left_inj.mp (h.injective he))) - · intro z hz - have hzH : G T (P z) - (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, - sub_zero] using hw.imag_positive T (P z) (hBD hz) - rw [CrosscutSides.straightened_real_zero_iff β h hH hstraight hzH] - constructor - · rintro ⟨t, ht, he⟩ - rw [hβ t ht] at he - have hcoord : guideCoordinate k x t = z := - hiP (hclosed t ht) (hBC hz) (hgInj (hD t ht) (hBD hz) (sub_left_inj.mp he)) - have hre := congrArg Complex.re hcoord - simp only [guideCoordinate_re] at hre - have htquarter : 1 / 4 < (t : ℝ) := by - linarith [(waypoint_ball_coordinates hz).1] - rw [← hcoord, guideCoordinate_im, max_eq_right (by linarith), mul_zero] - · intro him - let t : ℝ≥0 := ⟨z.re - k, by linarith [(waypoint_ball_coordinates hz).1]⟩ - have ht : 0 < t := by - change 0 < z.re - k - linarith [(waypoint_ball_coordinates hz).1] - have he : guideCoordinate k x t = z := by - apply Complex.ext - · rw [guideCoordinate_re] - change k + (z.re - k) = z.re - ring - · rw [guideCoordinate_im, max_eq_right, mul_zero, him] - change 1 - 4 * (z.re - k) ≤ 0 - linarith [(waypoint_ball_coordinates hz).1] - exact ⟨t, ht, by rw [hβ t ht, he]⟩ - -/-- The actual complete physical guide is proper because its tail is the -centerline; no compactness or properness of a parameter guide is assumed. -/ -theorem actual_ribbon_complete_guide - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) - (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) - (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) - (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : - ∃ β : ℝ≥0 → ℂ, Continuous β ∧ Function.Injective β ∧ β 0 = 0 ∧ - (∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) ∧ - Tendsto β atTop (cocompact ℂ) ∧ - ∀ t : ℝ≥0, 0 < t → β t = G T (P (guideCoordinate k x t)) - (U T : ℂ) := by - let α : ℝ≥0 → ℂ := fun t => P (guideCoordinate k x t) - have hgc (t : ℝ≥0) : guideCoordinate k x t ∈ closedRibbon := by - exact ⟨by simp only [guideCoordinate_re]; positivity, - (guideCoordinate_im_bound t.coe_nonneg).trans (hx.trans (by norm_num))⟩ - have hα : Continuous α := - hP.comp_continuous ((guideCoordinate_continuous k x).comp NNReal.continuous_coe) - hgc - have hiα : Function.Injective α := by - intro s t he - have hh := congrArg Complex.re (hiP (hgc s) (hgc t) he) - simp only [guideCoordinate_re] at hh - apply NNReal.eq - linarith - have hα0 : α 0 = γ T := by simpa only [α, NNReal.coe_zero, guideCoordinate_zero] using htip - have hαD (t : ℝ≥0) (ht : 0 < t) : α t ∈ survivingDomain γ T := - hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) - have htend : Tendsto (fun t : ℝ≥0 => k + (t : ℝ)) atTop atTop := - tendsto_atTop_add_const_left _ k (NNReal.tendsto_coe_atTop.mpr tendsto_id) - have hnorm : Tendsto (fun t => ‖α t‖) atTop atTop := by - apply (hescape.comp htend).congr' - filter_upwards [(NNReal.tendsto_coe_atTop.mpr tendsto_id).eventually_ge_atTop - (1 / 4 : ℝ)] with t ht - simp only [id_eq] at ht - change ‖P ((k + (t : ℝ) : ℝ) : ℂ)‖ = ‖P (guideCoordinate k x t)‖ - apply congrArg (fun z : ℂ => ‖P z‖) - apply Complex.ext - · simp [guideCoordinate] - · simp only [guideCoordinate_im, Complex.ofReal_im] - rw [max_eq_right (by linarith), mul_zero] - have hproper : Tendsto α atTop (cocompact ℂ) := by - apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 - simpa only [dist_zero_right] using hnorm - exact actual_complete_access_guide hw hU T hT hfresh α hα hiα hα0 hαD hproper - -theorem actual_ribbon_waypoint_straightening - (hpublished : PublishedHalfPlaneCrosscutStraightening) - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) - (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) - (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) - (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : - ∃ h : ℂ ≃ₜ ℂ, (∀ y : ℝ, h (y : ℂ) = (y : ℂ)) ∧ - MapsTo h upperHalfPlane upperHalfPlane ∧ - (∀ z ∈ survivingDomain γ T, (h (G T z - (U T : ℂ))).re = 0 → - ∃ u : ℝ, 0 ≤ u ∧ P (guideCoordinate k x u) = z) ∧ - let f := fun z => h (G T (P z) - (U T : ℂ)) - ContinuousOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ - InjOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ - ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4), (f z).re = 0 ↔ z.im = 0 := by - obtain ⟨β, hc, hi, hz, hH, hproper, heq⟩ := - actual_ribbon_complete_guide hw hU T hT hfresh P hP hiP hk hx htip hfuture hescape - obtain ⟨h, hreal, hhalf, hstraight⟩ := hpublished β hc hi hz hH hproper - refine ⟨h, hreal, hhalf, ?_, actual_waypoint_chart hw T P hP hiP hk hx hfuture β - heq h hhalf hstraight⟩ - intro z hz hzero - obtain ⟨u, hu, he⟩ := actual_straightened_zero_lies_on_guide hw T P hx hfuture - β heq h hhalf hstraight hz hzero - exact ⟨u, hu.le, he⟩ - -/-- Actual physical waypoint neighborhoods lie on opposite sides of the -complete stopped access guide. The orientation is selected by the local -chart and is shared by the whole pair of neighborhoods. -/ -theorem actual_physical_waypoint_tubes_opposite_sides - (hpublished : PublishedHalfPlaneCrosscutStraightening) - (hopen : PublishedPlanarDiskOpenness) - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) - (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) - (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) - {k x : ℝ} (hk : 1 ≤ k) (hx : |x| ≤ 1 / 4) - (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) - (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) - (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) - (K : ℝ≥0) (hK : 0 < (K : ℝ)) - (hlower : ∀ v ∈ crossingPatch k, ∀ w ∈ crossingPatch k, - dist v w ≤ K * dist (P v) (P w)) : - ∃ (h : ℂ ≃ₜ ℂ) (side : ℝ), - (∀ y : ℝ, h (y : ℂ) = (y : ℂ)) ∧ - MapsTo h upperHalfPlane upperHalfPlane ∧ - (side = -1 ∨ side = 1) ∧ - (∀ y : ℂ, dist y (P (waypointRoute k side (1 / 4))) < (1 / 32 : ℝ) / K → - 0 < (h (G T y - (U T : ℂ))).re) ∧ - (∀ y : ℂ, dist y (P (waypointRoute k (-side) (1 / 4))) < (1 / 32 : ℝ) / K → - (h (G T y - (U T : ℂ))).re < 0) := by - obtain ⟨h, hreal, hH, _, hc, hi, hzero⟩ := actual_ribbon_waypoint_straightening - hpublished hw hU T hT hfresh P hP hiP (by linarith) hx htip hfuture hescape - obtain ⟨side, hs, hp, hn⟩ := physical_waypoint_tubes_opposite_sides - hopen P hP hiP K hK hk hlower - (fun z => h (G T (P z) - (U T : ℂ))) - (fun y => (h (G T y - (U T : ℂ))).re) hc hi hzero (fun _ _ => rfl) - exact ⟨h, side, hreal, hH, hs, hp, hn⟩ - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompleteAccessGuide +import OAI.Probability.SLE.Geometry.WaypointCrosscutSides + +namespace OAI + +/-! The local straightening hypotheses at the actual ribbon waypoint. +The complete access guide and the inverse identities exclude nonlocal +pieces of the guide from the waypoint disk. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.RibbonCoordinates + +theorem waypoint_ball_coordinates {k : ℝ} {z : ℂ} + (hz : z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) : + k + 1 / 4 < z.re ∧ z.re < k + 3 / 4 ∧ |z.im| < 1 / 4 := by + have hd : ‖z - ((k + 1 / 2 : ℝ) : ℂ)‖ < 1 / 4 := by + simpa only [mem_ball, dist_eq_norm] using hz + have hr := (Complex.abs_re_le_norm (z - ((k + 1 / 2 : ℝ) : ℂ))).trans_lt hd + have hi := (Complex.abs_im_le_norm (z - ((k + 1 / 2 : ℝ) : ℂ))).trans_lt hd + simp only [Complex.sub_re, Complex.ofReal_re] at hr + simp only [Complex.sub_im, Complex.ofReal_im, sub_zero] at hi + exact ⟨by linarith [(abs_lt.mp hr).1], by linarith [(abs_lt.mp hr).2], hi⟩ + +theorem guideCoordinate_mem_future {k x : ℝ} (hx : |x| ≤ 1 / 4) + {t : ℝ} (ht : 0 < t) : guideCoordinate k x t ∈ futureRibbon k := by + exact ⟨by simp only [guideCoordinate_re]; linarith, + (guideCoordinate_im_bound ht.le).trans_lt (hx.trans_lt (by norm_num))⟩ + +/-- The complete straightening detects the whole physical access guide, +including points far outside the local coordinate disk. -/ +theorem actual_straightened_zero_lies_on_guide + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) + (P : ℂ → ℂ) {k x : ℝ} (hx : |x| ≤ 1 / 4) + (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) + (β : ℝ≥0 → ℂ) + (hβ : ∀ t : ℝ≥0, 0 < t → + β t = G T (P (guideCoordinate k x t)) - (U T : ℂ)) + (h : ℂ ≃ₜ ℂ) (hH : MapsTo h upperHalfPlane upperHalfPlane) + (hstraight : ∀ t : ℝ≥0, h (β t) = (t : ℂ) * Complex.I) + {z : ℂ} (hz : z ∈ survivingDomain γ T) + (hzero : (h (G T z - (U T : ℂ))).re = 0) : + ∃ t : ℝ, 0 < t ∧ P (guideCoordinate k x t) = z := by + have hzH : G T z - (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, + sub_zero] using hw.imag_positive T z hz + obtain ⟨t, ht, he⟩ := + (CrosscutSides.straightened_real_zero_iff β h hH hstraight hzH).mp hzero + have hD := hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) + have hg : G T (P (guideCoordinate k x t)) = G T z := by + rw [hβ t ht] at he + exact sub_left_inj.mp he + refine ⟨t, by exact_mod_cast ht, ?_⟩ + calc + P (guideCoordinate k x t) = F T (G T (P (guideCoordinate k x t))) := + (hw.2.2.2.2.2.2.2.1 T _ hD).symm + _ = F T (G T z) := congrArg (F T) hg + _ = z := hw.2.2.2.2.2.2.2.1 T z hz + +theorem actual_waypoint_chart + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) + (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) + (β : ℝ≥0 → ℂ) + (hβ : ∀ t : ℝ≥0, 0 < t → + β t = G T (P (guideCoordinate k x t)) - (U T : ℂ)) + (h : ℂ ≃ₜ ℂ) (hH : MapsTo h upperHalfPlane upperHalfPlane) + (hstraight : ∀ t : ℝ≥0, h (β t) = (t : ℂ) * Complex.I) : + let f := fun z => h (G T (P z) - (U T : ℂ)) + ContinuousOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ + InjOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ + ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4), (f z).re = 0 ↔ z.im = 0 := by + dsimp only + let B := ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4) + have hBF : B ⊆ futureRibbon k := by + intro z hz + obtain ⟨hr, _, him⟩ := waypoint_ball_coordinates hz + exact ⟨by linarith, him.trans (by norm_num)⟩ + have hBC := hBF.trans (futureRibbon_subset_closedRibbon hk) + have hBD : MapsTo P B (survivingDomain γ T) := by + intro z hz + exact hfuture (mem_image_of_mem P (hBF hz)) + have hclosed (t : ℝ≥0) (ht : 0 < t) : + guideCoordinate k x t ∈ closedRibbon := + futureRibbon_subset_closedRibbon hk (guideCoordinate_mem_future hx (by exact_mod_cast ht)) + have hD (t : ℝ≥0) (ht : 0 < t) : + P (guideCoordinate k x t) ∈ survivingDomain γ T := + hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) + have hgInj : InjOn (G T) (survivingDomain γ T) := by + intro z hz w hw' he + calc + z = F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hz).symm + _ = F T (G T w) := congrArg (F T) he + _ = w := hw.2.2.2.2.2.2.2.1 T w hw' + refine ⟨h.continuous.continuousOn.comp + (((hw.2.2.2.1 T).continuousOn.comp (hP.mono hBC) hBD).sub continuousOn_const) + (fun _ _ => mem_univ _), ?_, ?_⟩ + · intro z hz w hw' he + exact hiP (hBC hz) (hBC hw') + (hgInj (hBD hz) (hBD hw') (sub_left_inj.mp (h.injective he))) + · intro z hz + have hzH : G T (P z) - (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, + sub_zero] using hw.imag_positive T (P z) (hBD hz) + rw [CrosscutSides.straightened_real_zero_iff β h hH hstraight hzH] + constructor + · rintro ⟨t, ht, he⟩ + rw [hβ t ht] at he + have hcoord : guideCoordinate k x t = z := + hiP (hclosed t ht) (hBC hz) (hgInj (hD t ht) (hBD hz) (sub_left_inj.mp he)) + have hre := congrArg Complex.re hcoord + simp only [guideCoordinate_re] at hre + have htquarter : 1 / 4 < (t : ℝ) := by + linarith [(waypoint_ball_coordinates hz).1] + rw [← hcoord, guideCoordinate_im, max_eq_right (by linarith), mul_zero] + · intro him + let t : ℝ≥0 := ⟨z.re - k, by linarith [(waypoint_ball_coordinates hz).1]⟩ + have ht : 0 < t := by + change 0 < z.re - k + linarith [(waypoint_ball_coordinates hz).1] + have he : guideCoordinate k x t = z := by + apply Complex.ext + · rw [guideCoordinate_re] + change k + (z.re - k) = z.re + ring + · rw [guideCoordinate_im, max_eq_right, mul_zero, him] + change 1 - 4 * (z.re - k) ≤ 0 + linarith [(waypoint_ball_coordinates hz).1] + exact ⟨t, ht, by rw [hβ t ht, he]⟩ + +/-- The actual complete physical guide is proper because its tail is the +centerline; no compactness or properness of a parameter guide is assumed. -/ +theorem actual_ribbon_complete_guide + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) + (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) + (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) + (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : + ∃ β : ℝ≥0 → ℂ, Continuous β ∧ Function.Injective β ∧ β 0 = 0 ∧ + (∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) ∧ + Tendsto β atTop (cocompact ℂ) ∧ + ∀ t : ℝ≥0, 0 < t → β t = G T (P (guideCoordinate k x t)) - (U T : ℂ) := by + let α : ℝ≥0 → ℂ := fun t => P (guideCoordinate k x t) + have hgc (t : ℝ≥0) : guideCoordinate k x t ∈ closedRibbon := by + exact ⟨by simp only [guideCoordinate_re]; positivity, + (guideCoordinate_im_bound t.coe_nonneg).trans (hx.trans (by norm_num))⟩ + have hα : Continuous α := + hP.comp_continuous ((guideCoordinate_continuous k x).comp NNReal.continuous_coe) + hgc + have hiα : Function.Injective α := by + intro s t he + have hh := congrArg Complex.re (hiP (hgc s) (hgc t) he) + simp only [guideCoordinate_re] at hh + apply NNReal.eq + linarith + have hα0 : α 0 = γ T := by simpa only [α, NNReal.coe_zero, guideCoordinate_zero] using htip + have hαD (t : ℝ≥0) (ht : 0 < t) : α t ∈ survivingDomain γ T := + hfuture (mem_image_of_mem P (guideCoordinate_mem_future hx (by exact_mod_cast ht))) + have htend : Tendsto (fun t : ℝ≥0 => k + (t : ℝ)) atTop atTop := + tendsto_atTop_add_const_left _ k (NNReal.tendsto_coe_atTop.mpr tendsto_id) + have hnorm : Tendsto (fun t => ‖α t‖) atTop atTop := by + apply (hescape.comp htend).congr' + filter_upwards [(NNReal.tendsto_coe_atTop.mpr tendsto_id).eventually_ge_atTop + (1 / 4 : ℝ)] with t ht + simp only [id_eq] at ht + change ‖P ((k + (t : ℝ) : ℝ) : ℂ)‖ = ‖P (guideCoordinate k x t)‖ + apply congrArg (fun z : ℂ => ‖P z‖) + apply Complex.ext + · simp [guideCoordinate] + · simp only [guideCoordinate_im, Complex.ofReal_im] + rw [max_eq_right (by linarith), mul_zero] + have hproper : Tendsto α atTop (cocompact ℂ) := by + apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 + simpa only [dist_zero_right] using hnorm + exact actual_complete_access_guide hw hU T hT hfresh α hα hiα hα0 hαD hproper + +theorem actual_ribbon_waypoint_straightening + (hpublished : PublishedHalfPlaneCrosscutStraightening) + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) + (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {k x : ℝ} (hk : 0 ≤ k) (hx : |x| ≤ 1 / 4) + (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) + (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) : + ∃ h : ℂ ≃ₜ ℂ, (∀ y : ℝ, h (y : ℂ) = (y : ℂ)) ∧ + MapsTo h upperHalfPlane upperHalfPlane ∧ + (∀ z ∈ survivingDomain γ T, (h (G T z - (U T : ℂ))).re = 0 → + ∃ u : ℝ, 0 ≤ u ∧ P (guideCoordinate k x u) = z) ∧ + let f := fun z => h (G T (P z) - (U T : ℂ)) + ContinuousOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ + InjOn f (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)) ∧ + ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4), (f z).re = 0 ↔ z.im = 0 := by + obtain ⟨β, hc, hi, hz, hH, hproper, heq⟩ := + actual_ribbon_complete_guide hw hU T hT hfresh P hP hiP hk hx htip hfuture hescape + obtain ⟨h, hreal, hhalf, hstraight⟩ := hpublished β hc hi hz hH hproper + refine ⟨h, hreal, hhalf, ?_, actual_waypoint_chart hw T P hP hiP hk hx hfuture β + heq h hhalf hstraight⟩ + intro z hz hzero + obtain ⟨u, hu, he⟩ := actual_straightened_zero_lies_on_guide hw T P hx hfuture + β heq h hhalf hstraight hz hzero + exact ⟨u, hu.le, he⟩ + +/-- Actual physical waypoint neighborhoods lie on opposite sides of the +complete stopped access guide. The orientation is selected by the local +chart and is shared by the whole pair of neighborhoods. -/ +theorem actual_physical_waypoint_tubes_opposite_sides + (hpublished : PublishedHalfPlaneCrosscutStraightening) + (hopen : PublishedPlanarDiskOpenness) + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) + (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) + (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hiP : InjOn P closedRibbon) + {k x : ℝ} (hk : 1 ≤ k) (hx : |x| ≤ 1 / 4) + (htip : P ((k : ℂ) + (x : ℂ) * Complex.I) = γ T) + (hfuture : P '' futureRibbon k ⊆ survivingDomain γ T) + (hescape : Tendsto (fun s : ℝ => ‖P (s : ℂ)‖) atTop atTop) + (K : ℝ≥0) (hK : 0 < (K : ℝ)) + (hlower : ∀ v ∈ crossingPatch k, ∀ w ∈ crossingPatch k, + dist v w ≤ K * dist (P v) (P w)) : + ∃ (h : ℂ ≃ₜ ℂ) (side : ℝ), + (∀ y : ℝ, h (y : ℂ) = (y : ℂ)) ∧ + MapsTo h upperHalfPlane upperHalfPlane ∧ + (side = -1 ∨ side = 1) ∧ + (∀ y : ℂ, dist y (P (waypointRoute k side (1 / 4))) < (1 / 32 : ℝ) / K → + 0 < (h (G T y - (U T : ℂ))).re) ∧ + (∀ y : ℂ, dist y (P (waypointRoute k (-side) (1 / 4))) < (1 / 32 : ℝ) / K → + (h (G T y - (U T : ℂ))).re < 0) := by + obtain ⟨h, hreal, hH, _, hc, hi, hzero⟩ := actual_ribbon_waypoint_straightening + hpublished hw hU T hT hfresh P hP hiP (by linarith) hx htip hfuture hescape + obtain ⟨side, hs, hp, hn⟩ := physical_waypoint_tubes_opposite_sides + hopen P hP hiP K hK hk hlower + (fun z => h (G T (P z) - (U T : ℂ))) + (fun y => (h (G T y - (U T : ℂ))).re) hc hi hzero (fun _ _ => rfl) + exact ⟨h, side, hreal, hH, hs, hp, hn⟩ + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ActualWienerSupport.lean b/lean/OAI/Probability/SLE/Geometry/ActualWienerSupport.lean index a616317bc139795cec68b5b47983b971100b3d3c..59a18d530cbc35ff2090db88413e7a9b9bf1dabc 100644 --- a/lean/OAI/Probability/SLE/Geometry/ActualWienerSupport.lean +++ b/lean/OAI/Probability/SLE/Geometry/ActualWienerSupport.lean @@ -1,62 +1,62 @@ -import OAI.Probability.SLE.Stochastic.WienerRealSupport - -namespace OAI - -/-! Finite-window planar Wiener support from finite Gaussian increments, -independent interpolation residuals, and uniform continuity. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -def wienerUnitClamp (s : ℝ≥0) : unitInterval := - ⟨min (s:ℝ) 1,le_min s.2 zero_le_one,min_le_right _ _⟩ - -lemma continuous_wienerUnitClamp : Continuous wienerUnitClamp := by - unfold wienerUnitClamp - fun_prop - -@[simp] lemma wienerUnitClamp_unit (t : unitInterval) : - wienerUnitClamp ⟨t,t.2.1⟩ = t := by - apply Subtype.ext - exact min_eq_left t.2.2 - -@[simp] lemma wienerUnitClamp_zero : wienerUnitClamp 0=0 := by - apply Subtype.ext - simp [wienerUnitClamp] - -/-- The planar Wiener law gives positive probability to each uniform tube -about a continuous based path, as in Millet--Sanz-Sole (1994), Theorem 3.5. -/ -theorem actual_planarWienerUnitSupport : PublishedPlanarWienerUnitSupport := by - intro W hW f hf r hr - have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure - let fR : ℝ≥0 → ℝ := fun s => (f (wienerUnitClamp s)).re - let fI : ℝ≥0 → ℝ := fun s => (f (wienerUnitClamp s)).im - have hR : Continuous fR := Complex.continuous_re.comp (f.continuous.comp continuous_wienerUnitClamp) - have hI : Continuous fI := Complex.continuous_im.comp (f.continuous.comp continuous_wienerUnitClamp) - have hR0 : fR 0=0 := by simp [fR,hf] - have hI0 : fI 0=0 := by simp [fI,hf] - let A : Set Target.Driver := {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-fR ⟨t,t.2.1⟩| < r/4} - let B : Set Target.Driver := {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-fI ⟨t,t.2.1⟩| < r/4} - have hA : 0 < W A := wienerReal_uniform_tube_pos W hW fR hR hR0 (r/4) (by positivity) - have hB : 0 < W B := wienerReal_uniform_tube_pos W hW fI hI hI0 (r/4) (by positivity) - have hp : 0 < (W.prod W) (A ×ˢ B) := by - rw [Measure.prod_prod] - exact bot_lt_iff_ne_bot.mpr (mul_ne_zero hA.ne' hB.ne') - refine hp.trans_le (measure_mono ?_) - intro b hb - apply (BoundedContinuousFunction.dist_lt_iff_of_compact hr).mpr - intro t - rw [dist_eq_norm] - have hre : |(planarUnitPath b t-f t).re| < r/4 := by - simpa [planarUnitPath,fR] using hb.1 t - have him : |(planarUnitPath b t-f t).im| < r/4 := by - simpa [planarUnitPath,fI] using hb.2 t - exact (Complex.norm_le_abs_re_add_abs_im _).trans_lt (by linarith) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.WienerRealSupport + +namespace OAI + +/-! Finite-window planar Wiener support from finite Gaussian increments, +independent interpolation residuals, and uniform continuity. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +def wienerUnitClamp (s : ℝ≥0) : unitInterval := + ⟨min (s:ℝ) 1,le_min s.2 zero_le_one,min_le_right _ _⟩ + +lemma continuous_wienerUnitClamp : Continuous wienerUnitClamp := by + unfold wienerUnitClamp + fun_prop + +@[simp] lemma wienerUnitClamp_unit (t : unitInterval) : + wienerUnitClamp ⟨t,t.2.1⟩ = t := by + apply Subtype.ext + exact min_eq_left t.2.2 + +@[simp] lemma wienerUnitClamp_zero : wienerUnitClamp 0=0 := by + apply Subtype.ext + simp [wienerUnitClamp] + +/-- The planar Wiener law gives positive probability to each uniform tube +about a continuous based path, as in Millet--Sanz-Sole (1994), Theorem 3.5. -/ +theorem actual_planarWienerUnitSupport : PublishedPlanarWienerUnitSupport := by + intro W hW f hf r hr + have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure + let fR : ℝ≥0 → ℝ := fun s => (f (wienerUnitClamp s)).re + let fI : ℝ≥0 → ℝ := fun s => (f (wienerUnitClamp s)).im + have hR : Continuous fR := Complex.continuous_re.comp (f.continuous.comp continuous_wienerUnitClamp) + have hI : Continuous fI := Complex.continuous_im.comp (f.continuous.comp continuous_wienerUnitClamp) + have hR0 : fR 0=0 := by simp [fR,hf] + have hI0 : fI 0=0 := by simp [fI,hf] + let A : Set Target.Driver := {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-fR ⟨t,t.2.1⟩| < r/4} + let B : Set Target.Driver := {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-fI ⟨t,t.2.1⟩| < r/4} + have hA : 0 < W A := wienerReal_uniform_tube_pos W hW fR hR hR0 (r/4) (by positivity) + have hB : 0 < W B := wienerReal_uniform_tube_pos W hW fI hI hI0 (r/4) (by positivity) + have hp : 0 < (W.prod W) (A ×ˢ B) := by + rw [Measure.prod_prod] + exact bot_lt_iff_ne_bot.mpr (mul_ne_zero hA.ne' hB.ne') + refine hp.trans_le (measure_mono ?_) + intro b hb + apply (BoundedContinuousFunction.dist_lt_iff_of_compact hr).mpr + intro t + rw [dist_eq_norm] + have hre : |(planarUnitPath b t-f t).re| < r/4 := by + simpa [planarUnitPath,fR] using hb.1 t + have him : |(planarUnitPath b t-f t).im| < r/4 := by + simpa [planarUnitPath,fI] using hb.2 t + exact (Complex.norm_le_abs_re_add_abs_im _).trans_lt (by linarith) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/AnalyticCircleSquare.lean b/lean/OAI/Probability/SLE/Geometry/AnalyticCircleSquare.lean index 574ed5d8f4d97fb931f118a962dc1787df3d6e27..1e40125fca3890fd3a609346c42502567dd8c7f2 100644 --- a/lean/OAI/Probability/SLE/Geometry/AnalyticCircleSquare.lean +++ b/lean/OAI/Probability/SLE/Geometry/AnalyticCircleSquare.lean @@ -1,57 +1,57 @@ -import Mathlib - -namespace OAI - -/-! The square mean of a holomorphic function bounds its central value. -This is the local energy estimate needed for the first exterior coefficient. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real -open scoped Topology - -namespace SLEExactGauge - -lemma circleAverage_norm_sq_ge {f : ℂ → ℂ} {c : ℂ} {r : ℝ} - (hf : ContinuousOn f (sphere c |r|)) : - ‖circleAverage f c r‖^2≤circleAverage (fun z => ‖f z‖^2) c r := by - let a := circleAverage f c r - let L : ℂ →L[ℝ] ℝ := innerSL ℝ a - have hL : ContinuousOn (fun z => L (f z)) (sphere c |r|) := L.continuous.comp_continuousOn hf - have hI := hf.circleIntegrable' - have hN := (hf.norm.pow 2).circleIntegrable' - have hLin : circleAverage (fun z => L (f z)) c r=‖a‖^2 := by - have hh := L.circleAverage_comp_comm hI - change circleAverage (fun z => L (f z)) c r=L a at hh - rw [hh] - exact real_inner_self_eq_norm_sq a - have hleft : circleAverage (fun z => (2:ℝ)*L (f z)-‖a‖^2) c r=‖a‖^2 := by - rw [circleAverage_fun_sub (hL.const_mul 2).circleIntegrable' (circleIntegrable_const (‖a‖^2) c r)] - have hmul : circleAverage (fun z => (2:ℝ)*L (f z)) c r= - 2*circleAverage (fun z => L (f z)) c r := by - simpa only [smul_eq_mul] using (circleAverage_fun_smul (a := (2:ℝ)) (f := fun z => L (f z)) (c := c) (R := r)) - rw [hmul,hLin,circleAverage_const] - ring - rw [←hleft] - apply circleAverage_mono ((hL.const_mul 2).sub continuousOn_const).circleIntegrable' hN - intro z _ - have hn := norm_sub_sq_real (f z) a - have hp := sq_nonneg ‖f z-a‖ - have hLa : L (f z)=inner ℝ (f z) a := by - change inner ℝ a (f z)=inner ℝ (f z) a - exact real_inner_comm _ _ - dsimp only [Pi.sub_apply,Pi.pow_apply] - rw [hLa] - nlinarith - -lemma analytic_circleAverage_norm_sq_ge {f : ℂ → ℂ} {r : ℝ} - (hf : AnalyticOnNhd ℂ f (closedBall (0:ℂ) |r|)) : - ‖f 0‖^2≤circleAverage (fun z => ‖f z‖^2) 0 r := by - have hm : circleAverage f 0 r=f 0 := - (show HarmonicOnNhd f (closedBall (0:ℂ) |r|) from fun z hz => (hf z hz).harmonicAt).circleAverage_eq - rw [←hm] - exact circleAverage_norm_sq_ge (hf.continuousOn.mono sphere_subset_closedBall) - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! The square mean of a holomorphic function bounds its central value. +This is the local energy estimate needed for the first exterior coefficient. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real +open scoped Topology + +namespace SLEExactGauge + +lemma circleAverage_norm_sq_ge {f : ℂ → ℂ} {c : ℂ} {r : ℝ} + (hf : ContinuousOn f (sphere c |r|)) : + ‖circleAverage f c r‖^2≤circleAverage (fun z => ‖f z‖^2) c r := by + let a := circleAverage f c r + let L : ℂ →L[ℝ] ℝ := innerSL ℝ a + have hL : ContinuousOn (fun z => L (f z)) (sphere c |r|) := L.continuous.comp_continuousOn hf + have hI := hf.circleIntegrable' + have hN := (hf.norm.pow 2).circleIntegrable' + have hLin : circleAverage (fun z => L (f z)) c r=‖a‖^2 := by + have hh := L.circleAverage_comp_comm hI + change circleAverage (fun z => L (f z)) c r=L a at hh + rw [hh] + exact real_inner_self_eq_norm_sq a + have hleft : circleAverage (fun z => (2:ℝ)*L (f z)-‖a‖^2) c r=‖a‖^2 := by + rw [circleAverage_fun_sub (hL.const_mul 2).circleIntegrable' (circleIntegrable_const (‖a‖^2) c r)] + have hmul : circleAverage (fun z => (2:ℝ)*L (f z)) c r= + 2*circleAverage (fun z => L (f z)) c r := by + simpa only [smul_eq_mul] using (circleAverage_fun_smul (a := (2:ℝ)) (f := fun z => L (f z)) (c := c) (R := r)) + rw [hmul,hLin,circleAverage_const] + ring + rw [←hleft] + apply circleAverage_mono ((hL.const_mul 2).sub continuousOn_const).circleIntegrable' hN + intro z _ + have hn := norm_sub_sq_real (f z) a + have hp := sq_nonneg ‖f z-a‖ + have hLa : L (f z)=inner ℝ (f z) a := by + change inner ℝ a (f z)=inner ℝ (f z) a + exact real_inner_comm _ _ + dsimp only [Pi.sub_apply,Pi.pow_apply] + rw [hLa] + nlinarith + +lemma analytic_circleAverage_norm_sq_ge {f : ℂ → ℂ} {r : ℝ} + (hf : AnalyticOnNhd ℂ f (closedBall (0:ℂ) |r|)) : + ‖f 0‖^2≤circleAverage (fun z => ‖f z‖^2) 0 r := by + have hm : circleAverage f 0 r=f 0 := + (show HarmonicOnNhd f (closedBall (0:ℂ) |r|) from fun z hz => (hf z hz).harmonicAt).circleAverage_eq + rw [←hm] + exact circleAverage_norm_sq_ge (hf.continuousOn.mono sphere_subset_closedBall) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/AnalyticFluxEnergy.lean b/lean/OAI/Probability/SLE/Geometry/AnalyticFluxEnergy.lean index 1dec50c1d7ddc820b050ccc4554d4ec9a50d5794..291203366ca8611851215a0929567ab08c64564c 100644 --- a/lean/OAI/Probability/SLE/Geometry/AnalyticFluxEnergy.lean +++ b/lean/OAI/Probability/SLE/Geometry/AnalyticFluxEnergy.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Geometry.InteriorFluxBound -import OAI.Probability.SLE.Geometry.AnalyticCircleSquare -import OAI.Probability.SLE.Geometry.ExteriorLaplacianTest - -namespace OAI - -/-! The first Taylor coefficient is bounded by the signed area of a -holomorphic disk map. No injectivity of the disk map is needed. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma analytic_norm_sq_laplacian {f : ℂ → ℂ} {z : ℂ} (hf : AnalyticAt ℂ f z) : - (Δ (fun w => ‖f w‖^2)) z=4*‖deriv f z‖^2 := by - have hh := laplacian_holomorphic_comp hf - ((contDiff_id.norm_sq ℝ : ContDiff ℝ 2 (fun w : ℂ => ‖w‖^2)).contDiffAt) - simpa only [Function.comp_def,id_eq,complex_laplacian_norm_sq,mul_comm] using hh - -lemma analytic_flux_ge_first_coefficient {f : ℂ → ℂ} {R : ℝ} - (hR : 0 ‖f z‖^2) (R,θ) := by - have hu (z : ℂ) (hz : z∈closedBall (0:ℂ) R) : - ContDiffAt ℝ 2 (fun z => ‖f z‖^2) z := - (hf z hz).contDiffAt.restrict_scalars ℝ |>.norm_sq ℝ - have hAvg (r : ℝ) (hr : r∈Icc 0 R) : - 4*‖deriv f 0‖^2≤circleAverage (Δ (fun z => ‖f z‖^2)) 0 r := by - have hd : AnalyticOnNhd ℂ (deriv f) (closedBall (0:ℂ) |r|) := - hf.deriv.mono (closedBall_subset_closedBall (by rw [abs_of_nonneg hr.1]; exact hr.2)) - have hb := analytic_circleAverage_norm_sq_ge hd - have he : circleAverage (Δ (fun z => ‖f z‖^2)) 0 r= - circleAverage (fun z => (4:ℝ)*‖deriv f z‖^2) 0 r := by - apply circleAverage_congr_sphere - intro z hz - apply analytic_norm_sq_laplacian - exact hf z ((closedBall_subset_closedBall (by rw [abs_of_nonneg hr.1]; exact hr.2)) - (sphere_subset_closedBall hz)) - rw [he] - have hmul : circleAverage (fun z => (4:ℝ)*‖deriv f z‖^2) 0 r= - 4*circleAverage (fun z => ‖deriv f z‖^2) 0 r := by - simpa only [smul_eq_mul] using (circleAverage_fun_smul (a := (4:ℝ)) - (f := fun z => ‖deriv f z‖^2) (c := (0:ℂ)) (R := r)) - rw [hmul] - exact mul_le_mul_of_nonneg_left hb (by norm_num) - have hh := radial_flux_ge_of_circle_laplacian hR hu hAvg - convert! hh using 1; ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.InteriorFluxBound +import OAI.Probability.SLE.Geometry.AnalyticCircleSquare +import OAI.Probability.SLE.Geometry.ExteriorLaplacianTest + +namespace OAI + +/-! The first Taylor coefficient is bounded by the signed area of a +holomorphic disk map. No injectivity of the disk map is needed. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma analytic_norm_sq_laplacian {f : ℂ → ℂ} {z : ℂ} (hf : AnalyticAt ℂ f z) : + (Δ (fun w => ‖f w‖^2)) z=4*‖deriv f z‖^2 := by + have hh := laplacian_holomorphic_comp hf + ((contDiff_id.norm_sq ℝ : ContDiff ℝ 2 (fun w : ℂ => ‖w‖^2)).contDiffAt) + simpa only [Function.comp_def,id_eq,complex_laplacian_norm_sq,mul_comm] using hh + +lemma analytic_flux_ge_first_coefficient {f : ℂ → ℂ} {R : ℝ} + (hR : 0 ‖f z‖^2) (R,θ) := by + have hu (z : ℂ) (hz : z∈closedBall (0:ℂ) R) : + ContDiffAt ℝ 2 (fun z => ‖f z‖^2) z := + (hf z hz).contDiffAt.restrict_scalars ℝ |>.norm_sq ℝ + have hAvg (r : ℝ) (hr : r∈Icc 0 R) : + 4*‖deriv f 0‖^2≤circleAverage (Δ (fun z => ‖f z‖^2)) 0 r := by + have hd : AnalyticOnNhd ℂ (deriv f) (closedBall (0:ℂ) |r|) := + hf.deriv.mono (closedBall_subset_closedBall (by rw [abs_of_nonneg hr.1]; exact hr.2)) + have hb := analytic_circleAverage_norm_sq_ge hd + have he : circleAverage (Δ (fun z => ‖f z‖^2)) 0 r= + circleAverage (fun z => (4:ℝ)*‖deriv f z‖^2) 0 r := by + apply circleAverage_congr_sphere + intro z hz + apply analytic_norm_sq_laplacian + exact hf z ((closedBall_subset_closedBall (by rw [abs_of_nonneg hr.1]; exact hr.2)) + (sphere_subset_closedBall hz)) + rw [he] + have hmul : circleAverage (fun z => (4:ℝ)*‖deriv f z‖^2) 0 r= + 4*circleAverage (fun z => ‖deriv f z‖^2) 0 r := by + simpa only [smul_eq_mul] using (circleAverage_fun_smul (a := (4:ℝ)) + (f := fun z => ‖deriv f z‖^2) (c := (0:ℂ)) (R := r)) + rw [hmul] + exact mul_le_mul_of_nonneg_left hb (by norm_num) + have hh := radial_flux_ge_of_circle_laplacian hR hu hAvg + convert! hh using 1; ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/AnnulusPolarIntegral.lean b/lean/OAI/Probability/SLE/Geometry/AnnulusPolarIntegral.lean index 5a6b8811c9d81304ef3a8e326a13002546d52202..e4944895577cdd3599ae2edc24c3e659dea1133b 100644 --- a/lean/OAI/Probability/SLE/Geometry/AnnulusPolarIntegral.lean +++ b/lean/OAI/Probability/SLE/Geometry/AnnulusPolarIntegral.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.AngularLaws -import OAI.Probability.SLE.Geometry.PolarLaplacianFlux - -namespace OAI - -/-! Polar change of variables on a finite annulus. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma annulus_integral_polar_rectangle (g : ℂ → ℝ) {a R : ℝ} (ha : 0 ha.trans hr.1) subset_rfl - let f : ℂ → ℝ := A.indicator g - calc - _ = ∫ z, f z := (integral_indicator hA).symm - _ = ∫ p in T, p.1 • f (Complex.polarCoord.symm p) := - (Complex.integral_comp_polarCoord_symm f).symm - _ = ∫ p in T, S.indicator (fun p : ℝ × ℝ => p.1*g (polarPoint p.1 p.2)) p := by - apply setIntegral_congr_fun hT - intro p hp - have hn : ‖Complex.polarCoord.symm p‖=p.1 := by - rw [Complex.norm_polarCoord_symm,abs_of_pos hp.1] - by_cases hr : p.1∈Ioo a R - · have hz : Complex.polarCoord.symm p∈A := by - change a<‖_‖ ∧ ‖_‖ hr hs.1 - simp only [f,indicator_of_notMem hz,indicator_of_notMem hs,smul_zero] - _ = _ := by - rw [integral_indicator hS,Measure.restrict_restrict hS,inter_eq_left.mpr hST] - -lemma annulus_integral_polar (g : ℂ → ℝ) {a R : ℝ} (ha : 0 polarPoint p.1 p.2) - polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt - have hi : IntegrableOn G (Ioo a R ×ˢ Ioo (-Real.pi) Real.pi) := - (hGc.integrableOn_compact (isCompact_Icc.prod isCompact_Icc)).mono_set - (prod_mono Ioo_subset_Icc_self Ioo_subset_Icc_self) - rw [annulus_integral_polar_rectangle g ha,Measure.volume_eq_prod] - rw [Measure.volume_eq_prod] at hi - rw [setIntegral_prod _ hi] - simp_rw [←integral_Ioc_eq_integral_Ioo] - rw [←intervalIntegral.integral_of_le haR] - congr 1 - ext r - exact (intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -Real.pi≤Real.pi)).symm - -lemma annulus_integral_laplacian_flux {u : ℂ → ℝ} {a R : ℝ} - (ha : 0 plane_laplacian_continuousAt (hu z hz hzR))] - exact annulus_laplacian_flux ha haR hu - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.AngularLaws +import OAI.Probability.SLE.Geometry.PolarLaplacianFlux + +namespace OAI + +/-! Polar change of variables on a finite annulus. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma annulus_integral_polar_rectangle (g : ℂ → ℝ) {a R : ℝ} (ha : 0 ha.trans hr.1) subset_rfl + let f : ℂ → ℝ := A.indicator g + calc + _ = ∫ z, f z := (integral_indicator hA).symm + _ = ∫ p in T, p.1 • f (Complex.polarCoord.symm p) := + (Complex.integral_comp_polarCoord_symm f).symm + _ = ∫ p in T, S.indicator (fun p : ℝ × ℝ => p.1*g (polarPoint p.1 p.2)) p := by + apply setIntegral_congr_fun hT + intro p hp + have hn : ‖Complex.polarCoord.symm p‖=p.1 := by + rw [Complex.norm_polarCoord_symm,abs_of_pos hp.1] + by_cases hr : p.1∈Ioo a R + · have hz : Complex.polarCoord.symm p∈A := by + change a<‖_‖ ∧ ‖_‖ hr hs.1 + simp only [f,indicator_of_notMem hz,indicator_of_notMem hs,smul_zero] + _ = _ := by + rw [integral_indicator hS,Measure.restrict_restrict hS,inter_eq_left.mpr hST] + +lemma annulus_integral_polar (g : ℂ → ℝ) {a R : ℝ} (ha : 0 polarPoint p.1 p.2) + polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt + have hi : IntegrableOn G (Ioo a R ×ˢ Ioo (-Real.pi) Real.pi) := + (hGc.integrableOn_compact (isCompact_Icc.prod isCompact_Icc)).mono_set + (prod_mono Ioo_subset_Icc_self Ioo_subset_Icc_self) + rw [annulus_integral_polar_rectangle g ha,Measure.volume_eq_prod] + rw [Measure.volume_eq_prod] at hi + rw [setIntegral_prod _ hi] + simp_rw [←integral_Ioc_eq_integral_Ioo] + rw [←intervalIntegral.integral_of_le haR] + congr 1 + ext r + exact (intervalIntegral.integral_of_le (by linarith [Real.pi_pos] : -Real.pi≤Real.pi)).symm + +lemma annulus_integral_laplacian_flux {u : ℂ → ℝ} {a R : ℝ} + (ha : 0 plane_laplacian_continuousAt (hu z hz hzR))] + exact annulus_laplacian_flux ha haR hu + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ApproachIntervalEndpoint.lean b/lean/OAI/Probability/SLE/Geometry/ApproachIntervalEndpoint.lean index 24da392741604edade098fee0a5f4688b4593505..99dc9880dd89584cb99b3f570c89e2e144eb82cb 100644 --- a/lean/OAI/Probability/SLE/Geometry/ApproachIntervalEndpoint.lean +++ b/lean/OAI/Probability/SLE/Geometry/ApproachIntervalEndpoint.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Loewner.SlitSegmentParameter - -namespace OAI - -/-! An explicit sequence approaching the terminal parameter from below, -with every point still after the initial parameter. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -noncomputable def approachIntervalEndpoint {s t : unitInterval} (hst : s B t ω)) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b : ℝ) : C.limitingDiskTail hc b ≤ 1 := by - apply iSup_le - intro j - let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw - (4*(j:ℝ)/κ) (by positivity) - exact (measure_mono (subset_univ _)).trans_eq measure_univ - -lemma centeredDiskTail_le_limit - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b v : ℝ) (hv : 0≤v) : C.centeredDiskTail hc b v ≤ C.limitingDiskTail hc b := by - obtain ⟨n,hn⟩ := exists_nat_ge (v*κ/4) - have hnV : v≤4*(n:ℝ)/κ := (le_div_iff₀ hκ).mpr (by linarith) - exact (C.centeredDiskTail_mono hB hc h0 hκ hκ8 F hw b hv - (show 0≤4*(n:ℝ)/κ from by positivity) hnV).trans (le_iSup (fun j : ℕ => C.centeredDiskTail hc b (4*(j:ℝ)/κ)) n) - -lemma raw_angle_tail_average_le_limit - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (b v : ℝ) (hv : 0≤v) : - (∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v θ ∂Target.angleLaw κ) ≤ - C.limitingDiskTail hc b := by - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - rw [R.angleDiskTail_average _ κ hκ b v,C.raw_centeredDiskTail_eq hc hκ F hw R he v b hv] - exact C.centeredDiskTail_le_limit hB hc h0 hκ hκ8 F hw b v hv - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawDiskAngleTail +import OAI.Probability.SLE.Gauge.CanonicalDiskMomentTails + +namespace OAI + +/-! Arbitrary finite radius levels are bounded by the same countable tail +supremum already used in the prescribed-gauge budget. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint.CanonicalMass +local instance arbitraryTailMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance arbitraryTailBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma limitingDiskTail_le_one + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b : ℝ) : C.limitingDiskTail hc b ≤ 1 := by + apply iSup_le + intro j + let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw + (4*(j:ℝ)/κ) (by positivity) + exact (measure_mono (subset_univ _)).trans_eq measure_univ + +lemma centeredDiskTail_le_limit + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b v : ℝ) (hv : 0≤v) : C.centeredDiskTail hc b v ≤ C.limitingDiskTail hc b := by + obtain ⟨n,hn⟩ := exists_nat_ge (v*κ/4) + have hnV : v≤4*(n:ℝ)/κ := (le_div_iff₀ hκ).mpr (by linarith) + exact (C.centeredDiskTail_mono hB hc h0 hκ hκ8 F hw b hv + (show 0≤4*(n:ℝ)/κ from by positivity) hnV).trans (le_iSup (fun j : ℕ => C.centeredDiskTail hc b (4*(j:ℝ)/κ)) n) + +lemma raw_angle_tail_average_le_limit + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (b v : ℝ) (hv : 0≤v) : + (∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v θ ∂Target.angleLaw κ) ≤ + C.limitingDiskTail hc b := by + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + rw [R.angleDiskTail_average _ κ hκ b v,C.raw_centeredDiskTail_eq hc hκ F hw R he v b hv] + exact C.centeredDiskTail_le_limit hB hc h0 hκ hκ8 F hw b v hv + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ArcBoundaryReplacement.lean b/lean/OAI/Probability/SLE/Geometry/ArcBoundaryReplacement.lean index 8b571bd14b03ecff94a2e8cc5272e3deaa379e91..69e4f507ec59190d74a545782a5157f1b45a78b0 100644 --- a/lean/OAI/Probability/SLE/Geometry/ArcBoundaryReplacement.lean +++ b/lean/OAI/Probability/SLE/Geometry/ArcBoundaryReplacement.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Geometry.ClosedComplexUnion - -namespace OAI - -/-! Replacing one compact arc in a Jordan boundary while fixing the -other arc. The replacement retains the supplied interval parameter. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -def arcParameterHomeomorph (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) - (hi : InjOn p (Icc 0 1)) : (Icc (0:ℝ) 1) ≃ₜ (p '' Icc 0 1) := by - let e := Equiv.Set.imageOfInjOn p (Icc 0 1) hi - have he : Continuous e := hc.mapsToRestrict (mapsTo_image p (Icc 0 1)) - exact e.toHomeomorphOfContinuousClosed he he.isClosedMap - -@[simp] lemma arcParameterHomeomorph_apply - (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (s : Icc (0:ℝ) 1) : (arcParameterHomeomorph p hc hi s : ℂ)=p s := rfl - -def arcParameterReplacement (p q : ℝ → ℂ) - (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) : - (p '' Icc 0 1) ≃ₜ (q '' Icc 0 1) := - (arcParameterHomeomorph p hc hi).symm.trans (arcParameterHomeomorph q hqc hqi) - -lemma arcParameterReplacement_apply (p q : ℝ → ℂ) - (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) - (s : ℝ) (hs : s ∈ Icc 0 1) : - (arcParameterReplacement p q hc hi hqc hqi ⟨p s,mem_image_of_mem p hs⟩ : ℂ)=q s := by - have hh : arcParameterHomeomorph p hc hi ⟨s,hs⟩ = ⟨p s,mem_image_of_mem p hs⟩ := rfl - change (arcParameterHomeomorph q hqc hqi - ((arcParameterHomeomorph p hc hi).symm ⟨p s,mem_image_of_mem p hs⟩) : ℂ)=q s - rw [← hh,Homeomorph.symm_apply_apply] - rfl - -def fixedArcBoundaryReplacement (p q : ℝ → ℂ) (B : Set ℂ) - (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) - (hB : IsClosed B) (hzero : q 0=p 0) (hone : q 1=p 1) - (hpB : ∀ s ∈ Icc 0 1, p s ∈ B → s=0 ∨ s=1) - (hqB : ∀ s ∈ Icc 0 1, q s ∈ B → s=0 ∨ s=1) : - ↥(p '' Icc 0 1 ∪ B) ≃ₜ ↥(q '' Icc 0 1 ∪ B) := by - apply closedComplexUnionHomeomorph (isCompact_Icc.image_of_continuousOn hc).isClosed - (isCompact_Icc.image_of_continuousOn hqc).isClosed hB - (arcParameterReplacement p q hc hi hqc hqi) - · rintro ⟨z,⟨s,hs,rfl⟩⟩ hzB - rw [arcParameterReplacement_apply p q hc hi hqc hqi s hs] - rcases hpB s hs hzB with rfl | rfl - · exact hzero - · exact hone - · rintro ⟨z,⟨s,hs,rfl⟩⟩ hzB - change (arcParameterReplacement q p hqc hqi hc hi - ⟨q s,mem_image_of_mem q hs⟩ : ℂ)=q s - rw [arcParameterReplacement_apply q p hqc hqi hc hi s hs] - rcases hqB s hs hzB with rfl | rfl - · exact hzero.symm - · exact hone.symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ClosedComplexUnion + +namespace OAI + +/-! Replacing one compact arc in a Jordan boundary while fixing the +other arc. The replacement retains the supplied interval parameter. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +def arcParameterHomeomorph (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) + (hi : InjOn p (Icc 0 1)) : (Icc (0:ℝ) 1) ≃ₜ (p '' Icc 0 1) := by + let e := Equiv.Set.imageOfInjOn p (Icc 0 1) hi + have he : Continuous e := hc.mapsToRestrict (mapsTo_image p (Icc 0 1)) + exact e.toHomeomorphOfContinuousClosed he he.isClosedMap + +@[simp] lemma arcParameterHomeomorph_apply + (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (s : Icc (0:ℝ) 1) : (arcParameterHomeomorph p hc hi s : ℂ)=p s := rfl + +def arcParameterReplacement (p q : ℝ → ℂ) + (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) : + (p '' Icc 0 1) ≃ₜ (q '' Icc 0 1) := + (arcParameterHomeomorph p hc hi).symm.trans (arcParameterHomeomorph q hqc hqi) + +lemma arcParameterReplacement_apply (p q : ℝ → ℂ) + (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) + (s : ℝ) (hs : s ∈ Icc 0 1) : + (arcParameterReplacement p q hc hi hqc hqi ⟨p s,mem_image_of_mem p hs⟩ : ℂ)=q s := by + have hh : arcParameterHomeomorph p hc hi ⟨s,hs⟩ = ⟨p s,mem_image_of_mem p hs⟩ := rfl + change (arcParameterHomeomorph q hqc hqi + ((arcParameterHomeomorph p hc hi).symm ⟨p s,mem_image_of_mem p hs⟩) : ℂ)=q s + rw [← hh,Homeomorph.symm_apply_apply] + rfl + +def fixedArcBoundaryReplacement (p q : ℝ → ℂ) (B : Set ℂ) + (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (hqc : ContinuousOn q (Icc 0 1)) (hqi : InjOn q (Icc 0 1)) + (hB : IsClosed B) (hzero : q 0=p 0) (hone : q 1=p 1) + (hpB : ∀ s ∈ Icc 0 1, p s ∈ B → s=0 ∨ s=1) + (hqB : ∀ s ∈ Icc 0 1, q s ∈ B → s=0 ∨ s=1) : + ↥(p '' Icc 0 1 ∪ B) ≃ₜ ↥(q '' Icc 0 1 ∪ B) := by + apply closedComplexUnionHomeomorph (isCompact_Icc.image_of_continuousOn hc).isClosed + (isCompact_Icc.image_of_continuousOn hqc).isClosed hB + (arcParameterReplacement p q hc hi hqc hqi) + · rintro ⟨z,⟨s,hs,rfl⟩⟩ hzB + rw [arcParameterReplacement_apply p q hc hi hqc hqi s hs] + rcases hpB s hs hzB with rfl | rfl + · exact hzero + · exact hone + · rintro ⟨z,⟨s,hs,rfl⟩⟩ hzB + change (arcParameterReplacement q p hqc hqi hc hi + ⟨q s,mem_image_of_mem q hs⟩ : ℂ)=q s + rw [arcParameterReplacement_apply q p hqc hqi hc hi s hs] + rcases hqB s hs hzB with rfl | rfl + · exact hzero.symm + · exact hone.symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ArcHeightExtension.lean b/lean/OAI/Probability/SLE/Geometry/ArcHeightExtension.lean index d74725888e0cc80c8a3234837c9636d9462c0b22..4673cb5d9ad152f4783c812816cd22023759fd64 100644 --- a/lean/OAI/Probability/SLE/Geometry/ArcHeightExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ArcHeightExtension.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! Extending the time coordinate of a simple half-plane arc while fixing -the real boundary at time zero. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -theorem simple_arc_height_extension (p : C(unitInterval,ℂ)) (hi : Injective p) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ φ : C(ℂ,ℝ), (∀ z, φ z ∈ Icc (0 : ℝ) 1) ∧ - (∀ t : unitInterval, φ (p t)=(t : ℝ)) ∧ (∀ x : ℝ, φ (x : ℂ)=0) := by - classical - let K : Set ℂ := range p - let A : Set ℂ := K ∪ {z : ℂ | z.im=0} - have hK : IsClosed K := p.continuous.isClosedEmbedding hi |>.isClosed_range - have hR : IsClosed {z : ℂ | z.im=0} := isClosed_eq Complex.continuous_im continuous_const - have hA : IsClosed A := hK.union hR - let t : C(unitInterval,ℝ) := ⟨fun t => t,continuous_subtype_val⟩ - obtain ⟨g,hg⟩ := t.exists_extension' (p.continuous.isClosedEmbedding hi) - have hgp (t : unitInterval) : g (p t)=(t : ℝ) := congr_fun hg t - let f : ℂ → ℝ := K.piecewise g 0 - have hfK : EqOn f g K := fun z hz => piecewise_eq_of_mem K g 0 hz - have hfR (z : ℂ) (hz : z.im=0) : f z=0 := by - by_cases hzK : z ∈ K - · obtain ⟨s,rfl⟩ := hzK - have hs : s=0 := by - by_contra hs - have hh := hH s hs - change 0 < (p s).im at hh - linarith - rw [hfK (mem_range_self s),hgp,hs] - rfl - · exact piecewise_eq_of_notMem K g 0 hzK - have hf : ContinuousOn f A := - (g.continuous.continuousOn.congr hfK).union_of_isClosed - (continuousOn_const.congr (fun z hz => hfR z hz)) hK hR - let fA : C(A,ℝ) := ⟨fun z => f z,hf.domRestrict⟩ - have hfA (z : A) : fA z ∈ Icc (0 : ℝ) 1 := by - rcases z.property with hz | hz - · obtain ⟨s,hs⟩ := hz - change f z ∈ Icc (0 : ℝ) 1 - rw [hfK ⟨s,hs⟩,← hs,hgp] - exact s.property - · change f z ∈ Icc (0 : ℝ) 1 - rw [hfR z hz] - exact ⟨le_rfl,zero_le_one⟩ - obtain ⟨φ,hφ,hφf⟩ := fA.exists_restrict_eq_forall_mem_of_closed hfA - (nonempty_Icc.mpr zero_le_one) hA - have he (z : ℂ) (hz : z ∈ A) : φ z=f z := congr_fun (congrArg DFunLike.coe hφf) ⟨z,hz⟩ - refine ⟨φ,hφ,?_,?_⟩ - · intro s - rw [he _ (Or.inl (mem_range_self s)),hfK (mem_range_self s),hgp] - · intro x - rw [he _ (Or.inr (by simp)),hfR _ (by simp)] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! Extending the time coordinate of a simple half-plane arc while fixing +the real boundary at time zero. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +theorem simple_arc_height_extension (p : C(unitInterval,ℂ)) (hi : Injective p) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ φ : C(ℂ,ℝ), (∀ z, φ z ∈ Icc (0 : ℝ) 1) ∧ + (∀ t : unitInterval, φ (p t)=(t : ℝ)) ∧ (∀ x : ℝ, φ (x : ℂ)=0) := by + classical + let K : Set ℂ := range p + let A : Set ℂ := K ∪ {z : ℂ | z.im=0} + have hK : IsClosed K := p.continuous.isClosedEmbedding hi |>.isClosed_range + have hR : IsClosed {z : ℂ | z.im=0} := isClosed_eq Complex.continuous_im continuous_const + have hA : IsClosed A := hK.union hR + let t : C(unitInterval,ℝ) := ⟨fun t => t,continuous_subtype_val⟩ + obtain ⟨g,hg⟩ := t.exists_extension' (p.continuous.isClosedEmbedding hi) + have hgp (t : unitInterval) : g (p t)=(t : ℝ) := congr_fun hg t + let f : ℂ → ℝ := K.piecewise g 0 + have hfK : EqOn f g K := fun z hz => piecewise_eq_of_mem K g 0 hz + have hfR (z : ℂ) (hz : z.im=0) : f z=0 := by + by_cases hzK : z ∈ K + · obtain ⟨s,rfl⟩ := hzK + have hs : s=0 := by + by_contra hs + have hh := hH s hs + change 0 < (p s).im at hh + linarith + rw [hfK (mem_range_self s),hgp,hs] + rfl + · exact piecewise_eq_of_notMem K g 0 hzK + have hf : ContinuousOn f A := + (g.continuous.continuousOn.congr hfK).union_of_isClosed + (continuousOn_const.congr (fun z hz => hfR z hz)) hK hR + let fA : C(A,ℝ) := ⟨fun z => f z,hf.domRestrict⟩ + have hfA (z : A) : fA z ∈ Icc (0 : ℝ) 1 := by + rcases z.property with hz | hz + · obtain ⟨s,hs⟩ := hz + change f z ∈ Icc (0 : ℝ) 1 + rw [hfK ⟨s,hs⟩,← hs,hgp] + exact s.property + · change f z ∈ Icc (0 : ℝ) 1 + rw [hfR z hz] + exact ⟨le_rfl,zero_le_one⟩ + obtain ⟨φ,hφ,hφf⟩ := fA.exists_restrict_eq_forall_mem_of_closed hfA + (nonempty_Icc.mpr zero_le_one) hA + have he (z : ℂ) (hz : z ∈ A) : φ z=f z := congr_fun (congrArg DFunLike.coe hφf) ⟨z,hz⟩ + refine ⟨φ,hφ,?_,?_⟩ + · intro s + rw [he _ (Or.inl (mem_range_self s)),hfK (mem_range_self s),hgp] + · intro x + rw [he _ (Or.inr (by simp)),hfR _ (by simp)] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ArcRealRetraction.lean b/lean/OAI/Probability/SLE/Geometry/ArcRealRetraction.lean index fd07af3d3e465b358d39a964cf0e9f27d5b0f7d5..dd5dc74ab7c146dbe26b2391bcca9fa2f2a83745 100644 --- a/lean/OAI/Probability/SLE/Geometry/ArcRealRetraction.lean +++ b/lean/OAI/Probability/SLE/Geometry/ArcRealRetraction.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Geometry.ArcHeightExtension -import OAI.Probability.SLE.Stochastic.PlanarRetractComplement - -namespace OAI - -/-! A simple arc attached to the real axis is a retract together with -that axis. The explicit formula collapses the two coordinates onto the -three branches of the attached arc and real line. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -theorem simple_arc_real_retraction (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ r : C(ℂ,ℂ), (∀ z, r z ∈ range p ∪ {w : ℂ | w.im=0}) ∧ - ∀ z ∈ range p ∪ {w : ℂ | w.im=0}, r z=z := by - obtain ⟨φ,hφ,hφp,hφR⟩ := simple_arc_height_extension p hi hH - let v (z : ℂ) : unitInterval := ⟨φ z,hφ z⟩ - have hv : Continuous v := φ.continuous.subtype_mk _ - let b (z : ℂ) : ℝ := z.re-(p (v z)).re - have hb : Continuous b := Complex.continuous_re.sub - (Complex.continuous_re.comp (p.continuous.comp hv)) - let t (z : ℂ) : unitInterval := ⟨max (φ z-|b z|) 0, - le_max_right _ _,max_le (by linarith [(hφ z).2,abs_nonneg (b z)]) zero_le_one⟩ - have ht : Continuous t := ((φ.continuous.sub hb.abs).max continuous_const).subtype_mk _ - let x (z : ℂ) : ℝ := max (b z-φ z) 0-max (-b z-φ z) 0 - have hx : Continuous x := ((hb.sub φ.continuous).max continuous_const).sub - ((hb.neg.sub φ.continuous).max continuous_const) - let r : C(ℂ,ℂ) := ⟨fun z => p (t z)+(x z : ℂ), - (p.continuous.comp ht).add (Complex.continuous_ofReal.comp hx)⟩ - have ht0 (z : ℂ) (hz : φ z ≤ |b z|) : t z=0 := by - apply Subtype.ext - exact max_eq_right (sub_nonpos.mpr hz) - have hx0 (z : ℂ) (hz : |b z| ≤ φ z) : x z=0 := by - have hpos : b z-φ z ≤ 0 := sub_nonpos.mpr ((le_abs_self _).trans hz) - have hneg : -b z-φ z ≤ 0 := sub_nonpos.mpr ((neg_le_abs _).trans hz) - simp only [x,max_eq_right hpos,max_eq_right hneg,sub_self] - refine ⟨r,?_,?_⟩ - · intro z - by_cases hz : |b z| ≤ φ z - · left - exact ⟨t z,by simp [r,hx0 z hz]⟩ - · right - simp [r,ht0 z (le_of_not_ge hz),h0] - · intro z hz - rcases hz with ⟨s,rfl⟩ | hz - · have hvp : v (p s)=s := Subtype.ext (hφp s) - have hbp : b (p s)=0 := by simp [b,hvp] - have htp : t (p s)=s := by - apply Subtype.ext - simp only [t,hφp,hbp,abs_zero,sub_zero,max_eq_left s.property.1] - have hxp : x (p s)=0 := by - apply hx0 - simp only [hbp,abs_zero,hφp] - exact s.property.1 - simp [r,htp,hxp] - · change z.im=0 at hz - have hez : z=(z.re : ℂ) := by apply Complex.ext <;> simp [hz] - have hφz : φ z=0 := by rw [hez,hφR] - have hvz : v z=0 := Subtype.ext hφz - have hbz : b z=z.re := by simp [b,hvz,h0] - have htz : t z=0 := ht0 z (by rw [hφz]; exact abs_nonneg _) - have hxz : x z=z.re := by - change max (b z-φ z) 0-max (-b z-φ z) 0=z.re - rw [hbz,hφz,sub_zero,sub_zero] - rcases le_total 0 z.re with h | h - · rw [max_eq_left h,max_eq_right (neg_nonpos.mpr h),sub_zero] - · rw [max_eq_right h,max_eq_left (neg_nonneg.mpr h),zero_sub,neg_neg] - simpa only [r,ContinuousMap.coe_mk,htz,h0,zero_add,hxz] using hez.symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ArcHeightExtension +import OAI.Probability.SLE.Stochastic.PlanarRetractComplement + +namespace OAI + +/-! A simple arc attached to the real axis is a retract together with +that axis. The explicit formula collapses the two coordinates onto the +three branches of the attached arc and real line. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +theorem simple_arc_real_retraction (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ r : C(ℂ,ℂ), (∀ z, r z ∈ range p ∪ {w : ℂ | w.im=0}) ∧ + ∀ z ∈ range p ∪ {w : ℂ | w.im=0}, r z=z := by + obtain ⟨φ,hφ,hφp,hφR⟩ := simple_arc_height_extension p hi hH + let v (z : ℂ) : unitInterval := ⟨φ z,hφ z⟩ + have hv : Continuous v := φ.continuous.subtype_mk _ + let b (z : ℂ) : ℝ := z.re-(p (v z)).re + have hb : Continuous b := Complex.continuous_re.sub + (Complex.continuous_re.comp (p.continuous.comp hv)) + let t (z : ℂ) : unitInterval := ⟨max (φ z-|b z|) 0, + le_max_right _ _,max_le (by linarith [(hφ z).2,abs_nonneg (b z)]) zero_le_one⟩ + have ht : Continuous t := ((φ.continuous.sub hb.abs).max continuous_const).subtype_mk _ + let x (z : ℂ) : ℝ := max (b z-φ z) 0-max (-b z-φ z) 0 + have hx : Continuous x := ((hb.sub φ.continuous).max continuous_const).sub + ((hb.neg.sub φ.continuous).max continuous_const) + let r : C(ℂ,ℂ) := ⟨fun z => p (t z)+(x z : ℂ), + (p.continuous.comp ht).add (Complex.continuous_ofReal.comp hx)⟩ + have ht0 (z : ℂ) (hz : φ z ≤ |b z|) : t z=0 := by + apply Subtype.ext + exact max_eq_right (sub_nonpos.mpr hz) + have hx0 (z : ℂ) (hz : |b z| ≤ φ z) : x z=0 := by + have hpos : b z-φ z ≤ 0 := sub_nonpos.mpr ((le_abs_self _).trans hz) + have hneg : -b z-φ z ≤ 0 := sub_nonpos.mpr ((neg_le_abs _).trans hz) + simp only [x,max_eq_right hpos,max_eq_right hneg,sub_self] + refine ⟨r,?_,?_⟩ + · intro z + by_cases hz : |b z| ≤ φ z + · left + exact ⟨t z,by simp [r,hx0 z hz]⟩ + · right + simp [r,ht0 z (le_of_not_ge hz),h0] + · intro z hz + rcases hz with ⟨s,rfl⟩ | hz + · have hvp : v (p s)=s := Subtype.ext (hφp s) + have hbp : b (p s)=0 := by simp [b,hvp] + have htp : t (p s)=s := by + apply Subtype.ext + simp only [t,hφp,hbp,abs_zero,sub_zero,max_eq_left s.property.1] + have hxp : x (p s)=0 := by + apply hx0 + simp only [hbp,abs_zero,hφp] + exact s.property.1 + simp [r,htp,hxp] + · change z.im=0 at hz + have hez : z=(z.re : ℂ) := by apply Complex.ext <;> simp [hz] + have hφz : φ z=0 := by rw [hez,hφR] + have hvz : v z=0 := Subtype.ext hφz + have hbz : b z=z.re := by simp [b,hvz,h0] + have htz : t z=0 := ht0 z (by rw [hφz]; exact abs_nonneg _) + have hxz : x z=z.re := by + change max (b z-φ z) 0-max (-b z-φ z) 0=z.re + rw [hbz,hφz,sub_zero,sub_zero] + rcases le_total 0 z.re with h | h + · rw [max_eq_left h,max_eq_right (neg_nonpos.mpr h),sub_zero] + · rw [max_eq_right h,max_eq_left (neg_nonneg.mpr h),zero_sub,neg_neg] + simpa only [r,ContinuousMap.coe_mk,htz,h0,zero_add,hxz] using hez.symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/BoundaryNull.lean b/lean/OAI/Probability/SLE/Geometry/BoundaryNull.lean index b3d8b6663c0ae092071c5de6a361da8719b0206e..25fbe8670ba9b54cb23640c1544a3fbca72200c4 100644 --- a/lean/OAI/Probability/SLE/Geometry/BoundaryNull.lean +++ b/lean/OAI/Probability/SLE/Geometry/BoundaryNull.lean @@ -1,85 +1,85 @@ -import OAI.Probability.SLE.Gauge.GaugeCompletion - -namespace OAI - -/-! Real-boundary cleanup in the all-time upper Hausdorff estimate. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -/-- The log-log correction is smaller than every power lost from dimension. -/ -theorem hFormula_div_rpow_tendsto_zero {κ s : ℝ} - (hκ : 0 < κ) (hs : s < dimension κ) : - Tendsto (fun r => hFormula κ r / r ^ s) (𝓝[>] 0) (𝓝 0) := by - have hq : codimension κ / 2 ≤ 1 := by - dsimp [codimension, dimension] - linarith - have ht := (GaugeCompletion.tailProfile_tendsto_zero (sub_pos.mpr hs) hq).comp - GaugeCompletion.log_inv_tendsto_atTop - apply ht.congr' - filter_upwards [self_mem_nhdsWithin] with r hr - change GaugeCompletion.tailProfile (dimension κ - s) (codimension κ / 2) - (Real.log (1 / r)) = _ - rw [GaugeCompletion.tailProfile_log_inv hr] - unfold hFormula - rw [Real.rpow_sub hr] - ring_nf - -/-- An eventual power bound compares the diameter-defined gauge measure with -ordinary Hausdorff measure. -/ -theorem hausdorffGauge_le_hausdorffMeasure_of_power_bound - {h : ℝ → ℝ} (hh : IsGauge h) {s : ℝ} (hs : 0 ≤ s) - (hb : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r ≤ r ^ s) : - hausdorffGauge h ≤ (Measure.hausdorffMeasure s : Measure ℂ) := by - obtain ⟨δ, hδ, hbound⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hb - have ht : Tendsto ENNReal.toReal (𝓝 (0 : ℝ≥0∞)) (𝓝 (0 : ℝ)) := by - simpa using ENNReal.tendsto_toReal ENNReal.zero_ne_top - apply Measure.mkMetric_mono - filter_upwards [ht.eventually (gt_mem_nhds hδ)] with r hr - change ENNReal.ofReal (h r.toReal) ≤ r ^ s - by_cases hz : r.toReal = 0 - · simp only [hz, hh.zero, ENNReal.ofReal_zero] - exact bot_le - · have hp : 0 < r.toReal := lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm hz) - calc - ENNReal.ofReal (h r.toReal) ≤ ENNReal.ofReal (r.toReal ^ s) := - ENNReal.ofReal_le_ofReal (hbound ⟨hp, hr⟩) - _ = (ENNReal.ofReal r.toReal) ^ s := - (ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hs).symm - _ ≤ r ^ s := ENNReal.rpow_le_rpow ENNReal.ofReal_toReal_le hs - -/-- Every continuous completion of the explicit gauge gives zero measure to -the entire real axis, hence also to its trace points in any bounded disk. -/ -theorem explicit_gauge_real_axis_null {κ : ℝ} (hκ : 0 < κ) - {h : ℝ → ℝ} (hh : IsGauge h) - (hf : HasSmallRadiusFormula h (hFormula κ)) : - hausdorffGauge h (range ((↑) : ℝ → ℂ)) = 0 := by - let s : ℝ := (1 + dimension κ) / 2 - have hd : 1 < dimension κ := by dsimp [dimension]; linarith - have hs1 : 1 < s := by dsimp [s]; linarith - have hsd : s < dimension κ := by dsimp [s]; linarith - have ht : Tendsto (fun r => h r / r ^ s) (𝓝[>] 0) (𝓝 0) := by - apply (hFormula_div_rpow_tendsto_zero hκ hsd).congr' - filter_upwards [hf] with r hr - rw [hr] - have hb : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r ≤ r ^ s := by - filter_upwards [self_mem_nhdsWithin, ht.eventually (gt_mem_nhds zero_lt_one)] with r hr hb - exact ((div_lt_one (Real.rpow_pos_of_pos hr s)).mp hb).le - have hcomp := hausdorffGauge_le_hausdorffMeasure_of_power_bound hh - (zero_lt_one.trans hs1).le hb - apply le_antisymm _ bot_le - apply (hcomp _).trans - have he := Complex.isometry_ofReal.hausdorffMeasure_image - (Or.inl (zero_lt_one.trans hs1).le) (univ : Set ℝ) - rw [image_univ] at he - rw [he, Real.hausdorffMeasure_of_finrank_lt (by simpa using hs1)] - simp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.GaugeCompletion + +namespace OAI + +/-! Real-boundary cleanup in the all-time upper Hausdorff estimate. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +/-- The log-log correction is smaller than every power lost from dimension. -/ +theorem hFormula_div_rpow_tendsto_zero {κ s : ℝ} + (hκ : 0 < κ) (hs : s < dimension κ) : + Tendsto (fun r => hFormula κ r / r ^ s) (𝓝[>] 0) (𝓝 0) := by + have hq : codimension κ / 2 ≤ 1 := by + dsimp [codimension, dimension] + linarith + have ht := (GaugeCompletion.tailProfile_tendsto_zero (sub_pos.mpr hs) hq).comp + GaugeCompletion.log_inv_tendsto_atTop + apply ht.congr' + filter_upwards [self_mem_nhdsWithin] with r hr + change GaugeCompletion.tailProfile (dimension κ - s) (codimension κ / 2) + (Real.log (1 / r)) = _ + rw [GaugeCompletion.tailProfile_log_inv hr] + unfold hFormula + rw [Real.rpow_sub hr] + ring_nf + +/-- An eventual power bound compares the diameter-defined gauge measure with +ordinary Hausdorff measure. -/ +theorem hausdorffGauge_le_hausdorffMeasure_of_power_bound + {h : ℝ → ℝ} (hh : IsGauge h) {s : ℝ} (hs : 0 ≤ s) + (hb : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r ≤ r ^ s) : + hausdorffGauge h ≤ (Measure.hausdorffMeasure s : Measure ℂ) := by + obtain ⟨δ, hδ, hbound⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hb + have ht : Tendsto ENNReal.toReal (𝓝 (0 : ℝ≥0∞)) (𝓝 (0 : ℝ)) := by + simpa using ENNReal.tendsto_toReal ENNReal.zero_ne_top + apply Measure.mkMetric_mono + filter_upwards [ht.eventually (gt_mem_nhds hδ)] with r hr + change ENNReal.ofReal (h r.toReal) ≤ r ^ s + by_cases hz : r.toReal = 0 + · simp only [hz, hh.zero, ENNReal.ofReal_zero] + exact bot_le + · have hp : 0 < r.toReal := lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm hz) + calc + ENNReal.ofReal (h r.toReal) ≤ ENNReal.ofReal (r.toReal ^ s) := + ENNReal.ofReal_le_ofReal (hbound ⟨hp, hr⟩) + _ = (ENNReal.ofReal r.toReal) ^ s := + (ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hs).symm + _ ≤ r ^ s := ENNReal.rpow_le_rpow ENNReal.ofReal_toReal_le hs + +/-- Every continuous completion of the explicit gauge gives zero measure to +the entire real axis, hence also to its trace points in any bounded disk. -/ +theorem explicit_gauge_real_axis_null {κ : ℝ} (hκ : 0 < κ) + {h : ℝ → ℝ} (hh : IsGauge h) + (hf : HasSmallRadiusFormula h (hFormula κ)) : + hausdorffGauge h (range ((↑) : ℝ → ℂ)) = 0 := by + let s : ℝ := (1 + dimension κ) / 2 + have hd : 1 < dimension κ := by dsimp [dimension]; linarith + have hs1 : 1 < s := by dsimp [s]; linarith + have hsd : s < dimension κ := by dsimp [s]; linarith + have ht : Tendsto (fun r => h r / r ^ s) (𝓝[>] 0) (𝓝 0) := by + apply (hFormula_div_rpow_tendsto_zero hκ hsd).congr' + filter_upwards [hf] with r hr + rw [hr] + have hb : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r ≤ r ^ s := by + filter_upwards [self_mem_nhdsWithin, ht.eventually (gt_mem_nhds zero_lt_one)] with r hr hb + exact ((div_lt_one (Real.rpow_pos_of_pos hr s)).mp hb).le + have hcomp := hausdorffGauge_le_hausdorffMeasure_of_power_bound hh + (zero_lt_one.trans hs1).le hb + apply le_antisymm _ bot_le + apply (hcomp _).trans + have he := Complex.isometry_ofReal.hausdorffMeasure_image + (Or.inl (zero_lt_one.trans hs1).le) (univ : Set ℝ) + rw [image_univ] at he + rw [he, Real.hausdorffMeasure_of_finrank_lt (by simpa using hs1)] + simp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/BoundaryRadialDistance.lean b/lean/OAI/Probability/SLE/Geometry/BoundaryRadialDistance.lean index a8935f3a75d1980d91be217b123d8c22c712b7ad..4b8a706aaa4d4c3ced60e37d44ccba2a4fe51992 100644 --- a/lean/OAI/Probability/SLE/Geometry/BoundaryRadialDistance.lean +++ b/lean/OAI/Probability/SLE/Geometry/BoundaryRadialDistance.lean @@ -1,91 +1,91 @@ -import OAI.Probability.SLE.Loewner.ReverseConeContraction -import OAI.Probability.SLE.Loewner.ReverseLongContraction - -namespace OAI - -/-! A boundary-start reverse trajectory is the uniform limit of the -positive-height trajectories on each finite time interval. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -theorem exists_boundary_radial_distance_constant {T a L : ℝ} - (ha : 0 < a) (haT : a ≤ T) (hL : 0 ≤ L) (hsmall : L^2*a ≤ 1) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hqc : ContinuousOn q (Icc 0 T)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hcone : ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) : - ∃ K : ℝ, 0 ≤ K ∧ ∀ r : ℝ → ℂ, - (∀ t ∈ Icc 0 T, HasDerivWithinAt r - (-2/(r t-(u t : ℂ))) (Ici 0) t) → - (∀ t ∈ Icc 0 T, 0 < (r t).im) → (r 0).re=u 0 → - dist (q T) (r T) ≤ dist (q 0) (r 0)*K := by - have hT : 0 ≤ T := ha.le.trans haT - have hpost : Icc a T ⊆ Icc 0 T := Icc_subset_Icc_left ha.le - obtain ⟨s,hs,hsm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) - (Complex.continuous_im.comp_continuousOn (hqc.mono hpost)) - have hsp : s ∈ Ioc 0 T := ⟨ha.trans_le hs.1,hs.2⟩ - let η : ℝ := min (q s).im (Real.sqrt a) - have hη : 0 < η := lt_min (hqp s hsp) (Real.sqrt_pos.mpr ha) - let K : ℝ := Real.exp ((4*η⁻¹^2)*(T-a)) - refine ⟨K,(Real.exp_pos _).le,?_⟩ - intro r hr hrp hr0 - have hrd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt r - (-2/(r t-(u t : ℂ))) (Icc 0 T) t := (hr t ht).mono Icc_subset_Ici_self - have hrc : ContinuousOn r (Icc 0 T) := fun t ht => (hrd t ht).continuousWithinAt - have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT - have hrda (t : ℝ) (ht : t ∈ Icc 0 a) : HasDerivWithinAt r - (-2/(r t-(u t : ℂ))) (Icc 0 a) t := (hrd t (hsub ht)).mono hsub - have hua (t : ℝ) (ht : t ∈ Icc 0 a) : HasDerivWithinAt u - (u' t) (Icc 0 a) t := (hu t (hsub ht)).mono hsub - have hra := reverse_loewner_height_lower hL r u u' hrda hua - (fun t ht => hb t (hsub ht)) (fun t ht => hrp t (hsub ht)) hr0 hsmall - have hrx := (reverse_centered_coordinates hL r u u' hrda hua - (fun t ht => hb t (hsub ht)) (fun t ht => hrp t (hsub ht)) hr0).1 - have hrcone (t : ℝ) (ht : t ∈ Ioc 0 a) : ((r t).re-u t)^2 ≤ t := by - have ht' : t ∈ Icc 0 a := Ioc_subset_Icc_self ht - have hxx := (sq_le_sq₀ (abs_nonneg ((r t).re-u t)) (mul_nonneg hL ht.1.le)).mpr (hrx t ht') - rw [sq_abs] at hxx - have hl := (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall - nlinarith [mul_le_mul_of_nonneg_right hl ht.1.le] - have hshort := reverse_trajectories_contract_in_cones ha.le q r u - (hqc.mono hsub) (hrc.mono hsub) - (fun t ht => hq t ⟨ht.1,ht.2.trans haT⟩) - (fun t ht => (hr t (hsub (Ioc_subset_Icc_self ht))).mono (Ici_subset_Ici.mpr ht.1.le)) - (fun t ht => hqp t ⟨ht.1,ht.2.trans haT⟩) - (fun t ht => hrp t (hsub (Ioc_subset_Icc_self ht))) - (fun t ht => (hcone t (Ioc_subset_Icc_self ht)).1) hrcone - (fun t ht => by have hh := (hcone t (Ioc_subset_Icc_self ht)).2; linarith [ht.1]) - (fun t ht => by have hh := hra t (Ioc_subset_Icc_self ht); linarith [ht.1]) - have hQ (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (q t).im := - (min_le_left _ _).trans (hsm ht) - have hroot : Real.sqrt a ≤ (r a).im := by - have hh := hra a ⟨ha.le,le_rfl⟩ - nlinarith [Real.sq_sqrt ha.le,Real.sqrt_nonneg a,hrp a ⟨ha.le,haT⟩] - have hrmono := reverse_height_monotone r u hrd hrp - have hR (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (r t).im := - (min_le_right _ _).trans (hroot.trans (hrmono ⟨ha.le,haT⟩ (hpost ht) ht.1)) - have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q - (reverseField η (u t) (q t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using hq t ⟨ha.trans_le ht.1,ht.2.le⟩ - have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r - (reverseField η (u t) (r t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using - (hr t (hpost (Ico_subset_Icc_self ht))).mono (Ici_subset_Ici.mpr (hpost (Ico_subset_Icc_self ht)).1) - exact dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) - (hqc.mono hpost) hdQ (hrc.mono hpost) hdR - (hshort a ⟨ha.le,le_rfl⟩) T ⟨haT,le_rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReverseConeContraction +import OAI.Probability.SLE.Loewner.ReverseLongContraction + +namespace OAI + +/-! A boundary-start reverse trajectory is the uniform limit of the +positive-height trajectories on each finite time interval. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +theorem exists_boundary_radial_distance_constant {T a L : ℝ} + (ha : 0 < a) (haT : a ≤ T) (hL : 0 ≤ L) (hsmall : L^2*a ≤ 1) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hqc : ContinuousOn q (Icc 0 T)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hcone : ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) : + ∃ K : ℝ, 0 ≤ K ∧ ∀ r : ℝ → ℂ, + (∀ t ∈ Icc 0 T, HasDerivWithinAt r + (-2/(r t-(u t : ℂ))) (Ici 0) t) → + (∀ t ∈ Icc 0 T, 0 < (r t).im) → (r 0).re=u 0 → + dist (q T) (r T) ≤ dist (q 0) (r 0)*K := by + have hT : 0 ≤ T := ha.le.trans haT + have hpost : Icc a T ⊆ Icc 0 T := Icc_subset_Icc_left ha.le + obtain ⟨s,hs,hsm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) + (Complex.continuous_im.comp_continuousOn (hqc.mono hpost)) + have hsp : s ∈ Ioc 0 T := ⟨ha.trans_le hs.1,hs.2⟩ + let η : ℝ := min (q s).im (Real.sqrt a) + have hη : 0 < η := lt_min (hqp s hsp) (Real.sqrt_pos.mpr ha) + let K : ℝ := Real.exp ((4*η⁻¹^2)*(T-a)) + refine ⟨K,(Real.exp_pos _).le,?_⟩ + intro r hr hrp hr0 + have hrd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt r + (-2/(r t-(u t : ℂ))) (Icc 0 T) t := (hr t ht).mono Icc_subset_Ici_self + have hrc : ContinuousOn r (Icc 0 T) := fun t ht => (hrd t ht).continuousWithinAt + have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT + have hrda (t : ℝ) (ht : t ∈ Icc 0 a) : HasDerivWithinAt r + (-2/(r t-(u t : ℂ))) (Icc 0 a) t := (hrd t (hsub ht)).mono hsub + have hua (t : ℝ) (ht : t ∈ Icc 0 a) : HasDerivWithinAt u + (u' t) (Icc 0 a) t := (hu t (hsub ht)).mono hsub + have hra := reverse_loewner_height_lower hL r u u' hrda hua + (fun t ht => hb t (hsub ht)) (fun t ht => hrp t (hsub ht)) hr0 hsmall + have hrx := (reverse_centered_coordinates hL r u u' hrda hua + (fun t ht => hb t (hsub ht)) (fun t ht => hrp t (hsub ht)) hr0).1 + have hrcone (t : ℝ) (ht : t ∈ Ioc 0 a) : ((r t).re-u t)^2 ≤ t := by + have ht' : t ∈ Icc 0 a := Ioc_subset_Icc_self ht + have hxx := (sq_le_sq₀ (abs_nonneg ((r t).re-u t)) (mul_nonneg hL ht.1.le)).mpr (hrx t ht') + rw [sq_abs] at hxx + have hl := (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall + nlinarith [mul_le_mul_of_nonneg_right hl ht.1.le] + have hshort := reverse_trajectories_contract_in_cones ha.le q r u + (hqc.mono hsub) (hrc.mono hsub) + (fun t ht => hq t ⟨ht.1,ht.2.trans haT⟩) + (fun t ht => (hr t (hsub (Ioc_subset_Icc_self ht))).mono (Ici_subset_Ici.mpr ht.1.le)) + (fun t ht => hqp t ⟨ht.1,ht.2.trans haT⟩) + (fun t ht => hrp t (hsub (Ioc_subset_Icc_self ht))) + (fun t ht => (hcone t (Ioc_subset_Icc_self ht)).1) hrcone + (fun t ht => by have hh := (hcone t (Ioc_subset_Icc_self ht)).2; linarith [ht.1]) + (fun t ht => by have hh := hra t (Ioc_subset_Icc_self ht); linarith [ht.1]) + have hQ (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (q t).im := + (min_le_left _ _).trans (hsm ht) + have hroot : Real.sqrt a ≤ (r a).im := by + have hh := hra a ⟨ha.le,le_rfl⟩ + nlinarith [Real.sq_sqrt ha.le,Real.sqrt_nonneg a,hrp a ⟨ha.le,haT⟩] + have hrmono := reverse_height_monotone r u hrd hrp + have hR (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (r t).im := + (min_le_right _ _).trans (hroot.trans (hrmono ⟨ha.le,haT⟩ (hpost ht) ht.1)) + have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q + (reverseField η (u t) (q t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using hq t ⟨ha.trans_le ht.1,ht.2.le⟩ + have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r + (reverseField η (u t) (r t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using + (hr t (hpost (Ico_subset_Icc_self ht))).mono (Ici_subset_Ici.mpr (hpost (Ico_subset_Icc_self ht)).1) + exact dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) + (hqc.mono hpost) hdQ (hrc.mono hpost) hdR + (hshort a ⟨ha.le,le_rfl⟩) T ⟨haT,le_rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/BoundaryScaleLaw.lean b/lean/OAI/Probability/SLE/Geometry/BoundaryScaleLaw.lean index 21a8f418d712c3002f7debb3a452c7a917fd3530..cf87437ff8e9357b4c76f88fb90b315356032143 100644 --- a/lean/OAI/Probability/SLE/Geometry/BoundaryScaleLaw.lean +++ b/lean/OAI/Probability/SLE/Geometry/BoundaryScaleLaw.lean @@ -1,194 +1,194 @@ -import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension -import OAI.Probability.SLE.Geometry.FrontClockChange -import OAI.Probability.SLE.Loewner.ScaleTraceLaw - -namespace OAI - -/-! Finite boundary charts commute with normalization. Capacity scaling -changes only the clock of the physical image path, and therefore preserves -the first-front success-and-sausage event. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.BoundaryScaleLaw - -local instance : LocallyCompactSpace closedHalfPlane := - isClosed_closedHalfPlane.locallyCompactSpace -local instance boundaryScalePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance boundaryScalePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem finite_boundary_ext (f g : C(closedHalfPlane, ℂ)) - (heq : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f z = g z) : f = g := by - ext z - let zₙ (n : ℕ) : closedHalfPlane := - ⟨z + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I, by - change 0 ≤ (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, - mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] - exact add_nonneg z.2 (by positivity)⟩ - have hH (n : ℕ) : (zₙ n).1 ∈ upperHalfPlane := by - change 0 < (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, - mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] - exact add_pos_of_nonneg_of_pos z.2 (by positivity) - have hz : Tendsto zₙ atTop (𝓝 z) := by - apply tendsto_subtype_rng.mpr - simpa only [Function.comp_def, Complex.ofReal_zero, zero_mul, add_zero] using - tendsto_const_nhds.add (((Complex.continuous_ofReal.tendsto 0).comp - (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))).mul_const Complex.I) - exact tendsto_nhds_unique ((f.continuous.tendsto z).comp hz) - (((g.continuous.tendsto z).comp hz).congr (fun n => (heq (zₙ n) (hH n)).symm)) - -def scaleClosed (c : ℝ≥0) : C(closedHalfPlane, closedHalfPlane) := - ⟨fun z => ⟨(c : ℂ) * z, by - change 0 ≤ ((c : ℂ) * (z : ℂ)).im - simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, - add_zero] using mul_nonneg c.coe_nonneg z.2⟩, - (continuous_const.mul continuous_subtype_val).subtype_mk _⟩ - -theorem normalized_boundary_eq - (F : ℂ → ℂ) (u : ℝ) (c : ℝ≥0) (hc : 0 < c) - (f₁ fc : C(closedHalfPlane, ℂ)) - (h₁ : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f₁ z = F ((u : ℂ) + z)) - (hscaled : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → - fc z = F ((u : ℂ) + (c : ℂ) * z)) : - fc = f₁.comp (scaleClosed c) := by - apply finite_boundary_ext - intro z hz - change fc z = f₁ (scaleClosed c z) - rw [hscaled z hz, h₁] - · rfl - · change 0 < ((c : ℂ) * (z : ℂ)).im - simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, - add_zero] using mul_pos (show 0 < (c : ℝ) by exact_mod_cast hc) hz - -def clockPath (e : ℝ≥0 ≃o ℝ≥0) (p : C(ℝ≥0, ℂ)) : C(ℝ≥0, ℂ) := - ⟨fun t => p (e t), p.continuous.comp e.toHomeomorph.continuous⟩ - -theorem measurable_clockPath (e : ℝ≥0 ≃o ℝ≥0) : Measurable (clockPath e) := by - apply ContinuousMap.measurable_iff_eval.mpr - intro t - change Measurable (fun p : C(ℝ≥0, ℂ) => p (e t)) - exact (continuous_eval_const (e t)).measurable - -theorem boundary_scale_clock_identity - (c : ℝ≥0) (hc : 0 < c) (f : C(closedHalfPlane, ℂ)) - (p : C(ℝ≥0, ℂ)) (hp : ClosedPathProperty p) : - finiteBoundaryPath (f.comp (scaleClosed c)) p = - clockPath (OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)) - (finiteBoundaryPath f (ScaleTraceLaw.scalePath c p)) := by - have hpc : ClosedPathProperty (ScaleTraceLaw.scalePath c p) := by - intro t - change 0 ≤ ((c : ℂ) * p (sourceScaleTime c t)).im - simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, - add_zero] using mul_nonneg c.coe_nonneg (hp _) - ext t - rw [finiteBoundaryPath_apply _ _ hp] - change f (scaleClosed c ⟨p t, hp t⟩) = - finiteBoundaryPath f (ScaleTraceLaw.scalePath c p) ((c ^ 2) * t) - rw [finiteBoundaryPath_apply _ _ hpc] - congr 1 - apply Subtype.ext - change (c : ℂ) * p t = (c : ℂ) * p (sourceScaleTime c (c ^ 2 * t)) - congr 2 - unfold sourceScaleTime - field_simp - -/-- The normalized physical path has the same law as a deterministic -capacity-clock change of the unnormalized physical path. -/ -theorem boundary_scale_path_law - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (p : Ω → C(ℝ≥0, ℂ)) (hp : Measurable p) - (hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (p ω)) - (c : ℝ≥0) (hc : 0 < c) (f : C(closedHalfPlane, ℂ)) - (hlaw : P.map (fun ω => ScaleTraceLaw.scalePath c (p ω)) = P.map p) : - P.map (fun ω => finiteBoundaryPath (f.comp (scaleClosed c)) (p ω)) = - P.map (fun ω => clockPath (OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)) - (finiteBoundaryPath f (p ω))) := by - let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) - calc - _ = P.map (fun ω => clockPath e (finiteBoundaryPath f (ScaleTraceLaw.scalePath c (p ω)))) := by - apply Measure.map_congr - filter_upwards [hclosed] with ω hω - exact boundary_scale_clock_identity c hc f (p ω) hω - _ = (P.map (fun ω => ScaleTraceLaw.scalePath c (p ω))).map - (fun q => clockPath e (finiteBoundaryPath f q)) := by - exact (Measure.map_map ((measurable_clockPath e).comp (measurable_finiteBoundaryPath f)) - ((ScaleTraceLaw.measurable_scalePath c).comp hp)).symm - _ = (P.map p).map (fun q => clockPath e (finiteBoundaryPath f q)) := by rw [hlaw] - _ = _ := by - exact Measure.map_map ((measurable_clockPath e).comp (measurable_finiteBoundaryPath f)) hp - -open FirstFrontPassage PassageSuccess - -def frontMassEvent (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R : Set ℂ) : - Set C(ℝ≥0, ℂ) := - {γ | γ ∈ FrontSuccess (fun _ (γ : C(ℝ≥0, ℂ)) => (γ : ℝ≥0 → ℂ)) (fun _ => p) (fun _ => 0) - (fun _ => V) (fun _ => a) ε 0 ∧ - ENNReal.ofReal c ≤ segmentSausageCharge κ γ 0 - (frontTime (fun t => p (γ t)) 1) v R} - -theorem measurableSet_frontMassEvent - (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) - (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : - MeasurableSet (frontMassEvent p V a ε κ v c R) := by - let ℱ : Filtration ℝ≥0 (borel C(ℝ≥0, ℂ)) := - Filtration.const ℝ≥0 (borel C(ℝ≥0, ℂ)) le_rfl - have ha : Adapted ℱ (fun t (γ : C(ℝ≥0, ℂ)) => γ t) := by - intro t - change @Measurable C(ℝ≥0, ℂ) ℂ (borel C(ℝ≥0, ℂ)) _ (fun γ => γ t) - exact (continuous_eval_const t).measurable - exact FrontSuccessMeasurability.measurableSet_frontSuccess_mass - ℱ - (fun _ (γ : C(ℝ≥0, ℂ)) => (γ : ℝ≥0 → ℂ)) (fun _ => p) (fun _ => 0) - (fun _ => V) (fun _ => a) ε 0 - ha (fun γ => γ.continuous) - K hK hp measurable_const hV hVK κ v c R - -theorem mem_frontMassEvent_clock - (e : ℝ≥0 ≃o ℝ≥0) (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) - (ε κ v c : ℝ) (R : Set ℂ) (γ : C(ℝ≥0, ℂ)) : - clockPath e γ ∈ frontMassEvent p V a ε κ v c R ↔ - γ ∈ frontMassEvent p V a ε κ v c R := by - have he := congrArg (fun S : Set C(ℝ≥0, ℂ) => γ ∈ S) - (FrontClockChange.frontSuccess_mass_clock e (fun γ : C(ℝ≥0, ℂ) => (γ : ℝ≥0 → ℂ)) p (fun _ => 0) - V a ε κ v c R) - have he0 : e.symm 0 = 0 := e.symm.map_bot - simpa only [he0, frontMassEvent, FrontSuccess, clockPath, ContinuousMap.coe_mk, - mem_ofPred_eq] using eq_iff_iff.mp he - -/-- Normalize the inverse chart without changing the physical first-front -event probability. The scale law is the actual SLE path law. -/ -theorem frontMassEvent_boundary_scale_probability - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (γ : Ω → C(ℝ≥0, ℂ)) (hγ : Measurable γ) - (hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (γ ω)) - (ell : ℝ≥0) (hell : 0 < ell) (f : C(closedHalfPlane, ℂ)) - (hlaw : P.map (fun ω => ScaleTraceLaw.scalePath ell (γ ω)) = P.map γ) - (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) - (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : - P {ω | finiteBoundaryPath (f.comp (scaleClosed ell)) (γ ω) ∈ frontMassEvent p V a ε κ v c R} = - P {ω | finiteBoundaryPath f (γ ω) ∈ frontMassEvent p V a ε κ v c R} := by - let e := OrderIso.mulLeft₀ (ell ^ 2) (pow_pos hell 2) - let E := frontMassEvent p V a ε κ v c R - have hE : MeasurableSet E := measurableSet_frontMassEvent p V a ε κ v c R K hK hp hV hVK - have hn := (measurable_finiteBoundaryPath (f.comp (scaleClosed ell))).comp hγ - have hm := (measurable_clockPath e).comp ((measurable_finiteBoundaryPath f).comp hγ) - calc - _ = (P.map (fun ω => finiteBoundaryPath (f.comp (scaleClosed ell)) (γ ω))) E := - (Measure.map_apply hn hE).symm - _ = (P.map (fun ω => clockPath e (finiteBoundaryPath f (γ ω)))) E := by - rw [boundary_scale_path_law P γ hγ hclosed ell hell f hlaw] - _ = P {ω | clockPath e (finiteBoundaryPath f (γ ω)) ∈ E} := Measure.map_apply hm hE - _ = _ := by - congr 1 - ext ω - exact mem_frontMassEvent_clock e p V a ε κ v c R _ - -end SLEExactGauge.BoundaryScaleLaw - -end - -end OAI +import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension +import OAI.Probability.SLE.Geometry.FrontClockChange +import OAI.Probability.SLE.Loewner.ScaleTraceLaw + +namespace OAI + +/-! Finite boundary charts commute with normalization. Capacity scaling +changes only the clock of the physical image path, and therefore preserves +the first-front success-and-sausage event. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.BoundaryScaleLaw + +local instance : LocallyCompactSpace closedHalfPlane := + isClosed_closedHalfPlane.locallyCompactSpace +local instance boundaryScalePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance boundaryScalePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem finite_boundary_ext (f g : C(closedHalfPlane, ℂ)) + (heq : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f z = g z) : f = g := by + ext z + let zₙ (n : ℕ) : closedHalfPlane := + ⟨z + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I, by + change 0 ≤ (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, + mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] + exact add_nonneg z.2 (by positivity)⟩ + have hH (n : ℕ) : (zₙ n).1 ∈ upperHalfPlane := by + change 0 < (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, + mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] + exact add_pos_of_nonneg_of_pos z.2 (by positivity) + have hz : Tendsto zₙ atTop (𝓝 z) := by + apply tendsto_subtype_rng.mpr + simpa only [Function.comp_def, Complex.ofReal_zero, zero_mul, add_zero] using + tendsto_const_nhds.add (((Complex.continuous_ofReal.tendsto 0).comp + (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))).mul_const Complex.I) + exact tendsto_nhds_unique ((f.continuous.tendsto z).comp hz) + (((g.continuous.tendsto z).comp hz).congr (fun n => (heq (zₙ n) (hH n)).symm)) + +def scaleClosed (c : ℝ≥0) : C(closedHalfPlane, closedHalfPlane) := + ⟨fun z => ⟨(c : ℂ) * z, by + change 0 ≤ ((c : ℂ) * (z : ℂ)).im + simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, + add_zero] using mul_nonneg c.coe_nonneg z.2⟩, + (continuous_const.mul continuous_subtype_val).subtype_mk _⟩ + +theorem normalized_boundary_eq + (F : ℂ → ℂ) (u : ℝ) (c : ℝ≥0) (hc : 0 < c) + (f₁ fc : C(closedHalfPlane, ℂ)) + (h₁ : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f₁ z = F ((u : ℂ) + z)) + (hscaled : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → + fc z = F ((u : ℂ) + (c : ℂ) * z)) : + fc = f₁.comp (scaleClosed c) := by + apply finite_boundary_ext + intro z hz + change fc z = f₁ (scaleClosed c z) + rw [hscaled z hz, h₁] + · rfl + · change 0 < ((c : ℂ) * (z : ℂ)).im + simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, + add_zero] using mul_pos (show 0 < (c : ℝ) by exact_mod_cast hc) hz + +def clockPath (e : ℝ≥0 ≃o ℝ≥0) (p : C(ℝ≥0, ℂ)) : C(ℝ≥0, ℂ) := + ⟨fun t => p (e t), p.continuous.comp e.toHomeomorph.continuous⟩ + +theorem measurable_clockPath (e : ℝ≥0 ≃o ℝ≥0) : Measurable (clockPath e) := by + apply ContinuousMap.measurable_iff_eval.mpr + intro t + change Measurable (fun p : C(ℝ≥0, ℂ) => p (e t)) + exact (continuous_eval_const (e t)).measurable + +theorem boundary_scale_clock_identity + (c : ℝ≥0) (hc : 0 < c) (f : C(closedHalfPlane, ℂ)) + (p : C(ℝ≥0, ℂ)) (hp : ClosedPathProperty p) : + finiteBoundaryPath (f.comp (scaleClosed c)) p = + clockPath (OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)) + (finiteBoundaryPath f (ScaleTraceLaw.scalePath c p)) := by + have hpc : ClosedPathProperty (ScaleTraceLaw.scalePath c p) := by + intro t + change 0 ≤ ((c : ℂ) * p (sourceScaleTime c t)).im + simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, + add_zero] using mul_nonneg c.coe_nonneg (hp _) + ext t + rw [finiteBoundaryPath_apply _ _ hp] + change f (scaleClosed c ⟨p t, hp t⟩) = + finiteBoundaryPath f (ScaleTraceLaw.scalePath c p) ((c ^ 2) * t) + rw [finiteBoundaryPath_apply _ _ hpc] + congr 1 + apply Subtype.ext + change (c : ℂ) * p t = (c : ℂ) * p (sourceScaleTime c (c ^ 2 * t)) + congr 2 + unfold sourceScaleTime + field_simp + +/-- The normalized physical path has the same law as a deterministic +capacity-clock change of the unnormalized physical path. -/ +theorem boundary_scale_path_law + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (p : Ω → C(ℝ≥0, ℂ)) (hp : Measurable p) + (hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (p ω)) + (c : ℝ≥0) (hc : 0 < c) (f : C(closedHalfPlane, ℂ)) + (hlaw : P.map (fun ω => ScaleTraceLaw.scalePath c (p ω)) = P.map p) : + P.map (fun ω => finiteBoundaryPath (f.comp (scaleClosed c)) (p ω)) = + P.map (fun ω => clockPath (OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)) + (finiteBoundaryPath f (p ω))) := by + let e := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2) + calc + _ = P.map (fun ω => clockPath e (finiteBoundaryPath f (ScaleTraceLaw.scalePath c (p ω)))) := by + apply Measure.map_congr + filter_upwards [hclosed] with ω hω + exact boundary_scale_clock_identity c hc f (p ω) hω + _ = (P.map (fun ω => ScaleTraceLaw.scalePath c (p ω))).map + (fun q => clockPath e (finiteBoundaryPath f q)) := by + exact (Measure.map_map ((measurable_clockPath e).comp (measurable_finiteBoundaryPath f)) + ((ScaleTraceLaw.measurable_scalePath c).comp hp)).symm + _ = (P.map p).map (fun q => clockPath e (finiteBoundaryPath f q)) := by rw [hlaw] + _ = _ := by + exact Measure.map_map ((measurable_clockPath e).comp (measurable_finiteBoundaryPath f)) hp + +open FirstFrontPassage PassageSuccess + +def frontMassEvent (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R : Set ℂ) : + Set C(ℝ≥0, ℂ) := + {γ | γ ∈ FrontSuccess (fun _ (γ : C(ℝ≥0, ℂ)) => (γ : ℝ≥0 → ℂ)) (fun _ => p) (fun _ => 0) + (fun _ => V) (fun _ => a) ε 0 ∧ + ENNReal.ofReal c ≤ segmentSausageCharge κ γ 0 + (frontTime (fun t => p (γ t)) 1) v R} + +theorem measurableSet_frontMassEvent + (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) + (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : + MeasurableSet (frontMassEvent p V a ε κ v c R) := by + let ℱ : Filtration ℝ≥0 (borel C(ℝ≥0, ℂ)) := + Filtration.const ℝ≥0 (borel C(ℝ≥0, ℂ)) le_rfl + have ha : Adapted ℱ (fun t (γ : C(ℝ≥0, ℂ)) => γ t) := by + intro t + change @Measurable C(ℝ≥0, ℂ) ℂ (borel C(ℝ≥0, ℂ)) _ (fun γ => γ t) + exact (continuous_eval_const t).measurable + exact FrontSuccessMeasurability.measurableSet_frontSuccess_mass + ℱ + (fun _ (γ : C(ℝ≥0, ℂ)) => (γ : ℝ≥0 → ℂ)) (fun _ => p) (fun _ => 0) + (fun _ => V) (fun _ => a) ε 0 + ha (fun γ => γ.continuous) + K hK hp measurable_const hV hVK κ v c R + +theorem mem_frontMassEvent_clock + (e : ℝ≥0 ≃o ℝ≥0) (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) + (ε κ v c : ℝ) (R : Set ℂ) (γ : C(ℝ≥0, ℂ)) : + clockPath e γ ∈ frontMassEvent p V a ε κ v c R ↔ + γ ∈ frontMassEvent p V a ε κ v c R := by + have he := congrArg (fun S : Set C(ℝ≥0, ℂ) => γ ∈ S) + (FrontClockChange.frontSuccess_mass_clock e (fun γ : C(ℝ≥0, ℂ) => (γ : ℝ≥0 → ℂ)) p (fun _ => 0) + V a ε κ v c R) + have he0 : e.symm 0 = 0 := e.symm.map_bot + simpa only [he0, frontMassEvent, FrontSuccess, clockPath, ContinuousMap.coe_mk, + mem_ofPred_eq] using eq_iff_iff.mp he + +/-- Normalize the inverse chart without changing the physical first-front +event probability. The scale law is the actual SLE path law. -/ +theorem frontMassEvent_boundary_scale_probability + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (γ : Ω → C(ℝ≥0, ℂ)) (hγ : Measurable γ) + (hclosed : ∀ᵐ ω ∂P, ClosedPathProperty (γ ω)) + (ell : ℝ≥0) (hell : 0 < ell) (f : C(closedHalfPlane, ℂ)) + (hlaw : P.map (fun ω => ScaleTraceLaw.scalePath ell (γ ω)) = P.map γ) + (p : ℂ → ℝ) (V : Set ℂ) (a : ℂ) (ε κ v c : ℝ) (R K : Set ℂ) + (hK : IsClosed K) (hp : ContinuousOn p K) (hV : IsOpen V) (hVK : V ⊆ K) : + P {ω | finiteBoundaryPath (f.comp (scaleClosed ell)) (γ ω) ∈ frontMassEvent p V a ε κ v c R} = + P {ω | finiteBoundaryPath f (γ ω) ∈ frontMassEvent p V a ε κ v c R} := by + let e := OrderIso.mulLeft₀ (ell ^ 2) (pow_pos hell 2) + let E := frontMassEvent p V a ε κ v c R + have hE : MeasurableSet E := measurableSet_frontMassEvent p V a ε κ v c R K hK hp hV hVK + have hn := (measurable_finiteBoundaryPath (f.comp (scaleClosed ell))).comp hγ + have hm := (measurable_clockPath e).comp ((measurable_finiteBoundaryPath f).comp hγ) + calc + _ = (P.map (fun ω => finiteBoundaryPath (f.comp (scaleClosed ell)) (γ ω))) E := + (Measure.map_apply hn hE).symm + _ = (P.map (fun ω => clockPath e (finiteBoundaryPath f (γ ω)))) E := by + rw [boundary_scale_path_law P γ hγ hclosed ell hell f hlaw] + _ = P {ω | clockPath e (finiteBoundaryPath f (γ ω)) ∈ E} := Measure.map_apply hm hE + _ = _ := by + congr 1 + ext ω + exact mem_frontMassEvent_clock e p V a ε κ v c R _ + +end SLEExactGauge.BoundaryScaleLaw + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/BoundarySeparationRectangle.lean b/lean/OAI/Probability/SLE/Geometry/BoundarySeparationRectangle.lean index fa4e680f18247339f4ec6eee529715a4b073bd8f..826463c0d6cb12278195af60e9dc3b29169d7b1e 100644 --- a/lean/OAI/Probability/SLE/Geometry/BoundarySeparationRectangle.lean +++ b/lean/OAI/Probability/SLE/Geometry/BoundarySeparationRectangle.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Geometry.ThinCrossingEnergy - -namespace OAI - -/-! A nonconstant continuous boundary value remains separated on a -small upper rectangle. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -lemma boundary_separation_rectangle {B F : ℂ → ℂ} - (hB : ContinuousOn B {z : ℂ | 0≤z.im}) - (hBF : ∀ z∈upperHalfPlane, B z=F z) - {a b x : ℝ} (hx : x∈Ioo a b) {c : ℂ} (hne : B (x:ℂ)≠c) : - ∃ a' b' H δ : ℝ, a B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) : - Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun t ω => γ ω t) := by - apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ - (fun ω => continuous_const.mul (hc ω)) γ G F hw - intro t - exact measurable_const.mul - ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable - -theorem bounded_firstHeight_splice_positive - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) - (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc - let hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ - (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc - ∀ᵐ ω ∂P, ω ∈ FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc → - 0 < P⁻[(stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W).indicator - (fun _ => 1) | hσ.measurableSpace] ω := by - dsimp only - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc - have hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ - (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc - apply conditional_stopped_splice_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw - hboundary hsupport σ hσ B r hB hr - (FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc) _ - F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend - intro ω hω - exact (FirstHeightSupport.bounded_certificate_geometry - (sourceTracePath γ (fun ω => (hw ω).1) ω) - (hw ω).2.1 (hw ω).2.2.1 hB Tc hω).2 - -/-- Actual bounded-height splice support combined with varying early events -produces one positive probability for the whole sequence of charts. -/ -theorem bounded_firstHeight_splice_varying_probability - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) - (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ ω) (G ω) (F₀ ω)) - (hboundary : PublishedLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) - (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) - (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) - (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) - (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) - (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) - (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) - (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) - (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → - ‖F z - a‖ < r / 4) - (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc - let hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ - (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc - let E := FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc - let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W - ∀ H : ℕ → Set Ω, (∀ n, MeasurableSet[hσ.measurableSpace] (H n)) → - 0 < P E → (∀ n, P (E \ H n) ≤ P E / 2) → - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ n, q ≤ P (E ∩ H n ∩ C) := by - dsimp only - let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) - let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc - have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw - have hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ ha - (fun ω => (hw ω).1) B Tc - let E := FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc - let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W - intro H hH hEpos hmissing - have hE : MeasurableSet[hσ.measurableSpace] E := - FirstHeightSupport.boundedFirstHeight_certificate_measurable ℱ γ ha - (fun ω => (hw ω).1) B Tc - have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := - (ha t).mono (ℱ.le t) le_rfl - have hC : MeasurableSet C := measurableSet_stoppedPhysicalTube γ - (fun ω => (hw ω).1) hγm σ (stopping_time_measurable_nnreal hσ) - (parameterContinuationRegion F a r O) W - (parameterContinuationRegion_isOpen hF a r hO) hW - have hpass := bounded_firstHeight_splice_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw - hboundary hsupport B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH - hcross hafter hcircle hsector htail hend - exact varying_passage_events_lower_bound P hσ.measurableSpace hσ.measurableSpace_le - E C H hE hC hH hEpos hpass hmissing - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.StoppedSplicePassage +import OAI.Probability.SLE.Geometry.FirstHeightBounded + +namespace OAI + +/-! Passage from the actual bounded first-height event. The geometry of +the stopped past is derived from the measurable certificate, rather than +imposed separately on the stopped Loewner realization. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem brownian_capacity_trace_adapted + {Ω : Type*} [m : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) : + Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun t ω => γ ω t) := by + apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ + (fun ω => continuous_const.mul (hc ω)) γ G F hw + intro t + exact measurable_const.mul + ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable + +theorem bounded_firstHeight_splice_positive + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) + (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc + let hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ + (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc + ∀ᵐ ω ∂P, ω ∈ FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc → + 0 < P⁻[(stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W).indicator + (fun _ => 1) | hσ.measurableSpace] ω := by + dsimp only + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc + have hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ + (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc + apply conditional_stopped_splice_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw + hboundary hsupport σ hσ B r hB hr + (FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc) _ + F hF a O W hO hW β tB T htB htBT hβ hi hβH hcross hafter hcircle hsector htail hend + intro ω hω + exact (FirstHeightSupport.bounded_certificate_geometry + (sourceTracePath γ (fun ω => (hw ω).1) ω) + (hw ω).2.1 (hw ω).2.2.1 hB Tc hω).2 + +/-- Actual bounded-height splice support combined with varying early events +produces one positive probability for the whole sequence of charts. -/ +theorem bounded_firstHeight_splice_varying_probability + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P) + (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ ω) (G ω) (F₀ ω)) + (hboundary : PublishedLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (B r : ℝ) (hB : 0 < B) (hr : 0 < r) (Tc : ℝ≥0) + (F : ℂ → ℂ) (hF : ContinuousOn F upperHalfPlane) (a : ℂ) + (O W : Set ℂ) (hO : IsOpen O) (hW : IsOpen W) + (β : ℝ → ℂ) (tB T : ℝ) (htB : 0 < tB) (htBT : tB < T) + (hβ : ContinuousOn β (Icc 0 T)) (hi : InjOn β (Icc 0 T)) + (hβH : ∀ t ∈ Ioc 0 T, 0 < (β t).im) + (hcross : ‖β tB‖ = B) (hafter : ∀ t ∈ Ioc tB T, B < ‖β t‖) + (hcircle : ∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) + (hsector : ∀ z : ℂ, B / 8 ≤ ‖z‖ → ‖z‖ ≤ B → ‖z‖ / 2 ≤ z.im → + ‖F z - a‖ < r / 4) + (htail : F '' (β '' Icc tB T) ⊆ O) (hend : β T ∈ W) : + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc + let hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ + (brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw) (fun ω => (hw ω).1) B Tc + let E := FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc + let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W + ∀ H : ℕ → Set Ω, (∀ n, MeasurableSet[hσ.measurableSpace] (H n)) → + 0 < P E → (∀ n, P (E \ H n) ≤ P E / 2) → + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ n, q ≤ P (E ∩ H n ∩ C) := by + dsimp only + let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable) + let σ := fun ω => FirstHeightSupport.boundedFirstHeightTime (γ ω) B Tc + have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw + have hσ := FirstHeightSupport.boundedFirstHeightTime_isStoppingTime ℱ γ ha + (fun ω => (hw ω).1) B Tc + let E := FirstHeightSupport.stoppedNarrowCertificate γ σ B Tc + let C := stoppedPhysicalTube γ σ (parameterContinuationRegion F a r O) W + intro H hH hEpos hmissing + have hE : MeasurableSet[hσ.measurableSpace] E := + FirstHeightSupport.boundedFirstHeight_certificate_measurable ℱ γ ha + (fun ω => (hw ω).1) B Tc + have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := + (ha t).mono (ℱ.le t) le_rfl + have hC : MeasurableSet C := measurableSet_stoppedPhysicalTube γ + (fun ω => (hw ω).1) hγm σ (stopping_time_measurable_nnreal hσ) + (parameterContinuationRegion F a r O) W + (parameterContinuationRegion_isOpen hF a r hO) hW + have hpass := bounded_firstHeight_splice_positive κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw + hboundary hsupport B r hB hr Tc F hF a O W hO hW β tB T htB htBT hβ hi hβH + hcross hafter hcircle hsector htail hend + exact varying_passage_events_lower_bound P hσ.measurableSpace hσ.measurableSpace_le + E C H hE hC hH hEpos hpass hmissing + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/BoundedTerminalMean.lean b/lean/OAI/Probability/SLE/Geometry/BoundedTerminalMean.lean index 2036a37dda26e0f8de0be855a03d5f53551a7af3..e240f72e33418ca7436b137b13049facdcfee6f0 100644 --- a/lean/OAI/Probability/SLE/Geometry/BoundedTerminalMean.lean +++ b/lean/OAI/Probability/SLE/Geometry/BoundedTerminalMean.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Mass.RationalCanonicalMass -import OAI.Probability.SLE.SimpleTails - -namespace OAI - -/-! The canonical mass has finite expected mass on bounded physical sets, -including sets that approach the real boundary. Compactness inside the -half-plane is not needed for the upper-cover charge. -/ -noncomputable section -open Set Filter MeasureTheory Metric -open scoped ENNReal NNReal - -namespace SLEExactGauge.Multipoint - -lemma initialGreenMeasure_bounded_ball_lt_top {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) - (R : ℝ) : - initialGreenMeasure κ (Subtype.val ⁻¹' closedBall (0:ℂ) R) < ∞ := by - let A : Set upperHalfPlane := Subtype.val ⁻¹' closedBall (0:ℂ) R - have hA : MeasurableSet A := isClosed_closedBall.measurableSet.preimage - continuous_subtype_val.measurable - have hI : MeasurableSet ((Subtype.val : upperHalfPlane → ℂ) '' A) := - (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hsub : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ closedBall 0 R := by - rintro _ ⟨z,hz,rfl⟩ - exact hz - have hi : IntegrableOn (fun z : ℂ => (2:ℝ)^(-codimension κ)*|z.im|^(-codimension κ)) - (closedBall 0 R) volume := - (integrable_height_power_ball (codimension κ) R (codimension_lt_one κ hκ)).const_mul _ - change initialGreenMeasure κ A < ∞ - rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hI] - refine lt_of_le_of_lt ?_ (lintegral_ofReal_lt_top_of_integrable hi) - calc - (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' A, ENNReal.ofReal (initialGreen κ z)) ≤ - ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' A, - ENNReal.ofReal ((2:ℝ)^(-codimension κ)*|z.im|^(-codimension κ)) := by - apply setLIntegral_mono' hI - rintro z ⟨w,_,rfl⟩ - apply ENNReal.ofReal_le_ofReal - simpa only [abs_of_pos (show 0<(w:ℂ).im from w.property)] using - initialGreen_le_height κ hκ hκ8 w w.property - _ ≤ _ := lintegral_mono' (Measure.restrict_mono hsub le_rfl) le_rfl - -lemma RationalCanonicalMass.terminal_bounded_mean_lt_top - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8) - (R : ℝ) : - (∫⁻ ω, C.terminal ω (Subtype.val ⁻¹' closedBall (0:ℂ) R) ∂P) < ∞ := by - rw [← Measure.bind_apply - (isClosed_closedBall.measurableSet.preimage continuous_subtype_val.measurable) - C.terminal_measurable.aemeasurable,C.mean] - exact initialGreenMeasure_bounded_ball_lt_top hκ hκ8 R - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.RationalCanonicalMass +import OAI.Probability.SLE.SimpleTails + +namespace OAI + +/-! The canonical mass has finite expected mass on bounded physical sets, +including sets that approach the real boundary. Compactness inside the +half-plane is not needed for the upper-cover charge. -/ +noncomputable section +open Set Filter MeasureTheory Metric +open scoped ENNReal NNReal + +namespace SLEExactGauge.Multipoint + +lemma initialGreenMeasure_bounded_ball_lt_top {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) + (R : ℝ) : + initialGreenMeasure κ (Subtype.val ⁻¹' closedBall (0:ℂ) R) < ∞ := by + let A : Set upperHalfPlane := Subtype.val ⁻¹' closedBall (0:ℂ) R + have hA : MeasurableSet A := isClosed_closedBall.measurableSet.preimage + continuous_subtype_val.measurable + have hI : MeasurableSet ((Subtype.val : upperHalfPlane → ℂ) '' A) := + (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hsub : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ closedBall 0 R := by + rintro _ ⟨z,hz,rfl⟩ + exact hz + have hi : IntegrableOn (fun z : ℂ => (2:ℝ)^(-codimension κ)*|z.im|^(-codimension κ)) + (closedBall 0 R) volume := + (integrable_height_power_ball (codimension κ) R (codimension_lt_one κ hκ)).const_mul _ + change initialGreenMeasure κ A < ∞ + rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hI] + refine lt_of_le_of_lt ?_ (lintegral_ofReal_lt_top_of_integrable hi) + calc + (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' A, ENNReal.ofReal (initialGreen κ z)) ≤ + ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' A, + ENNReal.ofReal ((2:ℝ)^(-codimension κ)*|z.im|^(-codimension κ)) := by + apply setLIntegral_mono' hI + rintro z ⟨w,_,rfl⟩ + apply ENNReal.ofReal_le_ofReal + simpa only [abs_of_pos (show 0<(w:ℂ).im from w.property)] using + initialGreen_le_height κ hκ hκ8 w w.property + _ ≤ _ := lintegral_mono' (Measure.restrict_mono hsub le_rfl) le_rfl + +lemma RationalCanonicalMass.terminal_bounded_mean_lt_top + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8) + (R : ℝ) : + (∫⁻ ω, C.terminal ω (Subtype.val ⁻¹' closedBall (0:ℂ) R) ∂P) < ∞ := by + rw [← Measure.bind_apply + (isClosed_closedBall.measurableSet.preimage continuous_subtype_val.measurable) + C.terminal_measurable.aemeasurable,C.mean] + exact initialGreenMeasure_bounded_ball_lt_top hκ hκ8 R + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalBoundaryLimit.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalBoundaryLimit.lean index 7b2334bb659893e772bd9fe7dd2b682bf61e30aa..8b184a8ff883dbfb0e73b1a348b1e00bc25f9109 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalBoundaryLimit.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalBoundaryLimit.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Loewner.CanonicalTraceCandidate -import OAI.Probability.SLE.Loewner.TargetBoundaryLimit -import OAI.Probability.SLE.Geometry.FiniteParameterProjection - -namespace OAI - -/-! A continuous canonical trace modification with the exact inverse-tip -limit simultaneously at every time, obtained from an actual source witness. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance boundaryRealMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance boundaryRealBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ -local instance boundaryComplexMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _ -local instance boundaryComplexBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩ -local instance boundaryDriverPolish : PolishSpace Target.Driver := - ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace -local instance boundaryDriverMetric : MetricSpace Target.Driver := - TopologicalSpace.metrizableSpaceMetric Target.Driver - -theorem exists_canonical_all_time_boundary_limit - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (W : Target.WienerLaw) (hW : Target.IsWiener W) : - ∃ Ψ : Target.Driver → C(ℝ≥0,ℂ), Measurable Ψ ∧ - (∀ ω, Ψ (Target.normalizedDriver κ B hc h0 ω)=sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ - ∀ᵐ b ∂W, Ψ b 0=0 ∧ (∀ T, 0 ≤ (Ψ b T).im) ∧ - ∀ T, Tendsto (fun y => Target.rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (Ψ b T)) := by - have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure - obtain ⟨Ψ,hΨ,hΨeq,_⟩ := exists_canonical_trace_candidate_modification - κ hκ hκ8 B hB hm hc h0 γ G F hw W hW - let p (T : ℝ≥0) (b : Target.Driver) : Prop := - Ψ b 0=0 ∧ 0 ≤ (Ψ b T).im ∧ - Tendsto (fun y => Target.rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (Ψ b T)) - have hEval : Measurable (fun q : ℝ≥0 × Target.Driver => Ψ q.2 q.1) := - continuous_eval.measurable.comp ((hΨ.comp measurable_snd).prodMk measurable_fst) - have hEval0 : Measurable (fun q : ℝ≥0 × Target.Driver => Ψ q.2 0) := - ((continuous_eval_const 0).measurable.comp hΨ).comp measurable_snd - have hp : MeasurableSet {q : ℝ≥0 × Target.Driver | p q.1 q.2} := - (measurableSet_eq_fun hEval0 measurable_const).inter - ((measurableSet_le measurable_const hEval.im).inter - (Target.joint_measurableSet_rawTipAt_tendsto κ hκ _ hEval)) - have hLaw := Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW - have hsource : ∀ᵐ ω ∂P, ∀ T, p T (Target.normalizedDriver κ B hc h0 ω) := by - apply ae_of_all - intro ω T - have he (t : ℝ≥0) : Ψ (Target.normalizedDriver κ B hc h0 ω) t=γ ω t := by - have h := congrArg (fun e : C(ℝ≥0,ℂ) => e t) (hΨeq ω) - simpa only [sourceTracePath,ContinuousMap.coe_mk] using h - refine ⟨(he 0).trans (hw ω).2.1,?_,?_⟩ - · rw [he] - exact (hw ω).2.2.1 T - · simpa only [he] using Target.rawTipAt_tendsto_witness κ hκ - (Target.normalizedDriver κ B hc h0 ω) (fun t => Real.sqrt κ*B t ω) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) T - have hall := ae_forall_parameter_of_hasLaw hLaw p hp hsource - refine ⟨Ψ,hΨ,hΨeq,?_⟩ - filter_upwards [hall] with b hb - exact ⟨(hb 0).1,fun T => (hb T).2.1,fun T => (hb T).2.2⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.CanonicalTraceCandidate +import OAI.Probability.SLE.Loewner.TargetBoundaryLimit +import OAI.Probability.SLE.Geometry.FiniteParameterProjection + +namespace OAI + +/-! A continuous canonical trace modification with the exact inverse-tip +limit simultaneously at every time, obtained from an actual source witness. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance boundaryRealMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance boundaryRealBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ +local instance boundaryComplexMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _ +local instance boundaryComplexBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩ +local instance boundaryDriverPolish : PolishSpace Target.Driver := + ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace +local instance boundaryDriverMetric : MetricSpace Target.Driver := + TopologicalSpace.metrizableSpaceMetric Target.Driver + +theorem exists_canonical_all_time_boundary_limit + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (W : Target.WienerLaw) (hW : Target.IsWiener W) : + ∃ Ψ : Target.Driver → C(ℝ≥0,ℂ), Measurable Ψ ∧ + (∀ ω, Ψ (Target.normalizedDriver κ B hc h0 ω)=sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ + ∀ᵐ b ∂W, Ψ b 0=0 ∧ (∀ T, 0 ≤ (Ψ b T).im) ∧ + ∀ T, Tendsto (fun y => Target.rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (Ψ b T)) := by + have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure + obtain ⟨Ψ,hΨ,hΨeq,_⟩ := exists_canonical_trace_candidate_modification + κ hκ hκ8 B hB hm hc h0 γ G F hw W hW + let p (T : ℝ≥0) (b : Target.Driver) : Prop := + Ψ b 0=0 ∧ 0 ≤ (Ψ b T).im ∧ + Tendsto (fun y => Target.rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (Ψ b T)) + have hEval : Measurable (fun q : ℝ≥0 × Target.Driver => Ψ q.2 q.1) := + continuous_eval.measurable.comp ((hΨ.comp measurable_snd).prodMk measurable_fst) + have hEval0 : Measurable (fun q : ℝ≥0 × Target.Driver => Ψ q.2 0) := + ((continuous_eval_const 0).measurable.comp hΨ).comp measurable_snd + have hp : MeasurableSet {q : ℝ≥0 × Target.Driver | p q.1 q.2} := + (measurableSet_eq_fun hEval0 measurable_const).inter + ((measurableSet_le measurable_const hEval.im).inter + (Target.joint_measurableSet_rawTipAt_tendsto κ hκ _ hEval)) + have hLaw := Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW + have hsource : ∀ᵐ ω ∂P, ∀ T, p T (Target.normalizedDriver κ B hc h0 ω) := by + apply ae_of_all + intro ω T + have he (t : ℝ≥0) : Ψ (Target.normalizedDriver κ B hc h0 ω) t=γ ω t := by + have h := congrArg (fun e : C(ℝ≥0,ℂ) => e t) (hΨeq ω) + simpa only [sourceTracePath,ContinuousMap.coe_mk] using h + refine ⟨(he 0).trans (hw ω).2.1,?_,?_⟩ + · rw [he] + exact (hw ω).2.2.1 T + · simpa only [he] using Target.rawTipAt_tendsto_witness κ hκ + (Target.normalizedDriver κ B hc h0 ω) (fun t => Real.sqrt κ*B t ω) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) T + have hall := ae_forall_parameter_of_hasLaw hLaw p hp hsource + refine ⟨Ψ,hΨ,hΨeq,?_⟩ + filter_upwards [hall] with b hb + exact ⟨(hb 0).1,fun T => (hb T).2.1,fun T => (hb T).2.2⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalCoefficientIntegrability.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalCoefficientIntegrability.lean index f466239e604f33b63b350ad1b896c6b83c9f30d2..45b513b8b628e2794bb8f58e636c9129d75cdb35 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalCoefficientIntegrability.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalCoefficientIntegrability.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Gauge.DensityWeightedMoments -import OAI.Probability.SLE.Geometry.NormalizedStoppedDisk -import OAI.Probability.SLE.Geometry.OuterCoefficientLimit -import OAI.Probability.SLE.Gauge.MomentUnclipping - -namespace OAI - -/-! Integrability and removal of the exact coefficient's clipping, -using the canonical terminal disk moment as a dominator. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma density_disk_terminal_integrable - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) : - Integrable (fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * - (C.diskTerminalScalar θ hθ ω)^m) P := by - obtain ⟨M,hM,hbound⟩ := C.uniform_density_disk_terminal_moments hB hc h0 hκ hκ8 F hw hgrowth m - have hb := Target.normalizedDriver_measurable κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 - have hwM := (Target.measurable_fixedLevelDensity κ θ ℓ).comp hb - have hzM : Measurable (C.diskTerminalScalar θ hθ) := (compactWeightedMass_measurable _ _ (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp - C.rational.terminal_measurable - have hnonneg (ω) : 0≤Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * - (C.diskTerminalScalar θ hθ ω)^m := - mul_nonneg (Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ) - (pow_nonneg (compactWeightedMass_nonneg _ _ _) m) - apply (lintegral_ofReal_ne_top_iff_integrable (hwM.mul (hzM.pow_const m)).aestronglyMeasurable - (Eventually.of_forall hnonneg)).mp - exact ne_top_of_lt ((hbound θ hθ ℓ hℓ).trans_lt (ENNReal.mul_lt_top hM ENNReal.ofReal_lt_top)) - -theorem clippedCoefficient_unclip_limit [IsProbabilityMeasure P] - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : - Tendsto (fun n : ℕ => Target.clippedCoefficient κ W ℓ n m θ) atTop - (𝓝 (∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * - (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) - (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ))^m ∂P)) := by - let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) (T ω) - let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - have hT : Measurable T := normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ - have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T hT - have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp - (Target.normalizedDriver_measurable κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) - have hd : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤C.diskTerminalScalar θ hθ ω := by - filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω - exact ⟨Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ, - C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω,hω⟩ - have ht := weighted_unclip_pow_tendsto P Y (C.diskTerminalScalar θ hθ) w - hY.aestronglyMeasurable hwM.aestronglyMeasurable hd m - (C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ) - apply ht.congr' - filter_upwards [] with n - exact (C.clippedCoefficient_eq_mass hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ n (Nat.cast_nonneg n) m).symm - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Gauge.DensityWeightedMoments +import OAI.Probability.SLE.Geometry.NormalizedStoppedDisk +import OAI.Probability.SLE.Geometry.OuterCoefficientLimit +import OAI.Probability.SLE.Gauge.MomentUnclipping + +namespace OAI + +/-! Integrability and removal of the exact coefficient's clipping, +using the canonical terminal disk moment as a dominator. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma density_disk_terminal_integrable + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) : + Integrable (fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * + (C.diskTerminalScalar θ hθ ω)^m) P := by + obtain ⟨M,hM,hbound⟩ := C.uniform_density_disk_terminal_moments hB hc h0 hκ hκ8 F hw hgrowth m + have hb := Target.normalizedDriver_measurable κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 + have hwM := (Target.measurable_fixedLevelDensity κ θ ℓ).comp hb + have hzM : Measurable (C.diskTerminalScalar θ hθ) := (compactWeightedMass_measurable _ _ (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp + C.rational.terminal_measurable + have hnonneg (ω) : 0≤Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * + (C.diskTerminalScalar θ hθ ω)^m := + mul_nonneg (Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ) + (pow_nonneg (compactWeightedMass_nonneg _ _ _) m) + apply (lintegral_ofReal_ne_top_iff_integrable (hwM.mul (hzM.pow_const m)).aestronglyMeasurable + (Eventually.of_forall hnonneg)).mp + exact ne_top_of_lt ((hbound θ hθ ℓ hℓ).trans_lt (ENNReal.mul_lt_top hM ENNReal.ofReal_lt_top)) + +theorem clippedCoefficient_unclip_limit [IsProbabilityMeasure P] + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : + Tendsto (fun n : ℕ => Target.clippedCoefficient κ W ℓ n m θ) atTop + (𝓝 (∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ * + (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) + (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ))^m ∂P)) := by + let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) (T ω) + let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + have hT : Measurable T := normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ + have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T hT + have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp + (Target.normalizedDriver_measurable κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) + have hd : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤C.diskTerminalScalar θ hθ ω := by + filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω + exact ⟨Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ, + C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω,hω⟩ + have ht := weighted_unclip_pow_tendsto P Y (C.diskTerminalScalar θ hθ) w + hY.aestronglyMeasurable hwM.aestronglyMeasurable hd m + (C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ) + apply ht.congr' + filter_upwards [] with n + exact (C.clippedCoefficient_eq_mass hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ n (Nat.cast_nonneg n) m).symm + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalCompactPotential.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalCompactPotential.lean index eb93378c1e4fbd4ab136b5317ce47c719158bbe9..8e476a856dd427907b3af3d3d82e2515b064eb00 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalCompactPotential.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalCompactPotential.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Geometry.WeightedCompactKernel -import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit -import OAI.Probability.SLE.Geometry.CanonicalGreenPast - -namespace OAI - -/-! The current Green potential of a compact weighted mass is the -conditional expectation of its remaining terminal mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance compactPotentialDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance compactPotentialDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma compactWeightedMass_add - (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) - (μ ν : Measure upperHalfPlane) - [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] : - compactWeightedMass w K (μ+ν)=compactWeightedMass w K μ+compactWeightedMass w K ν := by - unfold compactWeightedMass - rw [Measure.restrict_add] - exact integral_add_measure (compact_weight_integrable w K hK μ) - (compact_weight_integrable w K hK ν) - -lemma halfPlaneGreenMeasure_measurable_past - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : - Measurable[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] - (halfPlaneGreenMeasure κ B γ G t) := by - let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) - have hp : Measurable[ℱ t,driverPathFiltration t] (sourceDriverPath B hc) := - driverPath_measurable_past_of_adapted ℱ _ - (fun s => (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable) s).measurable) t - have h := (canonicalGreenMass_measurable_past κ t).comp hp - convert h using 1 - funext ω - exact (canonicalGreenMass_eq κ (sourceDriverPath B hc ω) (hw ω) t).symm - -namespace CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma compact_potential_condExp - (hc : ∀ ω, Continuous (fun t => B t ω)) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) - (t : ℝ≥0) : - (fun ω => compactWeightedMass w K (halfPlaneGreenMeasure κ B γ G t ω)) =ᵐ[P] - P[(fun ω => compactWeightedMass w K (C.rational.terminal ω)- - compactWeightedMass w K (C.process t ω)) | - (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t] := by - let ℱ := Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) - let V := halfPlaneGreenMeasure κ B γ G t - have hVp : Measurable[ℱ t] V := halfPlaneGreenMeasure_measurable_past - (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hc κ γ G F hw t - have hVm : Measurable V := hVp.mono (ℱ.le t) le_rfl - have hk (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : - (P.restrict E).bind C.rational.terminal = - (P.restrict E).bind (C.process t)+(P.restrict E).bind V := by - have hs := isStoppingTime_const ℱ t - have hES : MeasurableSet[hs.measurableSpace] E := by - rw [IsStoppingTime.measurableSpace_const] - exact hE - have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs E hES - change (P.restrict E).bind C.rational.terminal = - (P.restrict E).bind (C.process t)+ - (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G (fun _ => (t:ℝ≥0∞))) at h - rwa [halfPlaneStoppingGreenMeasure_const] at h - have hVbound (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : - (P.restrict E).bind V ≤ initialGreenMeasure κ := by - apply le_trans _ (C.uniqueness_terminal_mean_le E) - rw [hk E hE] - exact Measure.le_add_left le_rfl - have hVL := compactWeightedMass_kernel_integrable P κ V hVm - (by simpa only [Measure.restrict_univ] using hVbound univ MeasurableSet.univ) w K hK - have hNL := compactWeightedMass_kernel_integrable P κ (C.process t) - C.progressive.adapted.measurable - (by simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ) w K hK - have hTL := compactWeightedMass_kernel_integrable P κ C.rational.terminal - C.rational.terminal_measurable (by rw [C.rational.mean]) w K hK - apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le t) (hTL.1.sub hNL.1) - (fun E _ _ => hVL.1.integrableOn) ?_ - (((compactWeightedMass_measurable w K hK.measurableSet).comp hVp).aestronglyMeasurable) - intro E hE _ - have hVE := (compactWeightedMass_kernel_integrable (P.restrict E) κ V hVm (hVbound E hE) w K hK).2 - have hNE := (compactWeightedMass_kernel_integrable (P.restrict E) κ (C.process t) - C.progressive.adapted.measurable (C.uniqueness_process_mean_le t E) w K hK).2 - have hTE := (compactWeightedMass_kernel_integrable (P.restrict E) κ C.rational.terminal - C.rational.terminal_measurable (C.uniqueness_terminal_mean_le E) w K hK).2 - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (C.process t)) := - ⟨fun A hA => lt_of_le_of_lt (C.uniqueness_process_mean_le t E A) hA.measure_lt_top⟩ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V) := - ⟨fun A hA => lt_of_le_of_lt (hVbound E hE A) hA.measure_lt_top⟩ - simp only [Pi.sub_apply] - rw [integral_sub hTL.1.integrableOn hNL.1.integrableOn,hVE,hTE,hNE,hk E hE, - compactWeightedMass_add w K hK] - ring - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.WeightedCompactKernel +import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit +import OAI.Probability.SLE.Geometry.CanonicalGreenPast + +namespace OAI + +/-! The current Green potential of a compact weighted mass is the +conditional expectation of its remaining terminal mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance compactPotentialDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance compactPotentialDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma compactWeightedMass_add + (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) + (μ ν : Measure upperHalfPlane) + [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] : + compactWeightedMass w K (μ+ν)=compactWeightedMass w K μ+compactWeightedMass w K ν := by + unfold compactWeightedMass + rw [Measure.restrict_add] + exact integral_add_measure (compact_weight_integrable w K hK μ) + (compact_weight_integrable w K hK ν) + +lemma halfPlaneGreenMeasure_measurable_past + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : + Measurable[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] + (halfPlaneGreenMeasure κ B γ G t) := by + let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) + have hp : Measurable[ℱ t,driverPathFiltration t] (sourceDriverPath B hc) := + driverPath_measurable_past_of_adapted ℱ _ + (fun s => (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable) s).measurable) t + have h := (canonicalGreenMass_measurable_past κ t).comp hp + convert h using 1 + funext ω + exact (canonicalGreenMass_eq κ (sourceDriverPath B hc ω) (hw ω) t).symm + +namespace CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma compact_potential_condExp + (hc : ∀ ω, Continuous (fun t => B t ω)) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) + (t : ℝ≥0) : + (fun ω => compactWeightedMass w K (halfPlaneGreenMeasure κ B γ G t ω)) =ᵐ[P] + P[(fun ω => compactWeightedMass w K (C.rational.terminal ω)- + compactWeightedMass w K (C.process t ω)) | + (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t] := by + let ℱ := Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) + let V := halfPlaneGreenMeasure κ B γ G t + have hVp : Measurable[ℱ t] V := halfPlaneGreenMeasure_measurable_past + (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hc κ γ G F hw t + have hVm : Measurable V := hVp.mono (ℱ.le t) le_rfl + have hk (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : + (P.restrict E).bind C.rational.terminal = + (P.restrict E).bind (C.process t)+(P.restrict E).bind V := by + have hs := isStoppingTime_const ℱ t + have hES : MeasurableSet[hs.measurableSpace] E := by + rw [IsStoppingTime.measurableSpace_const] + exact hE + have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs E hES + change (P.restrict E).bind C.rational.terminal = + (P.restrict E).bind (C.process t)+ + (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G (fun _ => (t:ℝ≥0∞))) at h + rwa [halfPlaneStoppingGreenMeasure_const] at h + have hVbound (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : + (P.restrict E).bind V ≤ initialGreenMeasure κ := by + apply le_trans _ (C.uniqueness_terminal_mean_le E) + rw [hk E hE] + exact Measure.le_add_left le_rfl + have hVL := compactWeightedMass_kernel_integrable P κ V hVm + (by simpa only [Measure.restrict_univ] using hVbound univ MeasurableSet.univ) w K hK + have hNL := compactWeightedMass_kernel_integrable P κ (C.process t) + C.progressive.adapted.measurable + (by simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ) w K hK + have hTL := compactWeightedMass_kernel_integrable P κ C.rational.terminal + C.rational.terminal_measurable (by rw [C.rational.mean]) w K hK + apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le t) (hTL.1.sub hNL.1) + (fun E _ _ => hVL.1.integrableOn) ?_ + (((compactWeightedMass_measurable w K hK.measurableSet).comp hVp).aestronglyMeasurable) + intro E hE _ + have hVE := (compactWeightedMass_kernel_integrable (P.restrict E) κ V hVm (hVbound E hE) w K hK).2 + have hNE := (compactWeightedMass_kernel_integrable (P.restrict E) κ (C.process t) + C.progressive.adapted.measurable (C.uniqueness_process_mean_le t E) w K hK).2 + have hTE := (compactWeightedMass_kernel_integrable (P.restrict E) κ C.rational.terminal + C.rational.terminal_measurable (C.uniqueness_terminal_mean_le E) w K hK).2 + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (C.process t)) := + ⟨fun A hA => lt_of_le_of_lt (C.uniqueness_process_mean_le t E A) hA.measure_lt_top⟩ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V) := + ⟨fun A hA => lt_of_le_of_lt (hVbound E hE A) hA.measure_lt_top⟩ + simp only [Pi.sub_apply] + rw [integral_sub hTL.1.integrableOn hNL.1.integrableOn,hVE,hTE,hNE,hk E hE, + compactWeightedMass_add w K hK] + ring + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPositive.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPositive.lean index 59f0621ec36179287750f4daafeb58b19436e0c3..209cf90e97f7b09f4d49b611b525baeae4dcedfa 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPositive.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPositive.lean @@ -1,112 +1,112 @@ -import OAI.Probability.SLE.Angle.TargetDiskVisit -import OAI.Probability.SLE.Mass.PositiveTraceMass -import OAI.Probability.SLE.Gauge.CenteredMomentJensen - -namespace OAI - -/-! Strict positivity of the exact finite-level disk coefficients. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma compactWeightedMass_pos (w : C(upperHalfPlane,ℝ≥0)) - (hw : ∀ z, 0 (w z).coe_nonneg) (compact_weight_integrable w K hK μ)] - have hs : Function.support (fun z => (w z:ℝ))=univ := by - ext z - simp only [Function.mem_support,mem_univ,iff_true] - exact ne_of_gt (show (0:ℝ) B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (u : ℝ) (hu : 0≤u) - (hsmall : Real.exp (-u)<1/8) : - ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u, - 0 B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) - (hsmall : Real.exp (-(4*ℓ/κ))<1/8) : - 0 (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0) - have hσ := actual_radiusLevelTime_stopping hB - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) - hBm hc h0 κ hκ hκ8 γ G F hw z (Target.diskMapInv_maps hθ (by simp)) u hu - have heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z u := Eventually.of_forall (fun _ => rfl) - let Q := finiteCenteredLaw P κ B γ G z u - let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F - (Eventually.of_forall hw) z (Target.diskMapInv_maps hθ (by simp)) u hu - obtain ⟨hi,he⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ 1 σ hσ heσ - rw [he] - simp only [pow_one] at hi ⊢ - have hfinite := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F - (Eventually.of_forall hw) z (Target.diskMapInv_maps hθ (by simp)) u u hu le_rfl σ - (Eventually.of_forall (fun _ => le_rfl)) - have hp := C.centered_diskScalar_positive hB hc h0 hκ hκ8 F hw hpublished θ hθ u hu hsmall - have hpos : ∀ᵐ ω ∂Q, 0 h.le)) hi).2 - have hevent : Q (Function.support (finiteStoppedScalar (C.diskScalar θ hθ) σ))=Q univ := - measure_congr (hpos.mono (fun ω h => propext - (show finiteStoppedScalar (C.diskScalar θ hθ) σ ω≠0 ↔ True from iff_true_intro h.ne'))) - rw [hevent,measure_univ] - exact zero_lt_one - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetDiskVisit +import OAI.Probability.SLE.Mass.PositiveTraceMass +import OAI.Probability.SLE.Gauge.CenteredMomentJensen + +namespace OAI + +/-! Strict positivity of the exact finite-level disk coefficients. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma compactWeightedMass_pos (w : C(upperHalfPlane,ℝ≥0)) + (hw : ∀ z, 0 (w z).coe_nonneg) (compact_weight_integrable w K hK μ)] + have hs : Function.support (fun z => (w z:ℝ))=univ := by + ext z + simp only [Function.mem_support,mem_univ,iff_true] + exact ne_of_gt (show (0:ℝ) B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (u : ℝ) (hu : 0≤u) + (hsmall : Real.exp (-u)<1/8) : + ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u, + 0 B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) + (hsmall : Real.exp (-(4*ℓ/κ))<1/8) : + 0 (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0) + have hσ := actual_radiusLevelTime_stopping hB + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) + hBm hc h0 κ hκ hκ8 γ G F hw z (Target.diskMapInv_maps hθ (by simp)) u hu + have heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z u := Eventually.of_forall (fun _ => rfl) + let Q := finiteCenteredLaw P κ B γ G z u + let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F + (Eventually.of_forall hw) z (Target.diskMapInv_maps hθ (by simp)) u hu + obtain ⟨hi,he⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ 1 σ hσ heσ + rw [he] + simp only [pow_one] at hi ⊢ + have hfinite := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F + (Eventually.of_forall hw) z (Target.diskMapInv_maps hθ (by simp)) u u hu le_rfl σ + (Eventually.of_forall (fun _ => le_rfl)) + have hp := C.centered_diskScalar_positive hB hc h0 hκ hκ8 F hw hpublished θ hθ u hu hsmall + have hpos : ∀ᵐ ω ∂Q, 0 h.le)) hi).2 + have hevent : Q (Function.support (finiteStoppedScalar (C.diskScalar θ hθ) σ))=Q univ := + measure_congr (hpos.mono (fun ω h => propext + (show finiteStoppedScalar (C.diskScalar θ hθ) σ ω≠0 ↔ True from iff_true_intro h.ne'))) + rw [hevent,measure_univ] + exact zero_lt_one + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPotential.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPotential.lean index eb7f27d2bba38524ac58dcfeed71744b0ade5d8f..ac55beeb4fe3f1a470211316a9dec79fdd446926 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPotential.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskPotential.lean @@ -1,73 +1,73 @@ -import OAI.Probability.SLE.Angle.TargetDiskMass -import OAI.Probability.SLE.Geometry.CanonicalCompactPotential - -namespace OAI - -/-! The source-prescribed disk potential is the remaining canonical disk -mass, with exactly the source's Brownian clock normalization. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -def diskScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) (ω : Ω) : ℝ := - compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) (C.process t ω) - -def diskTerminalScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ω : Ω) : ℝ := - compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) - (C.rational.terminal ω) - -theorem target_potential_condExp [IsProbabilityMeasure P] - (hκ : 0 < κ) (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b : Ω → Target.Driver) - (hU : ∀ ω (s : ℝ), 0 ≤ s → Real.sqrt κ*B s.toNNReal ω=Target.driverValue (b ω) (κ*s)) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : - (fun ω => (Target.potential κ θ (b ω) (κ*t)).toReal) =ᵐ[P] - P[C.diskTerminalScalar θ hθ-C.diskScalar θ hθ t | - (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t] := by - have he (ω : Ω) : (Target.potential κ θ (b ω) (κ*t)).toReal = - compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) - (halfPlaneGreenMeasure κ B γ G t ω) := by - exact Target.potential_toReal_eq_compactWeightedMass κ hκ (b ω) _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ hθ t - have heAE : (fun ω => (Target.potential κ θ (b ω) (κ*t)).toReal) =ᵐ[P] - (fun ω => compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) (halfPlaneGreenMeasure κ B γ G t ω)) := - Eventually.of_forall he - exact heAE.trans (C.compact_potential_condExp hc F hw _ _ - (Target.diskSourceHalfPlane_compact hθ) t) - -include hBm in -theorem target_potential_lintegral_ne_top - (hB : IsBrownianReal B P) (hκ : 0 < κ) (hκ8 : κ<8) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b : Ω → Target.Driver) - (hU : ∀ ω (s : ℝ), 0 ≤ s → Real.sqrt κ*B s.toNNReal ω=Target.driverValue (b ω) (κ*s)) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : - (∫⁻ ω, Target.potential κ θ (b ω) (κ*t) ∂P) ≠ ∞ := by - have he (ω : Ω) := Target.potential_eq_weighted_green κ hκ (b ω) _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ hθ t - simp_rw [he] - apply compactWeightedMass_kernel_lintegral_ne_top P κ (halfPlaneGreenMeasure κ B γ G t) - (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t) - (by simpa only [Measure.restrict_univ] using - halfPlaneGreenMeasure_mean_le hB hBm hc h0 hκ hκ8 γ G F hw t univ) - _ _ (Target.diskSourceHalfPlane_compact hθ) - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetDiskMass +import OAI.Probability.SLE.Geometry.CanonicalCompactPotential + +namespace OAI + +/-! The source-prescribed disk potential is the remaining canonical disk +mass, with exactly the source's Brownian clock normalization. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +def diskScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) (ω : Ω) : ℝ := + compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) (C.process t ω) + +def diskTerminalScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ω : Ω) : ℝ := + compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) + (C.rational.terminal ω) + +theorem target_potential_condExp [IsProbabilityMeasure P] + (hκ : 0 < κ) (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b : Ω → Target.Driver) + (hU : ∀ ω (s : ℝ), 0 ≤ s → Real.sqrt κ*B s.toNNReal ω=Target.driverValue (b ω) (κ*s)) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : + (fun ω => (Target.potential κ θ (b ω) (κ*t)).toReal) =ᵐ[P] + P[C.diskTerminalScalar θ hθ-C.diskScalar θ hθ t | + (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t] := by + have he (ω : Ω) : (Target.potential κ θ (b ω) (κ*t)).toReal = + compactWeightedMass (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) + (halfPlaneGreenMeasure κ B γ G t ω) := by + exact Target.potential_toReal_eq_compactWeightedMass κ hκ (b ω) _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ hθ t + have heAE : (fun ω => (Target.potential κ θ (b ω) (κ*t)).toReal) =ᵐ[P] + (fun ω => compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) (halfPlaneGreenMeasure κ B γ G t ω)) := + Eventually.of_forall he + exact heAE.trans (C.compact_potential_condExp hc F hw _ _ + (Target.diskSourceHalfPlane_compact hθ) t) + +include hBm in +theorem target_potential_lintegral_ne_top + (hB : IsBrownianReal B P) (hκ : 0 < κ) (hκ8 : κ<8) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b : Ω → Target.Driver) + (hU : ∀ ω (s : ℝ), 0 ≤ s → Real.sqrt κ*B s.toNNReal ω=Target.driverValue (b ω) (κ*s)) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : + (∫⁻ ω, Target.potential κ θ (b ω) (κ*t) ∂P) ≠ ∞ := by + have he (ω : Ω) := Target.potential_eq_weighted_green κ hκ (b ω) _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ hθ t + simp_rw [he] + apply compactWeightedMass_kernel_lintegral_ne_top P κ (halfPlaneGreenMeasure κ B γ G t) + (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t) + (by simpa only [Measure.restrict_univ] using + halfPlaneGreenMeasure_mean_le hB hBm hc h0 hκ hκ8 γ G F hw t univ) + _ _ (Target.diskSourceHalfPlane_compact hθ) + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskTails.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskTails.lean index 4760fcb9ad97dec3dc49fa0a13a4b0a5a06cd2a5..aa5f575a3b36881f6ae58fa94158dc2f9a2c6090 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalDiskTails.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalDiskTails.lean @@ -1,112 +1,112 @@ -import OAI.Probability.SLE.Geometry.CenteredDiskLaw -import OAI.Probability.SLE.Geometry.CenteredStoppedTails - -namespace OAI - -/-! The tails of the actual finite centered disk laws increase with the -radius level. Their supremum is sufficient; no infinite tilted path law -is postulated. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge - -lemma measure_joint_tail_of_lintegral {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] - (ν : Measure (α × β)) (η : Measure α) (Q : α → Measure β) - (h : ∀ f : α × β → ℝ≥0∞, Measurable f → - (∫⁻ p, f p ∂ν) = ∫⁻ a, ∫⁻ b, f (a,b) ∂Q a ∂η) - (X : α × β → ℝ) (hX : Measurable X) (c : ℝ) : - ν {p | c ≤ X p} = ∫⁻ a, Q a {b | c ≤ X (a,b)} ∂η := by - have hA : MeasurableSet {p | c ≤ X p} := measurableSet_le measurable_const hX - let f : α × β → ℝ≥0∞ := {p | c ≤ X p}.indicator 1 - have hf : Measurable f := (measurable_one : Measurable (1 : α × β → ℝ≥0∞)).indicator hA - calc - _ = ∫⁻ p, f p ∂ν := (lintegral_indicator_one hA).symm - _ = ∫⁻ a, ∫⁻ b, f (a,b) ∂Q a ∂η := h f hf - _ = _ := by - apply lintegral_congr - intro a - have hpair : Measurable (fun b : β => (a,b)) := measurable_const.prodMk measurable_id - have hAa : MeasurableSet {b | c ≤ X (a,b)} := hA.preimage hpair - change (∫⁻ b, {b | c ≤ X (a,b)}.indicator 1 b ∂Q a) = _ - exact lintegral_indicator_one hAa - -end SLEExactGauge - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -def centeredDiskTail (hc : ∀ ω, Continuous (fun t => B t ω)) (b u : ℝ) : ℝ≥0∞ := - centeredDiskLaw P κ B hc u {p | b ≤ C.diskAtRadius p.1 u p.2} - -lemma centeredDiskTail_eq_integral - (hc : ∀ ω, Continuous (fun t => B t ω)) - (hκ : 0<κ) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b u : ℝ) (hu : 0≤u) : - C.centeredDiskTail hc b u = ∫⁻ θ, - finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u - {ω | b ≤ C.diskAtRadius θ u ω} ∂Target.angleLaw κ := by - exact measure_joint_tail_of_lintegral (centeredDiskLaw P κ B hc u) (Target.angleLaw κ) - (fun θ => finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u) - (fun f hf => centeredDiskLaw_lintegral P κ hκ B - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw u hu f hf) - (fun p => C.diskAtRadius p.1 u p.2) (C.diskAtRadius_joint_measurable hc F hw u hu) b - -lemma diskAtRadius_tail_mono - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (b v V : ℝ) (hv : 0≤v) (hvV : v≤V) : - finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) v {ω | b ≤ C.diskAtRadius θ v ω} ≤ - finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) V {ω | b ≤ C.diskAtRadius θ V ω} := by - let z := Target.diskMapInv θ 0 - have hz : z ∈ upperHalfPlane := Target.diskMapInv_maps hθ (by simp) - let σ := fun ω => radiusLevelTime (γ ω) (G ω) z v - let τ := fun ω => radiusLevelTime (γ ω) (G ω) z V - have hσ := actual_radiusLevelTime_stopping hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0 κ hκ hκ8 γ G F hw z hz v hv - have hf (w : ℝ) (hw0 : 0≤w) := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz w w hw0 le_rfl - (fun ω => radiusLevelTime (γ ω) (G ω) z w) (Eventually.of_forall (fun _ => le_rfl)) - have he (w : ℝ) (hw0 : 0≤w) : - finiteCenteredLaw P κ B γ G z w {ω | b ≤ C.diskAtRadius θ w ω} = - finiteCenteredLaw P κ B γ G z w - {ω | b ≤ finiteStoppedScalar (C.diskScalar θ hθ) - (fun ω => radiusLevelTime (γ ω) (G ω) z w) ω} := by - apply measure_congr - filter_upwards [hf w hw0] with ω hω - rw [C.diskAtRadius_eq_finite θ hθ w - (fun ω => radiusLevelTime (γ ω) (G ω) z w) ω rfl hω] - rw [he v hv,he V (hv.trans hvV)] - exact finiteCenteredLaw_stoppedScalar_tail_mono hB hBm hc h0 κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz v V hv hvV (C.diskScalar θ hθ) - (C.diskScalar_progressive θ hθ) - ((C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) - (Target.diskSourceHalfPlane_compact hθ)).mono (fun _ h => h.1)) - σ τ hσ (Eventually.of_forall (fun _ => rfl)) (Eventually.of_forall (fun _ => rfl)) b - -lemma centeredDiskTail_mono - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (b : ℝ) : MonotoneOn (C.centeredDiskTail hc b) (Ici 0) := by - intro v hv V hV hvV - rw [C.centeredDiskTail_eq_integral hc hκ F hw b v hv, - C.centeredDiskTail_eq_integral hc hκ F hw b V hV] - apply lintegral_mono_ae - filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ - exact C.diskAtRadius_tail_mono hB hc h0 hκ hκ8 F hw θ hθ b v V hv hvV - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.CenteredDiskLaw +import OAI.Probability.SLE.Geometry.CenteredStoppedTails + +namespace OAI + +/-! The tails of the actual finite centered disk laws increase with the +radius level. Their supremum is sufficient; no infinite tilted path law +is postulated. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge + +lemma measure_joint_tail_of_lintegral {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] + (ν : Measure (α × β)) (η : Measure α) (Q : α → Measure β) + (h : ∀ f : α × β → ℝ≥0∞, Measurable f → + (∫⁻ p, f p ∂ν) = ∫⁻ a, ∫⁻ b, f (a,b) ∂Q a ∂η) + (X : α × β → ℝ) (hX : Measurable X) (c : ℝ) : + ν {p | c ≤ X p} = ∫⁻ a, Q a {b | c ≤ X (a,b)} ∂η := by + have hA : MeasurableSet {p | c ≤ X p} := measurableSet_le measurable_const hX + let f : α × β → ℝ≥0∞ := {p | c ≤ X p}.indicator 1 + have hf : Measurable f := (measurable_one : Measurable (1 : α × β → ℝ≥0∞)).indicator hA + calc + _ = ∫⁻ p, f p ∂ν := (lintegral_indicator_one hA).symm + _ = ∫⁻ a, ∫⁻ b, f (a,b) ∂Q a ∂η := h f hf + _ = _ := by + apply lintegral_congr + intro a + have hpair : Measurable (fun b : β => (a,b)) := measurable_const.prodMk measurable_id + have hAa : MeasurableSet {b | c ≤ X (a,b)} := hA.preimage hpair + change (∫⁻ b, {b | c ≤ X (a,b)}.indicator 1 b ∂Q a) = _ + exact lintegral_indicator_one hAa + +end SLEExactGauge + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +def centeredDiskTail (hc : ∀ ω, Continuous (fun t => B t ω)) (b u : ℝ) : ℝ≥0∞ := + centeredDiskLaw P κ B hc u {p | b ≤ C.diskAtRadius p.1 u p.2} + +lemma centeredDiskTail_eq_integral + (hc : ∀ ω, Continuous (fun t => B t ω)) + (hκ : 0<κ) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b u : ℝ) (hu : 0≤u) : + C.centeredDiskTail hc b u = ∫⁻ θ, + finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u + {ω | b ≤ C.diskAtRadius θ u ω} ∂Target.angleLaw κ := by + exact measure_joint_tail_of_lintegral (centeredDiskLaw P κ B hc u) (Target.angleLaw κ) + (fun θ => finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u) + (fun f hf => centeredDiskLaw_lintegral P κ hκ B + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw u hu f hf) + (fun p => C.diskAtRadius p.1 u p.2) (C.diskAtRadius_joint_measurable hc F hw u hu) b + +lemma diskAtRadius_tail_mono + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (b v V : ℝ) (hv : 0≤v) (hvV : v≤V) : + finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) v {ω | b ≤ C.diskAtRadius θ v ω} ≤ + finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) V {ω | b ≤ C.diskAtRadius θ V ω} := by + let z := Target.diskMapInv θ 0 + have hz : z ∈ upperHalfPlane := Target.diskMapInv_maps hθ (by simp) + let σ := fun ω => radiusLevelTime (γ ω) (G ω) z v + let τ := fun ω => radiusLevelTime (γ ω) (G ω) z V + have hσ := actual_radiusLevelTime_stopping hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0 κ hκ hκ8 γ G F hw z hz v hv + have hf (w : ℝ) (hw0 : 0≤w) := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz w w hw0 le_rfl + (fun ω => radiusLevelTime (γ ω) (G ω) z w) (Eventually.of_forall (fun _ => le_rfl)) + have he (w : ℝ) (hw0 : 0≤w) : + finiteCenteredLaw P κ B γ G z w {ω | b ≤ C.diskAtRadius θ w ω} = + finiteCenteredLaw P κ B γ G z w + {ω | b ≤ finiteStoppedScalar (C.diskScalar θ hθ) + (fun ω => radiusLevelTime (γ ω) (G ω) z w) ω} := by + apply measure_congr + filter_upwards [hf w hw0] with ω hω + rw [C.diskAtRadius_eq_finite θ hθ w + (fun ω => radiusLevelTime (γ ω) (G ω) z w) ω rfl hω] + rw [he v hv,he V (hv.trans hvV)] + exact finiteCenteredLaw_stoppedScalar_tail_mono hB hBm hc h0 κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz v V hv hvV (C.diskScalar θ hθ) + (C.diskScalar_progressive θ hθ) + ((C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) + (Target.diskSourceHalfPlane_compact hθ)).mono (fun _ h => h.1)) + σ τ hσ (Eventually.of_forall (fun _ => rfl)) (Eventually.of_forall (fun _ => rfl)) b + +lemma centeredDiskTail_mono + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (b : ℝ) : MonotoneOn (C.centeredDiskTail hc b) (Ici 0) := by + intro v hv V hV hvV + rw [C.centeredDiskTail_eq_integral hc hκ F hw b v hv, + C.centeredDiskTail_eq_integral hc hκ F hw b V hV] + apply lintegral_mono_ae + filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ + exact C.diskAtRadius_tail_mono hB hc h0 hκ hκ8 F hw θ hθ b v V hv hvV + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalGreenPast.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalGreenPast.lean index 7deb081bd79b1768e8ac9ba1a9b004fee837d6a2..118046720e4546f3856c8c82539ded1161deba0a 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalGreenPast.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalGreenPast.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel -import OAI.Probability.SLE.Geometry.SpatialPastKernels -import OAI.Probability.SLE.Stochastic.DriverPastFactorization -import OAI.Probability.SLE.Mass.GreenMeasureSupport - -namespace OAI - -/-! The canonical Green measure at a deterministic time uses only the -continuous driver prefix. This is a pathwise ODE fact, before any law is fixed. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance greenPastDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance greenPastDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma canonicalGreen_eq_of_agree (κ : ℝ) (T : ℝ≥0) - (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ T, p s=q s) (z : ℂ) : - canonicalGreen κ (T,z,p)=canonicalGreen κ (T,z,q) := by - let c := canonicalRadiusFamily κ - have hQ (n : ℕ) : c.Q n T (z,p)=c.Q n T (z,q) := by - apply clipped_flow_eqOn_prefix (-(n:ℝ)) n _ T - (fun s hs => congrArg (fun x => Real.sqrt κ*x) (hpq s hs)) - (c.derivative n z p) (c.derivative n z q) - (by rw [c.initial,c.initial]) - exact ⟨T.coe_nonneg,le_rfl⟩ - have hv (n : ℕ) : c.valid n T z p ↔ c.valid n T z q := by - simp only [SpatialClippedFamily.valid,hQ] - change c.jointGreen κ (fun s r => r s) (T,z,p) = - c.jointGreen κ (fun s r => r s) (T,z,q) - unfold SpatialClippedFamily.jointGreen - congr 1 - funext n - simp only [SpatialClippedFamily.greenApprox,hv,hQ,hpq T le_rfl] - -def canonicalGreenMass (κ : ℝ) (t : ℝ≥0) (p : C(ℝ≥0,ℝ)) : - Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (canonicalGreen κ (t,z,p)))).comap Subtype.val - -lemma canonicalGreenMass_measurable (κ : ℝ) (t : ℝ≥0) : - Measurable (canonicalGreenMass κ t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [canonicalGreenMass,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - have hg : Measurable (fun x : C(ℝ≥0,ℝ) × ℂ => canonicalGreen κ (t,x.2,x.1)) := - (canonicalGreen_measurable κ).comp - (measurable_const.prodMk (measurable_snd.prodMk measurable_fst)) - exact hg.ennreal_ofReal.lintegral_prod_right' - -lemma canonicalGreenMass_eq_of_agree (κ : ℝ) (t : ℝ≥0) - (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ t, p s=q s) : - canonicalGreenMass κ t p=canonicalGreenMass κ t q := by - unfold canonicalGreenMass - congr 2 - funext z - rw [canonicalGreen_eq_of_agree κ t p q hpq z] - -lemma canonicalGreenMass_measurable_past (κ : ℝ) (t : ℝ≥0) : - Measurable[driverPathFiltration t] (canonicalGreenMass κ t) := by - have h := (canonicalGreenMass_measurable κ t).comp (driverPathStopped_measurable_past t) - convert h using 1 - funext p - apply canonicalGreenMass_eq_of_agree - intro s hs - simp only [driverPathStopped,ContinuousMap.coe_mk,min_eq_left hs] - -lemma canonicalGreenMass_eq (κ : ℝ) (p : C(ℝ≥0,ℝ)) - {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) γ G F) (t : ℝ≥0) : - canonicalGreenMass κ t p = - (volume.withDensity (fun z => ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ*p s) γ G t z))).comap Subtype.val := by - unfold canonicalGreenMass - congr 2 - funext z - rw [canonicalGreen_eq κ p hw t z] - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel +import OAI.Probability.SLE.Geometry.SpatialPastKernels +import OAI.Probability.SLE.Stochastic.DriverPastFactorization +import OAI.Probability.SLE.Mass.GreenMeasureSupport + +namespace OAI + +/-! The canonical Green measure at a deterministic time uses only the +continuous driver prefix. This is a pathwise ODE fact, before any law is fixed. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance greenPastDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance greenPastDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma canonicalGreen_eq_of_agree (κ : ℝ) (T : ℝ≥0) + (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ T, p s=q s) (z : ℂ) : + canonicalGreen κ (T,z,p)=canonicalGreen κ (T,z,q) := by + let c := canonicalRadiusFamily κ + have hQ (n : ℕ) : c.Q n T (z,p)=c.Q n T (z,q) := by + apply clipped_flow_eqOn_prefix (-(n:ℝ)) n _ T + (fun s hs => congrArg (fun x => Real.sqrt κ*x) (hpq s hs)) + (c.derivative n z p) (c.derivative n z q) + (by rw [c.initial,c.initial]) + exact ⟨T.coe_nonneg,le_rfl⟩ + have hv (n : ℕ) : c.valid n T z p ↔ c.valid n T z q := by + simp only [SpatialClippedFamily.valid,hQ] + change c.jointGreen κ (fun s r => r s) (T,z,p) = + c.jointGreen κ (fun s r => r s) (T,z,q) + unfold SpatialClippedFamily.jointGreen + congr 1 + funext n + simp only [SpatialClippedFamily.greenApprox,hv,hQ,hpq T le_rfl] + +def canonicalGreenMass (κ : ℝ) (t : ℝ≥0) (p : C(ℝ≥0,ℝ)) : + Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (canonicalGreen κ (t,z,p)))).comap Subtype.val + +lemma canonicalGreenMass_measurable (κ : ℝ) (t : ℝ≥0) : + Measurable (canonicalGreenMass κ t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [canonicalGreenMass,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + have hg : Measurable (fun x : C(ℝ≥0,ℝ) × ℂ => canonicalGreen κ (t,x.2,x.1)) := + (canonicalGreen_measurable κ).comp + (measurable_const.prodMk (measurable_snd.prodMk measurable_fst)) + exact hg.ennreal_ofReal.lintegral_prod_right' + +lemma canonicalGreenMass_eq_of_agree (κ : ℝ) (t : ℝ≥0) + (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ t, p s=q s) : + canonicalGreenMass κ t p=canonicalGreenMass κ t q := by + unfold canonicalGreenMass + congr 2 + funext z + rw [canonicalGreen_eq_of_agree κ t p q hpq z] + +lemma canonicalGreenMass_measurable_past (κ : ℝ) (t : ℝ≥0) : + Measurable[driverPathFiltration t] (canonicalGreenMass κ t) := by + have h := (canonicalGreenMass_measurable κ t).comp (driverPathStopped_measurable_past t) + convert h using 1 + funext p + apply canonicalGreenMass_eq_of_agree + intro s hs + simp only [driverPathStopped,ContinuousMap.coe_mk,min_eq_left hs] + +lemma canonicalGreenMass_eq (κ : ℝ) (p : C(ℝ≥0,ℝ)) + {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) γ G F) (t : ℝ≥0) : + canonicalGreenMass κ t p = + (volume.withDensity (fun z => ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ*p s) γ G t z))).comap Subtype.val := by + unfold canonicalGreenMass + congr 2 + funext z + rw [canonicalGreen_eq κ p hw t z] + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalInteriorBound.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalInteriorBound.lean index 714d09e8d92f535643d2b91639c8d39fc639ded1..cc0b590dc03004cee692274e47897eb362b723c1 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalInteriorBound.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalInteriorBound.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Gauge.PrescribedCanonicalTransfer -import OAI.Probability.SLE.Gauge.PrescribedUniformGeometry - -namespace OAI - -/-! The canonical finite-law failure estimate gives the exact uniform - interior bound used in the prescribed upper cover. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (M : CanonicalMass P κ B hBm γ G) - -def prescribedCenteredFailure (W : Target.WienerLaw) (n : ℕ) {J : ℕ} - (e : Fin J → ℤ×ℕ) (C c : ℝ) (z : ℂ) : ℝ≥0∞ := - finiteCenteredLaw P κ B γ G z (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z) - (M.prescribedTestFailures W n e C c z - (fun i ω => radiusLevelTime (γ ω) (G ω) z - (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+ - 2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) - -theorem eventually_prescribed_interior_bound - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (W : Target.WienerLaw) (hm1 : 0 < Target.moment κ W 1) - (hJ : ∀ n, 1≤n → Target.moment κ W 1^n ≤ Target.moment κ W n) - (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ) - (he : ∀ n i, (e n i).2 < Target.multiplicity κ W n (e n i).1) - (C c R : ℝ) (K : Set ℂ) - (hfailure : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, - Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im → - M.prescribedCenteredFailure W n (e n) C c - (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) ≤ - ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16)))) : - ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, - Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im → - P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (γ ω)∩K) - (M.rational.physicalTerminal ω)} ≤ - ENNReal.ofReal ((uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ* - (2:ℝ)^(-codimension κ))*(Target.entryRadius κ W (n,none))^codimension κ* - (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)* - Real.exp (-((2:ℝ)^n/16))) := by - filter_upwards [hfailure,Target.eventually_prescribed_uniform_geometry hκ W hm1 hJ, - Target.eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 1] - with n hn hg hl - intro ij hij hy - let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij - obtain ⟨hlevels,hsmall,hends⟩ := hg z hy - let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z - have hV : 1 radiusLevelTime (γ ω) (G ω) z + (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+ + 2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) + +theorem eventually_prescribed_interior_bound + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (W : Target.WienerLaw) (hm1 : 0 < Target.moment κ W 1) + (hJ : ∀ n, 1≤n → Target.moment κ W 1^n ≤ Target.moment κ W n) + (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ) + (he : ∀ n i, (e n i).2 < Target.multiplicity κ W n (e n i).1) + (C c R : ℝ) (K : Set ℂ) + (hfailure : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, + Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im → + M.prescribedCenteredFailure W n (e n) C c + (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) ≤ + ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16)))) : + ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, + Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im → + P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (γ ω)∩K) + (M.rational.physicalTerminal ω)} ≤ + ENNReal.ofReal ((uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ* + (2:ℝ)^(-codimension κ))*(Target.entryRadius κ W (n,none))^codimension κ* + (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)* + Real.exp (-((2:ℝ)^n/16))) := by + filter_upwards [hfailure,Target.eventually_prescribed_uniform_geometry hκ W hm1 hJ, + Target.eventually_entryPhysicalLevel_large hκ W (by positivity : 0<2048*Real.pi) 1] + with n hn hg hl + intro ij hij hy + let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij + obtain ⟨hlevels,hsmall,hends⟩ := hg z hy + let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z + have hV : 1 diskMassValue κ (θ x) (C.atTime (T x) (g x))) := by - apply diskMassValue_measurable κ θ hθ - · exact ((_root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable C.progressive g hg _ hT.ennreal_toNNReal).ite - (measurableSet_lt hT measurable_const) (C.rational.terminal_measurable.comp hg)) - · intro x - exact (C.atTime_radon (T x) (g x)).2 - -/-- The actual compact disk mass at the indicated logarithmic radius level. -Infinite radius times use the already constructed terminal measure. -/ -def diskAtRadius (θ u : ℝ) (ω : Ω) : ℝ := - (diskMassValue κ θ (C.atTime - (radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u) ω)).toReal - -lemma diskAtRadius_nonneg (θ u : ℝ) (ω : Ω) : 0 ≤ C.diskAtRadius θ u ω := - ENNReal.toReal_nonneg - -lemma diskAtRadius_joint_measurable - (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) : - Measurable (fun p : ℝ × Ω => C.diskAtRadius p.1 u p.2) := by - have hz : Measurable (fun p : ℝ × Ω => Target.diskMapInv p.1 0) := by - unfold Target.diskMapInv - fun_prop - have ht : Measurable (fun p : ℝ × Ω => - radiusLevelTime (γ p.2) (G p.2) (Target.diskMapInv p.1 0) u) := by - have hj : Measurable (fun p : Ω × ℂ => radiusLevelTime (γ p.1) (G p.1) p.2 u) := - actual_radiusLevel_joint_measurable - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc κ u hu γ G F hw - exact @Measurable.comp (ℝ × Ω) (Ω × ℂ) ℝ≥0∞ - inferInstance inferInstance inferInstance - (fun p => radiusLevelTime (γ p.1) (G p.1) p.2 u) - (fun p => (p.2,Target.diskMapInv p.1 0)) hj (measurable_snd.prodMk hz) - exact (C.disk_at_parameter_measurable _ measurable_fst _ measurable_snd _ ht).ennreal_toReal - -lemma diskAtRadius_eq_finite (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (u : ℝ) (σ : Ω → WithTop ℝ≥0) (ω : Ω) - (he : σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u) - (hf : σ ω≠⊤) : - C.diskAtRadius θ u ω = finiteStoppedScalar (C.diskScalar θ hθ) σ ω := by - rw [diskAtRadius,← he,diskMassValue_toReal_eq κ θ hθ] - have hfin : (σ ω : ℝ≥0∞) < ∞ := lt_top_iff_ne_top.mpr hf - simp only [atTime,extendedCanonicalMass,finiteStoppedScalar,hf,ite_false] - split_ifs with hs - · unfold diskScalar - lift σ ω to ℝ≥0 using hf with t - rfl - · exact False.elim (hs hfin) - -/-- The moment identity uses the same jointly measurable disk variable -which enters tail events, with exactly the prescribed inner-clock scaling. -/ -lemma levelCoefficient_eq_diskAtRadius_moment [IsProbabilityMeasure P] - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : - ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) = - ∫⁻ ω, ENNReal.ofReal (C.diskAtRadius θ (4*ℓ/κ) ω)^m - ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) := by - let u := 4*ℓ/κ - have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le - let σ := fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u - have hz : Target.diskMapInv θ 0 ∈ upperHalfPlane := Target.diskMapInv_maps hθ (by simp) - have hσ := actual_radiusLevelTime_stopping hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0 - κ hκ hκ8 γ G F hw (Target.diskMapInv θ 0) hz u hu - have hf := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F - (Eventually.of_forall hw) (Target.diskMapInv θ 0) hz u u hu le_rfl σ - (Eventually.of_forall (fun _ => le_rfl)) - rw [C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW - θ hθ ℓ hℓ m σ hσ (Eventually.of_forall (fun _ => rfl))] - apply lintegral_congr_ae - filter_upwards [hf] with ω hω - rw [C.diskAtRadius_eq_finite θ hθ u σ ω rfl hω] - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Mass.JointDiskMass +import OAI.Probability.SLE.Mass.CanonicalMassRemaining +import OAI.Probability.SLE.Geometry.CenteredDiskCoefficient + +namespace OAI + +/-! Joint angle and sample measurability of actual canonical disk masses +at radius stops. No angle-dependent exceptional set is selected. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma disk_at_parameter_measurable + {Ξ : Type*} [MeasurableSpace Ξ] (θ : Ξ → ℝ) (hθ : Measurable θ) + (g : Ξ → Ω) (hg : Measurable g) (T : Ξ → ℝ≥0∞) (hT : Measurable T) : + Measurable (fun x => diskMassValue κ (θ x) (C.atTime (T x) (g x))) := by + apply diskMassValue_measurable κ θ hθ + · exact ((_root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable C.progressive g hg _ hT.ennreal_toNNReal).ite + (measurableSet_lt hT measurable_const) (C.rational.terminal_measurable.comp hg)) + · intro x + exact (C.atTime_radon (T x) (g x)).2 + +/-- The actual compact disk mass at the indicated logarithmic radius level. +Infinite radius times use the already constructed terminal measure. -/ +def diskAtRadius (θ u : ℝ) (ω : Ω) : ℝ := + (diskMassValue κ θ (C.atTime + (radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u) ω)).toReal + +lemma diskAtRadius_nonneg (θ u : ℝ) (ω : Ω) : 0 ≤ C.diskAtRadius θ u ω := + ENNReal.toReal_nonneg + +lemma diskAtRadius_joint_measurable + (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) : + Measurable (fun p : ℝ × Ω => C.diskAtRadius p.1 u p.2) := by + have hz : Measurable (fun p : ℝ × Ω => Target.diskMapInv p.1 0) := by + unfold Target.diskMapInv + fun_prop + have ht : Measurable (fun p : ℝ × Ω => + radiusLevelTime (γ p.2) (G p.2) (Target.diskMapInv p.1 0) u) := by + have hj : Measurable (fun p : Ω × ℂ => radiusLevelTime (γ p.1) (G p.1) p.2 u) := + actual_radiusLevel_joint_measurable + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc κ u hu γ G F hw + exact @Measurable.comp (ℝ × Ω) (Ω × ℂ) ℝ≥0∞ + inferInstance inferInstance inferInstance + (fun p => radiusLevelTime (γ p.1) (G p.1) p.2 u) + (fun p => (p.2,Target.diskMapInv p.1 0)) hj (measurable_snd.prodMk hz) + exact (C.disk_at_parameter_measurable _ measurable_fst _ measurable_snd _ ht).ennreal_toReal + +lemma diskAtRadius_eq_finite (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (u : ℝ) (σ : Ω → WithTop ℝ≥0) (ω : Ω) + (he : σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u) + (hf : σ ω≠⊤) : + C.diskAtRadius θ u ω = finiteStoppedScalar (C.diskScalar θ hθ) σ ω := by + rw [diskAtRadius,← he,diskMassValue_toReal_eq κ θ hθ] + have hfin : (σ ω : ℝ≥0∞) < ∞ := lt_top_iff_ne_top.mpr hf + simp only [atTime,extendedCanonicalMass,finiteStoppedScalar,hf,ite_false] + split_ifs with hs + · unfold diskScalar + lift σ ω to ℝ≥0 using hf with t + rfl + · exact False.elim (hs hfin) + +/-- The moment identity uses the same jointly measurable disk variable +which enters tail events, with exactly the prescribed inner-clock scaling. -/ +lemma levelCoefficient_eq_diskAtRadius_moment [IsProbabilityMeasure P] + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : + ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) = + ∫⁻ ω, ENNReal.ofReal (C.diskAtRadius θ (4*ℓ/κ) ω)^m + ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) := by + let u := 4*ℓ/κ + have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le + let σ := fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u + have hz : Target.diskMapInv θ 0 ∈ upperHalfPlane := Target.diskMapInv_maps hθ (by simp) + have hσ := actual_radiusLevelTime_stopping hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0 + κ hκ hκ8 γ G F hw (Target.diskMapInv θ 0) hz u hu + have hf := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F + (Eventually.of_forall hw) (Target.diskMapInv θ 0) hz u u hu le_rfl σ + (Eventually.of_forall (fun _ => le_rfl)) + rw [C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW + θ hθ ℓ hℓ m σ hσ (Eventually.of_forall (fun _ => rfl))] + apply lintegral_congr_ae + filter_upwards [hf] with ω hω + rw [C.diskAtRadius_eq_finite θ hθ u σ ω rfl hω] + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalLowerSegments.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalLowerSegments.lean index 8b09038d7c963b74502af1ed7361c721c4ecc390..761e31b97d918f2513d41a9fb53d39a50fa39e42 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalLowerSegments.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalLowerSegments.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Gauge.CanonicalGridLower -import OAI.Probability.SLE.Mass.PositiveTraceMass - -namespace OAI - -/-! Compact lower bounds give positivity simultaneously on every trace segment. -Only a countable compact exhaustion is used before choosing the time interval. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma exists_compact_covering_positive_inter (μ : Measure upperHalfPlane) - (A : Set upperHalfPlane) (hA : 0 < μ A) : - ∃ n : ℕ, 0 < μ (compactCovering upperHalfPlane n ∩ A) := by - by_contra! hn - have he : (⋃ n : ℕ, compactCovering upperHalfPlane n ∩ A) = A := by - rw [← iUnion_inter,iUnion_compactCovering,univ_inter] - have hz : μ A = 0 := by - apply le_antisymm ?_ bot_le - rw [← he] - calc - _ ≤ ∑' n : ℕ, μ (compactCovering upperHalfPlane n ∩ A) := measure_iUnion_le _ - _ = 0 := by simp only [fun n => le_zero_iff.mp (hn n),tsum_zero] - exact hA.ne' hz - -namespace CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - -include C hB hc h0 hκ F hw hpublished - -/-- A countable compact lower criterion and the proved positivity of every -canonical increment imply the required simultaneous segment lower bound. -/ -theorem positive_all_segments_of_compact_lower (h : ℝ → ℝ) - (hlower : ∀ n : ℕ, ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → - 0 < C.rational.terminal ω (compactCovering upperHalfPlane n ∩ Subtype.val ⁻¹' E) → - 0 < Target.hausdorff h E) : - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s < t → - 0 < Target.hausdorff h (γ ω '' Icc s t) := by - have hp := C.positive_all_times hB hc h0 F hw - have hi := C.positive_increment_all_times hB hc hκ F hw hpublished hp - filter_upwards [ae_all_iff.mpr hlower,C.paths,hi] with ω hω hpath hinc - intro s t hst - let E := γ ω '' Icc s t - have hE : MeasurableSet E := (isCompact_Icc.image (hw ω).1).measurableSet - let A : Set upperHalfPlane := Subtype.val ⁻¹' E - have hA : MeasurableSet A := hE.preimage measurable_subtype_coe - have hsupp : (C.process t ω-C.process s ω) Aᶜ = 0 := by - have hh := hpath.increment_support s t hst.le - have he : (fun r : ℝ => γ ω r.toNNReal) '' Icc (s:ℝ) t = E := by - ext z - constructor - · rintro ⟨r,hr,rfl⟩ - exact ⟨r.toNNReal,⟨by simpa using Real.toNNReal_mono hr.1, - by simpa using Real.toNNReal_mono hr.2⟩,rfl⟩ - · rintro ⟨r,hr,rfl⟩ - exact ⟨r,⟨by exact_mod_cast hr.1,by exact_mod_cast hr.2⟩,by simp⟩ - change (C.process t ω-C.process s ω) {z : upperHalfPlane | (z:ℂ) ∉ E} = 0 - rw [he] at hh - exact hh - have hpos : 0 < (C.process t ω-C.process s ω) A := by - have he := measure_add_measure_compl (μ := C.process t ω-C.process s ω) hA - rw [hsupp,add_zero] at he - rw [he] - exact Measure.measure_univ_pos.mpr (hinc s t hst) - have hterminal : 0 < C.rational.terminal ω A := - hpos.trans_le ((Measure.sub_le.trans (hpath.dominated t)) A) - obtain ⟨n,hn⟩ := exists_compact_covering_positive_inter (C.rational.terminal ω) A hterminal - exact hω n E hE hn - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.CanonicalGridLower +import OAI.Probability.SLE.Mass.PositiveTraceMass + +namespace OAI + +/-! Compact lower bounds give positivity simultaneously on every trace segment. +Only a countable compact exhaustion is used before choosing the time interval. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma exists_compact_covering_positive_inter (μ : Measure upperHalfPlane) + (A : Set upperHalfPlane) (hA : 0 < μ A) : + ∃ n : ℕ, 0 < μ (compactCovering upperHalfPlane n ∩ A) := by + by_contra! hn + have he : (⋃ n : ℕ, compactCovering upperHalfPlane n ∩ A) = A := by + rw [← iUnion_inter,iUnion_compactCovering,univ_inter] + have hz : μ A = 0 := by + apply le_antisymm ?_ bot_le + rw [← he] + calc + _ ≤ ∑' n : ℕ, μ (compactCovering upperHalfPlane n ∩ A) := measure_iUnion_le _ + _ = 0 := by simp only [fun n => le_zero_iff.mp (hn n),tsum_zero] + exact hA.ne' hz + +namespace CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + +include C hB hc h0 hκ F hw hpublished + +/-- A countable compact lower criterion and the proved positivity of every +canonical increment imply the required simultaneous segment lower bound. -/ +theorem positive_all_segments_of_compact_lower (h : ℝ → ℝ) + (hlower : ∀ n : ℕ, ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → + 0 < C.rational.terminal ω (compactCovering upperHalfPlane n ∩ Subtype.val ⁻¹' E) → + 0 < Target.hausdorff h E) : + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s < t → + 0 < Target.hausdorff h (γ ω '' Icc s t) := by + have hp := C.positive_all_times hB hc h0 F hw + have hi := C.positive_increment_all_times hB hc hκ F hw hpublished hp + filter_upwards [ae_all_iff.mpr hlower,C.paths,hi] with ω hω hpath hinc + intro s t hst + let E := γ ω '' Icc s t + have hE : MeasurableSet E := (isCompact_Icc.image (hw ω).1).measurableSet + let A : Set upperHalfPlane := Subtype.val ⁻¹' E + have hA : MeasurableSet A := hE.preimage measurable_subtype_coe + have hsupp : (C.process t ω-C.process s ω) Aᶜ = 0 := by + have hh := hpath.increment_support s t hst.le + have he : (fun r : ℝ => γ ω r.toNNReal) '' Icc (s:ℝ) t = E := by + ext z + constructor + · rintro ⟨r,hr,rfl⟩ + exact ⟨r.toNNReal,⟨by simpa using Real.toNNReal_mono hr.1, + by simpa using Real.toNNReal_mono hr.2⟩,rfl⟩ + · rintro ⟨r,hr,rfl⟩ + exact ⟨r,⟨by exact_mod_cast hr.1,by exact_mod_cast hr.2⟩,by simp⟩ + change (C.process t ω-C.process s ω) {z : upperHalfPlane | (z:ℂ) ∉ E} = 0 + rw [he] at hh + exact hh + have hpos : 0 < (C.process t ω-C.process s ω) A := by + have he := measure_add_measure_compl (μ := C.process t ω-C.process s ω) hA + rw [hsupp,add_zero] at he + rw [he] + exact Measure.measure_univ_pos.mpr (hinc s t hst) + have hterminal : 0 < C.rational.terminal ω A := + hpos.trans_le ((Measure.sub_le.trans (hpath.dominated t)) A) + obtain ⟨n,hn⟩ := exists_compact_covering_positive_inter (C.rational.terminal ω) A hterminal + exact hω n E hE hn + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalNormalizedPotential.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalNormalizedPotential.lean index 0100468db5dd32905653f355f6b3eb1d8abffb41..60757e6309b7ffdb96766a8cc79e86ed381e082c 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalNormalizedPotential.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalNormalizedPotential.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Geometry.CanonicalDiskPotential -import OAI.Probability.SLE.Angle.TargetNormalizedFiltration -import OAI.Probability.SLE.Angle.TargetLevelTime - -namespace OAI - -/-! Canonical remaining mass in the exact clock and natural filtration -of the source-prescribed Brownian driver. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -theorem normalized_potential_condExp [IsProbabilityMeasure P] - (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : - (fun ω => (Target.potential κ θ (Target.normalizedDriver κ B hc h0 ω) t).toReal) =ᵐ[P] - P[C.diskTerminalScalar θ hθ-C.diskScalar θ hθ (t/κ.toNNReal) | - Target.normalizedFiltration κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 t] := by - have ht : κ*((t/κ.toNNReal:ℝ≥0):ℝ)=(t:ℝ) := by - rw [NNReal.coe_div,Real.coe_toNNReal _ hκ.le,mul_div_cancel₀ _ hκ.ne'] - erw [Target.normalizedFiltration_at κ hκ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 t] - simpa only [ht] using C.target_potential_condExp hκ hc F hw - (Target.normalizedDriver κ B hc h0) (Target.normalizedDriver_matches κ hκ B hc h0) - θ hθ (t/κ.toNNReal) - -include hBm in -theorem normalized_potential_lintegral_ne_top - (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : - (∫⁻ ω, Target.potential κ θ (Target.normalizedDriver κ B hc h0 ω) t ∂P) ≠ ∞ := by - have ht : κ*((t/κ.toNNReal:ℝ≥0):ℝ)=(t:ℝ) := by - rw [NNReal.coe_div,Real.coe_toNNReal _ hκ.le,mul_div_cancel₀ _ hκ.ne'] - simpa only [ht] using target_potential_lintegral_ne_top (hBm := hBm) - hB hκ hκ8 hc h0 F hw (Target.normalizedDriver κ B hc h0) - (Target.normalizedDriver_matches κ hκ B hc h0) θ hθ (t/κ.toNNReal) - -include hBm in -theorem normalized_levelTime_measurable - (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) : - Measurable (fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ) := by - apply Target.measurable_levelTime_of_radius_kernel κ hκ _ _ - (fun ω => (hc ω).const_mul _) γ G F hw - (Target.normalizedDriver_matches κ hκ B hc h0) θ ℓ hℓ - have hu : 0 ≤ 4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le - have hm := measurable_canonicalRadiusLevel_random_state κ - (source_path_measurable (sourceDriverPath B hc) - (fun t => hBm.comp (measurable_const.prodMk measurable_id))) - (u := fun _ => 4*ℓ/κ) measurable_const - (z := fun _ => Target.diskMapInv θ 0) measurable_const - convert hm using 1 - funext ω - exact (canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) _ hu _).symm - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalDiskPotential +import OAI.Probability.SLE.Angle.TargetNormalizedFiltration +import OAI.Probability.SLE.Angle.TargetLevelTime + +namespace OAI + +/-! Canonical remaining mass in the exact clock and natural filtration +of the source-prescribed Brownian driver. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +theorem normalized_potential_condExp [IsProbabilityMeasure P] + (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : + (fun ω => (Target.potential κ θ (Target.normalizedDriver κ B hc h0 ω) t).toReal) =ᵐ[P] + P[C.diskTerminalScalar θ hθ-C.diskScalar θ hθ (t/κ.toNNReal) | + Target.normalizedFiltration κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 t] := by + have ht : κ*((t/κ.toNNReal:ℝ≥0):ℝ)=(t:ℝ) := by + rw [NNReal.coe_div,Real.coe_toNNReal _ hκ.le,mul_div_cancel₀ _ hκ.ne'] + erw [Target.normalizedFiltration_at κ hκ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 t] + simpa only [ht] using C.target_potential_condExp hκ hc F hw + (Target.normalizedDriver κ B hc h0) (Target.normalizedDriver_matches κ hκ B hc h0) + θ hθ (t/κ.toNNReal) + +include hBm in +theorem normalized_potential_lintegral_ne_top + (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (t : ℝ≥0) : + (∫⁻ ω, Target.potential κ θ (Target.normalizedDriver κ B hc h0 ω) t ∂P) ≠ ∞ := by + have ht : κ*((t/κ.toNNReal:ℝ≥0):ℝ)=(t:ℝ) := by + rw [NNReal.coe_div,Real.coe_toNNReal _ hκ.le,mul_div_cancel₀ _ hκ.ne'] + simpa only [ht] using target_potential_lintegral_ne_top (hBm := hBm) + hB hκ hκ8 hc h0 F hw (Target.normalizedDriver κ B hc h0) + (Target.normalizedDriver_matches κ hκ B hc h0) θ hθ (t/κ.toNNReal) + +include hBm in +theorem normalized_levelTime_measurable + (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) : + Measurable (fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ) := by + apply Target.measurable_levelTime_of_radius_kernel κ hκ _ _ + (fun ω => (hc ω).const_mul _) γ G F hw + (Target.normalizedDriver_matches κ hκ B hc h0) θ ℓ hℓ + have hu : 0 ≤ 4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le + have hm := measurable_canonicalRadiusLevel_random_state κ + (source_path_measurable (sourceDriverPath B hc) + (fun t => hBm.comp (measurable_const.prodMk measurable_id))) + (u := fun _ => 4*ℓ/κ) measurable_const + (z := fun _ => Target.diskMapInv θ 0) measurable_const + convert hm using 1 + funext ω + exact (canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) _ hu _).symm + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalRawMean.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalRawMean.lean index a6cf48bdb6d6ba34e75627eff59ea9c4a0fa7f28..34b5c648c73003577cf7c6588d10632f4c57a185 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalRawMean.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalRawMean.lean @@ -1,99 +1,99 @@ -import OAI.Probability.SLE.Geometry.CanonicalRawTerminal -import OAI.Probability.SLE.Mass.RestartRandomMassTesting - -namespace OAI - -/-! The fixed raw terminal functional has the exact Green mean under the -driver law. Strong Markov then allows arbitrary stopped-past spatial weights -in the fresh terminal mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.raw_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω) : - (P.map (sourceDriverPath B hBc)).bind R.terminal = initialGreenMeasure κ := by - have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) - calc - _ = P.bind (fun ω => R.terminal (sourceDriverPath B hBc ω)) := by - unfold Measure.bind - rw [Measure.map_map R.terminal_measurable hp] - rfl - _ = P.bind C.rational.terminal := Measure.bind_congr_right he - _ = initialGreenMeasure κ := C.rational.mean - -theorem CanonicalMass.restarted_raw_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) : - P.bind (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω))=initialGreenMeasure κ := by - have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ - (fun t => (measurable_pi_apply t).comp - (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) - have hlaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by - simpa only [Measure.restrict_univ,measure_univ,one_smul] using - source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ - calc - _ = (P.map (sourceRestartDriverPath B hc τ)).bind R.terminal := by - unfold Measure.bind - rw [Measure.map_map R.terminal_measurable hY] - rfl - _ = initialGreenMeasure κ := by rw [hlaw,C.raw_terminal_mean hc R he] - - -theorem CanonicalMass.fresh_terminal_random_test - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (f : Ω × upperHalfPlane → ℝ≥0∞) - (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) : - (∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.terminal (sourceRestartDriverPath B hBc τ ω)) ∂P) = - ∫⁻ ω, ∫⁻ z, f (ω,z) ∂initialGreenMeasure κ ∂P := by - have hm (t : ℝ≥0) : Measurable (B t) := - hBm.comp (measurable_const.prodMk measurable_id) - have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm - let : IsProbabilityMeasure (P.map (sourceDriverPath B hBc)) := - (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance - have hY : Measurable (sourceRestartDriverPath B hBc τ) := - source_path_measurable _ (fun t => (measurable_pi_apply t).comp - (brownianRestart_measurable hm hBc (stopping_time_measurable_nnreal hτ))) - have h := restart_random_mass_test P (P.map (sourceDriverPath B hBc)) - hτ.measurableSpace hτ.measurableSpace_le (sourceRestartDriverPath B hBc τ) hY - (fun A hA => source_restart_continuous_driver_law hB hm hBc τ hτ hA) - R.terminal R.terminal_measurable (fun p => (R.terminal_radon p).2) f hf - rwa [C.raw_terminal_mean hBc R he] at h - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalRawTerminal +import OAI.Probability.SLE.Mass.RestartRandomMassTesting + +namespace OAI + +/-! The fixed raw terminal functional has the exact Green mean under the +driver law. Strong Markov then allows arbitrary stopped-past spatial weights +in the fresh terminal mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.raw_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω) : + (P.map (sourceDriverPath B hBc)).bind R.terminal = initialGreenMeasure κ := by + have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) + calc + _ = P.bind (fun ω => R.terminal (sourceDriverPath B hBc ω)) := by + unfold Measure.bind + rw [Measure.map_map R.terminal_measurable hp] + rfl + _ = P.bind C.rational.terminal := Measure.bind_congr_right he + _ = initialGreenMeasure κ := C.rational.mean + +theorem CanonicalMass.restarted_raw_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) : + P.bind (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω))=initialGreenMeasure κ := by + have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ + (fun t => (measurable_pi_apply t).comp + (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) + have hlaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by + simpa only [Measure.restrict_univ,measure_univ,one_smul] using + source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ + calc + _ = (P.map (sourceRestartDriverPath B hc τ)).bind R.terminal := by + unfold Measure.bind + rw [Measure.map_map R.terminal_measurable hY] + rfl + _ = initialGreenMeasure κ := by rw [hlaw,C.raw_terminal_mean hc R he] + + +theorem CanonicalMass.fresh_terminal_random_test + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (f : Ω × upperHalfPlane → ℝ≥0∞) + (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) : + (∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.terminal (sourceRestartDriverPath B hBc τ ω)) ∂P) = + ∫⁻ ω, ∫⁻ z, f (ω,z) ∂initialGreenMeasure κ ∂P := by + have hm (t : ℝ≥0) : Measurable (B t) := + hBm.comp (measurable_const.prodMk measurable_id) + have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm + let : IsProbabilityMeasure (P.map (sourceDriverPath B hBc)) := + (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance + have hY : Measurable (sourceRestartDriverPath B hBc τ) := + source_path_measurable _ (fun t => (measurable_pi_apply t).comp + (brownianRestart_measurable hm hBc (stopping_time_measurable_nnreal hτ))) + have h := restart_random_mass_test P (P.map (sourceDriverPath B hBc)) + hτ.measurableSpace hτ.measurableSpace_le (sourceRestartDriverPath B hBc τ) hY + (fun A hA => source_restart_continuous_driver_law hB hm hBc τ hτ hA) + R.terminal R.terminal_measurable (fun p => (R.terminal_radon p).2) f hf + rwa [C.raw_terminal_mean hBc R he] at h + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalRawTerminal.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalRawTerminal.lean index 1c968e0aaa40f656142bc1f4bc64135442b02015..5a09692ac007867ede3940de37ec4ed916d3c7d2 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalRawTerminal.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalRawTerminal.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Mass.CanonicalRawMass -import OAI.Probability.SLE.Mass.TerminalMassTestLimit - -namespace OAI - -/-! Terminal extension of the fixed raw canonical mass functional. Its -integer-time vague limit agrees with the locally exhausted terminal mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -structure RawCanonicalMass where - process : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane - terminal : C(ℝ≥0,ℝ) → Measure upperHalfPlane - progressive : IsProgressive driverPathFiltration process - radon : ∀ t p, (process t p).Regular ∧ IsFiniteMeasureOnCompacts (process t p) - terminal_measurable : Measurable terminal - terminal_radon : ∀ p, (terminal p).Regular ∧ IsFiniteMeasureOnCompacts (terminal p) - nonanticipating : ∀ t p q, (∀ s ≤ t, p s=q s) → process t p=process t q - -def rawTerminalMass (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (p : C(ℝ≥0,ℝ)) : Measure upperHalfPlane := - canonicalRadonLimit (fun n : ℕ => R n p) - -lemma rawTerminalMass_measurable - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (hRp : IsProgressive driverPathFiltration R) - (hRr : ∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) : - Measurable (rawTerminalMass R) := by - apply canonicalRadonLimit_measurable (fun n : ℕ => R n) - (fun n p => (hRr n p).2) - intro n f - exact measurable_halfPlane_compact_test_integral (R n) - (hRp.adapted.measurable (i := (n:ℝ≥0))) (fun p => (hRr n p).1) f - -lemma rawTerminalMass_radon - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) (p : C(ℝ≥0,ℝ)) : - (rawTerminalMass R p).Regular ∧ IsFiniteMeasureOnCompacts (rawTerminalMass R p) := by - have : (rawTerminalMass R p).Regular := canonicalRadonLimit_regular _ - exact ⟨this,inferInstance⟩ - -theorem CanonicalMass.rawTerminal_realizes - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hBc : ∀ ω, Continuous (fun t => B t ω)) - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (he : ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω) = C.process t ω) : - ∀ᵐ ω ∂P, rawTerminalMass R (sourceDriverPath B hBc ω) = C.rational.terminal ω := by - filter_upwards [he,C.paths,C.exhaustion] with ω he hp hlim - apply canonicalRadonLimit_identified _ _ - refine ⟨(C.rational.terminal_radon ω).1,?_⟩ - intro f - let : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 - have ht := terminal_mass_test_tendsto (C.rational.terminal ω) (fun t => C.process t ω) - (fun t => (C.radon t ω).1) hp.dominated hlim f - simpa only [Function.comp_def,he] using ht.comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) - -theorem CanonicalMass.exists_raw_canonical - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hBc : ∀ ω, Continuous (fun t => B t ω)) : - ∃ R : RawCanonicalMass, ∀ᵐ ω ∂P, - (∀ t, R.process t (sourceDriverPath B hBc ω) = C.process t ω) ∧ - R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω := by - obtain ⟨R,hRp,hRr,hRn,he⟩ := C.exists_raw_nonanticipating_functional hBc - refine ⟨⟨R,rawTerminalMass R,hRp,hRr,rawTerminalMass_measurable R hRp hRr, - rawTerminalMass_radon R,hRn⟩,?_⟩ - filter_upwards [he,C.rawTerminal_realizes hBc R he] with ω hf ht - exact ⟨hf,ht⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalRawMass +import OAI.Probability.SLE.Mass.TerminalMassTestLimit + +namespace OAI + +/-! Terminal extension of the fixed raw canonical mass functional. Its +integer-time vague limit agrees with the locally exhausted terminal mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +structure RawCanonicalMass where + process : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane + terminal : C(ℝ≥0,ℝ) → Measure upperHalfPlane + progressive : IsProgressive driverPathFiltration process + radon : ∀ t p, (process t p).Regular ∧ IsFiniteMeasureOnCompacts (process t p) + terminal_measurable : Measurable terminal + terminal_radon : ∀ p, (terminal p).Regular ∧ IsFiniteMeasureOnCompacts (terminal p) + nonanticipating : ∀ t p q, (∀ s ≤ t, p s=q s) → process t p=process t q + +def rawTerminalMass (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (p : C(ℝ≥0,ℝ)) : Measure upperHalfPlane := + canonicalRadonLimit (fun n : ℕ => R n p) + +lemma rawTerminalMass_measurable + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (hRp : IsProgressive driverPathFiltration R) + (hRr : ∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) : + Measurable (rawTerminalMass R) := by + apply canonicalRadonLimit_measurable (fun n : ℕ => R n) + (fun n p => (hRr n p).2) + intro n f + exact measurable_halfPlane_compact_test_integral (R n) + (hRp.adapted.measurable (i := (n:ℝ≥0))) (fun p => (hRr n p).1) f + +lemma rawTerminalMass_radon + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) (p : C(ℝ≥0,ℝ)) : + (rawTerminalMass R p).Regular ∧ IsFiniteMeasureOnCompacts (rawTerminalMass R p) := by + have : (rawTerminalMass R p).Regular := canonicalRadonLimit_regular _ + exact ⟨this,inferInstance⟩ + +theorem CanonicalMass.rawTerminal_realizes + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hBc : ∀ ω, Continuous (fun t => B t ω)) + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (he : ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω) = C.process t ω) : + ∀ᵐ ω ∂P, rawTerminalMass R (sourceDriverPath B hBc ω) = C.rational.terminal ω := by + filter_upwards [he,C.paths,C.exhaustion] with ω he hp hlim + apply canonicalRadonLimit_identified _ _ + refine ⟨(C.rational.terminal_radon ω).1,?_⟩ + intro f + let : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 + have ht := terminal_mass_test_tendsto (C.rational.terminal ω) (fun t => C.process t ω) + (fun t => (C.radon t ω).1) hp.dominated hlim f + simpa only [Function.comp_def,he] using ht.comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) + +theorem CanonicalMass.exists_raw_canonical + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hBc : ∀ ω, Continuous (fun t => B t ω)) : + ∃ R : RawCanonicalMass, ∀ᵐ ω ∂P, + (∀ t, R.process t (sourceDriverPath B hBc ω) = C.process t ω) ∧ + R.terminal (sourceDriverPath B hBc ω) = C.rational.terminal ω := by + obtain ⟨R,hRp,hRr,hRn,he⟩ := C.exists_raw_nonanticipating_functional hBc + refine ⟨⟨R,rawTerminalMass R,hRp,hRr,rawTerminalMass_measurable R hRp hRr, + rawTerminalMass_radon R,hRn⟩,?_⟩ + filter_upwards [he,C.rawTerminal_realizes hBc R he] with ω hf ht + exact ⟨hf,ht⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalRestartDischarge.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalRestartDischarge.lean index 400721b87ff71c07a72c7b6ab5104c08381e9087..84bd6fd4644e7337d130cb1c9c84c0e3e4686ac6 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalRestartDischarge.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalRestartDischarge.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Gauge.GaugeGrowthDischarge -import OAI.Probability.SLE.Mass.CanonicalMassImageRestart - -namespace OAI - -/-! Finite and terminal stopping-time restart identities using a capacity -trace witness for the canonical driver law. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance restartDischargeDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance restartDischargeDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -theorem CanonicalMass.exists_raw_all_stopping_restart_of_brownian_witness - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hκ8 : κ < 8) (hB0 : ∀ ω, B 0 ω = 0) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ - ∀ (S : Ω → WithTop ℝ≥0) - (_hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), - ∀ᵐ ω ∂P, S ω < ⊤ → ∀ u : ℝ≥0∞, - extendedCanonicalMass C.process C.rational.terminal - ((((S ω).untopD (0 : ℝ≥0) : ℝ≥0) : ℝ≥0∞)+u) ω-C.process ((S ω).untopD (0 : ℝ≥0)) ω = - weightedImageMass - (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) - (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) - (extendedCanonicalMass R.process R.terminal u - (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0 : ℝ≥0)) ω)) := by - exact C.exists_raw_all_stopping_restart hB hc hκ F hw - (canonical_driver_trace_generation_of_witness κ hκ hκ8 B hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hB0 γ G F hw) - -theorem CanonicalMass.exists_raw_all_image_restart_of_brownian_witness - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hκ8 : κ < 8) (hB0 : ∀ ω, B 0 ω = 0) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ - ∀ (S : Ω → WithTop ℝ≥0) - (_hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), - ∀ᵐ ω ∂P, S ω < ⊤ → ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane,ℝ≥0)) (u : ℝ≥0∞), - conformalMass J w (C.atTime ((((S ω).untopD (0:ℝ≥0):ℝ≥0):ℝ≥0∞)+u) ω)- - conformalMass J w (C.process ((S ω).untopD (0:ℝ≥0)) ω) = - conformalMass J w (weightedImageMass - (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω) - (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω) - (extendedCanonicalMass R.process R.terminal u - (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω))) := by - exact C.exists_raw_all_image_restart hB hc hκ F hw - (canonical_driver_trace_generation_of_witness κ hκ hκ8 B hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hB0 γ G F hw) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.GaugeGrowthDischarge +import OAI.Probability.SLE.Mass.CanonicalMassImageRestart + +namespace OAI + +/-! Finite and terminal stopping-time restart identities using a capacity +trace witness for the canonical driver law. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance restartDischargeDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance restartDischargeDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +theorem CanonicalMass.exists_raw_all_stopping_restart_of_brownian_witness + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hκ8 : κ < 8) (hB0 : ∀ ω, B 0 ω = 0) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ + ∀ (S : Ω → WithTop ℝ≥0) + (_hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), + ∀ᵐ ω ∂P, S ω < ⊤ → ∀ u : ℝ≥0∞, + extendedCanonicalMass C.process C.rational.terminal + ((((S ω).untopD (0 : ℝ≥0) : ℝ≥0) : ℝ≥0∞)+u) ω-C.process ((S ω).untopD (0 : ℝ≥0)) ω = + weightedImageMass + (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) + (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) + (extendedCanonicalMass R.process R.terminal u + (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0 : ℝ≥0)) ω)) := by + exact C.exists_raw_all_stopping_restart hB hc hκ F hw + (canonical_driver_trace_generation_of_witness κ hκ hκ8 B hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hB0 γ G F hw) + +theorem CanonicalMass.exists_raw_all_image_restart_of_brownian_witness + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hκ8 : κ < 8) (hB0 : ∀ ω, B 0 ω = 0) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ + ∀ (S : Ω → WithTop ℝ≥0) + (_hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), + ∀ᵐ ω ∂P, S ω < ⊤ → ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane,ℝ≥0)) (u : ℝ≥0∞), + conformalMass J w (C.atTime ((((S ω).untopD (0:ℝ≥0):ℝ≥0):ℝ≥0∞)+u) ω)- + conformalMass J w (C.process ((S ω).untopD (0:ℝ≥0)) ω) = + conformalMass J w (weightedImageMass + (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω) + (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω) + (extendedCanonicalMass R.process R.terminal u + (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω))) := by + exact C.exists_raw_all_image_restart hB hc hκ F hw + (canonical_driver_trace_generation_of_witness κ hκ hκ8 B hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hB0 γ G F hw) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CanonicalRestartWitness.lean b/lean/OAI/Probability/SLE/Geometry/CanonicalRestartWitness.lean index f0f4c7d893493d576a8abfef5887caa6c4c780ba..0434cc6ed9e7fd5df9e6fe03e139ebfe098e1372 100644 --- a/lean/OAI/Probability/SLE/Geometry/CanonicalRestartWitness.lean +++ b/lean/OAI/Probability/SLE/Geometry/CanonicalRestartWitness.lean @@ -1,172 +1,172 @@ -import OAI.Probability.SLE.Geometry.PublishedInputs -import OAI.Probability.SLE.Loewner.TraceWitnessUniqueness -import OAI.Probability.SLE.Geometry.FiniteBoundaryRestart - -namespace OAI - -/-! Actual capacity witnesses for the fresh driver path. The trace-generation -input is Rohde--Schramm's existing input on the single canonical driver law. -Deterministic uniqueness identifies the resulting measurable trace factor -with the factor in the proved stopped trace law. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance canonicalDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance canonicalDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance canonicalTraceMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance canonicalTraceBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -/-- Evaluation on the canonical continuous-driver law is the same Brownian -motion law, so the trace-generation input below is applied to a Brownian driver. -/ -theorem canonical_driver_isBrownian - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) : - IsBrownianReal (fun (t : ℝ≥0) (p : C(ℝ≥0, ℝ)) => p t) - (P.map (sourceDriverPath B hc)) := by - have hbp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm - refine ⟨⟨fun I => ?_⟩, Eventually.of_forall (fun p => p.continuous)⟩ - have hI : Measurable (fun p : C(ℝ≥0, ℝ) => I.restrict (fun t => p t)) := - Measurable.of_eval (fun i => (continuous_eval_const (i : ℝ≥0)).measurable) - refine ⟨hI.aemeasurable, ?_⟩ - rw [Measure.map_map hI hbp] - exact (hB.hasLaw I).map_eq - -theorem exists_measurable_canonical_witness - (ν : Measure C(ℝ≥0, ℝ)) (κ : ℝ) - (hpublished : PublishedTraceGeneration ν κ (fun t p => p t)) : - ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) - (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ), Measurable Φ ∧ - ∀ᵐ p ∂ν, IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p) := by - classical - obtain ⟨γ, hm, htrace⟩ := hpublished - obtain ⟨S, hSa, hSm, hS⟩ := htrace.exists_measurable_mem - choose G F hw using fun p : S => hS p p.property - let g : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ := fun p t => if p ∈ S then γ p t else 0 - have hg (p : C(ℝ≥0, ℝ)) : Continuous (g p) := by - by_cases hp : p ∈ S - · simpa only [g, ite_eq_left hp] using (hw ⟨p, hp⟩).1 - · simpa only [g, ite_eq_right hp] using (continuous_const : Continuous (fun _ : ℝ≥0 => (0 : ℂ))) - let Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ) := fun p => ⟨g p, hg p⟩ - let Gc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ := fun p => - if hp : p ∈ S then G ⟨p, hp⟩ else fun _ _ => 0 - let Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ := fun p => - if hp : p ∈ S then F ⟨p, hp⟩ else fun _ _ => 0 - refine ⟨Φ, Gc, Fc, ?_, ?_⟩ - · apply source_path_measurable - intro t - change Measurable (fun p => if p ∈ S then γ p t else 0) - exact Measurable.ite hSm (hm t) measurable_const - · filter_upwards [hSa] with p hp - change IsCapacityTwoWitness _ (g p) (Gc p) (Fc p) - simpa only [g, Gc, Fc, ite_eq_left hp, dite_eq_left hp] using hw ⟨p, hp⟩ - -theorem canonical_witness_pullback - {ν : Measure C(ℝ≥0, ℝ)} {κ : ℝ} - {Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)} - {Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ} - (hw : ∀ᵐ p ∂ν, IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable Y) (hlaw : P.map Y = ν) : - ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * Y ω t) - (Φ (Y ω)) (Gc (Y ω)) (Fc (Y ω)) := by - apply ae_of_ae_map (p := fun p : C(ℝ≥0, ℝ) => - IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) hY.aemeasurable - rwa [hlaw] - -theorem canonical_witness_original_trace - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : - ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) - (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ), Measurable Φ ∧ - (∀ᵐ p ∂P.map (sourceDriverPath B hc), - IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) ∧ - ∀ᵐ ω ∂P, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω := by - obtain ⟨Φ, Gc, Fc, hΦm, hΦw⟩ := exists_measurable_canonical_witness _ κ hpublished - refine ⟨Φ, Gc, Fc, hΦm, hΦw, ?_⟩ - have hbp := source_path_measurable (sourceDriverPath B hc) hm - filter_upwards [ae_of_ae_map hbp.aemeasurable hΦw] with ω hω - have hω' : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (Φ (sourceDriverPath B hc ω)) (Gc (sourceDriverPath B hc ω)) (Fc (sourceDriverPath B hc ω)) := hω - have he := hω'.trace_unique (hw ω) - ext t - exact congrFun he t - -/-- The same fresh path has an actual capacity witness and realizes the -all-time finite-boundary restart identity. The only new instance of trace -generation is its explicit specialization to the original driver path law. -/ -theorem source_canonical_witness_boundary_restart - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) : - ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) - (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ) (Fb : Ω → C(closedHalfPlane, ℂ)), - Measurable Φ ∧ Measurable[hτ.measurableSpace] Fb ∧ - (∀ᵐ p ∂P.map (sourceDriverPath B hc), - IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) ∧ - (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → - Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ - (∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω)) ∧ - (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → - (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) = - P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧ - ∀ᵐ ω ∂P, - IsCapacityTwoWitness (fun t => Real.sqrt κ * sourceRestartDriverPath B hc τ ω t) - (Φ (sourceRestartDriverPath B hc τ ω)) - (Gc (sourceRestartDriverPath B hc τ ω)) (Fc (sourceRestartDriverPath B hc τ ω)) ∧ - ∀ t : ℝ≥0, γ ω (τ ω + t) = - finiteBoundaryPath (Fb ω) (Φ (sourceRestartDriverPath B hc τ ω)) t := by - obtain ⟨Φ, Gc, Fc, hΦm, hΦw, hΦorig⟩ := - canonical_witness_original_trace P κ B hm hc γ G F hw hpublished - obtain ⟨Ψ, Fb, hΨm, hΨorig, hFbm, hFb, hFb0, hΨlaw, hΨrestart⟩ := - source_measurable_finite_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ - let bp := sourceDriverPath B hc - let rp := sourceRestartDriverPath B hc τ - have hbp : Measurable bp := source_path_measurable bp hm - have hrp : Measurable rp := source_path_measurable rp (fun t => - (measurable_pi_apply t).comp (brownianRestart_measurable hm hc - (stopping_time_measurable_nnreal hτ))) - have heq : ∀ᵐ p ∂P.map bp, Φ p = Ψ p := by - apply (ae_map_iff hbp.aemeasurable (measurableSet_eq_fun hΦm hΨm)).mpr - filter_upwards [hΦorig] with ω hω - exact hω.trans (hΨorig ω).symm - have hlaw : P.map rp = P.map bp := by - simpa only [Measure.restrict_univ, measure_univ, one_smul] using - source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ - have hrEq : ∀ᵐ ω ∂P, Φ (rp ω) = Ψ (rp ω) := by - apply ae_of_ae_map (p := fun p : C(ℝ≥0, ℝ) => Φ p = Ψ p) hrp.aemeasurable - rwa [hlaw] - refine ⟨Φ, Gc, Fc, Fb, hΦm, hFbm, hΦw, hFb, hFb0, ?_, ?_⟩ - · intro A hA - calc - _ = (P.restrict A).map (fun ω => Ψ (rp ω)) := - Measure.map_congr (hrEq.filter_mono (ae_mono Measure.restrict_le_self)) - _ = _ := hΨlaw A hA - · have hrw := canonical_witness_pullback hΦw rp hrp hlaw - filter_upwards [hrw, hrEq, hΨrestart] with ω hω heω hrestart - refine ⟨hω, ?_⟩ - intro t - rw [heω] - exact hrestart t - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.PublishedInputs +import OAI.Probability.SLE.Loewner.TraceWitnessUniqueness +import OAI.Probability.SLE.Geometry.FiniteBoundaryRestart + +namespace OAI + +/-! Actual capacity witnesses for the fresh driver path. The trace-generation +input is Rohde--Schramm's existing input on the single canonical driver law. +Deterministic uniqueness identifies the resulting measurable trace factor +with the factor in the proved stopped trace law. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance canonicalDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance canonicalDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance canonicalTraceMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance canonicalTraceBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +/-- Evaluation on the canonical continuous-driver law is the same Brownian +motion law, so the trace-generation input below is applied to a Brownian driver. -/ +theorem canonical_driver_isBrownian + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) : + IsBrownianReal (fun (t : ℝ≥0) (p : C(ℝ≥0, ℝ)) => p t) + (P.map (sourceDriverPath B hc)) := by + have hbp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm + refine ⟨⟨fun I => ?_⟩, Eventually.of_forall (fun p => p.continuous)⟩ + have hI : Measurable (fun p : C(ℝ≥0, ℝ) => I.restrict (fun t => p t)) := + Measurable.of_eval (fun i => (continuous_eval_const (i : ℝ≥0)).measurable) + refine ⟨hI.aemeasurable, ?_⟩ + rw [Measure.map_map hI hbp] + exact (hB.hasLaw I).map_eq + +theorem exists_measurable_canonical_witness + (ν : Measure C(ℝ≥0, ℝ)) (κ : ℝ) + (hpublished : PublishedTraceGeneration ν κ (fun t p => p t)) : + ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) + (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ), Measurable Φ ∧ + ∀ᵐ p ∂ν, IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p) := by + classical + obtain ⟨γ, hm, htrace⟩ := hpublished + obtain ⟨S, hSa, hSm, hS⟩ := htrace.exists_measurable_mem + choose G F hw using fun p : S => hS p p.property + let g : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ := fun p t => if p ∈ S then γ p t else 0 + have hg (p : C(ℝ≥0, ℝ)) : Continuous (g p) := by + by_cases hp : p ∈ S + · simpa only [g, ite_eq_left hp] using (hw ⟨p, hp⟩).1 + · simpa only [g, ite_eq_right hp] using (continuous_const : Continuous (fun _ : ℝ≥0 => (0 : ℂ))) + let Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ) := fun p => ⟨g p, hg p⟩ + let Gc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ := fun p => + if hp : p ∈ S then G ⟨p, hp⟩ else fun _ _ => 0 + let Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ := fun p => + if hp : p ∈ S then F ⟨p, hp⟩ else fun _ _ => 0 + refine ⟨Φ, Gc, Fc, ?_, ?_⟩ + · apply source_path_measurable + intro t + change Measurable (fun p => if p ∈ S then γ p t else 0) + exact Measurable.ite hSm (hm t) measurable_const + · filter_upwards [hSa] with p hp + change IsCapacityTwoWitness _ (g p) (Gc p) (Fc p) + simpa only [g, Gc, Fc, ite_eq_left hp, dite_eq_left hp] using hw ⟨p, hp⟩ + +theorem canonical_witness_pullback + {ν : Measure C(ℝ≥0, ℝ)} {κ : ℝ} + {Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)} + {Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ} + (hw : ∀ᵐ p ∂ν, IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable Y) (hlaw : P.map Y = ν) : + ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * Y ω t) + (Φ (Y ω)) (Gc (Y ω)) (Fc (Y ω)) := by + apply ae_of_ae_map (p := fun p : C(ℝ≥0, ℝ) => + IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) hY.aemeasurable + rwa [hlaw] + +theorem canonical_witness_original_trace + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : + ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) + (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ), Measurable Φ ∧ + (∀ᵐ p ∂P.map (sourceDriverPath B hc), + IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) ∧ + ∀ᵐ ω ∂P, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω := by + obtain ⟨Φ, Gc, Fc, hΦm, hΦw⟩ := exists_measurable_canonical_witness _ κ hpublished + refine ⟨Φ, Gc, Fc, hΦm, hΦw, ?_⟩ + have hbp := source_path_measurable (sourceDriverPath B hc) hm + filter_upwards [ae_of_ae_map hbp.aemeasurable hΦw] with ω hω + have hω' : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (Φ (sourceDriverPath B hc ω)) (Gc (sourceDriverPath B hc ω)) (Fc (sourceDriverPath B hc ω)) := hω + have he := hω'.trace_unique (hw ω) + ext t + exact congrFun he t + +/-- The same fresh path has an actual capacity witness and realizes the +all-time finite-boundary restart identity. The only new instance of trace +generation is its explicit specialization to the original driver path law. -/ +theorem source_canonical_witness_boundary_restart + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) : + ∃ (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) + (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ) (Fb : Ω → C(closedHalfPlane, ℂ)), + Measurable Φ ∧ Measurable[hτ.measurableSpace] Fb ∧ + (∀ᵐ p ∂P.map (sourceDriverPath B hc), + IsCapacityTwoWitness (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) ∧ + (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → + Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ + (∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω)) ∧ + (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → + (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) = + P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧ + ∀ᵐ ω ∂P, + IsCapacityTwoWitness (fun t => Real.sqrt κ * sourceRestartDriverPath B hc τ ω t) + (Φ (sourceRestartDriverPath B hc τ ω)) + (Gc (sourceRestartDriverPath B hc τ ω)) (Fc (sourceRestartDriverPath B hc τ ω)) ∧ + ∀ t : ℝ≥0, γ ω (τ ω + t) = + finiteBoundaryPath (Fb ω) (Φ (sourceRestartDriverPath B hc τ ω)) t := by + obtain ⟨Φ, Gc, Fc, hΦm, hΦw, hΦorig⟩ := + canonical_witness_original_trace P κ B hm hc γ G F hw hpublished + obtain ⟨Ψ, Fb, hΨm, hΨorig, hFbm, hFb, hFb0, hΨlaw, hΨrestart⟩ := + source_measurable_finite_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ + let bp := sourceDriverPath B hc + let rp := sourceRestartDriverPath B hc τ + have hbp : Measurable bp := source_path_measurable bp hm + have hrp : Measurable rp := source_path_measurable rp (fun t => + (measurable_pi_apply t).comp (brownianRestart_measurable hm hc + (stopping_time_measurable_nnreal hτ))) + have heq : ∀ᵐ p ∂P.map bp, Φ p = Ψ p := by + apply (ae_map_iff hbp.aemeasurable (measurableSet_eq_fun hΦm hΨm)).mpr + filter_upwards [hΦorig] with ω hω + exact hω.trans (hΨorig ω).symm + have hlaw : P.map rp = P.map bp := by + simpa only [Measure.restrict_univ, measure_univ, one_smul] using + source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ + have hrEq : ∀ᵐ ω ∂P, Φ (rp ω) = Ψ (rp ω) := by + apply ae_of_ae_map (p := fun p : C(ℝ≥0, ℝ) => Φ p = Ψ p) hrp.aemeasurable + rwa [hlaw] + refine ⟨Φ, Gc, Fc, Fb, hΦm, hFbm, hΦw, hFb, hFb0, ?_, ?_⟩ + · intro A hA + calc + _ = (P.restrict A).map (fun ω => Ψ (rp ω)) := + Measure.map_congr (hrEq.filter_mono (ae_mono Measure.restrict_le_self)) + _ = _ := hΨlaw A hA + · have hrw := canonical_witness_pullback hΦw rp hrp hlaw + filter_upwards [hrw, hrEq, hΨrestart] with ω hω heω hrestart + refine ⟨hω, ?_⟩ + intro t + rw [heω] + exact hrestart t + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredCellTail.lean b/lean/OAI/Probability/SLE/Geometry/CenteredCellTail.lean index 44000a8d6f253f8077c264df61581508f0420087..806ca98b3e917dc66eb06c94b3a52ee84b418c37 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredCellTail.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredCellTail.lean @@ -1,113 +1,113 @@ -import OAI.Probability.SLE.Geometry.RadiusCellTail -import OAI.Probability.SLE.Geometry.FinitePastCellTail -import OAI.Probability.SLE.Angle.RadiusAngleBounds -import OAI.Probability.SLE.Geometry.ArbitraryDiskTail - -namespace OAI - -/-! Actual centered cell tails, and finite strict-past tests, are bounded -by the prescribed limiting disk tail after angular mixing at the start. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance centeredCellMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance centeredCellBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.centered_cell_tail_of_angle_bound - {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) - (hu : 0 radiusLevelTime (γ ω) (G ω) z (u+v)) K - (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) ≤ - 2*C.limitingDiskTail hc b := by - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let ν := P.map (sourceDriverPath B hc) - let : IsProbabilityMeasure ν := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - let Q := finiteCenteredLaw P κ B γ G z (u+v) - let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz (u+v) (by linarith) - let f := R.angleDiskTail ν κ b v - have hf : Measurable f := R.angleDiskTail_measurable ν κ b v - have hh := C.cell_mass_disk_tail_at_radius hB hc h0 hκ hκ8 F hw R he D hD hDf hDp - hgrowth hpublished z hz u v r b hu.le hv hr hR K hK hKball univ MeasurableSet.univ - simp only [univ_inter,setLIntegral_univ] at hh - have hi := (radius_angle_average_bounds hB hBm hc h0 hκ hκ8 γ G F hw hpublished - z hz 0 u (u+v) (by norm_num) hu (by linarith) f hf 0 - (2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) (fun θ hθ => ⟨bot_le,hangle θ hθ⟩) - univ MeasurableSet.univ).2 - simp only [zero_add,setLIntegral_univ,measure_univ,mul_one] at hi - exact hh.trans (hi.trans (mul_le_mul' le_rfl - (C.raw_angle_tail_average_le_limit hB hc h0 hκ hκ8 F hw R he b v hv.le))) - -theorem CanonicalMass.centered_strict_past_cell_tail_of_angle_bound - {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) - (hu : 0 {ω | (((32768*Real.pi)*r)^dimension κ)*b < - (C.process t ω (Subtype.val ⁻¹' K)).toReal}) - (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))) ≤ - 2*C.limitingDiskTail hc b := by - let Q := finiteCenteredLaw P κ B γ G z (u+v) - let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) - have hfinite : ∀ᵐ ω ∂Q, T ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw) - z hz (u+v) (u+v) (by linarith) le_rfl T (Eventually.of_forall fun _ => le_rfl) - have hh := C.finite_strict_past_le_cell_test Q (withDensity_absolutelyContinuous P _) T - hfinite J K ((((32768*Real.pi)*r)^dimension κ)*b) - rw [ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _)] at hh - exact hh.trans (C.centered_cell_tail_of_angle_bound hB hc h0 hκ hκ8 F hw R he D hD hDf hDp - hgrowth hpublished z hz u v r b hu hv hr hR K hK hKball hangle) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.RadiusCellTail +import OAI.Probability.SLE.Geometry.FinitePastCellTail +import OAI.Probability.SLE.Angle.RadiusAngleBounds +import OAI.Probability.SLE.Geometry.ArbitraryDiskTail + +namespace OAI + +/-! Actual centered cell tails, and finite strict-past tests, are bounded +by the prescribed limiting disk tail after angular mixing at the start. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance centeredCellMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance centeredCellBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.centered_cell_tail_of_angle_bound + {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) + (hu : 0 radiusLevelTime (γ ω) (G ω) z (u+v)) K + (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) ≤ + 2*C.limitingDiskTail hc b := by + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let ν := P.map (sourceDriverPath B hc) + let : IsProbabilityMeasure ν := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + let Q := finiteCenteredLaw P κ B γ G z (u+v) + let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz (u+v) (by linarith) + let f := R.angleDiskTail ν κ b v + have hf : Measurable f := R.angleDiskTail_measurable ν κ b v + have hh := C.cell_mass_disk_tail_at_radius hB hc h0 hκ hκ8 F hw R he D hD hDf hDp + hgrowth hpublished z hz u v r b hu.le hv hr hR K hK hKball univ MeasurableSet.univ + simp only [univ_inter,setLIntegral_univ] at hh + have hi := (radius_angle_average_bounds hB hBm hc h0 hκ hκ8 γ G F hw hpublished + z hz 0 u (u+v) (by norm_num) hu (by linarith) f hf 0 + (2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) (fun θ hθ => ⟨bot_le,hangle θ hθ⟩) + univ MeasurableSet.univ).2 + simp only [zero_add,setLIntegral_univ,measure_univ,mul_one] at hi + exact hh.trans (hi.trans (mul_le_mul' le_rfl + (C.raw_angle_tail_average_le_limit hB hc h0 hκ hκ8 F hw R he b v hv.le))) + +theorem CanonicalMass.centered_strict_past_cell_tail_of_angle_bound + {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) + (hu : 0 {ω | (((32768*Real.pi)*r)^dimension κ)*b < + (C.process t ω (Subtype.val ⁻¹' K)).toReal}) + (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))) ≤ + 2*C.limitingDiskTail hc b := by + let Q := finiteCenteredLaw P κ B γ G z (u+v) + let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) + have hfinite : ∀ᵐ ω ∂Q, T ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw) + z hz (u+v) (u+v) (by linarith) le_rfl T (Eventually.of_forall fun _ => le_rfl) + have hh := C.finite_strict_past_le_cell_test Q (withDensity_absolutelyContinuous P _) T + hfinite J K ((((32768*Real.pi)*r)^dimension κ)*b) + rw [ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _)] at hh + exact hh.trans (C.centered_cell_tail_of_angle_bound hB hc h0 hκ hκ8 F hw R he D hD hDf hDp + hgrowth hpublished z hz u v r b hu hv hr hR K hK hKball hangle) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredDiskCoefficient.lean b/lean/OAI/Probability/SLE/Geometry/CenteredDiskCoefficient.lean index 3413648506ea0135ba7288652f7e70b954c228ca..12a8aa53b366086f8d0df59fb20349c02bcf099a 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredDiskCoefficient.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredDiskCoefficient.lean @@ -1,125 +1,125 @@ -import OAI.Probability.SLE.Geometry.CenteredStoppedMonotonicity -import OAI.Probability.SLE.Gauge.FiniteMomentBounds - -namespace OAI - -/-! The exact prescribed coefficient is a stopped canonical-mass moment -under the finite centered law, including paths with an infinite level time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -omit [IsProbabilityMeasure P] in -lemma diskScalar_progressive (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) : - IsProgressive (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) - (C.diskScalar θ hθ) := by - intro t - exact (compactWeightedMass_measurable _ _ - (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp (C.progressive t) - -omit [IsProbabilityMeasure P] in -lemma normalizedStoppedDisk_eq_finite - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) - (σ : Ω → WithTop ℝ≥0) (ω : Ω) - (hs : σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) - (hf : σ ω≠⊤) : - massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) - (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ) = - finiteStoppedScalar (C.diskScalar θ hθ) σ ω := by - have he := Target.levelTime_eq_radiusLevelTime κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ ℓ hℓ - rw [← hs] at he - rw [he, massAtExtendedTime, ite_eq_right (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hf), - finiteStoppedScalar, ite_eq_right hf, normalizedDiskScalar] - congr 1 - lift σ ω to ℝ≥0 using hf with s hs' - simp only [ENNReal.ofReal, WithTop.untopD_coe] - exact mul_div_cancel_left₀ s (Real.toNNReal_pos.mpr hκ).ne' - -lemma levelCoefficient_eq_centered_moment - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) - (σ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) - (heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) : - ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) = - ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar (C.diskScalar θ hθ) σ ω)^m - ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) := by - let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) (T ω) - have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp - (Target.normalizedDriver_measurable κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) - have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T - (normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ) - have hw0 (ω) : 0≤w ω := Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ - have hY0 (ω) : 0≤Y ω := C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω - have hi : Integrable (fun ω => w ω*(Y ω)^m) P := by - apply (C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ).mono' - (hwM.mul (hY.pow_const m)).aestronglyMeasurable - filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω - change ‖w ω*(Y ω)^m‖ ≤ _ - rw [Real.norm_of_nonneg (mul_nonneg (hw0 ω) (pow_nonneg (hY0 ω) m))] - exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (hY0 ω) hω m) (hw0 ω) - have hQ : finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) = - P.withDensity (fun ω => ENNReal.ofReal (w ω)) := by - unfold finiteCenteredLaw - congr 1 - funext ω - dsimp only [w] - rw [Target.levelDensity_eq_stoppedGreen κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ] - rw [C.levelCoefficient_eq_mass hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m] - change ENNReal.ofReal (∫ ω, w ω*(Y ω)^m ∂P)=_ - rw [ofReal_integral_eq_lintegral_ofReal hi - (Eventually.of_forall (fun ω => mul_nonneg (hw0 ω) (pow_nonneg (hY0 ω) m))),hQ, - lintegral_withDensity_eq_lintegral_mul P hwM.ennreal_ofReal - (((finiteStoppedScalar_measurable _ _ (C.diskScalar_progressive θ hθ) σ hσ).mono - hσ.measurableSpace_le le_rfl).ennreal_ofReal.pow_const m)] - apply lintegral_congr_ae - filter_upwards [heσ] with ω hs - rw [ENNReal.ofReal_mul (hw0 ω),ENNReal.ofReal_pow (hY0 ω)] - change ENNReal.ofReal (w ω)*ENNReal.ofReal (Y ω)^m = - ENNReal.ofReal (w ω)*ENNReal.ofReal (finiteStoppedScalar (C.diskScalar θ hθ) σ ω)^m - by_cases hf : σ ω=⊤ - · have hT : T ω=∞ := by - dsimp only [T] - rw [Target.levelTime_eq_radiusLevelTime κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ ℓ hℓ, - ← hs,hf] - change ENNReal.ofReal κ*(⊤:ℝ≥0∞)=∞ - exact ENNReal.mul_top (ENNReal.ofReal_pos.mpr hκ).ne' - have hwz : w ω=0 := by - change Target.levelDensity κ θ _ ℓ=0 - have hT' : Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ=∞ := hT - simp only [Target.levelDensity,hT',lt_self_iff_false,ite_false] - simp only [hwz,ENNReal.ofReal_zero,zero_mul] - · dsimp only [Y,T] - rw [C.normalizedStoppedDisk_eq_finite hc h0 hκ F hw θ hθ ℓ hℓ σ ω hs hf] - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.CenteredStoppedMonotonicity +import OAI.Probability.SLE.Gauge.FiniteMomentBounds + +namespace OAI + +/-! The exact prescribed coefficient is a stopped canonical-mass moment +under the finite centered law, including paths with an infinite level time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +omit [IsProbabilityMeasure P] in +lemma diskScalar_progressive (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) : + IsProgressive (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) + (C.diskScalar θ hθ) := by + intro t + exact (compactWeightedMass_measurable _ _ + (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp (C.progressive t) + +omit [IsProbabilityMeasure P] in +lemma normalizedStoppedDisk_eq_finite + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) + (σ : Ω → WithTop ℝ≥0) (ω : Ω) + (hs : σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) + (hf : σ ω≠⊤) : + massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) + (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ) = + finiteStoppedScalar (C.diskScalar θ hθ) σ ω := by + have he := Target.levelTime_eq_radiusLevelTime κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ ℓ hℓ + rw [← hs] at he + rw [he, massAtExtendedTime, ite_eq_right (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hf), + finiteStoppedScalar, ite_eq_right hf, normalizedDiskScalar] + congr 1 + lift σ ω to ℝ≥0 using hf with s hs' + simp only [ENNReal.ofReal, WithTop.untopD_coe] + exact mul_div_cancel_left₀ s (Real.toNNReal_pos.mpr hκ).ne' + +lemma levelCoefficient_eq_centered_moment + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) + (σ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) + (heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) : + ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) = + ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar (C.diskScalar θ hθ) σ ω)^m + ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) := by + let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) (T ω) + have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp + (Target.normalizedDriver_measurable κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) + have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T + (normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ) + have hw0 (ω) : 0≤w ω := Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ + have hY0 (ω) : 0≤Y ω := C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω + have hi : Integrable (fun ω => w ω*(Y ω)^m) P := by + apply (C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ).mono' + (hwM.mul (hY.pow_const m)).aestronglyMeasurable + filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω + change ‖w ω*(Y ω)^m‖ ≤ _ + rw [Real.norm_of_nonneg (mul_nonneg (hw0 ω) (pow_nonneg (hY0 ω) m))] + exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (hY0 ω) hω m) (hw0 ω) + have hQ : finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ) = + P.withDensity (fun ω => ENNReal.ofReal (w ω)) := by + unfold finiteCenteredLaw + congr 1 + funext ω + dsimp only [w] + rw [Target.levelDensity_eq_stoppedGreen κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ] + rw [C.levelCoefficient_eq_mass hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m] + change ENNReal.ofReal (∫ ω, w ω*(Y ω)^m ∂P)=_ + rw [ofReal_integral_eq_lintegral_ofReal hi + (Eventually.of_forall (fun ω => mul_nonneg (hw0 ω) (pow_nonneg (hY0 ω) m))),hQ, + lintegral_withDensity_eq_lintegral_mul P hwM.ennreal_ofReal + (((finiteStoppedScalar_measurable _ _ (C.diskScalar_progressive θ hθ) σ hσ).mono + hσ.measurableSpace_le le_rfl).ennreal_ofReal.pow_const m)] + apply lintegral_congr_ae + filter_upwards [heσ] with ω hs + rw [ENNReal.ofReal_mul (hw0 ω),ENNReal.ofReal_pow (hY0 ω)] + change ENNReal.ofReal (w ω)*ENNReal.ofReal (Y ω)^m = + ENNReal.ofReal (w ω)*ENNReal.ofReal (finiteStoppedScalar (C.diskScalar θ hθ) σ ω)^m + by_cases hf : σ ω=⊤ + · have hT : T ω=∞ := by + dsimp only [T] + rw [Target.levelTime_eq_radiusLevelTime κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ ℓ hℓ, + ← hs,hf] + change ENNReal.ofReal κ*(⊤:ℝ≥0∞)=∞ + exact ENNReal.mul_top (ENNReal.ofReal_pos.mpr hκ).ne' + have hwz : w ω=0 := by + change Target.levelDensity κ θ _ ℓ=0 + have hT' : Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ=∞ := hT + simp only [Target.levelDensity,hT',lt_self_iff_false,ite_false] + simp only [hwz,ENNReal.ofReal_zero,zero_mul] + · dsimp only [Y,T] + rw [C.normalizedStoppedDisk_eq_finite hc h0 hκ F hw θ hθ ℓ hℓ σ ω hs hf] + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredDiskLaw.lean b/lean/OAI/Probability/SLE/Geometry/CenteredDiskLaw.lean index a4e28b44df5218ad43e1076f2182f801348e9d9d..3bbf8b6dcb07f26c40063c17e21fffa382058a40 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredDiskLaw.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredDiskLaw.lean @@ -1,131 +1,131 @@ -import OAI.Probability.SLE.Geometry.CanonicalJointDisk -import OAI.Probability.SLE.Gauge.ExactMomentGauge - -namespace OAI - -/-! The actual finite centered law with the prescribed initial angle mixture. -Its moments are exactly the finite coefficients in the gauge construction. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def centeredDiskDensity {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) (p : ℝ × Ω) : ℝ≥0∞ := - canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,sourceDriverPath B hc p.2) - -def centeredDiskLaw {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) - (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) : - Measure (ℝ × Ω) := - ((Target.angleLaw κ).prod P).withDensity (centeredDiskDensity κ B hc u) - -lemma centeredDiskDensity_measurable {Ω : Type*} [MeasurableSpace Ω] - (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) : - Measurable (centeredDiskDensity κ B hc u) := by - apply (canonicalFiniteDensity_measurable κ).comp - apply measurable_const.prodMk - apply Measurable.prodMk - · unfold Target.diskMapInv - fun_prop - · exact (source_path_measurable (sourceDriverPath B hc) hm).comp measurable_snd - -lemma centeredDiskLaw_lintegral {Ω : Type*} [MeasurableSpace Ω] - (P : Measure Ω) [SFinite P] (κ : ℝ) (hκ : 0 < κ) - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (f : ℝ × Ω → ℝ≥0∞) (hf : Measurable f) : - (∫⁻ p, f p ∂centeredDiskLaw P κ B hc u) = - ∫⁻ θ, ∫⁻ ω, f (θ,ω) ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u - ∂Target.angleLaw κ := by - let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ - have hd := centeredDiskDensity_measurable κ B hm hc u - rw [centeredDiskLaw,lintegral_withDensity_eq_lintegral_mul _ hd hf, - lintegral_prod _ (hd.mul hf).aemeasurable] - apply lintegral_congr - intro θ - have hdθ : Measurable (fun ω => ENNReal.ofReal - (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) - (Target.diskMapInv θ 0) u / initialGreen κ (Target.diskMapInv θ 0))) := by - have hmθ : Measurable (fun ω : Ω => (θ,ω)) := measurable_const.prodMk measurable_id - have h : Measurable (fun ω : Ω => centeredDiskDensity κ B hc u (θ,ω)) := hd.comp hmθ - convert h using 1 - funext ω - exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) u hu _).symm - have hfθ : Measurable (fun ω : Ω => f (θ,ω)) := - hf.comp (measurable_const.prodMk measurable_id) - rw [finiteCenteredLaw,lintegral_withDensity_eq_lintegral_mul _ hdθ hfθ] - apply lintegral_congr - intro ω - change centeredDiskDensity κ B hc u (θ,ω) * f (θ,ω) = _ - dsimp only [centeredDiskDensity] - rw [canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) u hu] - rfl - -lemma centeredDiskLaw_isProbability {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (hκ : 0<κ) (hκ8 : κ<8) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0≤u) : IsProbabilityMeasure (centeredDiskLaw P κ B hc u) := by - let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ - constructor - rw [← lintegral_one,centeredDiskLaw_lintegral P κ hκ B - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw u hu - (fun _ => 1) measurable_const] - calc - _ = ∫⁻ _θ, (1 : ℝ≥0∞) ∂Target.angleLaw κ := by - apply lintegral_congr_ae - filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ - let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) - (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu - simp - _ = 1 := by simp - -namespace Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma centeredDiskLaw_moment - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) - (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : - (∫⁻ p, ENNReal.ofReal (C.diskAtRadius p.1 (4*ℓ/κ) p.2)^m - ∂centeredDiskLaw P κ B hc (4*ℓ/κ)) = ENNReal.ofReal (Target.finiteMoment κ W m ℓ) := by - have hu : 0 ≤ 4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le - rw [centeredDiskLaw_lintegral P κ hκ B - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw _ hu _ - ((C.diskAtRadius_joint_measurable hc F hw _ hu).ennreal_ofReal.pow_const m)] - calc - _ = ∫⁻ θ, ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) ∂Target.angleLaw κ := by - apply lintegral_congr_ae - filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ - exact (C.levelCoefficient_eq_diskAtRadius_moment hB hc h0 hκ hκ8 F hw hgrowth W hW - θ hθ ℓ hℓ m).symm - _ = _ := by - rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m] - symm - apply ofReal_integral_eq_lintegral_ofReal - (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).2 - filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ - exact (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).1 θ hθ - -end Multipoint.CanonicalMass -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalJointDisk +import OAI.Probability.SLE.Gauge.ExactMomentGauge + +namespace OAI + +/-! The actual finite centered law with the prescribed initial angle mixture. +Its moments are exactly the finite coefficients in the gauge construction. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +def centeredDiskDensity {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) (p : ℝ × Ω) : ℝ≥0∞ := + canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,sourceDriverPath B hc p.2) + +def centeredDiskLaw {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) + (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) : + Measure (ℝ × Ω) := + ((Target.angleLaw κ).prod P).withDensity (centeredDiskDensity κ B hc u) + +lemma centeredDiskDensity_measurable {Ω : Type*} [MeasurableSpace Ω] + (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) (u : ℝ) : + Measurable (centeredDiskDensity κ B hc u) := by + apply (canonicalFiniteDensity_measurable κ).comp + apply measurable_const.prodMk + apply Measurable.prodMk + · unfold Target.diskMapInv + fun_prop + · exact (source_path_measurable (sourceDriverPath B hc) hm).comp measurable_snd + +lemma centeredDiskLaw_lintegral {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [SFinite P] (κ : ℝ) (hκ : 0 < κ) + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (f : ℝ × Ω → ℝ≥0∞) (hf : Measurable f) : + (∫⁻ p, f p ∂centeredDiskLaw P κ B hc u) = + ∫⁻ θ, ∫⁻ ω, f (θ,ω) ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u + ∂Target.angleLaw κ := by + let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ + have hd := centeredDiskDensity_measurable κ B hm hc u + rw [centeredDiskLaw,lintegral_withDensity_eq_lintegral_mul _ hd hf, + lintegral_prod _ (hd.mul hf).aemeasurable] + apply lintegral_congr + intro θ + have hdθ : Measurable (fun ω => ENNReal.ofReal + (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) + (Target.diskMapInv θ 0) u / initialGreen κ (Target.diskMapInv θ 0))) := by + have hmθ : Measurable (fun ω : Ω => (θ,ω)) := measurable_const.prodMk measurable_id + have h : Measurable (fun ω : Ω => centeredDiskDensity κ B hc u (θ,ω)) := hd.comp hmθ + convert h using 1 + funext ω + exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) u hu _).symm + have hfθ : Measurable (fun ω : Ω => f (θ,ω)) := + hf.comp (measurable_const.prodMk measurable_id) + rw [finiteCenteredLaw,lintegral_withDensity_eq_lintegral_mul _ hdθ hfθ] + apply lintegral_congr + intro ω + change centeredDiskDensity κ B hc u (θ,ω) * f (θ,ω) = _ + dsimp only [centeredDiskDensity] + rw [canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) u hu] + rfl + +lemma centeredDiskLaw_isProbability {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (hκ : 0<κ) (hκ8 : κ<8) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0≤u) : IsProbabilityMeasure (centeredDiskLaw P κ B hc u) := by + let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ + constructor + rw [← lintegral_one,centeredDiskLaw_lintegral P κ hκ B + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw u hu + (fun _ => 1) measurable_const] + calc + _ = ∫⁻ _θ, (1 : ℝ≥0∞) ∂Target.angleLaw κ := by + apply lintegral_congr_ae + filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ + let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) + (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu + simp + _ = 1 := by simp + +namespace Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma centeredDiskLaw_moment + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) + (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) : + (∫⁻ p, ENNReal.ofReal (C.diskAtRadius p.1 (4*ℓ/κ) p.2)^m + ∂centeredDiskLaw P κ B hc (4*ℓ/κ)) = ENNReal.ofReal (Target.finiteMoment κ W m ℓ) := by + have hu : 0 ≤ 4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le + rw [centeredDiskLaw_lintegral P κ hκ B + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc γ G F hw _ hu _ + ((C.diskAtRadius_joint_measurable hc F hw _ hu).ennreal_ofReal.pow_const m)] + calc + _ = ∫⁻ θ, ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ) ∂Target.angleLaw κ := by + apply lintegral_congr_ae + filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ + exact (C.levelCoefficient_eq_diskAtRadius_moment hB hc h0 hκ hκ8 F hw hgrowth W hW + θ hθ ℓ hℓ m).symm + _ = _ := by + rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m] + symm + apply ofReal_integral_eq_lintegral_ofReal + (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).2 + filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ + exact (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).1 θ hθ + +end Multipoint.CanonicalMass +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredFutureSlit.lean b/lean/OAI/Probability/SLE/Geometry/CenteredFutureSlit.lean index f18de79dc90246374c02cbabcd739261cb7054ea..7711c7127907736576898d5242fdf846a52fbc78 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredFutureSlit.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredFutureSlit.lean @@ -1,59 +1,59 @@ -import OAI.Probability.SLE.Loewner.MappedFutureSimple -import OAI.Probability.SLE.Loewner.SlitSegmentParameter - -namespace OAI - -/-! Centering at the proved driving value makes every future segment an -ordinary simple attached slit, to which the existing normalized chart -theorem applies. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -noncomputable def centeredFutureArc (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s centeredFutureArc_upper p hi h0 hH hst ht hu)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.MappedFutureSimple +import OAI.Probability.SLE.Loewner.SlitSegmentParameter + +namespace OAI + +/-! Centering at the proved driving value makes every future segment an +ordinary simple attached slit, to which the existing normalized chart +theorem applies. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +noncomputable def centeredFutureArc (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s centeredFutureArc_upper p hi h0 hH hst ht hu)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredStoppedMonotonicity.lean b/lean/OAI/Probability/SLE/Geometry/CenteredStoppedMonotonicity.lean index 5dbd2e8fbfd5fb7ce61086bb208048ec2fb00223..61a12ac790f927973bf71be53208c9727bae171d 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredStoppedMonotonicity.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredStoppedMonotonicity.lean @@ -1,70 +1,70 @@ -import OAI.Probability.SLE.Geometry.FiniteStoppedScalar -import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency - -namespace OAI - -/-! The moments of an increasing progressive mass, evaluated at radius -stops, increase with the finite centered law's terminal level. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -theorem finiteCenteredLaw_stoppedScalar_moment_mono - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z∈upperHalfPlane) (v V : ℝ) (hv : 0≤v) (hvV : v≤V) - (A : ℝ≥0 → Ω → ℝ) - (hA : IsProgressive (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) A) - (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) - (σ τ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) - (_hτ : IsStoppingTime (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) τ) - (hσeq : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z v) - (hτeq : ∀ᵐ ω ∂P, τ ω=radiusLevelTime (γ ω) (G ω) z V) (m : ℕ) : - (∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂finiteCenteredLaw P κ B γ G z v) ≤ - ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A τ ω)^m ∂finiteCenteredLaw P κ B γ G z V := by - let Q := finiteCenteredLaw P κ B γ G z V - have hσf : ∀ᵐ ω ∂Q, σ ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz v V hv hvV σ - (hσeq.mono (fun _ h => h.le)) - have hτf : ∀ᵐ ω ∂Q, τ ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz V V - (hv.trans hvV) le_rfl τ (hτeq.mono (fun _ h => h.le)) - have horder : ∀ᵐ ω ∂P, σ ω≤τ ω := by - filter_upwards [hw,hσeq,hτeq] with ω hω hs ht - rw [hs,ht] - exact radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV - have habs : Q.AbsolutelyContinuous P := withDensity_absolutelyContinuous P _ - have heq : (∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂Q) = - ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂finiteCenteredLaw P κ B γ G z v := by - apply lintegral_eq_of_stopped_events Q (finiteCenteredLaw P κ B γ G z v) - hσ.measurableSpace hσ.measurableSpace_le - · intro E hE - exact finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F hw - z hz v V hv hvV σ hσ hσeq E hE - · exact ((finiteStoppedScalar_measurable _ A hA σ hσ).ennreal_ofReal).pow_const m - rw [← heq] - apply lintegral_mono_ae - filter_upwards [hσf,hτf,habs.ae_le horder,habs.ae_le hmono] with ω hs ht hst hm - apply pow_le_pow_left' - apply ENNReal.ofReal_le_ofReal - simp only [finiteStoppedScalar,hs,ht,ite_false] - apply hm - exact WithTop.untopD_mono ht hst - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteStoppedScalar +import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency + +namespace OAI + +/-! The moments of an increasing progressive mass, evaluated at radius +stops, increase with the finite centered law's terminal level. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +theorem finiteCenteredLaw_stoppedScalar_moment_mono + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z∈upperHalfPlane) (v V : ℝ) (hv : 0≤v) (hvV : v≤V) + (A : ℝ≥0 → Ω → ℝ) + (hA : IsProgressive (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) A) + (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) + (σ τ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) + (_hτ : IsStoppingTime (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) τ) + (hσeq : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z v) + (hτeq : ∀ᵐ ω ∂P, τ ω=radiusLevelTime (γ ω) (G ω) z V) (m : ℕ) : + (∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂finiteCenteredLaw P κ B γ G z v) ≤ + ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A τ ω)^m ∂finiteCenteredLaw P κ B γ G z V := by + let Q := finiteCenteredLaw P κ B γ G z V + have hσf : ∀ᵐ ω ∂Q, σ ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz v V hv hvV σ + (hσeq.mono (fun _ h => h.le)) + have hτf : ∀ᵐ ω ∂Q, τ ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz V V + (hv.trans hvV) le_rfl τ (hτeq.mono (fun _ h => h.le)) + have horder : ∀ᵐ ω ∂P, σ ω≤τ ω := by + filter_upwards [hw,hσeq,hτeq] with ω hω hs ht + rw [hs,ht] + exact radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV + have habs : Q.AbsolutelyContinuous P := withDensity_absolutelyContinuous P _ + have heq : (∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂Q) = + ∫⁻ ω, ENNReal.ofReal (finiteStoppedScalar A σ ω)^m ∂finiteCenteredLaw P κ B γ G z v := by + apply lintegral_eq_of_stopped_events Q (finiteCenteredLaw P κ B γ G z v) + hσ.measurableSpace hσ.measurableSpace_le + · intro E hE + exact finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F hw + z hz v V hv hvV σ hσ hσeq E hE + · exact ((finiteStoppedScalar_measurable _ A hA σ hσ).ennreal_ofReal).pow_const m + rw [← heq] + apply lintegral_mono_ae + filter_upwards [hσf,hτf,habs.ae_le horder,habs.ae_le hmono] with ω hs ht hst hm + apply pow_le_pow_left' + apply ENNReal.ofReal_le_ofReal + simp only [finiteStoppedScalar,hs,ht,ite_false] + apply hm + exact WithTop.untopD_mono ht hst + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CenteredStoppedTails.lean b/lean/OAI/Probability/SLE/Geometry/CenteredStoppedTails.lean index ea2b9498f80e032f5f64f625116b6b5f70e5b940..d52b9df89b383c93e5e0c60bf3b961587749979b 100644 --- a/lean/OAI/Probability/SLE/Geometry/CenteredStoppedTails.lean +++ b/lean/OAI/Probability/SLE/Geometry/CenteredStoppedTails.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Geometry.CenteredStoppedMonotonicity - -namespace OAI - -/-! Tail events of increasing canonical scalars under consistent finite laws. -Only the stopped past is transported when the terminal level is increased. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -theorem finiteCenteredLaw_stoppedScalar_tail_mono - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z∈upperHalfPlane) (v V : ℝ) (hv : 0≤v) (hvV : v≤V) - (A : ℝ≥0 → Ω → ℝ) - (hA : IsProgressive (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) A) - (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) - (σ τ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) - (hσeq : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z v) - (hτeq : ∀ᵐ ω ∂P, τ ω=radiusLevelTime (γ ω) (G ω) z V) (b : ℝ) : - finiteCenteredLaw P κ B γ G z v {ω | b ≤ finiteStoppedScalar A σ ω} ≤ - finiteCenteredLaw P κ B γ G z V {ω | b ≤ finiteStoppedScalar A τ ω} := by - let Q := finiteCenteredLaw P κ B γ G z V - have hσf : ∀ᵐ ω ∂Q, σ ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz v V hv hvV σ - (hσeq.mono (fun _ h => h.le)) - have hτf : ∀ᵐ ω ∂Q, τ ω≠⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz V V - (hv.trans hvV) le_rfl τ (hτeq.mono (fun _ h => h.le)) - have horder : ∀ᵐ ω ∂P, σ ω≤τ ω := by - filter_upwards [hw,hσeq,hτeq] with ω hω hs ht - rw [hs,ht] - exact radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV - have habs : Q.AbsolutelyContinuous P := withDensity_absolutelyContinuous P _ - have he := finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F hw - z hz v V hv hvV σ hσ hσeq {ω | b ≤ finiteStoppedScalar A σ ω} - (measurableSet_le measurable_const (finiteStoppedScalar_measurable _ A hA σ hσ)) - rw [← he] - apply measure_mono_ae - filter_upwards [hσf,hτf,habs.ae_le horder,habs.ae_le hmono] with ω hs ht hst hm - intro hb - apply hb.trans - simp only [finiteStoppedScalar,hs,ht,ite_false] - exact hm (WithTop.untopD_mono ht hst) - -/-- A finite-law moment gives a tail bound without constructing an infinite -centered path law. This form is convenient before taking the level supremum. -/ -lemma stoppedScalar_tail_moment_bound - {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω) - (X : Ω → ℝ) (hX : Measurable X) - (b : ℝ) (hb : 0 ≤ b) (n : ℕ) : - ENNReal.ofReal (b^n)*Q {ω | b ≤ X ω} ≤ ∫⁻ ω, ENNReal.ofReal (X ω)^n ∂Q := by - have he : ENNReal.ofReal (b^n)=ENNReal.ofReal b^n := ENNReal.ofReal_pow hb n - rw [he] - apply le_trans ?_ (mul_meas_ge_le_lintegral (hX.ennreal_ofReal.pow_const n) (ENNReal.ofReal b^n)) - apply mul_le_mul' le_rfl - apply measure_mono - intro ω hω - exact pow_le_pow_left' (ENNReal.ofReal_le_ofReal hω) n - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CenteredStoppedMonotonicity + +namespace OAI + +/-! Tail events of increasing canonical scalars under consistent finite laws. +Only the stopped past is transported when the terminal level is increased. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +theorem finiteCenteredLaw_stoppedScalar_tail_mono + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z∈upperHalfPlane) (v V : ℝ) (hv : 0≤v) (hvV : v≤V) + (A : ℝ≥0 → Ω → ℝ) + (hA : IsProgressive (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) A) + (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) + (σ τ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ) + (hσeq : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) z v) + (hτeq : ∀ᵐ ω ∂P, τ ω=radiusLevelTime (γ ω) (G ω) z V) (b : ℝ) : + finiteCenteredLaw P κ B γ G z v {ω | b ≤ finiteStoppedScalar A σ ω} ≤ + finiteCenteredLaw P κ B γ G z V {ω | b ≤ finiteStoppedScalar A τ ω} := by + let Q := finiteCenteredLaw P κ B γ G z V + have hσf : ∀ᵐ ω ∂Q, σ ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz v V hv hvV σ + (hσeq.mono (fun _ h => h.le)) + have hτf : ∀ᵐ ω ∂Q, τ ω≠⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw z hz V V + (hv.trans hvV) le_rfl τ (hτeq.mono (fun _ h => h.le)) + have horder : ∀ᵐ ω ∂P, σ ω≤τ ω := by + filter_upwards [hw,hσeq,hτeq] with ω hω hs ht + rw [hs,ht] + exact radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV + have habs : Q.AbsolutelyContinuous P := withDensity_absolutelyContinuous P _ + have he := finiteCenteredLaw_consistent_at_radius_stop hB hBm hc h0 κ hκ hκ8 γ G F hw + z hz v V hv hvV σ hσ hσeq {ω | b ≤ finiteStoppedScalar A σ ω} + (measurableSet_le measurable_const (finiteStoppedScalar_measurable _ A hA σ hσ)) + rw [← he] + apply measure_mono_ae + filter_upwards [hσf,hτf,habs.ae_le horder,habs.ae_le hmono] with ω hs ht hst hm + intro hb + apply hb.trans + simp only [finiteStoppedScalar,hs,ht,ite_false] + exact hm (WithTop.untopD_mono ht hst) + +/-- A finite-law moment gives a tail bound without constructing an infinite +centered path law. This form is convenient before taking the level supremum. -/ +lemma stoppedScalar_tail_moment_bound + {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω) + (X : Ω → ℝ) (hX : Measurable X) + (b : ℝ) (hb : 0 ≤ b) (n : ℕ) : + ENNReal.ofReal (b^n)*Q {ω | b ≤ X ω} ≤ ∫⁻ ω, ENNReal.ofReal (X ω)^n ∂Q := by + have he : ENNReal.ofReal (b^n)=ENNReal.ofReal b^n := ENNReal.ofReal_pow hb n + rw [he] + apply le_trans ?_ (mul_meas_ge_le_lintegral (hX.ennreal_ofReal.pow_const n) (ENNReal.ofReal b^n)) + apply mul_le_mul' le_rfl + apply measure_mono + intro ω hω + exact pow_le_pow_left' (ENNReal.ofReal_le_ofReal hω) n + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CircleRadialFlux.lean b/lean/OAI/Probability/SLE/Geometry/CircleRadialFlux.lean index 6e6295faccc7e2c80bc30c73d7791bd3f61465c8..b87f6895e718ff6fc536f42b821150035bd31743 100644 --- a/lean/OAI/Probability/SLE/Geometry/CircleRadialFlux.lean +++ b/lean/OAI/Probability/SLE/Geometry/CircleRadialFlux.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Geometry.PolarLaplacianFlux - -namespace OAI - -/-! The ordinary angular integral of radial flux is a circle average of -the derivative applied to the position vector. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma circleAverage_eq_neg_pi_integral {E : Type*} [NormedAddCommGroup E] - [NormedSpace ℝ E] (f : ℂ → E) (r : ℝ) : - circleAverage f 0 r=(2*Real.pi)⁻¹ • ∫ θ in -Real.pi..Real.pi, f (polarPoint r θ) := by - have h := circleAverage_eq_integral_add (f := f) (c := (0:ℂ)) (R := r) (-Real.pi) - have hs := intervalIntegral.integral_comp_add_right (fun θ => f (circleMap 0 r θ)) - (a := (0:ℝ)) (b := 2*Real.pi) (-Real.pi) - rw [hs] at h - have hend : 2*Real.pi+ -Real.pi=Real.pi := by ring - simpa only [zero_add,hend,←polarPoint_eq_circleMap] using h - -lemma polar_integral_eq_circleAverage (g : ℂ → ℝ) (r : ℝ) : - (∫ θ in -Real.pi..Real.pi, r*g (polarPoint r θ))= - 2*Real.pi*r*circleAverage g 0 r := by - rw [intervalIntegral.integral_const_mul,circleAverage_eq_neg_pi_integral,smul_eq_mul] - field_simp - -lemma radial_flux_eq_circleAverage (u : ℂ → ℝ) (r : ℝ) : - (∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (r,θ))= - 2*Real.pi*circleAverage (fun z => fderiv ℝ u z z) 0 r := by - rw [circleAverage_eq_neg_pi_integral,smul_eq_mul] - have hpoint (θ : ℝ) : radialLaplacianFlux u (r,θ)= - fderiv ℝ u (polarPoint r θ) (polarPoint r θ) := by - unfold radialLaplacianFlux - conv_rhs => arg 2; rw [polarPoint] - simp only [map_smul,smul_eq_mul] - simp_rw [hpoint] - field_simp - -lemma radial_flux_continuousOn {u : ℂ → ℝ} {R : ℝ} - (hu : ∀ z∈closedBall (0:ℂ) R, ContDiffAt ℝ 2 u z) : - ContinuousOn (fun r => ∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (r,θ)) (Icc 0 R) := by - have hc : ContinuousOn (fun z => fderiv ℝ u z z) - {z : ℂ | ‖z-(0:ℂ)‖∈Icc 0 R} := by - intro z hz - have hzR : z∈closedBall (0:ℂ) R := by - rw [mem_closedBall,dist_eq_norm] - exact hz.2 - have hh := ((hu z hzR).fderiv_right (m := 1) (by norm_num)).continuousAt - exact (hh.clm_apply continuousAt_id).continuousWithinAt - have hav := hc.circleAverage (fun r hr => hr.1) - simp_rw [radial_flux_eq_circleAverage] - exact hav.const_mul _ - -@[simp] lemma radial_flux_zero (u : ℂ → ℝ) : - (∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (0,θ))=0 := by - simp [radialLaplacianFlux] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.PolarLaplacianFlux + +namespace OAI + +/-! The ordinary angular integral of radial flux is a circle average of +the derivative applied to the position vector. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma circleAverage_eq_neg_pi_integral {E : Type*} [NormedAddCommGroup E] + [NormedSpace ℝ E] (f : ℂ → E) (r : ℝ) : + circleAverage f 0 r=(2*Real.pi)⁻¹ • ∫ θ in -Real.pi..Real.pi, f (polarPoint r θ) := by + have h := circleAverage_eq_integral_add (f := f) (c := (0:ℂ)) (R := r) (-Real.pi) + have hs := intervalIntegral.integral_comp_add_right (fun θ => f (circleMap 0 r θ)) + (a := (0:ℝ)) (b := 2*Real.pi) (-Real.pi) + rw [hs] at h + have hend : 2*Real.pi+ -Real.pi=Real.pi := by ring + simpa only [zero_add,hend,←polarPoint_eq_circleMap] using h + +lemma polar_integral_eq_circleAverage (g : ℂ → ℝ) (r : ℝ) : + (∫ θ in -Real.pi..Real.pi, r*g (polarPoint r θ))= + 2*Real.pi*r*circleAverage g 0 r := by + rw [intervalIntegral.integral_const_mul,circleAverage_eq_neg_pi_integral,smul_eq_mul] + field_simp + +lemma radial_flux_eq_circleAverage (u : ℂ → ℝ) (r : ℝ) : + (∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (r,θ))= + 2*Real.pi*circleAverage (fun z => fderiv ℝ u z z) 0 r := by + rw [circleAverage_eq_neg_pi_integral,smul_eq_mul] + have hpoint (θ : ℝ) : radialLaplacianFlux u (r,θ)= + fderiv ℝ u (polarPoint r θ) (polarPoint r θ) := by + unfold radialLaplacianFlux + conv_rhs => arg 2; rw [polarPoint] + simp only [map_smul,smul_eq_mul] + simp_rw [hpoint] + field_simp + +lemma radial_flux_continuousOn {u : ℂ → ℝ} {R : ℝ} + (hu : ∀ z∈closedBall (0:ℂ) R, ContDiffAt ℝ 2 u z) : + ContinuousOn (fun r => ∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (r,θ)) (Icc 0 R) := by + have hc : ContinuousOn (fun z => fderiv ℝ u z z) + {z : ℂ | ‖z-(0:ℂ)‖∈Icc 0 R} := by + intro z hz + have hzR : z∈closedBall (0:ℂ) R := by + rw [mem_closedBall,dist_eq_norm] + exact hz.2 + have hh := ((hu z hzR).fderiv_right (m := 1) (by norm_num)).continuousAt + exact (hh.clm_apply continuousAt_id).continuousWithinAt + have hav := hc.circleAverage (fun r hr => hr.1) + simp_rw [radial_flux_eq_circleAverage] + exact hav.const_mul _ + +@[simp] lemma radial_flux_zero (u : ℂ → ℝ) : + (∫ θ in -Real.pi..Real.pi, radialLaplacianFlux u (0,θ))=0 := by + simp [radialLaplacianFlux] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ClosedArcCrossing.lean b/lean/OAI/Probability/SLE/Geometry/ClosedArcCrossing.lean index f3875bf73e99594b11aa9aa3c384dc46c5e5c100..19bd4b285e0435a604bf7e5c7ebbc552519a88ed 100644 --- a/lean/OAI/Probability/SLE/Geometry/ClosedArcCrossing.lean +++ b/lean/OAI/Probability/SLE/Geometry/ClosedArcCrossing.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Geometry.TransverseCurveDifference - -namespace OAI - -/-! Closing an arc by a path outside the domain preserves its unique -interior crossing. This connects the small-loop obstruction to inverse -Loewner semicircles. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma path_trans_first_strict_deriv {a b d : ℂ} (c : Path a b) (q : Path b d) - {t : ℝ} (ht : t<1) {α : ℂ} (hc : HasStrictDerivAt c.extend α t) : - HasStrictDerivAt (c.trans q).extend ((2:ℝ) • α) (t/2) := by - have hd : HasStrictDerivAt (fun u : ℝ => 2*u) 2 (t/2) := by - simpa using (hasStrictDerivAt_id (t/2)).const_mul 2 - have hcomp := hc.scomp_of_eq (t/2) hd (by ring) - apply hcomp.congr_of_eventuallyEq - filter_upwards [eventually_lt_nhds (show t/2<(1:ℝ)/2 by linarith)] with u hu - exact (Path.extend_trans_of_le_half c q hu.le).symm - -lemma small_closed_arc_no_transverse_crossing {D : Set ℂ} {a b y z : ℂ} - (c : Path a b) (q : Path b a) (p : Path y z) - (hpin : ∀ u, p u∈D) (hqout : ∀ u, q u∉D) - (w : ℂ) (r : ℝ) (hr : 0 linarith [t.property.1,t.property.2]⟩ - have ht' : 0<(t':ℝ) ∧ (t':ℝ)<1 := ⟨half_pos ht.1,by dsimp [t']; linarith [ht.2]⟩ - have hlball (u : unitInterval) : l u∈ball w r := by - have hmem : l u∈range l := mem_range_self u - rw [Path.trans_range] at hmem - rcases hmem with ⟨v,hv⟩ | ⟨v,hv⟩ - · exact hv ▸ hcball v - · exact hv ▸ hqball v - have he' : l t'=p s := by - rw [Path.trans_apply,dite_eq_left (show (t':ℝ)≤1/2 by dsimp [t']; linarith [t.property.2])] - convert he using 1 - congr 1 - apply Subtype.ext - dsimp [t'] - ring - have hu' (u v : unitInterval) (hev : l v=p u) : u=s ∧ v=t' := by - rw [Path.trans_apply] at hev - split_ifs at hev with hv - · obtain ⟨hu,hv'⟩ := hu _ _ hev - refine ⟨hu,Subtype.ext ?_⟩ - have hvR := congrArg (fun a : unitInterval => (a:ℝ)) hv' - dsimp [t'] - dsimp at hvR - linarith - · exact False.elim ((hqout _) (hev ▸ hpin u)) - have htrans' (u v : ℝ) (heq : u • ((2:ℝ) • α)-v • β=0) : u=0 ∧ v=0 := by - have heq' : (2*u) • α-v • β=0 := by simpa only [smul_smul,mul_comm] using heq - obtain ⟨hu,hv⟩ := htrans _ _ heq' - exact ⟨by linarith,hv⟩ - have hloc := transverse_difference_locally_injective - (path_trans_first_strict_deriv c q ht.2 hc) hp htrans' - apply small_loop_no_single_crossing (l : C(unitInterval,ℂ)) (p : C(unitInterval,ℂ)) - w r hr hlball (by simpa using hy) (by simpa using hz) s t' hs ht' he' hu' - · exact hloc - · simp [l] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.TransverseCurveDifference + +namespace OAI + +/-! Closing an arc by a path outside the domain preserves its unique +interior crossing. This connects the small-loop obstruction to inverse +Loewner semicircles. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma path_trans_first_strict_deriv {a b d : ℂ} (c : Path a b) (q : Path b d) + {t : ℝ} (ht : t<1) {α : ℂ} (hc : HasStrictDerivAt c.extend α t) : + HasStrictDerivAt (c.trans q).extend ((2:ℝ) • α) (t/2) := by + have hd : HasStrictDerivAt (fun u : ℝ => 2*u) 2 (t/2) := by + simpa using (hasStrictDerivAt_id (t/2)).const_mul 2 + have hcomp := hc.scomp_of_eq (t/2) hd (by ring) + apply hcomp.congr_of_eventuallyEq + filter_upwards [eventually_lt_nhds (show t/2<(1:ℝ)/2 by linarith)] with u hu + exact (Path.extend_trans_of_le_half c q hu.le).symm + +lemma small_closed_arc_no_transverse_crossing {D : Set ℂ} {a b y z : ℂ} + (c : Path a b) (q : Path b a) (p : Path y z) + (hpin : ∀ u, p u∈D) (hqout : ∀ u, q u∉D) + (w : ℂ) (r : ℝ) (hr : 0 linarith [t.property.1,t.property.2]⟩ + have ht' : 0<(t':ℝ) ∧ (t':ℝ)<1 := ⟨half_pos ht.1,by dsimp [t']; linarith [ht.2]⟩ + have hlball (u : unitInterval) : l u∈ball w r := by + have hmem : l u∈range l := mem_range_self u + rw [Path.trans_range] at hmem + rcases hmem with ⟨v,hv⟩ | ⟨v,hv⟩ + · exact hv ▸ hcball v + · exact hv ▸ hqball v + have he' : l t'=p s := by + rw [Path.trans_apply,dite_eq_left (show (t':ℝ)≤1/2 by dsimp [t']; linarith [t.property.2])] + convert he using 1 + congr 1 + apply Subtype.ext + dsimp [t'] + ring + have hu' (u v : unitInterval) (hev : l v=p u) : u=s ∧ v=t' := by + rw [Path.trans_apply] at hev + split_ifs at hev with hv + · obtain ⟨hu,hv'⟩ := hu _ _ hev + refine ⟨hu,Subtype.ext ?_⟩ + have hvR := congrArg (fun a : unitInterval => (a:ℝ)) hv' + dsimp [t'] + dsimp at hvR + linarith + · exact False.elim ((hqout _) (hev ▸ hpin u)) + have htrans' (u v : ℝ) (heq : u • ((2:ℝ) • α)-v • β=0) : u=0 ∧ v=0 := by + have heq' : (2*u) • α-v • β=0 := by simpa only [smul_smul,mul_comm] using heq + obtain ⟨hu,hv⟩ := htrans _ _ heq' + exact ⟨by linarith,hv⟩ + have hloc := transverse_difference_locally_injective + (path_trans_first_strict_deriv c q ht.2 hc) hp htrans' + apply small_loop_no_single_crossing (l : C(unitInterval,ℂ)) (p : C(unitInterval,ℂ)) + w r hr hlball (by simpa using hy) (by simpa using hz) s t' hs ht' he' hu' + · exact hloc + · simp [l] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ClosedComplexPair.lean b/lean/OAI/Probability/SLE/Geometry/ClosedComplexPair.lean index acff636faa895ef1f124232defc7cdf624867416..e391c832cdefa3d61ef54ffe7722db149c524a2b 100644 --- a/lean/OAI/Probability/SLE/Geometry/ClosedComplexPair.lean +++ b/lean/OAI/Probability/SLE/Geometry/ClosedComplexPair.lean @@ -1,84 +1,84 @@ -import OAI.Probability.SLE.Geometry.ClosedComplexUnion - -namespace OAI - -/-! Gluing two prescribed homeomorphisms on closed pieces. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -def closedComplexPairHomeomorph {A A' B B' : Set ℂ} - (hA : IsClosed A) (hA' : IsClosed A') (hB : IsClosed B) (hB' : IsClosed B') - (e : A ≃ₜ A') (d : B ≃ₜ B') - (he : ∀ (x : A) (hx : (x:ℂ) ∈ B), (e x:ℂ)=d ⟨x,hx⟩) - (hi : ∀ (x : A') (hx : (x:ℂ) ∈ B'), (e.symm x:ℂ)=d.symm ⟨x,hx⟩) : - ↥(A ∪ B) ≃ₜ ↥(A' ∪ B') := by - classical - let f (x : ℂ) := if ha : x ∈ A then (e ⟨x,ha⟩:ℂ) - else if hb : x ∈ B then (d ⟨x,hb⟩:ℂ) else x - let g (x : ℂ) := if ha : x ∈ A' then (e.symm ⟨x,ha⟩:ℂ) - else if hb : x ∈ B' then (d.symm ⟨x,hb⟩:ℂ) else x - have hfA (x : A) : f x=(e x:ℂ) := by simp only [f,dite_eq_left x.property] - have hfB (x : B) : f x=(d x:ℂ) := by - dsimp [f] - by_cases ha : (x:ℂ) ∈ A - · rw [dite_eq_left ha] - exact he ⟨x,ha⟩ x.property - · rw [dite_eq_right ha,dite_eq_left x.property] - have hgA (x : A') : g x=(e.symm x:ℂ) := by simp only [g,dite_eq_left x.property] - have hgB (x : B') : g x=(d.symm x:ℂ) := by - dsimp [g] - by_cases ha : (x:ℂ) ∈ A' - · rw [dite_eq_left ha] - exact hi ⟨x,ha⟩ x.property - · rw [dite_eq_right ha,dite_eq_left x.property] - have hfAc : ContinuousOn f A := by - rw [continuousOn_iff_continuous_domRestrict] - exact (continuous_subtype_val.comp e.continuous).congr (fun x => (hfA x).symm) - have hfBc : ContinuousOn f B := by - rw [continuousOn_iff_continuous_domRestrict] - exact (continuous_subtype_val.comp d.continuous).congr (fun x => (hfB x).symm) - have hgAc : ContinuousOn g A' := by - rw [continuousOn_iff_continuous_domRestrict] - exact (continuous_subtype_val.comp e.symm.continuous).congr (fun x => (hgA x).symm) - have hgBc : ContinuousOn g B' := by - rw [continuousOn_iff_continuous_domRestrict] - exact (continuous_subtype_val.comp d.symm.continuous).congr (fun x => (hgB x).symm) - have hfm : MapsTo f (A ∪ B) (A' ∪ B') := by - intro x hx - rcases hx with hx|hx - · rw [hfA ⟨x,hx⟩]; exact Or.inl (e ⟨x,hx⟩).property - · rw [hfB ⟨x,hx⟩]; exact Or.inr (d ⟨x,hx⟩).property - have hgm : MapsTo g (A' ∪ B') (A ∪ B) := by - intro x hx - rcases hx with hx|hx - · rw [hgA ⟨x,hx⟩]; exact Or.inl (e.symm ⟨x,hx⟩).property - · rw [hgB ⟨x,hx⟩]; exact Or.inr (d.symm ⟨x,hx⟩).property - have hgf (x : ℂ) (hx : x ∈ A ∪ B) : g (f x)=x := by - rcases hx with hx|hx - · rw [hfA ⟨x,hx⟩,hgA (e ⟨x,hx⟩)] - exact congrArg Subtype.val (e.symm_apply_apply ⟨x,hx⟩) - · rw [hfB ⟨x,hx⟩,hgB (d ⟨x,hx⟩)] - exact congrArg Subtype.val (d.symm_apply_apply ⟨x,hx⟩) - have hfg (x : ℂ) (hx : x ∈ A' ∪ B') : f (g x)=x := by - rcases hx with hx|hx - · rw [hgA ⟨x,hx⟩,hfA (e.symm ⟨x,hx⟩)] - exact congrArg Subtype.val (e.apply_symm_apply ⟨x,hx⟩) - · rw [hgB ⟨x,hx⟩,hfB (d.symm ⟨x,hx⟩)] - exact congrArg Subtype.val (d.apply_symm_apply ⟨x,hx⟩) - exact - { toFun := fun x => ⟨f x,hfm x.property⟩ - invFun := fun x => ⟨g x,hgm x.property⟩ - left_inv := fun x => Subtype.ext (hgf x x.property) - right_inv := fun x => Subtype.ext (hfg x x.property) - continuous_toFun := ((hfAc.union_of_isClosed hfBc hA hB).comp_continuous - continuous_subtype_val (fun x => x.property)).subtype_mk _ - continuous_invFun := ((hgAc.union_of_isClosed hgBc hA' hB').comp_continuous - continuous_subtype_val (fun x => x.property)).subtype_mk _ } - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ClosedComplexUnion + +namespace OAI + +/-! Gluing two prescribed homeomorphisms on closed pieces. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +def closedComplexPairHomeomorph {A A' B B' : Set ℂ} + (hA : IsClosed A) (hA' : IsClosed A') (hB : IsClosed B) (hB' : IsClosed B') + (e : A ≃ₜ A') (d : B ≃ₜ B') + (he : ∀ (x : A) (hx : (x:ℂ) ∈ B), (e x:ℂ)=d ⟨x,hx⟩) + (hi : ∀ (x : A') (hx : (x:ℂ) ∈ B'), (e.symm x:ℂ)=d.symm ⟨x,hx⟩) : + ↥(A ∪ B) ≃ₜ ↥(A' ∪ B') := by + classical + let f (x : ℂ) := if ha : x ∈ A then (e ⟨x,ha⟩:ℂ) + else if hb : x ∈ B then (d ⟨x,hb⟩:ℂ) else x + let g (x : ℂ) := if ha : x ∈ A' then (e.symm ⟨x,ha⟩:ℂ) + else if hb : x ∈ B' then (d.symm ⟨x,hb⟩:ℂ) else x + have hfA (x : A) : f x=(e x:ℂ) := by simp only [f,dite_eq_left x.property] + have hfB (x : B) : f x=(d x:ℂ) := by + dsimp [f] + by_cases ha : (x:ℂ) ∈ A + · rw [dite_eq_left ha] + exact he ⟨x,ha⟩ x.property + · rw [dite_eq_right ha,dite_eq_left x.property] + have hgA (x : A') : g x=(e.symm x:ℂ) := by simp only [g,dite_eq_left x.property] + have hgB (x : B') : g x=(d.symm x:ℂ) := by + dsimp [g] + by_cases ha : (x:ℂ) ∈ A' + · rw [dite_eq_left ha] + exact hi ⟨x,ha⟩ x.property + · rw [dite_eq_right ha,dite_eq_left x.property] + have hfAc : ContinuousOn f A := by + rw [continuousOn_iff_continuous_domRestrict] + exact (continuous_subtype_val.comp e.continuous).congr (fun x => (hfA x).symm) + have hfBc : ContinuousOn f B := by + rw [continuousOn_iff_continuous_domRestrict] + exact (continuous_subtype_val.comp d.continuous).congr (fun x => (hfB x).symm) + have hgAc : ContinuousOn g A' := by + rw [continuousOn_iff_continuous_domRestrict] + exact (continuous_subtype_val.comp e.symm.continuous).congr (fun x => (hgA x).symm) + have hgBc : ContinuousOn g B' := by + rw [continuousOn_iff_continuous_domRestrict] + exact (continuous_subtype_val.comp d.symm.continuous).congr (fun x => (hgB x).symm) + have hfm : MapsTo f (A ∪ B) (A' ∪ B') := by + intro x hx + rcases hx with hx|hx + · rw [hfA ⟨x,hx⟩]; exact Or.inl (e ⟨x,hx⟩).property + · rw [hfB ⟨x,hx⟩]; exact Or.inr (d ⟨x,hx⟩).property + have hgm : MapsTo g (A' ∪ B') (A ∪ B) := by + intro x hx + rcases hx with hx|hx + · rw [hgA ⟨x,hx⟩]; exact Or.inl (e.symm ⟨x,hx⟩).property + · rw [hgB ⟨x,hx⟩]; exact Or.inr (d.symm ⟨x,hx⟩).property + have hgf (x : ℂ) (hx : x ∈ A ∪ B) : g (f x)=x := by + rcases hx with hx|hx + · rw [hfA ⟨x,hx⟩,hgA (e ⟨x,hx⟩)] + exact congrArg Subtype.val (e.symm_apply_apply ⟨x,hx⟩) + · rw [hfB ⟨x,hx⟩,hgB (d ⟨x,hx⟩)] + exact congrArg Subtype.val (d.symm_apply_apply ⟨x,hx⟩) + have hfg (x : ℂ) (hx : x ∈ A' ∪ B') : f (g x)=x := by + rcases hx with hx|hx + · rw [hgA ⟨x,hx⟩,hfA (e.symm ⟨x,hx⟩)] + exact congrArg Subtype.val (e.apply_symm_apply ⟨x,hx⟩) + · rw [hgB ⟨x,hx⟩,hfB (d.symm ⟨x,hx⟩)] + exact congrArg Subtype.val (d.apply_symm_apply ⟨x,hx⟩) + exact + { toFun := fun x => ⟨f x,hfm x.property⟩ + invFun := fun x => ⟨g x,hgm x.property⟩ + left_inv := fun x => Subtype.ext (hgf x x.property) + right_inv := fun x => Subtype.ext (hfg x x.property) + continuous_toFun := ((hfAc.union_of_isClosed hfBc hA hB).comp_continuous + continuous_subtype_val (fun x => x.property)).subtype_mk _ + continuous_invFun := ((hgAc.union_of_isClosed hgBc hA' hB').comp_continuous + continuous_subtype_val (fun x => x.property)).subtype_mk _ } + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ClosedComplexUnion.lean b/lean/OAI/Probability/SLE/Geometry/ClosedComplexUnion.lean index 53ee338102cffad0900dd0938f08fe8aa47aa94f..a71775bc7b0a22b346dfc6a5966b49b3c7662852 100644 --- a/lean/OAI/Probability/SLE/Geometry/ClosedComplexUnion.lean +++ b/lean/OAI/Probability/SLE/Geometry/ClosedComplexUnion.lean @@ -1,97 +1,97 @@ -import OAI.Probability.SLE.Geometry.ComplexJordanExtension - -namespace OAI - -/-! Glue a homeomorphism on one closed piece to the identity on a second -closed piece, retaining the prescribed values on their intersection. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -def closedComplexUnionHomeomorph {A A' B : Set ℂ} - (hA : IsClosed A) (hA' : IsClosed A') (hB : IsClosed B) - (e : A ≃ₜ A') - (he : ∀ x : A, (x : ℂ) ∈ B → (e x : ℂ)=x) - (hi : ∀ x : A', (x : ℂ) ∈ B → (e.symm x : ℂ)=x) : - ↥(A ∪ B) ≃ₜ ↥(A' ∪ B) := by - classical - let f (x : ℂ) := if hx : x ∈ A then (e ⟨x,hx⟩ : ℂ) else x - let g (x : ℂ) := if hx : x ∈ A' then (e.symm ⟨x,hx⟩ : ℂ) else x - have hfA : ContinuousOn f A := by - rw [continuousOn_iff_continuous_domRestrict] - convert continuous_subtype_val.comp e.continuous using 1 - funext x - exact dite_eq_left x.property - have hgA : ContinuousOn g A' := by - rw [continuousOn_iff_continuous_domRestrict] - convert continuous_subtype_val.comp e.symm.continuous using 1 - funext x - exact dite_eq_left x.property - have hfB : ContinuousOn f B := by - apply continuousOn_id.congr - intro x hx - dsimp [f] - split_ifs with ha - · exact he ⟨x,ha⟩ hx - · rfl - have hgB : ContinuousOn g B := by - apply continuousOn_id.congr - intro x hx - dsimp [g] - split_ifs with ha - · exact hi ⟨x,ha⟩ hx - · rfl - have hf : ContinuousOn f (A ∪ B) := hfA.union_of_isClosed hfB hA hB - have hg : ContinuousOn g (A' ∪ B) := hgA.union_of_isClosed hgB hA' hB - have hfm : MapsTo f (A ∪ B) (A' ∪ B) := by - intro x hx - dsimp [f] - split_ifs with ha - · exact Or.inl (e ⟨x,ha⟩).property - · exact Or.inr (hx.resolve_left ha) - have hgm : MapsTo g (A' ∪ B) (A ∪ B) := by - intro x hx - dsimp [g] - split_ifs with ha - · exact Or.inl (e.symm ⟨x,ha⟩).property - · exact Or.inr (hx.resolve_left ha) - have hgf (x : ℂ) (hx : x ∈ A ∪ B) : g (f x)=x := by - by_cases ha : x ∈ A - · have hfa : f x=(e ⟨x,ha⟩ : ℂ) := dite_eq_left ha - rw [hfa] - dsimp [g] - rw [dite_eq_left (e ⟨x,ha⟩).property] - exact congrArg Subtype.val (e.symm_apply_apply ⟨x,ha⟩) - · rw [show f x=x from dite_eq_right ha] - dsimp [g] - split_ifs with ha' - · exact hi ⟨x,ha'⟩ (hx.resolve_left ha) - · rfl - have hfg (x : ℂ) (hx : x ∈ A' ∪ B) : f (g x)=x := by - by_cases ha : x ∈ A' - · have hga : g x=(e.symm ⟨x,ha⟩ : ℂ) := dite_eq_left ha - rw [hga] - dsimp [f] - rw [dite_eq_left (e.symm ⟨x,ha⟩).property] - exact congrArg Subtype.val (e.apply_symm_apply ⟨x,ha⟩) - · rw [show g x=x from dite_eq_right ha] - dsimp [f] - split_ifs with ha' - · exact he ⟨x,ha'⟩ (hx.resolve_left ha) - · rfl - exact - { toFun := fun x => ⟨f x,hfm x.property⟩ - invFun := fun x => ⟨g x,hgm x.property⟩ - left_inv := fun x => Subtype.ext (hgf x x.property) - right_inv := fun x => Subtype.ext (hfg x x.property) - continuous_toFun := (hf.comp_continuous continuous_subtype_val - (fun x => x.property)).subtype_mk _ - continuous_invFun := (hg.comp_continuous continuous_subtype_val - (fun x => x.property)).subtype_mk _ } - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexJordanExtension + +namespace OAI + +/-! Glue a homeomorphism on one closed piece to the identity on a second +closed piece, retaining the prescribed values on their intersection. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +def closedComplexUnionHomeomorph {A A' B : Set ℂ} + (hA : IsClosed A) (hA' : IsClosed A') (hB : IsClosed B) + (e : A ≃ₜ A') + (he : ∀ x : A, (x : ℂ) ∈ B → (e x : ℂ)=x) + (hi : ∀ x : A', (x : ℂ) ∈ B → (e.symm x : ℂ)=x) : + ↥(A ∪ B) ≃ₜ ↥(A' ∪ B) := by + classical + let f (x : ℂ) := if hx : x ∈ A then (e ⟨x,hx⟩ : ℂ) else x + let g (x : ℂ) := if hx : x ∈ A' then (e.symm ⟨x,hx⟩ : ℂ) else x + have hfA : ContinuousOn f A := by + rw [continuousOn_iff_continuous_domRestrict] + convert continuous_subtype_val.comp e.continuous using 1 + funext x + exact dite_eq_left x.property + have hgA : ContinuousOn g A' := by + rw [continuousOn_iff_continuous_domRestrict] + convert continuous_subtype_val.comp e.symm.continuous using 1 + funext x + exact dite_eq_left x.property + have hfB : ContinuousOn f B := by + apply continuousOn_id.congr + intro x hx + dsimp [f] + split_ifs with ha + · exact he ⟨x,ha⟩ hx + · rfl + have hgB : ContinuousOn g B := by + apply continuousOn_id.congr + intro x hx + dsimp [g] + split_ifs with ha + · exact hi ⟨x,ha⟩ hx + · rfl + have hf : ContinuousOn f (A ∪ B) := hfA.union_of_isClosed hfB hA hB + have hg : ContinuousOn g (A' ∪ B) := hgA.union_of_isClosed hgB hA' hB + have hfm : MapsTo f (A ∪ B) (A' ∪ B) := by + intro x hx + dsimp [f] + split_ifs with ha + · exact Or.inl (e ⟨x,ha⟩).property + · exact Or.inr (hx.resolve_left ha) + have hgm : MapsTo g (A' ∪ B) (A ∪ B) := by + intro x hx + dsimp [g] + split_ifs with ha + · exact Or.inl (e.symm ⟨x,ha⟩).property + · exact Or.inr (hx.resolve_left ha) + have hgf (x : ℂ) (hx : x ∈ A ∪ B) : g (f x)=x := by + by_cases ha : x ∈ A + · have hfa : f x=(e ⟨x,ha⟩ : ℂ) := dite_eq_left ha + rw [hfa] + dsimp [g] + rw [dite_eq_left (e ⟨x,ha⟩).property] + exact congrArg Subtype.val (e.symm_apply_apply ⟨x,ha⟩) + · rw [show f x=x from dite_eq_right ha] + dsimp [g] + split_ifs with ha' + · exact hi ⟨x,ha'⟩ (hx.resolve_left ha) + · rfl + have hfg (x : ℂ) (hx : x ∈ A' ∪ B) : f (g x)=x := by + by_cases ha : x ∈ A' + · have hga : g x=(e.symm ⟨x,ha⟩ : ℂ) := dite_eq_left ha + rw [hga] + dsimp [f] + rw [dite_eq_left (e.symm ⟨x,ha⟩).property] + exact congrArg Subtype.val (e.apply_symm_apply ⟨x,ha⟩) + · rw [show g x=x from dite_eq_right ha] + dsimp [f] + split_ifs with ha' + · exact he ⟨x,ha'⟩ (hx.resolve_left ha) + · rfl + exact + { toFun := fun x => ⟨f x,hfm x.property⟩ + invFun := fun x => ⟨g x,hgm x.property⟩ + left_inv := fun x => Subtype.ext (hgf x x.property) + right_inv := fun x => Subtype.ext (hfg x x.property) + continuous_toFun := (hf.comp_continuous continuous_subtype_val + (fun x => x.property)).subtype_mk _ + continuous_invFun := (hg.comp_continuous continuous_subtype_val + (fun x => x.property)).subtype_mk _ } + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneCayley.lean b/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneCayley.lean index f04a94c52a0e3c2fac3a1134638a7687d5d09cf6..16e521d9925df1226d6f76cf6a7d8d18706e41d8 100644 --- a/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneCayley.lean +++ b/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneCayley.lean @@ -1,28 +1,28 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneCayleyInverse - -namespace OAI - -/-! The fixed Cayley chart as a homeomorphism of the closed half-plane -onto the closed disk with the point at infinity removed. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -def closedHalfPlaneCayley : - {z : ℂ // 0 ≤ z.im} ≃ₜ {w : ℂ // ‖w‖ ≤ 1 ∧ w ≠ 1} where - toFun z := ⟨halfPlaneCayley z,halfPlaneCayley_norm_le z.property, - halfPlaneCayley_ne_one z.property⟩ - invFun w := ⟨halfPlaneCayleyInv w,halfPlaneCayleyInv_closed w.property.1⟩ - left_inv z := Subtype.ext (halfPlaneCayleyInv_cayley z.property) - right_inv w := Subtype.ext (halfPlaneCayley_cayleyInv w.property.2) - continuous_toFun := (halfPlaneCayley_continuousOn.comp_continuous continuous_subtype_val - (fun z => z.property)).subtype_mk _ - continuous_invFun := (halfPlaneCayleyInv_continuousOn.comp_continuous continuous_subtype_val - (fun w => w.property.2)).subtype_mk _ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfPlaneCayleyInverse + +namespace OAI + +/-! The fixed Cayley chart as a homeomorphism of the closed half-plane +onto the closed disk with the point at infinity removed. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +def closedHalfPlaneCayley : + {z : ℂ // 0 ≤ z.im} ≃ₜ {w : ℂ // ‖w‖ ≤ 1 ∧ w ≠ 1} where + toFun z := ⟨halfPlaneCayley z,halfPlaneCayley_norm_le z.property, + halfPlaneCayley_ne_one z.property⟩ + invFun w := ⟨halfPlaneCayleyInv w,halfPlaneCayleyInv_closed w.property.1⟩ + left_inv z := Subtype.ext (halfPlaneCayleyInv_cayley z.property) + right_inv w := Subtype.ext (halfPlaneCayley_cayleyInv w.property.2) + continuous_toFun := (halfPlaneCayley_continuousOn.comp_continuous continuous_subtype_val + (fun z => z.property)).subtype_mk _ + continuous_invFun := (halfPlaneCayleyInv_continuousOn.comp_continuous continuous_subtype_val + (fun w => w.property.2)).subtype_mk _ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneExtension.lean b/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneExtension.lean index c9919c916b82b74e92e64f530ab3cf5cf06cff40..6724e791e042f9a5808dc9594c4b51b164896529 100644 --- a/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ClosedHalfPlaneExtension.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Geometry.ClosedComplexUnion - -namespace OAI - -/-! Extend a closed half-plane homeomorphism fixing its boundary by the -identity on the other half-plane. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -theorem closedHalfPlane_extension - (e : {z : ℂ // 0 ≤ z.im} ≃ₜ {z : ℂ // 0 ≤ z.im}) - (he : ∀ (z : {z : ℂ // 0 ≤ z.im}), z.val.im=0 → (e z:ℂ)=z) : - ∃ F : ℂ ≃ₜ ℂ, - (∀ (z : ℂ) (hz : 0 ≤ z.im), F z=(e ⟨z,hz⟩:ℂ)) ∧ - (∀ x : ℝ, F (x:ℂ)=(x:ℂ)) ∧ - MapsTo F {z : ℂ | 0 < z.im} {z : ℂ | 0 < z.im} := by - classical - let A : Set ℂ := {z | 0 ≤ z.im} - let B : Set ℂ := {z | z.im ≤ 0} - have hA : IsClosed A := isClosed_le continuous_const Complex.continuous_im - have hB : IsClosed B := isClosed_le Complex.continuous_im continuous_const - have hfix (z : A) (hz : (z:ℂ) ∈ B) : (e z:ℂ)=z := - he z (le_antisymm hz z.property) - have hfixi (z : A) (hz : (z:ℂ) ∈ B) : (e.symm z:ℂ)=z := by - have hh : e z=z := Subtype.ext (hfix z hz) - have hh' := congrArg e.symm hh - rw [e.symm_apply_apply] at hh' - exact congrArg Subtype.val hh'.symm - let E := closedComplexUnionHomeomorph hA hA hB e hfix hfixi - have hU : A ∪ B=univ := by - ext z - exact iff_true_intro (le_total 0 z.im) - let j : ℂ ≃ₜ ↥(A ∪ B) := (Homeomorph.Set.univ ℂ).symm.trans - (Homeomorph.setCongr hU.symm) - let F : ℂ ≃ₜ ℂ := (j.trans E).trans j.symm - have hF (z : ℂ) (hz : 0 ≤ z.im) : F z=(e ⟨z,hz⟩:ℂ) := by - change (if hx : z ∈ A then (e ⟨z,hx⟩:ℂ) else z)=_ - rw [dite_eq_left (show z ∈ A from hz)] - have hFr (x : ℝ) : F (x:ℂ)=(x:ℂ) := by - rw [hF _ (by simp)] - exact he _ (by simp) - refine ⟨F,hF,hFr,?_⟩ - intro z hz - have hn : 0 ≤ (F z).im := by - rw [hF z hz.le] - exact (e ⟨z,hz.le⟩).property - apply lt_of_le_of_ne hn - intro hzero - have hx : F z=((F z).re:ℂ) := Complex.ext rfl (by simpa using hzero.symm) - have hsame : z=((F z).re:ℂ) := F.injective (hx.trans (hFr _).symm) - have hz0 := congrArg Complex.im hsame - simpa only [Complex.ofReal_im] using hz.ne' hz0 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ClosedComplexUnion + +namespace OAI + +/-! Extend a closed half-plane homeomorphism fixing its boundary by the +identity on the other half-plane. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +theorem closedHalfPlane_extension + (e : {z : ℂ // 0 ≤ z.im} ≃ₜ {z : ℂ // 0 ≤ z.im}) + (he : ∀ (z : {z : ℂ // 0 ≤ z.im}), z.val.im=0 → (e z:ℂ)=z) : + ∃ F : ℂ ≃ₜ ℂ, + (∀ (z : ℂ) (hz : 0 ≤ z.im), F z=(e ⟨z,hz⟩:ℂ)) ∧ + (∀ x : ℝ, F (x:ℂ)=(x:ℂ)) ∧ + MapsTo F {z : ℂ | 0 < z.im} {z : ℂ | 0 < z.im} := by + classical + let A : Set ℂ := {z | 0 ≤ z.im} + let B : Set ℂ := {z | z.im ≤ 0} + have hA : IsClosed A := isClosed_le continuous_const Complex.continuous_im + have hB : IsClosed B := isClosed_le Complex.continuous_im continuous_const + have hfix (z : A) (hz : (z:ℂ) ∈ B) : (e z:ℂ)=z := + he z (le_antisymm hz z.property) + have hfixi (z : A) (hz : (z:ℂ) ∈ B) : (e.symm z:ℂ)=z := by + have hh : e z=z := Subtype.ext (hfix z hz) + have hh' := congrArg e.symm hh + rw [e.symm_apply_apply] at hh' + exact congrArg Subtype.val hh'.symm + let E := closedComplexUnionHomeomorph hA hA hB e hfix hfixi + have hU : A ∪ B=univ := by + ext z + exact iff_true_intro (le_total 0 z.im) + let j : ℂ ≃ₜ ↥(A ∪ B) := (Homeomorph.Set.univ ℂ).symm.trans + (Homeomorph.setCongr hU.symm) + let F : ℂ ≃ₜ ℂ := (j.trans E).trans j.symm + have hF (z : ℂ) (hz : 0 ≤ z.im) : F z=(e ⟨z,hz⟩:ℂ) := by + change (if hx : z ∈ A then (e ⟨z,hx⟩:ℂ) else z)=_ + rw [dite_eq_left (show z ∈ A from hz)] + have hFr (x : ℝ) : F (x:ℂ)=(x:ℂ) := by + rw [hF _ (by simp)] + exact he _ (by simp) + refine ⟨F,hF,hFr,?_⟩ + intro z hz + have hn : 0 ≤ (F z).im := by + rw [hF z hz.le] + exact (e ⟨z,hz.le⟩).property + apply lt_of_le_of_ne hn + intro hzero + have hx : F z=((F z).re:ℂ) := Complex.ext rfl (by simpa using hzero.symm) + have hsame : z=((F z).re:ℂ) := F.injective (hx.trans (hFr _).symm) + have hz0 := congrArg Complex.im hsame + simpa only [Complex.ofReal_im] using hz.ne' hz0 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CoefficientAngularBound.lean b/lean/OAI/Probability/SLE/Geometry/CoefficientAngularBound.lean index 8fe19777f5c31c61d712961b281d436771176484..782148d552832073829c35956d85d41f3f8cfe99 100644 --- a/lean/OAI/Probability/SLE/Geometry/CoefficientAngularBound.lean +++ b/lean/OAI/Probability/SLE/Geometry/CoefficientAngularBound.lean @@ -1,70 +1,70 @@ -import OAI.Probability.SLE.Geometry.CanonicalCoefficientIntegrability - -namespace OAI - -/-! One integrable angular majorant for every clipping and center level -in the exact Wiener coefficient formula. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -include C in -theorem uniform_clippedCoefficient_bound - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) (m : ℕ) : - ∃ M : ℝ, 0≤M ∧ ∀ θ∈Ioo (0:ℝ) Real.pi, ∀ ℓ : ℝ, 0≤ℓ → ∀ u : ℝ, 0≤u → - 0≤Target.clippedCoefficient κ W ℓ u m θ ∧ - Target.clippedCoefficient κ W ℓ u m θ≤M/(Real.sin θ)^greenExponent κ := by - obtain ⟨M,hM,hbound⟩ := C.uniform_density_disk_terminal_moments hB hc h0 hκ hκ8 F hw hgrowth m - refine ⟨M.toReal,ENNReal.toReal_nonneg,?_⟩ - intro θ hθ ℓ hℓ u hu - let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) (T ω) - let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ - have hT : Measurable T := normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ - have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T hT - have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp - (Target.normalizedDriver_measurable κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) - have hw0 (ω) : 0≤w ω := Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) - (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ - have hd : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤C.diskTerminalScalar θ hθ ω := by - filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω - exact ⟨hw0 ω,C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω,hω⟩ - have hi := C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ - have hb := weighted_clip_integral_bound P Y (C.diskTerminalScalar θ hθ) w - hY.aestronglyMeasurable hwM.aestronglyMeasurable hd m hi u hu - rw [C.clippedCoefficient_eq_mass hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu m] - refine ⟨hb.1,hb.2.trans ?_⟩ - have hs : 0 mul_nonneg (hw0 ω) (pow_nonneg (compactWeightedMass_nonneg _ _ _) m)) - have hEn : ENNReal.ofReal (∫ ω, w ω*(C.diskTerminalScalar θ hθ ω)^m ∂P) ≤ - M*ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) := by - rw [ofReal_integral_eq_lintegral_ofReal hi] - · exact hbound θ hθ ℓ hℓ - · exact Eventually.of_forall (fun ω => mul_nonneg (hw0 ω) - (pow_nonneg (compactWeightedMass_nonneg _ _ _) m)) - have ht := ENNReal.toReal_mono - (ne_top_of_lt (ENNReal.mul_lt_top hM ENNReal.ofReal_lt_top)) hEn - rw [ENNReal.toReal_ofReal hI0,ENNReal.toReal_mul,ENNReal.toReal_ofReal - (one_div_nonneg.mpr (Real.rpow_nonneg hs.le _))] at ht - simpa only [div_eq_mul_inv,mul_one,one_mul] using ht - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalCoefficientIntegrability + +namespace OAI + +/-! One integrable angular majorant for every clipping and center level +in the exact Wiener coefficient formula. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +include C in +theorem uniform_clippedCoefficient_bound + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) (m : ℕ) : + ∃ M : ℝ, 0≤M ∧ ∀ θ∈Ioo (0:ℝ) Real.pi, ∀ ℓ : ℝ, 0≤ℓ → ∀ u : ℝ, 0≤u → + 0≤Target.clippedCoefficient κ W ℓ u m θ ∧ + Target.clippedCoefficient κ W ℓ u m θ≤M/(Real.sin θ)^greenExponent κ := by + obtain ⟨M,hM,hbound⟩ := C.uniform_density_disk_terminal_moments hB hc h0 hκ hκ8 F hw hgrowth m + refine ⟨M.toReal,ENNReal.toReal_nonneg,?_⟩ + intro θ hθ ℓ hℓ u hu + let T := fun ω => Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + let Y := fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) (T ω) + let w := fun ω => Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ + have hT : Measurable T := normalized_levelTime_measurable (hBm:=hBm) hκ hc h0 F hw θ ℓ hℓ + have hY : Measurable Y := C.normalizedStoppedDisk_measurable θ hθ T hT + have hwM : Measurable w := (Target.measurable_fixedLevelDensity κ θ ℓ).comp + (Target.normalizedDriver_measurable κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) + have hw0 (ω) : 0≤w ω := Target.levelDensity_nonneg_of_witness κ hκ _ _ ((hc ω).const_mul _) + (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ + have hd : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤C.diskTerminalScalar θ hθ ω := by + filter_upwards [C.normalizedStoppedDisk_le_terminal hκ θ hθ T] with ω hω + exact ⟨hw0 ω,C.normalizedStoppedDisk_nonneg θ hθ (T ω) ω,hω⟩ + have hi := C.density_disk_terminal_integrable hB hc h0 hκ hκ8 F hw hgrowth m θ hθ ℓ hℓ + have hb := weighted_clip_integral_bound P Y (C.diskTerminalScalar θ hθ) w + hY.aestronglyMeasurable hwM.aestronglyMeasurable hd m hi u hu + rw [C.clippedCoefficient_eq_mass hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu m] + refine ⟨hb.1,hb.2.trans ?_⟩ + have hs : 0 mul_nonneg (hw0 ω) (pow_nonneg (compactWeightedMass_nonneg _ _ _) m)) + have hEn : ENNReal.ofReal (∫ ω, w ω*(C.diskTerminalScalar θ hθ ω)^m ∂P) ≤ + M*ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) := by + rw [ofReal_integral_eq_lintegral_ofReal hi] + · exact hbound θ hθ ℓ hℓ + · exact Eventually.of_forall (fun ω => mul_nonneg (hw0 ω) + (pow_nonneg (compactWeightedMass_nonneg _ _ _) m)) + have ht := ENNReal.toReal_mono + (ne_top_of_lt (ENNReal.mul_lt_top hM ENNReal.ofReal_lt_top)) hEn + rw [ENNReal.toReal_ofReal hI0,ENNReal.toReal_mul,ENNReal.toReal_ofReal + (one_div_nonneg.mpr (Real.rpow_nonneg hs.le _))] at ht + simpa only [div_eq_mul_inv,mul_one,one_mul] using ht + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactCellTail.lean b/lean/OAI/Probability/SLE/Geometry/CompactCellTail.lean index 5e7858aefc34afc4d9783731d2c83075c93d5c66..defbf79a771837d4b0ca9e33f077536a359ade5a 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactCellTail.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactCellTail.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Gauge.PrescribedCellNormalization -import OAI.Probability.SLE.Mass.CompactEntryLevels -import OAI.Probability.SLE.Gauge.CanonicalLateGridLower - -namespace OAI - -/-! The actual finite centered cell bounds, integrated over a compact -spatial cell, supply the late-grid hypothesis for the prescribed gauge. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance compactCellTailMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance compactCellTailBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.compact_cell_tail_of_angle_bound - {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (W : Target.WienerLaw) (e : Target.GaugeIndex κ W) - (δ : ℝ) (hδ : 0<δ) - (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → - ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤ - 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) - (K : Set upperHalfPlane) (hK : IsCompact K) (c : ℂ) - (hKball : (Subtype.val : upperHalfPlane → ℂ) '' K ⊆ - closedBall c (2*Target.radii κ W e)) - (V : ℝ) - (hbase : ∀ z∈K, δ≤Target.entryPhysicalLevel κ W e.val (4096*Real.pi) z) - (hV : ∀ z∈K, Target.entryPhysicalLevel κ W e.val (4096*Real.pi) z B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (W : Target.WienerLaw) (e : Target.GaugeIndex κ W) + (δ : ℝ) (hδ : 0<δ) + (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f → + ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤ + 2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) + (K : Set upperHalfPlane) (hK : IsCompact K) (c : ℂ) + (hKball : (Subtype.val : upperHalfPlane → ℂ) '' K ⊆ + closedBall c (2*Target.radii κ W e)) + (V : ℝ) + (hbase : ∀ z∈K, δ≤Target.entryPhysicalLevel κ W e.val (4096*Real.pi) z) + (hV : ∀ z∈K, Target.entryPhysicalLevel κ W e.val (4096*Real.pi) z hV a) - refine ⟨δ,hδ,fun x y hxy => ?_⟩ - obtain ⟨a,ha⟩ := hcover x (mem_univ x) - have hsub : ball x δ⊆V a := by - intro z hz - apply ha - change dist x z<δ - simpa only [mem_ball,dist_comm] using hz - refine ⟨V a,hc a,hsub (mem_ball_self hδ),hsub (by simpa only [mem_ball,dist_comm] using hxy),?_⟩ - intro u hu v hv - have hu' := mem_ball.mp (hs a hu) - have hv' := mem_ball.mp (hs a hv) - have := dist_triangle u a v - rw [dist_comm a v] at this - linarith - -lemma finiteTrace_uniform_connection {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) - (T : ℝ≥0) (ε : ℝ) (hε : 0<ε) : - ∃ δ : ℝ, 0<δ ∧ ∀ x∈γ '' Icc 0 T, ∀ y∈γ '' Icc 0 T, dist x y<δ → - ∃ C : Set ℂ, C⊆γ '' Icc 0 T ∧ IsPreconnected C ∧ x∈C ∧ y∈C ∧ - ∀ u∈C, ∀ v∈C, dist u v<ε := by - have : CompactSpace (γ '' Icc 0 T) := isCompact_iff_compactSpace.mp (finiteTrace_isCompact hγ T) - have : LocallyConnectedSpace (γ '' Icc 0 T) := finiteTrace_locallyConnected hγ T - obtain ⟨δ,hδ,hconn⟩ := compact_locallyConnected_uniform_connection - (A := γ '' Icc 0 T) ε hε - refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ - obtain ⟨V,hV,hxV,hyV,hd⟩ := hconn ⟨x,hx⟩ ⟨y,hy⟩ hxy - refine ⟨Subtype.val '' V,?_,hV.image _ continuous_subtype_val.continuousOn, - ⟨⟨x,hx⟩,hxV,rfl⟩,⟨⟨y,hy⟩,hyV,rfl⟩,?_⟩ - · rintro _ ⟨z,_,rfl⟩ - exact z.property - · rintro _ ⟨u,hu,rfl⟩ _ ⟨v,hv,rfl⟩ - exact hd u hu v hv - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteTraceTopology + +namespace OAI + +/-! Uniform small connected neighborhoods on compact locally connected + metric spaces, specialized below to finite trace segments. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma compact_locallyConnected_uniform_connection {A : Type*} [MetricSpace A] + [CompactSpace A] [LocallyConnectedSpace A] (ε : ℝ) (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ x y : A, dist x y<δ → + ∃ V : Set A, IsPreconnected V ∧ x∈V ∧ y∈V ∧ + ∀ u∈V, ∀ v∈V, dist u v<ε := by + classical + have hlocal (a : A) : ∃ V : Set A, V∈𝓝 a ∧ IsPreconnected V ∧ + V⊆ball a (ε/2) := + (locallyConnectedSpace_iff_connected_subsets.mp inferInstance) a _ + (ball_mem_nhds a (half_pos hε)) + choose V hV hc hs using hlocal + obtain ⟨δ,hδ,hcover⟩ := Metric.uniformity_basis_dist.lebesgue_number_lemma_nhds + (K := (univ : Set A)) isCompact_univ (fun a _ => hV a) + refine ⟨δ,hδ,fun x y hxy => ?_⟩ + obtain ⟨a,ha⟩ := hcover x (mem_univ x) + have hsub : ball x δ⊆V a := by + intro z hz + apply ha + change dist x z<δ + simpa only [mem_ball,dist_comm] using hz + refine ⟨V a,hc a,hsub (mem_ball_self hδ),hsub (by simpa only [mem_ball,dist_comm] using hxy),?_⟩ + intro u hu v hv + have hu' := mem_ball.mp (hs a hu) + have hv' := mem_ball.mp (hs a hv) + have := dist_triangle u a v + rw [dist_comm a v] at this + linarith + +lemma finiteTrace_uniform_connection {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) + (T : ℝ≥0) (ε : ℝ) (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ x∈γ '' Icc 0 T, ∀ y∈γ '' Icc 0 T, dist x y<δ → + ∃ C : Set ℂ, C⊆γ '' Icc 0 T ∧ IsPreconnected C ∧ x∈C ∧ y∈C ∧ + ∀ u∈C, ∀ v∈C, dist u v<ε := by + have : CompactSpace (γ '' Icc 0 T) := isCompact_iff_compactSpace.mp (finiteTrace_isCompact hγ T) + have : LocallyConnectedSpace (γ '' Icc 0 T) := finiteTrace_locallyConnected hγ T + obtain ⟨δ,hδ,hconn⟩ := compact_locallyConnected_uniform_connection + (A := γ '' Icc 0 T) ε hε + refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ + obtain ⟨V,hV,hxV,hyV,hd⟩ := hconn ⟨x,hx⟩ ⟨y,hy⟩ hxy + refine ⟨Subtype.val '' V,?_,hV.image _ continuous_subtype_val.continuousOn, + ⟨⟨x,hx⟩,hxV,rfl⟩,⟨⟨y,hy⟩,hyV,rfl⟩,?_⟩ + · rintro _ ⟨z,_,rfl⟩ + exact z.property + · rintro _ ⟨u,hu,rfl⟩ _ ⟨v,hv,rfl⟩ + exact hd u hu v hv + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactDerivativeIntegral.lean b/lean/OAI/Probability/SLE/Geometry/CompactDerivativeIntegral.lean index 6a3a5e8b54aeff4b7ff591e3955eaae24882f15c..2d61eaccfb61bb2a401aff9cdca32699b4926050 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactDerivativeIntegral.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactDerivativeIntegral.lean @@ -1,57 +1,57 @@ -import Mathlib - -namespace OAI - -/-! Compactly supported derivative integrals used in the exterior -area calculation. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -lemma integral_deriv_compact_eq_zero {f : ℝ → ℝ} - (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : ∫ x, deriv f x=0 := by - have hi : Integrable (deriv f) := (hf.continuous_deriv le_rfl).integrable_of_hasCompactSupport hs.deriv - rw [←integral_add_compl measurableSet_Iic hi,compl_Iic, - hs.integral_Iic_deriv_eq hf 0,hs.integral_Ioi_deriv_eq hf 0] - ring - -lemma integral_complex_real_partial_eq_zero {f : ℂ → ℝ} - (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : - ∫ z, fderiv ℝ f z 1=0 := by - let D (z : ℂ) := fderiv ℝ f z 1 - have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const - have hDi : Integrable D := hD.integrable_of_hasCompactSupport (hs.fderiv_apply ℝ 1) - have hline (y : ℝ) : ∫ x : ℝ, D ((x:ℂ)+(y:ℂ)*Complex.I)=0 := by - let L (x : ℝ) := (x:ℂ)+(y:ℂ)*Complex.I - have hLc : ContDiff ℝ 1 L := Complex.ofRealCLM.contDiff.add contDiff_const - have hLi : Isometry L := by - apply isometry_iff_dist_eq.mpr - intro x x' - simpa only [L,dist_add_right] using Complex.isometry_ofReal.dist_eq x x' - have hcs : HasCompactSupport (f ∘ L) := hs.comp_isClosedEmbedding hLi.isClosedEmbedding - have he (x : ℝ) : deriv (f ∘ L) x=D (L x) := by - have hd := (hf.differentiable (by norm_num) (L x)).hasFDerivAt.comp_hasDerivAt x - ((Complex.ofRealCLM.hasDerivAt (x := x)).add_const ((y:ℂ)*Complex.I)) - simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using hd.deriv - simpa only [he] using integral_deriv_compact_eq_zero (hf.comp hLc) hcs - let e := Complex.measurableEquivRealProd.symm - have hp := Complex.volume_preserving_equiv_real_prod.symm - have hprod : Integrable (fun p : ℝ × ℝ => D (e p)) := - (hp.integrable_comp_emb e.measurableEmbedding).mpr hDi - rw [←hp.integral_comp e.measurableEmbedding] - change (∫ p : ℝ × ℝ, D (e p))=0 - rw [Measure.volume_eq_prod] at hprod ⊢ - rw [integral_prod_symm _ hprod] - have he (y : ℝ) : (∫ x : ℝ, D (e (x,y)))=0 := by - have heq (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by - apply Complex.ext <;> simp [e,Complex.measurableEquivRealProd_symm_apply] - simpa only [heq] using hline y - simp only [he,integral_zero] - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! Compactly supported derivative integrals used in the exterior +area calculation. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +lemma integral_deriv_compact_eq_zero {f : ℝ → ℝ} + (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : ∫ x, deriv f x=0 := by + have hi : Integrable (deriv f) := (hf.continuous_deriv le_rfl).integrable_of_hasCompactSupport hs.deriv + rw [←integral_add_compl measurableSet_Iic hi,compl_Iic, + hs.integral_Iic_deriv_eq hf 0,hs.integral_Ioi_deriv_eq hf 0] + ring + +lemma integral_complex_real_partial_eq_zero {f : ℂ → ℝ} + (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : + ∫ z, fderiv ℝ f z 1=0 := by + let D (z : ℂ) := fderiv ℝ f z 1 + have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const + have hDi : Integrable D := hD.integrable_of_hasCompactSupport (hs.fderiv_apply ℝ 1) + have hline (y : ℝ) : ∫ x : ℝ, D ((x:ℂ)+(y:ℂ)*Complex.I)=0 := by + let L (x : ℝ) := (x:ℂ)+(y:ℂ)*Complex.I + have hLc : ContDiff ℝ 1 L := Complex.ofRealCLM.contDiff.add contDiff_const + have hLi : Isometry L := by + apply isometry_iff_dist_eq.mpr + intro x x' + simpa only [L,dist_add_right] using Complex.isometry_ofReal.dist_eq x x' + have hcs : HasCompactSupport (f ∘ L) := hs.comp_isClosedEmbedding hLi.isClosedEmbedding + have he (x : ℝ) : deriv (f ∘ L) x=D (L x) := by + have hd := (hf.differentiable (by norm_num) (L x)).hasFDerivAt.comp_hasDerivAt x + ((Complex.ofRealCLM.hasDerivAt (x := x)).add_const ((y:ℂ)*Complex.I)) + simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using hd.deriv + simpa only [he] using integral_deriv_compact_eq_zero (hf.comp hLc) hcs + let e := Complex.measurableEquivRealProd.symm + have hp := Complex.volume_preserving_equiv_real_prod.symm + have hprod : Integrable (fun p : ℝ × ℝ => D (e p)) := + (hp.integrable_comp_emb e.measurableEmbedding).mpr hDi + rw [←hp.integral_comp e.measurableEmbedding] + change (∫ p : ℝ × ℝ, D (e p))=0 + rw [Measure.volume_eq_prod] at hprod ⊢ + rw [integral_prod_symm _ hprod] + have he (y : ℝ) : (∫ x : ℝ, D (e (x,y)))=0 := by + have heq (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by + apply Complex.ext <;> simp [e,Complex.measurableEquivRealProd_symm_apply] + simpa only [heq] using hline y + simp only [he,integral_zero] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactLaplacianIntegral.lean b/lean/OAI/Probability/SLE/Geometry/CompactLaplacianIntegral.lean index 07e44dc949cee7de17b978b1adc12822daa5d87b..cfaa845dde950a3c03a54e7e6c71a0d51e956eea 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactLaplacianIntegral.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactLaplacianIntegral.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Geometry.CompactDerivativeIntegral -import Mathlib.Analysis.InnerProductSpace.Laplacian - -namespace OAI - -/-! A compactly supported smooth planar test has zero total Laplacian. -This lets its integral over a conformal image be evaluated on the -compact omitted set instead. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma integral_complex_imag_partial_eq_zero {f : ℂ → ℝ} - (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : - ∫ z, fderiv ℝ f z Complex.I=0 := by - let D (z : ℂ) := fderiv ℝ f z Complex.I - have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const - have hDi : Integrable D := hD.integrable_of_hasCompactSupport (hs.fderiv_apply ℝ Complex.I) - have hline (x : ℝ) : ∫ y : ℝ, D ((x:ℂ)+(y:ℂ)*Complex.I)=0 := by - let L (y : ℝ) := (x:ℂ)+(y:ℂ)*Complex.I - have hLc : ContDiff ℝ 1 L := contDiff_const.add (Complex.ofRealCLM.contDiff.mul contDiff_const) - have hLi : Isometry L := by - apply isometry_iff_dist_eq.mpr - intro y y' - rw [dist_eq_norm,show L y-L y'=((y-y':ℝ):ℂ)*Complex.I by dsimp [L]; push_cast; ring] - simp only [norm_mul,Complex.norm_I,mul_one,Complex.norm_real,dist_eq_norm] - have hcs : HasCompactSupport (f ∘ L) := hs.comp_isClosedEmbedding hLi.isClosedEmbedding - have he (y : ℝ) : deriv (f ∘ L) y=D (L y) := by - have hd := (hf.differentiable (by norm_num) (L y)).hasFDerivAt.comp_hasDerivAt y - (((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ)) - simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul] using hd.deriv - simpa only [he] using integral_deriv_compact_eq_zero (hf.comp hLc) hcs - let e := Complex.measurableEquivRealProd.symm - have hp := Complex.volume_preserving_equiv_real_prod.symm - have hprod : Integrable (fun p : ℝ × ℝ => D (e p)) := - (hp.integrable_comp_emb e.measurableEmbedding).mpr hDi - rw [←hp.integral_comp e.measurableEmbedding] - change (∫ p : ℝ × ℝ, D (e p))=0 - rw [Measure.volume_eq_prod] at hprod ⊢ - rw [integral_prod _ hprod] - have he (x : ℝ) : (∫ y : ℝ, D (e (x,y)))=0 := by - have heq (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by - apply Complex.ext <;> simp [e,Complex.measurableEquivRealProd_symm_apply] - simpa only [heq] using hline x - simp only [he,integral_zero] - -lemma integral_laplacian_compact_eq_zero {f : ℂ → ℝ} - (hf : ContDiff ℝ 2 f) (hs : HasCompactSupport f) : ∫ z, (Δ f) z=0 := by - let A (z : ℂ) := fderiv ℝ f z 1 - let B (z : ℂ) := fderiv ℝ f z Complex.I - have hDf : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num) - have hA : ContDiff ℝ 1 A := hDf.clm_apply contDiff_const - have hB : ContDiff ℝ 1 B := hDf.clm_apply contDiff_const - have hsA : HasCompactSupport A := hs.fderiv_apply ℝ 1 - have hsB : HasCompactSupport B := hs.fderiv_apply ℝ Complex.I - have hAA (z : ℂ) : fderiv ℝ A z 1=fderiv ℝ (fderiv ℝ f) z 1 1 := by - have h := (hDf.differentiable (by norm_num) z).hasFDerivAt.clm_apply - (hasFDerivAt_const (1:ℂ) z) - rw [h.fderiv] - simp - have hBB (z : ℂ) : fderiv ℝ B z Complex.I=fderiv ℝ (fderiv ℝ f) z Complex.I Complex.I := by - have h := (hDf.differentiable (by norm_num) z).hasFDerivAt.clm_apply - (hasFDerivAt_const Complex.I z) - rw [h.fderiv] - simp - have hpoint (z : ℂ) : (Δ f) z=fderiv ℝ A z 1+fderiv ℝ B z Complex.I := by - rw [hAA,hBB,laplacian_eq_iteratedFDeriv_complexPlane] - simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, - Matrix.cons_val_one,Matrix.cons_val_fin_one] - simp_rw [hpoint] - rw [integral_add - (((hA.continuous_fderiv (by norm_num)).clm_apply continuous_const).integrable_of_hasCompactSupport (hsA.fderiv_apply ℝ 1)) - (((hB.continuous_fderiv (by norm_num)).clm_apply continuous_const).integrable_of_hasCompactSupport (hsB.fderiv_apply ℝ Complex.I)), - integral_complex_real_partial_eq_zero hA hsA,integral_complex_imag_partial_eq_zero hB hsB,add_zero] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactDerivativeIntegral +import Mathlib.Analysis.InnerProductSpace.Laplacian + +namespace OAI + +/-! A compactly supported smooth planar test has zero total Laplacian. +This lets its integral over a conformal image be evaluated on the +compact omitted set instead. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma integral_complex_imag_partial_eq_zero {f : ℂ → ℝ} + (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : + ∫ z, fderiv ℝ f z Complex.I=0 := by + let D (z : ℂ) := fderiv ℝ f z Complex.I + have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const + have hDi : Integrable D := hD.integrable_of_hasCompactSupport (hs.fderiv_apply ℝ Complex.I) + have hline (x : ℝ) : ∫ y : ℝ, D ((x:ℂ)+(y:ℂ)*Complex.I)=0 := by + let L (y : ℝ) := (x:ℂ)+(y:ℂ)*Complex.I + have hLc : ContDiff ℝ 1 L := contDiff_const.add (Complex.ofRealCLM.contDiff.mul contDiff_const) + have hLi : Isometry L := by + apply isometry_iff_dist_eq.mpr + intro y y' + rw [dist_eq_norm,show L y-L y'=((y-y':ℝ):ℂ)*Complex.I by dsimp [L]; push_cast; ring] + simp only [norm_mul,Complex.norm_I,mul_one,Complex.norm_real,dist_eq_norm] + have hcs : HasCompactSupport (f ∘ L) := hs.comp_isClosedEmbedding hLi.isClosedEmbedding + have he (y : ℝ) : deriv (f ∘ L) y=D (L y) := by + have hd := (hf.differentiable (by norm_num) (L y)).hasFDerivAt.comp_hasDerivAt y + (((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ)) + simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul] using hd.deriv + simpa only [he] using integral_deriv_compact_eq_zero (hf.comp hLc) hcs + let e := Complex.measurableEquivRealProd.symm + have hp := Complex.volume_preserving_equiv_real_prod.symm + have hprod : Integrable (fun p : ℝ × ℝ => D (e p)) := + (hp.integrable_comp_emb e.measurableEmbedding).mpr hDi + rw [←hp.integral_comp e.measurableEmbedding] + change (∫ p : ℝ × ℝ, D (e p))=0 + rw [Measure.volume_eq_prod] at hprod ⊢ + rw [integral_prod _ hprod] + have he (x : ℝ) : (∫ y : ℝ, D (e (x,y)))=0 := by + have heq (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by + apply Complex.ext <;> simp [e,Complex.measurableEquivRealProd_symm_apply] + simpa only [heq] using hline x + simp only [he,integral_zero] + +lemma integral_laplacian_compact_eq_zero {f : ℂ → ℝ} + (hf : ContDiff ℝ 2 f) (hs : HasCompactSupport f) : ∫ z, (Δ f) z=0 := by + let A (z : ℂ) := fderiv ℝ f z 1 + let B (z : ℂ) := fderiv ℝ f z Complex.I + have hDf : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num) + have hA : ContDiff ℝ 1 A := hDf.clm_apply contDiff_const + have hB : ContDiff ℝ 1 B := hDf.clm_apply contDiff_const + have hsA : HasCompactSupport A := hs.fderiv_apply ℝ 1 + have hsB : HasCompactSupport B := hs.fderiv_apply ℝ Complex.I + have hAA (z : ℂ) : fderiv ℝ A z 1=fderiv ℝ (fderiv ℝ f) z 1 1 := by + have h := (hDf.differentiable (by norm_num) z).hasFDerivAt.clm_apply + (hasFDerivAt_const (1:ℂ) z) + rw [h.fderiv] + simp + have hBB (z : ℂ) : fderiv ℝ B z Complex.I=fderiv ℝ (fderiv ℝ f) z Complex.I Complex.I := by + have h := (hDf.differentiable (by norm_num) z).hasFDerivAt.clm_apply + (hasFDerivAt_const Complex.I z) + rw [h.fderiv] + simp + have hpoint (z : ℂ) : (Δ f) z=fderiv ℝ A z 1+fderiv ℝ B z Complex.I := by + rw [hAA,hBB,laplacian_eq_iteratedFDeriv_complexPlane] + simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, + Matrix.cons_val_one,Matrix.cons_val_fin_one] + simp_rw [hpoint] + rw [integral_add + (((hA.continuous_fderiv (by norm_num)).clm_apply continuous_const).integrable_of_hasCompactSupport (hsA.fderiv_apply ℝ 1)) + (((hB.continuous_fderiv (by norm_num)).clm_apply continuous_const).integrable_of_hasCompactSupport (hsB.fderiv_apply ℝ Complex.I)), + integral_complex_real_partial_eq_zero hA hsA,integral_complex_imag_partial_eq_zero hB hsB,add_zero] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactPathNeighborhoods.lean b/lean/OAI/Probability/SLE/Geometry/CompactPathNeighborhoods.lean index cf717503938fbfc306e3d3a860d8efd9a8a85016..37c0ad46616dc8feaa68f54be62a9953504c3385 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactPathNeighborhoods.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactPathNeighborhoods.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Geometry.CompactConnectedNeighborhoods -import Mathlib.Topology.Connected.LocallyPathConnected - -namespace OAI - -/-! Compact trace sets admit small paths, including after adjoining a -compact interval of the real boundary. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma finiteTrace_locallyPathConnected {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (T : ℝ≥0) : - LocallyPathConnectedSpace (γ '' Icc 0 T) := by - have : LocallyPathConnectedSpace (Icc (0:ℝ) (T:ℝ)) := - (isQuotientMap_projIcc (a := (0:ℝ)) (b := (T:ℝ)) (h := T.coe_nonneg)).locallyPathConnectedSpace - let p : Icc (0:ℝ) (T:ℝ) → γ '' Icc 0 T := fun t => - ⟨γ t.1.toNNReal,⟨t.1.toNNReal,⟨by simp,(Real.toNNReal_le_iff_le_coe).mpr t.2.2⟩,rfl⟩⟩ - have hp : Continuous p := (hγ.comp (continuous_real_toNNReal.comp continuous_subtype_val)).subtype_mk _ - have hs : Surjective p := by - rintro ⟨z,t,ht,rfl⟩ - refine ⟨⟨t,⟨t.coe_nonneg,by exact_mod_cast ht.2⟩⟩,?_⟩ - apply Subtype.ext - simp [p] - exact (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace - -lemma compact_union_locallyPathConnected {X : Type*} [TopologicalSpace X] [T2Space X] - {A B : Set X} (hA : IsCompact A) (hB : IsCompact B) - [LocallyPathConnectedSpace A] [LocallyPathConnectedSpace B] : - LocallyPathConnectedSpace ↥(A∪B) := by - have : CompactSpace A := isCompact_iff_compactSpace.mp hA - have : CompactSpace B := isCompact_iff_compactSpace.mp hB - let p : A⊕B → ↥(A∪B) := Sum.elim - (fun a => ⟨a.1,Or.inl a.2⟩) (fun b => ⟨b.1,Or.inr b.2⟩) - have hp : Continuous p := continuous_sum_dom.mpr - ⟨continuous_subtype_val.subtype_mk _,continuous_subtype_val.subtype_mk _⟩ - have hs : Surjective p := by - rintro ⟨x,hx⟩ - rcases hx with hx | hx - · exact ⟨Sum.inl ⟨x,hx⟩,rfl⟩ - · exact ⟨Sum.inr ⟨x,hx⟩,rfl⟩ - exact (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace - -lemma compact_locallyPathConnected_uniform_path {A : Type*} [MetricSpace A] - [CompactSpace A] [LocallyPathConnectedSpace A] (ε : ℝ) (hε : 0<ε) : - ∃ δ : ℝ, 0<δ ∧ ∀ x y : A, dist x y<δ → - ∃ p : Path x y, ∀ t, dist (p t) x<ε := by - classical - have hlocal (a : A) : ∃ V : Set A, V∈𝓝 a ∧ IsPathConnected V ∧ V⊆ball a (ε/2) := by - obtain ⟨V,⟨hV,hpc⟩,hsub⟩ := (path_connected_basis a).mem_iff.mp - (ball_mem_nhds a (half_pos hε)) - exact ⟨V,hV,hpc,hsub⟩ - choose V hV hc hs using hlocal - obtain ⟨δ,hδ,hcover⟩ := Metric.uniformity_basis_dist.lebesgue_number_lemma_nhds - (K := (univ : Set A)) isCompact_univ (fun a _ => hV a) - refine ⟨δ,hδ,fun x y hxy => ?_⟩ - obtain ⟨a,ha⟩ := hcover x (mem_univ x) - have hsub : ball x δ⊆V a := by - intro z hz - apply ha - change dist x z<δ - simpa only [mem_ball,dist_comm] using hz - have hx := hsub (mem_ball_self hδ) - have hy := hsub (by simpa only [mem_ball,dist_comm] using hxy) - obtain ⟨p,hp⟩ := hc a |>.joinedIn x hx y hy - refine ⟨p,fun t => ?_⟩ - have ht := mem_ball.mp (hs a (hp t)) - have hx' := mem_ball.mp (hs a hx) - have htri := dist_triangle (p t) a x - rw [dist_comm a x] at htri - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactConnectedNeighborhoods +import Mathlib.Topology.Connected.LocallyPathConnected + +namespace OAI + +/-! Compact trace sets admit small paths, including after adjoining a +compact interval of the real boundary. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma finiteTrace_locallyPathConnected {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (T : ℝ≥0) : + LocallyPathConnectedSpace (γ '' Icc 0 T) := by + have : LocallyPathConnectedSpace (Icc (0:ℝ) (T:ℝ)) := + (isQuotientMap_projIcc (a := (0:ℝ)) (b := (T:ℝ)) (h := T.coe_nonneg)).locallyPathConnectedSpace + let p : Icc (0:ℝ) (T:ℝ) → γ '' Icc 0 T := fun t => + ⟨γ t.1.toNNReal,⟨t.1.toNNReal,⟨by simp,(Real.toNNReal_le_iff_le_coe).mpr t.2.2⟩,rfl⟩⟩ + have hp : Continuous p := (hγ.comp (continuous_real_toNNReal.comp continuous_subtype_val)).subtype_mk _ + have hs : Surjective p := by + rintro ⟨z,t,ht,rfl⟩ + refine ⟨⟨t,⟨t.coe_nonneg,by exact_mod_cast ht.2⟩⟩,?_⟩ + apply Subtype.ext + simp [p] + exact (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace + +lemma compact_union_locallyPathConnected {X : Type*} [TopologicalSpace X] [T2Space X] + {A B : Set X} (hA : IsCompact A) (hB : IsCompact B) + [LocallyPathConnectedSpace A] [LocallyPathConnectedSpace B] : + LocallyPathConnectedSpace ↥(A∪B) := by + have : CompactSpace A := isCompact_iff_compactSpace.mp hA + have : CompactSpace B := isCompact_iff_compactSpace.mp hB + let p : A⊕B → ↥(A∪B) := Sum.elim + (fun a => ⟨a.1,Or.inl a.2⟩) (fun b => ⟨b.1,Or.inr b.2⟩) + have hp : Continuous p := continuous_sum_dom.mpr + ⟨continuous_subtype_val.subtype_mk _,continuous_subtype_val.subtype_mk _⟩ + have hs : Surjective p := by + rintro ⟨x,hx⟩ + rcases hx with hx | hx + · exact ⟨Sum.inl ⟨x,hx⟩,rfl⟩ + · exact ⟨Sum.inr ⟨x,hx⟩,rfl⟩ + exact (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace + +lemma compact_locallyPathConnected_uniform_path {A : Type*} [MetricSpace A] + [CompactSpace A] [LocallyPathConnectedSpace A] (ε : ℝ) (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ x y : A, dist x y<δ → + ∃ p : Path x y, ∀ t, dist (p t) x<ε := by + classical + have hlocal (a : A) : ∃ V : Set A, V∈𝓝 a ∧ IsPathConnected V ∧ V⊆ball a (ε/2) := by + obtain ⟨V,⟨hV,hpc⟩,hsub⟩ := (path_connected_basis a).mem_iff.mp + (ball_mem_nhds a (half_pos hε)) + exact ⟨V,hV,hpc,hsub⟩ + choose V hV hc hs using hlocal + obtain ⟨δ,hδ,hcover⟩ := Metric.uniformity_basis_dist.lebesgue_number_lemma_nhds + (K := (univ : Set A)) isCompact_univ (fun a _ => hV a) + refine ⟨δ,hδ,fun x y hxy => ?_⟩ + obtain ⟨a,ha⟩ := hcover x (mem_univ x) + have hsub : ball x δ⊆V a := by + intro z hz + apply ha + change dist x z<δ + simpa only [mem_ball,dist_comm] using hz + have hx := hsub (mem_ball_self hδ) + have hy := hsub (by simpa only [mem_ball,dist_comm] using hxy) + obtain ⟨p,hp⟩ := hc a |>.joinedIn x hx y hy + refine ⟨p,fun t => ?_⟩ + have ht := mem_ball.mp (hs a (hp t)) + have hx' := mem_ball.mp (hs a hx) + have htri := dist_triangle (p t) a x + rw [dist_comm a x] at htri + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactPlanarGaussian.lean b/lean/OAI/Probability/SLE/Geometry/CompactPlanarGaussian.lean index 3621c3bd8602684b714b933fddf302a60e0d96b2..66f1b63676638461c3e67c5dbd957987c53dcb0a 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactPlanarGaussian.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactPlanarGaussian.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Stochastic.PlanarGaussianMean -import OAI.Probability.SLE.ClippedCoordinates - -namespace OAI - -/-! Compact smooth planar tests have one uniform Gaussian-generator -error constant. These are the tests needed for Brownian localization. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma compact_planar_test_bounds - (f : ℝ × ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) : - ∃ K : ℝ≥0, - (∀ a b j, j ≤ 3 → |iteratedDeriv j (fun x => f (x,b)) a| ≤ K) ∧ - (∀ a b j, j ≤ 3 → |iteratedDeriv j (fun y => f (a,y)) b| ≤ K) ∧ - ∀ a, LipschitzWith K (fun b => iteratedDeriv 2 (fun x => f (x,b)) a) := by - let g : ℝ × ℝ → ℝ := f ∘ Prod.swap - have hg : ContDiff ℝ (⊤ : ℕ∞) g := hf.comp (contDiff_snd.prodMk contDiff_fst) - have hgc : HasCompactSupport g := hc.comp_homeomorph (Homeomorph.prodComm ℝ ℝ) - have hx (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial g j) - (spatialPartial_smooth g hg j) (spatialPartial_compact g hgc j) - have hy (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial f j) - (spatialPartial_smooth f hf j) (spatialPartial_compact f hc j) - choose C hCb hCl using hx - choose D hDb hDl using hy - let K : ℝ≥0 := ∑ j ∈ Finset.range 4, (C j+D j) - have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ K := by - have hle : C j ≤ C j+D j := le_add_of_nonneg_right (show 0 ≤ D j from zero_le) - exact hle.trans (Finset.single_le_sum (s := Finset.range 4) (f := fun i => C i+D i) - (fun i _ => (show 0 ≤ C i+D i from zero_le)) - (Finset.mem_range.mpr (by omega))) - have hD (j : ℕ) (hj : j ≤ 3) : D j ≤ K := by - have hle : D j ≤ C j+D j := le_add_of_nonneg_left (show 0 ≤ C j from zero_le) - exact hle.trans (Finset.single_le_sum (s := Finset.range 4) (f := fun i => C i+D i) - (fun i _ => (show 0 ≤ C i+D i from zero_le)) - (Finset.mem_range.mpr (by omega))) - refine ⟨K,?_,?_,?_⟩ - · intro a b j hj - have hh := (hCb j (b,a)).trans (show (C j:ℝ)≤K by exact_mod_cast hC j hj) - simpa only [spatialPartial_eq_iteratedDeriv g hg, g, Function.comp_apply, - Prod.swap_prod_mk, Real.norm_eq_abs] using hh - · intro a b j hj - have hh := (hDb j (a,b)).trans (show (D j:ℝ)≤K by exact_mod_cast hD j hj) - simpa only [spatialPartial_eq_iteratedDeriv f hf, Real.norm_eq_abs] using hh - · intro a - apply LipschitzWith.of_dist_le_mul - intro b c - have hh := ((hCl 2).weaken (hC 2 (by decide))).dist_le_mul (b,a) (c,a) - simpa only [spatialPartial_eq_iteratedDeriv g hg, g, Function.comp_apply, - Prod.swap_prod_mk, Prod.dist_eq,dist_self,max_eq_left dist_nonneg] using hh - -def planarGenerator (f : ℝ × ℝ → ℝ) (z : ℝ × ℝ) : ℝ := - (1/2:ℝ)*(iteratedDeriv 2 (fun x => f (x,z.2)) z.1+ - iteratedDeriv 2 (fun y => f (z.1,y)) z.2) - -lemma compact_planar_gaussian_generator - (f : ℝ × ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) : - ∃ C : ℝ≥0, ∀ (δ : ℝ≥0) (z : ℝ × ℝ), - |planarGaussianMean δ f z.1 z.2-f z-(δ:ℝ)*planarGenerator f z| ≤ - (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by - obtain ⟨K,hx,hy,hxy⟩ := compact_planar_test_bounds f hf hc - let M := 2*standardGaussianAbsMoment 3+(1/2:ℝ)*standardGaussianAbsMoment 1 - have hM : 0≤M := by - dsimp [M] - have h3 := standardGaussianAbsMoment_nonneg 3 - have h1 := standardGaussianAbsMoment_nonneg 1 - positivity - refine ⟨K*⟨M,hM⟩,?_⟩ - intro δ z - have h1 (b : ℝ) : ContDiff ℝ 3 (fun a => f (a,b)) := - (hf.comp (contDiff_id.prodMk contDiff_const)).of_le (by simp) - have h2 (a : ℝ) : ContDiff ℝ 3 (fun b => f (a,b)) := - (hf.comp (contDiff_const.prodMk contDiff_id)).of_le (by simp) - have h := planar_gaussian_generator_error f hf.continuous K h1 h2 hx hy hxy δ z.1 z.2 - calc - _ = |planarGaussianMean δ f z.1 z.2-f z-(1/2:ℝ)*(δ:ℝ)* - (iteratedDeriv 2 (fun x => f (x,z.2)) z.1+ - iteratedDeriv 2 (fun y => f (z.1,y)) z.2)| := by - congr 1 - dsimp [planarGenerator] - ring - _ ≤ (K:ℝ)*(δ:ℝ)*Real.sqrt δ*M := h - _ = _ := by - change (K:ℝ)*(δ:ℝ)*Real.sqrt δ*M=(K:ℝ)*M*(δ:ℝ)*Real.sqrt δ - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.PlanarGaussianMean +import OAI.Probability.SLE.ClippedCoordinates + +namespace OAI + +/-! Compact smooth planar tests have one uniform Gaussian-generator +error constant. These are the tests needed for Brownian localization. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma compact_planar_test_bounds + (f : ℝ × ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) : + ∃ K : ℝ≥0, + (∀ a b j, j ≤ 3 → |iteratedDeriv j (fun x => f (x,b)) a| ≤ K) ∧ + (∀ a b j, j ≤ 3 → |iteratedDeriv j (fun y => f (a,y)) b| ≤ K) ∧ + ∀ a, LipschitzWith K (fun b => iteratedDeriv 2 (fun x => f (x,b)) a) := by + let g : ℝ × ℝ → ℝ := f ∘ Prod.swap + have hg : ContDiff ℝ (⊤ : ℕ∞) g := hf.comp (contDiff_snd.prodMk contDiff_fst) + have hgc : HasCompactSupport g := hc.comp_homeomorph (Homeomorph.prodComm ℝ ℝ) + have hx (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial g j) + (spatialPartial_smooth g hg j) (spatialPartial_compact g hgc j) + have hy (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial f j) + (spatialPartial_smooth f hf j) (spatialPartial_compact f hc j) + choose C hCb hCl using hx + choose D hDb hDl using hy + let K : ℝ≥0 := ∑ j ∈ Finset.range 4, (C j+D j) + have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ K := by + have hle : C j ≤ C j+D j := le_add_of_nonneg_right (show 0 ≤ D j from zero_le) + exact hle.trans (Finset.single_le_sum (s := Finset.range 4) (f := fun i => C i+D i) + (fun i _ => (show 0 ≤ C i+D i from zero_le)) + (Finset.mem_range.mpr (by omega))) + have hD (j : ℕ) (hj : j ≤ 3) : D j ≤ K := by + have hle : D j ≤ C j+D j := le_add_of_nonneg_left (show 0 ≤ C j from zero_le) + exact hle.trans (Finset.single_le_sum (s := Finset.range 4) (f := fun i => C i+D i) + (fun i _ => (show 0 ≤ C i+D i from zero_le)) + (Finset.mem_range.mpr (by omega))) + refine ⟨K,?_,?_,?_⟩ + · intro a b j hj + have hh := (hCb j (b,a)).trans (show (C j:ℝ)≤K by exact_mod_cast hC j hj) + simpa only [spatialPartial_eq_iteratedDeriv g hg, g, Function.comp_apply, + Prod.swap_prod_mk, Real.norm_eq_abs] using hh + · intro a b j hj + have hh := (hDb j (a,b)).trans (show (D j:ℝ)≤K by exact_mod_cast hD j hj) + simpa only [spatialPartial_eq_iteratedDeriv f hf, Real.norm_eq_abs] using hh + · intro a + apply LipschitzWith.of_dist_le_mul + intro b c + have hh := ((hCl 2).weaken (hC 2 (by decide))).dist_le_mul (b,a) (c,a) + simpa only [spatialPartial_eq_iteratedDeriv g hg, g, Function.comp_apply, + Prod.swap_prod_mk, Prod.dist_eq,dist_self,max_eq_left dist_nonneg] using hh + +def planarGenerator (f : ℝ × ℝ → ℝ) (z : ℝ × ℝ) : ℝ := + (1/2:ℝ)*(iteratedDeriv 2 (fun x => f (x,z.2)) z.1+ + iteratedDeriv 2 (fun y => f (z.1,y)) z.2) + +lemma compact_planar_gaussian_generator + (f : ℝ × ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) : + ∃ C : ℝ≥0, ∀ (δ : ℝ≥0) (z : ℝ × ℝ), + |planarGaussianMean δ f z.1 z.2-f z-(δ:ℝ)*planarGenerator f z| ≤ + (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by + obtain ⟨K,hx,hy,hxy⟩ := compact_planar_test_bounds f hf hc + let M := 2*standardGaussianAbsMoment 3+(1/2:ℝ)*standardGaussianAbsMoment 1 + have hM : 0≤M := by + dsimp [M] + have h3 := standardGaussianAbsMoment_nonneg 3 + have h1 := standardGaussianAbsMoment_nonneg 1 + positivity + refine ⟨K*⟨M,hM⟩,?_⟩ + intro δ z + have h1 (b : ℝ) : ContDiff ℝ 3 (fun a => f (a,b)) := + (hf.comp (contDiff_id.prodMk contDiff_const)).of_le (by simp) + have h2 (a : ℝ) : ContDiff ℝ 3 (fun b => f (a,b)) := + (hf.comp (contDiff_const.prodMk contDiff_id)).of_le (by simp) + have h := planar_gaussian_generator_error f hf.continuous K h1 h2 hx hy hxy δ z.1 z.2 + calc + _ = |planarGaussianMean δ f z.1 z.2-f z-(1/2:ℝ)*(δ:ℝ)* + (iteratedDeriv 2 (fun x => f (x,z.2)) z.1+ + iteratedDeriv 2 (fun y => f (z.1,y)) z.2)| := by + congr 1 + dsimp [planarGenerator] + ring + _ ≤ (K:ℝ)*(δ:ℝ)*Real.sqrt δ*M := h + _ = _ := by + change (K:ℝ)*(δ:ℝ)*Real.sqrt δ*M=(K:ℝ)*M*(δ:ℝ)*Real.sqrt δ + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactSmoothExtension.lean b/lean/OAI/Probability/SLE/Geometry/CompactSmoothExtension.lean index 1de159fe6044f53005c7d153fecd10a88b1a205e..5645e52c12db1bb7faf6228e413e9d5a42b24494 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactSmoothExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactSmoothExtension.lean @@ -1,61 +1,61 @@ -import Mathlib - -namespace OAI - -/-! A compactly supported smooth extension agreeing near a prescribed -compact subset of an open set. Its pointwise bound preserves bounded -harmonic tests during Brownian localization. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma exists_compact_smooth_extension - {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] - (U K : Set E) (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) - (u : E → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U) : - ∃ f : E → ℝ, ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ - tsupport f ⊆ U ∧ (∀ x∈K, f =ᶠ[𝓝 x] u) ∧ - (∀ x, |f x|≤|u x|) ∧ ∀ x∉U, f x=0 := by - obtain ⟨V,hV,hKV,hVU,hVc⟩ := exists_open_between_and_isCompact_closure hK hU hKU - obtain ⟨O,hO,hVO,hOU,hOc⟩ := exists_open_between_and_isCompact_closure hVc hU hVU - obtain ⟨χ,hχ,hχrange,hχsupport,hχone⟩ := - exists_contDiff_support_eq_eq_one_iff (n := (⊤ : ℕ∞)) hO isClosed_closure hVO - have hχc : HasCompactSupport χ := by - change IsCompact (closure (support χ)) - rw [hχsupport] - exact hOc - have hχU : tsupport χ ⊆ U := by simpa only [tsupport,hχsupport] using hOU - have hχb (x) : 0≤χ x ∧ χ x≤1 := hχrange (mem_range_self x) - let f : E → ℝ := fun x => χ x*u x - have hf : ContDiff ℝ (⊤ : ℕ∞) f := by - apply contDiff_iff_contDiffAt.mpr - intro x - by_cases hx : x∈U - · exact hχ.contDiffAt.mul ((hu x hx).contDiffAt (hU.mem_nhds hx)) - · have hn : x∉tsupport χ := fun h => hx (hχU h) - have he : f =ᶠ[𝓝 x] (fun _ => (0:ℝ)) := by - filter_upwards [(isClosed_tsupport χ).isOpen_compl.mem_nhds hn] with y hy - have hz : χ y=0 := notMem_support.mp (fun hs => hy ((subset_tsupport χ) hs)) - simp only [f,hz,zero_mul] - exact contDiffAt_const.congr_of_eventuallyEq he - refine ⟨f,hf,hχc.mul_right,?_,?_,?_,?_⟩ - · exact (tsupport_mul_subset_left (f := χ) (g := u)).trans hχU - · intro x hx - filter_upwards [hV.mem_nhds (hKV hx)] with y hy - have he : χ y=1 := (hχone y).mp (subset_closure hy) - simp only [f,he,one_mul] - · intro x - dsimp only [f] - rw [abs_mul,abs_of_nonneg (hχb x).1] - exact mul_le_of_le_one_left (abs_nonneg _) (hχb x).2 - · intro x hx - have hz : χ x=0 := notMem_support.mp (fun hs => hx (hχU ((subset_tsupport χ) hs))) - simp only [f,hz,zero_mul] - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! A compactly supported smooth extension agreeing near a prescribed +compact subset of an open set. Its pointwise bound preserves bounded +harmonic tests during Brownian localization. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma exists_compact_smooth_extension + {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + (U K : Set E) (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) + (u : E → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U) : + ∃ f : E → ℝ, ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ + tsupport f ⊆ U ∧ (∀ x∈K, f =ᶠ[𝓝 x] u) ∧ + (∀ x, |f x|≤|u x|) ∧ ∀ x∉U, f x=0 := by + obtain ⟨V,hV,hKV,hVU,hVc⟩ := exists_open_between_and_isCompact_closure hK hU hKU + obtain ⟨O,hO,hVO,hOU,hOc⟩ := exists_open_between_and_isCompact_closure hVc hU hVU + obtain ⟨χ,hχ,hχrange,hχsupport,hχone⟩ := + exists_contDiff_support_eq_eq_one_iff (n := (⊤ : ℕ∞)) hO isClosed_closure hVO + have hχc : HasCompactSupport χ := by + change IsCompact (closure (support χ)) + rw [hχsupport] + exact hOc + have hχU : tsupport χ ⊆ U := by simpa only [tsupport,hχsupport] using hOU + have hχb (x) : 0≤χ x ∧ χ x≤1 := hχrange (mem_range_self x) + let f : E → ℝ := fun x => χ x*u x + have hf : ContDiff ℝ (⊤ : ℕ∞) f := by + apply contDiff_iff_contDiffAt.mpr + intro x + by_cases hx : x∈U + · exact hχ.contDiffAt.mul ((hu x hx).contDiffAt (hU.mem_nhds hx)) + · have hn : x∉tsupport χ := fun h => hx (hχU h) + have he : f =ᶠ[𝓝 x] (fun _ => (0:ℝ)) := by + filter_upwards [(isClosed_tsupport χ).isOpen_compl.mem_nhds hn] with y hy + have hz : χ y=0 := notMem_support.mp (fun hs => hy ((subset_tsupport χ) hs)) + simp only [f,hz,zero_mul] + exact contDiffAt_const.congr_of_eventuallyEq he + refine ⟨f,hf,hχc.mul_right,?_,?_,?_,?_⟩ + · exact (tsupport_mul_subset_left (f := χ) (g := u)).trans hχU + · intro x hx + filter_upwards [hV.mem_nhds (hKV hx)] with y hy + have he : χ y=1 := (hχone y).mp (subset_closure hy) + simp only [f,he,one_mul] + · intro x + dsimp only [f] + rw [abs_mul,abs_of_nonneg (hχb x).1] + exact mul_le_of_le_one_left (abs_nonneg _) (hχb x).2 + · intro x hx + have hz : χ x=0 := notMem_support.mp (fun hs => hx (hχU ((subset_tsupport χ) hs))) + simp only [f,hz,zero_mul] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactTubeProbability.lean b/lean/OAI/Probability/SLE/Geometry/CompactTubeProbability.lean index a14126f349eb6ba965fe279ef74fafc7037c18ca..180fc371d60009d22c8e24f94b3f5d95515d0be9 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactTubeProbability.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactTubeProbability.lean @@ -1,98 +1,98 @@ -import Mathlib -import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli - -namespace OAI - -/-! A finite-cover argument makes individual tube probabilities uniform on -a compact family of routes. No quantitative support estimate is assumed. -/ - -noncomputable section -open Set MeasureTheory Metric -open scoped ENNReal NNReal BoundedContinuousFunction - -namespace SLEExactGauge - -theorem finite_positive_lower_bound {X : Type*} (S : Set X) (hS : S.Finite) - (f : X → ℝ≥0∞) (hf : ∀ x ∈ S, 0 < f x) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ x ∈ S, q ≤ f x := by - classical - induction S, hS using Set.Finite.induction_on with - | empty => exact ⟨1, by norm_num, by simp⟩ - | @insert x S hx hS ih => - obtain ⟨q, hq, hbound⟩ := ih (fun y hy => hf y (mem_insert_of_mem x hy)) - refine ⟨min (f x) q, lt_min (hf x (mem_insert x S)) hq, ?_⟩ - intro y hy - rcases mem_insert_iff.mp hy with rfl | hy - · exact min_le_left _ _ - · exact (min_le_right _ _).trans (hbound y hy) - -/-- Pointwise positivity on half-radius balls gives one positive lower -bound for full-radius balls centered anywhere in a compact set. -/ -theorem compact_tube_probability_lower_bound - {X : Type*} [PseudoMetricSpace X] [MeasurableSpace X] - (P : Measure X) (K : Set X) (hK : IsCompact K) {ε : ℝ} (hε : 0 < ε) - (hpositive : ∀ x ∈ K, 0 < P (ball x (ε / 2))) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ x ∈ K, q ≤ P (ball x ε) := by - obtain ⟨S, hSK, hS, hcover⟩ := hK.finite_cover_balls (half_pos hε) - obtain ⟨q, hq, hbound⟩ := finite_positive_lower_bound S hS - (fun x => P (ball x (ε / 2))) (fun x hx => hpositive x (hSK hx)) - refine ⟨q, hq, ?_⟩ - intro x hx - obtain ⟨y, hy, hxy⟩ := mem_iUnion₂.mp (hcover hx) - refine (hbound y hy).trans (measure_mono ?_) - intro z hz - have hd := dist_triangle z y x - have hxy' : dist y x < ε / 2 := by simpa only [mem_ball, dist_comm] using hxy - have hzy : dist z y < ε / 2 := hz - change dist z x < ε - linarith - -end SLEExactGauge - -namespace SLEExactGauge.CompactRouteTubes - -local instance compactRouteMeasurableSpace : MeasurableSpace (unitInterval →ᵇ ℂ) := borel _ -local instance compactRouteBorelSpace : BorelSpace (unitInterval →ᵇ ℂ) := ⟨rfl⟩ - -/-- The usual pointwise full-support statement for continuous paths based -at zero is enough for a uniform tube probability over all routes with a -fixed speed bound. Arzela--Ascoli and a finite cover supply the uniformity. -/ -theorem uniform_tube_probability_of_based_support - (P : Measure (unitInterval →ᵇ ℂ)) (C : ℝ≥0) {ε : ℝ} (hε : 0 < ε) - (hsupport : ∀ f : unitInterval →ᵇ ℂ, f 0 = 0 → ∀ r : ℝ, 0 < r → - 0 < P (ball f r)) : - ∃ q : ℝ≥0∞, 0 < q ∧ ∀ f : unitInterval →ᵇ ℂ, - LipschitzWith C f → f 0 = 0 → - q ≤ P {g | ∀ t : unitInterval, dist (g t) (f t) < ε} := by - let S : Set (unitInterval →ᵇ ℂ) := {f | LipschitzWith C f ∧ f 0 = 0} - have heq : Equicontinuous ((↑) : S → unitInterval → ℂ) := by - apply (LipschitzWith.uniformEquicontinuous _ C ?_).equicontinuous - intro f - exact f.property.1 - have hbound : ∀ f ∈ S, ∀ t : unitInterval, f t ∈ closedBall (0 : ℂ) C := by - intro f hf t - have hh := hf.1.dist_le_mul t 0 - have ht : dist t (0 : unitInterval) ≤ 1 := by - change |(t : ℝ) - 0| ≤ 1 - simpa only [sub_zero, abs_of_nonneg t.property.1] using t.property.2 - rw [hf.2] at hh - exact hh.trans (by simpa only [mul_one] using mul_le_mul_of_nonneg_left ht C.coe_nonneg) - have hcompact : IsCompact (closure S) := BoundedContinuousFunction.arzela_ascoli - (closedBall (0 : ℂ) C) (isCompact_closedBall _ _) S (fun f t hf => hbound f hf t) heq - have hbasedClosed : IsClosed {f : unitInterval →ᵇ ℂ | f 0 = 0} := - isClosed_eq (BoundedContinuousFunction.lipschitz_eval_const 0).continuous continuous_const - have hbased : closure S ⊆ {f : unitInterval →ᵇ ℂ | f 0 = 0} := - closure_minimal (fun _ hf => hf.2) hbasedClosed - obtain ⟨q, hq, hqbound⟩ := compact_tube_probability_lower_bound P (closure S) hcompact hε - (fun f hf => hsupport f (hbased hf) (ε / 2) (half_pos hε)) - refine ⟨q, hq, ?_⟩ - intro f hf hf0 - refine (hqbound f (subset_closure ⟨hf, hf0⟩)).trans (measure_mono ?_) - intro g hg t - exact (BoundedContinuousFunction.dist_coe_le_dist t).trans_lt hg - -end SLEExactGauge.CompactRouteTubes - -end - -end OAI +import Mathlib +import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli + +namespace OAI + +/-! A finite-cover argument makes individual tube probabilities uniform on +a compact family of routes. No quantitative support estimate is assumed. -/ + +noncomputable section +open Set MeasureTheory Metric +open scoped ENNReal NNReal BoundedContinuousFunction + +namespace SLEExactGauge + +theorem finite_positive_lower_bound {X : Type*} (S : Set X) (hS : S.Finite) + (f : X → ℝ≥0∞) (hf : ∀ x ∈ S, 0 < f x) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ x ∈ S, q ≤ f x := by + classical + induction S, hS using Set.Finite.induction_on with + | empty => exact ⟨1, by norm_num, by simp⟩ + | @insert x S hx hS ih => + obtain ⟨q, hq, hbound⟩ := ih (fun y hy => hf y (mem_insert_of_mem x hy)) + refine ⟨min (f x) q, lt_min (hf x (mem_insert x S)) hq, ?_⟩ + intro y hy + rcases mem_insert_iff.mp hy with rfl | hy + · exact min_le_left _ _ + · exact (min_le_right _ _).trans (hbound y hy) + +/-- Pointwise positivity on half-radius balls gives one positive lower +bound for full-radius balls centered anywhere in a compact set. -/ +theorem compact_tube_probability_lower_bound + {X : Type*} [PseudoMetricSpace X] [MeasurableSpace X] + (P : Measure X) (K : Set X) (hK : IsCompact K) {ε : ℝ} (hε : 0 < ε) + (hpositive : ∀ x ∈ K, 0 < P (ball x (ε / 2))) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ x ∈ K, q ≤ P (ball x ε) := by + obtain ⟨S, hSK, hS, hcover⟩ := hK.finite_cover_balls (half_pos hε) + obtain ⟨q, hq, hbound⟩ := finite_positive_lower_bound S hS + (fun x => P (ball x (ε / 2))) (fun x hx => hpositive x (hSK hx)) + refine ⟨q, hq, ?_⟩ + intro x hx + obtain ⟨y, hy, hxy⟩ := mem_iUnion₂.mp (hcover hx) + refine (hbound y hy).trans (measure_mono ?_) + intro z hz + have hd := dist_triangle z y x + have hxy' : dist y x < ε / 2 := by simpa only [mem_ball, dist_comm] using hxy + have hzy : dist z y < ε / 2 := hz + change dist z x < ε + linarith + +end SLEExactGauge + +namespace SLEExactGauge.CompactRouteTubes + +local instance compactRouteMeasurableSpace : MeasurableSpace (unitInterval →ᵇ ℂ) := borel _ +local instance compactRouteBorelSpace : BorelSpace (unitInterval →ᵇ ℂ) := ⟨rfl⟩ + +/-- The usual pointwise full-support statement for continuous paths based +at zero is enough for a uniform tube probability over all routes with a +fixed speed bound. Arzela--Ascoli and a finite cover supply the uniformity. -/ +theorem uniform_tube_probability_of_based_support + (P : Measure (unitInterval →ᵇ ℂ)) (C : ℝ≥0) {ε : ℝ} (hε : 0 < ε) + (hsupport : ∀ f : unitInterval →ᵇ ℂ, f 0 = 0 → ∀ r : ℝ, 0 < r → + 0 < P (ball f r)) : + ∃ q : ℝ≥0∞, 0 < q ∧ ∀ f : unitInterval →ᵇ ℂ, + LipschitzWith C f → f 0 = 0 → + q ≤ P {g | ∀ t : unitInterval, dist (g t) (f t) < ε} := by + let S : Set (unitInterval →ᵇ ℂ) := {f | LipschitzWith C f ∧ f 0 = 0} + have heq : Equicontinuous ((↑) : S → unitInterval → ℂ) := by + apply (LipschitzWith.uniformEquicontinuous _ C ?_).equicontinuous + intro f + exact f.property.1 + have hbound : ∀ f ∈ S, ∀ t : unitInterval, f t ∈ closedBall (0 : ℂ) C := by + intro f hf t + have hh := hf.1.dist_le_mul t 0 + have ht : dist t (0 : unitInterval) ≤ 1 := by + change |(t : ℝ) - 0| ≤ 1 + simpa only [sub_zero, abs_of_nonneg t.property.1] using t.property.2 + rw [hf.2] at hh + exact hh.trans (by simpa only [mul_one] using mul_le_mul_of_nonneg_left ht C.coe_nonneg) + have hcompact : IsCompact (closure S) := BoundedContinuousFunction.arzela_ascoli + (closedBall (0 : ℂ) C) (isCompact_closedBall _ _) S (fun f t hf => hbound f hf t) heq + have hbasedClosed : IsClosed {f : unitInterval →ᵇ ℂ | f 0 = 0} := + isClosed_eq (BoundedContinuousFunction.lipschitz_eval_const 0).continuous continuous_const + have hbased : closure S ⊆ {f : unitInterval →ᵇ ℂ | f 0 = 0} := + closure_minimal (fun _ hf => hf.2) hbasedClosed + obtain ⟨q, hq, hqbound⟩ := compact_tube_probability_lower_bound P (closure S) hcompact hε + (fun f hf => hsupport f (hbased hf) (ε / 2) (half_pos hε)) + refine ⟨q, hq, ?_⟩ + intro f hf hf0 + refine (hqbound f (subset_closure ⟨hf, hf0⟩)).trans (measure_mono ?_) + intro g hg t + exact (BoundedContinuousFunction.dist_coe_le_dist t).trans_lt hg + +end SLEExactGauge.CompactRouteTubes + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompactifiedCrosscut.lean b/lean/OAI/Probability/SLE/Geometry/CompactifiedCrosscut.lean index 8bf85a42d1a5e76c62a53a842bdcee8eb6805b0d..ec3654bc23423b3135d2a1cc86ae2058c9f74f81 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompactifiedCrosscut.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompactifiedCrosscut.lean @@ -1,120 +1,120 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneCayleyCompactification - -namespace OAI - -/-! A proper half-plane ray becomes an injective compact disk crosscut -from `-1` to `1`. The endpoint at infinity is filled by the proved Cayley -limit, preserving the prescribed ray parameter. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -def crosscutParameter (s : ℝ) : ℝ≥0 := (s/(1-s)).toNNReal - -lemma crosscutParameter_injectiveOn : InjOn crosscutParameter (Ico 0 1) := by - intro s hs t ht he - have he' := congrArg (fun r : ℝ≥0 => (r : ℝ)) he - have hsval : (crosscutParameter s : ℝ)=s/(1-s) := - Real.coe_toNNReal _ (div_nonneg hs.1 (sub_pos.mpr hs.2).le) - have htval : (crosscutParameter t : ℝ)=t/(1-t) := - Real.coe_toNNReal _ (div_nonneg ht.1 (sub_pos.mpr ht.2).le) - rw [hsval,htval] at he' - have hh := (div_eq_div_iff (sub_pos.mpr hs.2).ne' (sub_pos.mpr ht.2).ne').mp he' - nlinarith - -lemma crosscutParameter_continuousOn : ContinuousOn crosscutParameter (Ico 0 1) := by - apply continuous_real_toNNReal.comp_continuousOn - exact continuousOn_id.div (continuousOn_const.sub continuousOn_id) - (fun _ hs => (sub_pos.mpr hs.2).ne') - -lemma crosscutParameter_tendsto : Tendsto crosscutParameter (𝓝[<] 1) atTop := by - have hid : Tendsto (fun s : ℝ => s) (𝓝[<] 1) (𝓝 1) := - tendsto_id.mono_left nhdsWithin_le_nhds - have hsub : Tendsto (fun s : ℝ => 1-s) (𝓝[<] 1) (𝓝[>] 0) := by - apply tendsto_nhdsWithin_iff.mpr - refine ⟨?_,?_⟩ - · simpa only [sub_self] using - (tendsto_const_nhds : Tendsto (fun _ : ℝ => (1:ℝ)) (𝓝[<] 1) (𝓝 1)).sub hid - · filter_upwards [self_mem_nhdsWithin] with s hs - change 0 < 1-s - exact sub_pos.mpr hs - have hh := hid.pos_mul_atTop zero_lt_one (tendsto_inv_nhdsGT_zero.comp hsub) - exact tendsto_real_toNNReal_atTop.comp - (by simpa only [div_eq_mul_inv,Function.comp_apply] using hh) - -def compactifiedCrosscut (β : ℝ≥0 → ℂ) : ℝ → ℂ := - Function.update (fun s => halfPlaneCayley (β (crosscutParameter s))) 1 1 - -lemma compactifiedCrosscut_one (β : ℝ≥0 → ℂ) : compactifiedCrosscut β 1=1 := - Function.update_self .. - -lemma compactifiedCrosscut_eq (β : ℝ≥0 → ℂ) {s : ℝ} (hs : s < 1) : - compactifiedCrosscut β s=halfPlaneCayley (β (crosscutParameter s)) := - Function.update_of_ne hs.ne _ _ - -lemma compactifiedCrosscut_zero (β : ℝ≥0 → ℂ) (h0 : β 0=0) : - compactifiedCrosscut β 0 = -1 := by - rw [compactifiedCrosscut_eq β zero_lt_one] - simp only [crosscutParameter,zero_div,Real.toNNReal_zero,h0,halfPlaneCayley_zero] - -theorem compactifiedCrosscut_properties - (β : ℝ≥0 → ℂ) (hc : Continuous β) (hi : Function.Injective β) (h0 : β 0=0) - (hH : ∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) - (hinfty : Tendsto β atTop (cocompact ℂ)) : - ContinuousOn (compactifiedCrosscut β) (Icc 0 1) ∧ - InjOn (compactifiedCrosscut β) (Icc 0 1) ∧ - ∀ s ∈ Ioo 0 1, ‖compactifiedCrosscut β s‖ < 1 := by - have hH' (t : ℝ≥0) : 0 ≤ (β t).im := by - by_cases ht : t=0 - · simp [ht,h0] - · exact (hH t (pos_iff_ne_zero.mpr ht)).le - have hbase : ContinuousOn (fun s => halfPlaneCayley (β (crosscutParameter s))) - (Ico 0 1) := by - apply halfPlaneCayley_continuousOn.comp - (hc.comp_continuousOn crosscutParameter_continuousOn) - intro s _ - exact hH' _ - have hceq : Icc (0:ℝ) 1 \ {1} = Ico 0 1 := by - ext s - simp only [Set.mem_sdiff,mem_Icc,mem_singleton_iff,mem_Ico] - constructor - · rintro ⟨⟨hs,hs1⟩,hn⟩ - exact ⟨hs,lt_of_le_of_ne hs1 hn⟩ - · rintro ⟨hs,hs1⟩ - exact ⟨⟨hs,hs1.le⟩,hs1.ne⟩ - refine ⟨?_,?_,?_⟩ - · apply continuousOn_update_iff.mpr - rw [hceq] - refine ⟨hbase,fun _ => ?_⟩ - exact (halfPlaneCayley_tendsto_infinity.comp (hinfty.comp crosscutParameter_tendsto)).mono_left - (nhdsWithin_mono _ Ico_subset_Iio_self) - · intro s hs t ht he - by_cases hs1 : s=1 - · subst s - by_contra ht1 - rw [compactifiedCrosscut_one, - compactifiedCrosscut_eq β (lt_of_le_of_ne ht.2 (fun h => ht1 h.symm))] at he - exact halfPlaneCayley_ne_one (hH' _) he.symm - · by_cases ht1 : t=1 - · subst t - rw [compactifiedCrosscut_one,compactifiedCrosscut_eq β (lt_of_le_of_ne hs.2 hs1)] at he - exact False.elim (halfPlaneCayley_ne_one (hH' _) he) - · have hslt := lt_of_le_of_ne hs.2 hs1 - have htlt := lt_of_le_of_ne ht.2 ht1 - rw [compactifiedCrosscut_eq β hslt,compactifiedCrosscut_eq β htlt] at he - exact crosscutParameter_injectiveOn ⟨hs.1,hslt⟩ ⟨ht.1,htlt⟩ - (hi (halfPlaneCayley_injOn (hH' _) (hH' _) he)) - · intro s hs - rw [compactifiedCrosscut_eq β hs.2] - apply halfPlaneCayley_norm_lt - apply hH - change 0 < (s/(1-s)).toNNReal - exact Real.toNNReal_pos.mpr (div_pos hs.1 (sub_pos.mpr hs.2)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfPlaneCayleyCompactification + +namespace OAI + +/-! A proper half-plane ray becomes an injective compact disk crosscut +from `-1` to `1`. The endpoint at infinity is filled by the proved Cayley +limit, preserving the prescribed ray parameter. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +def crosscutParameter (s : ℝ) : ℝ≥0 := (s/(1-s)).toNNReal + +lemma crosscutParameter_injectiveOn : InjOn crosscutParameter (Ico 0 1) := by + intro s hs t ht he + have he' := congrArg (fun r : ℝ≥0 => (r : ℝ)) he + have hsval : (crosscutParameter s : ℝ)=s/(1-s) := + Real.coe_toNNReal _ (div_nonneg hs.1 (sub_pos.mpr hs.2).le) + have htval : (crosscutParameter t : ℝ)=t/(1-t) := + Real.coe_toNNReal _ (div_nonneg ht.1 (sub_pos.mpr ht.2).le) + rw [hsval,htval] at he' + have hh := (div_eq_div_iff (sub_pos.mpr hs.2).ne' (sub_pos.mpr ht.2).ne').mp he' + nlinarith + +lemma crosscutParameter_continuousOn : ContinuousOn crosscutParameter (Ico 0 1) := by + apply continuous_real_toNNReal.comp_continuousOn + exact continuousOn_id.div (continuousOn_const.sub continuousOn_id) + (fun _ hs => (sub_pos.mpr hs.2).ne') + +lemma crosscutParameter_tendsto : Tendsto crosscutParameter (𝓝[<] 1) atTop := by + have hid : Tendsto (fun s : ℝ => s) (𝓝[<] 1) (𝓝 1) := + tendsto_id.mono_left nhdsWithin_le_nhds + have hsub : Tendsto (fun s : ℝ => 1-s) (𝓝[<] 1) (𝓝[>] 0) := by + apply tendsto_nhdsWithin_iff.mpr + refine ⟨?_,?_⟩ + · simpa only [sub_self] using + (tendsto_const_nhds : Tendsto (fun _ : ℝ => (1:ℝ)) (𝓝[<] 1) (𝓝 1)).sub hid + · filter_upwards [self_mem_nhdsWithin] with s hs + change 0 < 1-s + exact sub_pos.mpr hs + have hh := hid.pos_mul_atTop zero_lt_one (tendsto_inv_nhdsGT_zero.comp hsub) + exact tendsto_real_toNNReal_atTop.comp + (by simpa only [div_eq_mul_inv,Function.comp_apply] using hh) + +def compactifiedCrosscut (β : ℝ≥0 → ℂ) : ℝ → ℂ := + Function.update (fun s => halfPlaneCayley (β (crosscutParameter s))) 1 1 + +lemma compactifiedCrosscut_one (β : ℝ≥0 → ℂ) : compactifiedCrosscut β 1=1 := + Function.update_self .. + +lemma compactifiedCrosscut_eq (β : ℝ≥0 → ℂ) {s : ℝ} (hs : s < 1) : + compactifiedCrosscut β s=halfPlaneCayley (β (crosscutParameter s)) := + Function.update_of_ne hs.ne _ _ + +lemma compactifiedCrosscut_zero (β : ℝ≥0 → ℂ) (h0 : β 0=0) : + compactifiedCrosscut β 0 = -1 := by + rw [compactifiedCrosscut_eq β zero_lt_one] + simp only [crosscutParameter,zero_div,Real.toNNReal_zero,h0,halfPlaneCayley_zero] + +theorem compactifiedCrosscut_properties + (β : ℝ≥0 → ℂ) (hc : Continuous β) (hi : Function.Injective β) (h0 : β 0=0) + (hH : ∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) + (hinfty : Tendsto β atTop (cocompact ℂ)) : + ContinuousOn (compactifiedCrosscut β) (Icc 0 1) ∧ + InjOn (compactifiedCrosscut β) (Icc 0 1) ∧ + ∀ s ∈ Ioo 0 1, ‖compactifiedCrosscut β s‖ < 1 := by + have hH' (t : ℝ≥0) : 0 ≤ (β t).im := by + by_cases ht : t=0 + · simp [ht,h0] + · exact (hH t (pos_iff_ne_zero.mpr ht)).le + have hbase : ContinuousOn (fun s => halfPlaneCayley (β (crosscutParameter s))) + (Ico 0 1) := by + apply halfPlaneCayley_continuousOn.comp + (hc.comp_continuousOn crosscutParameter_continuousOn) + intro s _ + exact hH' _ + have hceq : Icc (0:ℝ) 1 \ {1} = Ico 0 1 := by + ext s + simp only [Set.mem_sdiff,mem_Icc,mem_singleton_iff,mem_Ico] + constructor + · rintro ⟨⟨hs,hs1⟩,hn⟩ + exact ⟨hs,lt_of_le_of_ne hs1 hn⟩ + · rintro ⟨hs,hs1⟩ + exact ⟨⟨hs,hs1.le⟩,hs1.ne⟩ + refine ⟨?_,?_,?_⟩ + · apply continuousOn_update_iff.mpr + rw [hceq] + refine ⟨hbase,fun _ => ?_⟩ + exact (halfPlaneCayley_tendsto_infinity.comp (hinfty.comp crosscutParameter_tendsto)).mono_left + (nhdsWithin_mono _ Ico_subset_Iio_self) + · intro s hs t ht he + by_cases hs1 : s=1 + · subst s + by_contra ht1 + rw [compactifiedCrosscut_one, + compactifiedCrosscut_eq β (lt_of_le_of_ne ht.2 (fun h => ht1 h.symm))] at he + exact halfPlaneCayley_ne_one (hH' _) he.symm + · by_cases ht1 : t=1 + · subst t + rw [compactifiedCrosscut_one,compactifiedCrosscut_eq β (lt_of_le_of_ne hs.2 hs1)] at he + exact False.elim (halfPlaneCayley_ne_one (hH' _) he) + · have hslt := lt_of_le_of_ne hs.2 hs1 + have htlt := lt_of_le_of_ne ht.2 ht1 + rw [compactifiedCrosscut_eq β hslt,compactifiedCrosscut_eq β htlt] at he + exact crosscutParameter_injectiveOn ⟨hs.1,hslt⟩ ⟨ht.1,htlt⟩ + (hi (halfPlaneCayley_injOn (hH' _) (hH' _) he)) + · intro s hs + rw [compactifiedCrosscut_eq β hs.2] + apply halfPlaneCayley_norm_lt + apply hH + change 0 < (s/(1-s)).toNNReal + exact Real.toNNReal_pos.mpr (div_pos hs.1 (sub_pos.mpr hs.2)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CompleteAccessGuide.lean b/lean/OAI/Probability/SLE/Geometry/CompleteAccessGuide.lean index d767c76e0f6fb28a93fef9366ee662d176099ac4..35a9e722899e041be9dde12dfaaa52531f6d84d4 100644 --- a/lean/OAI/Probability/SLE/Geometry/CompleteAccessGuide.lean +++ b/lean/OAI/Probability/SLE/Geometry/CompleteAccessGuide.lean @@ -1,140 +1,140 @@ -import OAI.Probability.SLE.Geometry.NormalizedAccessGuide -import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening - -namespace OAI - -/-! The complete physical access guide maps to a proper half-plane -crosscut based at the actual driving prime end. Properness at infinity -is proved from the hydrodynamic normalization in the capacity witness. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem IsCapacityTwoWitness.centered_chart_escapes - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) - (α : ℝ≥0 → ℂ) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) - (hescape : Tendsto α atTop (cocompact ℂ)) : - Tendsto (fun t => G T (α t) - (U T : ℂ)) atTop (cocompact ℂ) := by - have hnorm := hw - rcases hnorm with ⟨_, _, _, _, _, _, _, _, _, _, _, hnorm, _⟩ - have hD : ∀ᶠ t in atTop, α t ∈ survivingDomain γ T := by - filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with t ht - exact hαD t ht - have hmap : Tendsto α atTop (cocompact ℂ ⊓ 𝓟 (survivingDomain γ T)) := - by simpa only [inf_idem] using hescape.inf (tendsto_principal.mpr hD) - have hproduct := (hnorm T).comp hmap - have hinv : Tendsto (fun t => (α t)⁻¹) atTop (𝓝 (0 : ℂ)) := - tendsto_inv₀_cobounded.comp (hescape.mono_right cobounded_eq_cocompact.ge) - have hprod0 := hproduct.mul hinv - have hαnorm : Tendsto (fun t => ‖α t‖) atTop atTop := - tendsto_norm_cocompact_atTop.comp hescape - have hne : ∀ᶠ t in atTop, α t ≠ 0 := by - filter_upwards [hαnorm.eventually_gt_atTop 0] with t ht - exact norm_pos_iff.mp ht - have hdiff : Tendsto (fun t => G T (α t) - α t) atTop (𝓝 (0 : ℂ)) := by - have he : (fun t => (α t * (G T (α t) - α t)) * (α t)⁻¹) =ᶠ[atTop] - (fun t => G T (α t) - α t) := by - filter_upwards [hne] with t ht - field_simp - simpa only [mul_zero] using hprod0.congr' he - apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 - simp only [dist_zero_right] - apply Filter.tendsto_atTop.mpr - intro R - have hsmall : ∀ᶠ t in atTop, ‖G T (α t) - α t‖ < 1 := by - simpa only [Metric.mem_ball, dist_zero_right] using hdiff.eventually (ball_mem_nhds 0 zero_lt_one) - filter_upwards [hαnorm.eventually_ge_atTop (R + |U T| + 1), hsmall] with t ht hs - have htri : ‖α t‖ ≤ ‖α t - (G T (α t) - (U T : ℂ))‖ + - ‖G T (α t) - (U T : ℂ)‖ := by - simpa only [add_comm] using norm_le_insert' (α t) (G T (α t) - (U T : ℂ)) - have he : α t - (G T (α t) - (U T : ℂ)) = - -(G T (α t) - α t) + (U T : ℂ) := by ring - have hb : ‖α t - (G T (α t) - (U T : ℂ))‖ ≤ - ‖G T (α t) - α t‖ + |U T| := by - rw [he] - simpa only [norm_neg, Complex.norm_real, Real.norm_eq_abs] using - norm_add_le (-(G T (α t) - α t)) (U T : ℂ) - linarith - -/-- The continuous complete parameter guide has the exact capacity-map -formula at every positive time, including its asymptotic endpoint. -/ -theorem actual_complete_access_guide - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) - (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) - (α : ℝ≥0 → ℂ) (hα : Continuous α) (hiα : Function.Injective α) - (hα0 : α 0 = γ T) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) - (hescape : Tendsto α atTop (cocompact ℂ)) : - ∃ β : ℝ≥0 → ℂ, Continuous β ∧ Function.Injective β ∧ β 0 = 0 ∧ - (∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) ∧ - Tendsto β atTop (cocompact ℂ) ∧ - ∀ t : ℝ≥0, 0 < t → β t = G T (α t) - (U T : ℂ) := by - let D := survivingDomain γ T - have htip : γ T ∉ D := fun hh => hh.1.2 ⟨T, ⟨bot_le, le_rfl⟩, rfl⟩ - let gb := Function.update (G T) (γ T) (U T : ℂ) - have hs : insert (γ T) D \ {γ T} = D := by - ext z - simp only [Set.mem_sdiff, mem_insert_iff, mem_singleton_iff] - constructor - · rintro ⟨rfl | hz, hn⟩ - · exact False.elim (hn rfl) - · exact hz - · intro hz - exact ⟨Or.inr hz, fun he => htip (he ▸ hz)⟩ - have hgb : ContinuousOn gb (insert (γ T) D) := by - rw [continuousOn_update_iff, hs] - exact ⟨(hw.2.2.2.1 T).continuousOn, - fun _ => hw.fresh_tip_chart_tendsto hU hT hfresh⟩ - have hαS : MapsTo α univ (insert (γ T) D) := by - intro t _ - by_cases ht : t = 0 - · simpa only [ht, hα0] using mem_insert (γ T) D - · exact mem_insert_of_mem _ (hαD t (pos_iff_ne_zero.mpr ht)) - let β : ℝ≥0 → ℂ := fun t => gb (α t) - (U T : ℂ) - have hβc : Continuous β := - (hgb.comp_continuous hα (fun t => hαS (mem_univ t))).sub continuous_const - have hβ0 : β 0 = 0 := by simp [β, gb, hα0] - have hβeq (t : ℝ≥0) (ht : 0 < t) : β t = G T (α t) - (U T : ℂ) := by - have hn : α t ≠ γ T := fun he => htip (he ▸ hαD t ht) - exact congrArg (fun z : ℂ => z - (U T : ℂ)) (Function.update_of_ne hn _ _) - have hβH (t : ℝ≥0) (ht : 0 < t) : β t ∈ upperHalfPlane := by - rw [hβeq t ht] - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] - using hw.imag_positive T (α t) (hαD t ht) - have hβi : Function.Injective β := by - intro s t he - by_cases hs0 : s = 0 - · have ht0 : t = 0 := by - by_contra ht0 - have hh : 0 < (β t).im := hβH t (pos_iff_ne_zero.mpr ht0) - rw [← he, hs0, hβ0, Complex.zero_im] at hh - exact (lt_irrefl 0) hh - exact hs0.trans ht0.symm - · by_cases ht0 : t = 0 - · have hh : 0 < (β s).im := hβH s (pos_iff_ne_zero.mpr hs0) - rw [he, ht0, hβ0, Complex.zero_im] at hh - exact False.elim ((lt_irrefl 0) hh) - · apply hiα - rw [hβeq s (pos_iff_ne_zero.mpr hs0), hβeq t (pos_iff_ne_zero.mpr ht0)] at he - have hg : G T (α s) = G T (α t) := sub_left_inj.mp he - calc - α s = F T (G T (α s)) := - (hw.2.2.2.2.2.2.2.1 T _ (hαD s (pos_iff_ne_zero.mpr hs0))).symm - _ = F T (G T (α t)) := congrArg (F T) hg - _ = α t := hw.2.2.2.2.2.2.2.1 T _ (hαD t (pos_iff_ne_zero.mpr ht0)) - have hβescape : Tendsto β atTop (cocompact ℂ) := by - apply (hw.centered_chart_escapes T α hαD hescape).congr' - filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with t ht - exact (hβeq t ht).symm - exact ⟨β, hβc, hβi, hβ0, hβH, hβescape, hβeq⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.NormalizedAccessGuide +import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening + +namespace OAI + +/-! The complete physical access guide maps to a proper half-plane +crosscut based at the actual driving prime end. Properness at infinity +is proved from the hydrodynamic normalization in the capacity witness. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem IsCapacityTwoWitness.centered_chart_escapes + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) + (α : ℝ≥0 → ℂ) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) + (hescape : Tendsto α atTop (cocompact ℂ)) : + Tendsto (fun t => G T (α t) - (U T : ℂ)) atTop (cocompact ℂ) := by + have hnorm := hw + rcases hnorm with ⟨_, _, _, _, _, _, _, _, _, _, _, hnorm, _⟩ + have hD : ∀ᶠ t in atTop, α t ∈ survivingDomain γ T := by + filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with t ht + exact hαD t ht + have hmap : Tendsto α atTop (cocompact ℂ ⊓ 𝓟 (survivingDomain γ T)) := + by simpa only [inf_idem] using hescape.inf (tendsto_principal.mpr hD) + have hproduct := (hnorm T).comp hmap + have hinv : Tendsto (fun t => (α t)⁻¹) atTop (𝓝 (0 : ℂ)) := + tendsto_inv₀_cobounded.comp (hescape.mono_right cobounded_eq_cocompact.ge) + have hprod0 := hproduct.mul hinv + have hαnorm : Tendsto (fun t => ‖α t‖) atTop atTop := + tendsto_norm_cocompact_atTop.comp hescape + have hne : ∀ᶠ t in atTop, α t ≠ 0 := by + filter_upwards [hαnorm.eventually_gt_atTop 0] with t ht + exact norm_pos_iff.mp ht + have hdiff : Tendsto (fun t => G T (α t) - α t) atTop (𝓝 (0 : ℂ)) := by + have he : (fun t => (α t * (G T (α t) - α t)) * (α t)⁻¹) =ᶠ[atTop] + (fun t => G T (α t) - α t) := by + filter_upwards [hne] with t ht + field_simp + simpa only [mul_zero] using hprod0.congr' he + apply tendsto_cocompact_of_tendsto_dist_comp_atTop 0 + simp only [dist_zero_right] + apply Filter.tendsto_atTop.mpr + intro R + have hsmall : ∀ᶠ t in atTop, ‖G T (α t) - α t‖ < 1 := by + simpa only [Metric.mem_ball, dist_zero_right] using hdiff.eventually (ball_mem_nhds 0 zero_lt_one) + filter_upwards [hαnorm.eventually_ge_atTop (R + |U T| + 1), hsmall] with t ht hs + have htri : ‖α t‖ ≤ ‖α t - (G T (α t) - (U T : ℂ))‖ + + ‖G T (α t) - (U T : ℂ)‖ := by + simpa only [add_comm] using norm_le_insert' (α t) (G T (α t) - (U T : ℂ)) + have he : α t - (G T (α t) - (U T : ℂ)) = + -(G T (α t) - α t) + (U T : ℂ) := by ring + have hb : ‖α t - (G T (α t) - (U T : ℂ))‖ ≤ + ‖G T (α t) - α t‖ + |U T| := by + rw [he] + simpa only [norm_neg, Complex.norm_real, Real.norm_eq_abs] using + norm_add_le (-(G T (α t) - α t)) (U T : ℂ) + linarith + +/-- The continuous complete parameter guide has the exact capacity-map +formula at every positive time, including its asymptotic endpoint. -/ +theorem actual_complete_access_guide + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) + (hfresh : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) + (α : ℝ≥0 → ℂ) (hα : Continuous α) (hiα : Function.Injective α) + (hα0 : α 0 = γ T) (hαD : ∀ t : ℝ≥0, 0 < t → α t ∈ survivingDomain γ T) + (hescape : Tendsto α atTop (cocompact ℂ)) : + ∃ β : ℝ≥0 → ℂ, Continuous β ∧ Function.Injective β ∧ β 0 = 0 ∧ + (∀ t : ℝ≥0, 0 < t → β t ∈ upperHalfPlane) ∧ + Tendsto β atTop (cocompact ℂ) ∧ + ∀ t : ℝ≥0, 0 < t → β t = G T (α t) - (U T : ℂ) := by + let D := survivingDomain γ T + have htip : γ T ∉ D := fun hh => hh.1.2 ⟨T, ⟨bot_le, le_rfl⟩, rfl⟩ + let gb := Function.update (G T) (γ T) (U T : ℂ) + have hs : insert (γ T) D \ {γ T} = D := by + ext z + simp only [Set.mem_sdiff, mem_insert_iff, mem_singleton_iff] + constructor + · rintro ⟨rfl | hz, hn⟩ + · exact False.elim (hn rfl) + · exact hz + · intro hz + exact ⟨Or.inr hz, fun he => htip (he ▸ hz)⟩ + have hgb : ContinuousOn gb (insert (γ T) D) := by + rw [continuousOn_update_iff, hs] + exact ⟨(hw.2.2.2.1 T).continuousOn, + fun _ => hw.fresh_tip_chart_tendsto hU hT hfresh⟩ + have hαS : MapsTo α univ (insert (γ T) D) := by + intro t _ + by_cases ht : t = 0 + · simpa only [ht, hα0] using mem_insert (γ T) D + · exact mem_insert_of_mem _ (hαD t (pos_iff_ne_zero.mpr ht)) + let β : ℝ≥0 → ℂ := fun t => gb (α t) - (U T : ℂ) + have hβc : Continuous β := + (hgb.comp_continuous hα (fun t => hαS (mem_univ t))).sub continuous_const + have hβ0 : β 0 = 0 := by simp [β, gb, hα0] + have hβeq (t : ℝ≥0) (ht : 0 < t) : β t = G T (α t) - (U T : ℂ) := by + have hn : α t ≠ γ T := fun he => htip (he ▸ hαD t ht) + exact congrArg (fun z : ℂ => z - (U T : ℂ)) (Function.update_of_ne hn _ _) + have hβH (t : ℝ≥0) (ht : 0 < t) : β t ∈ upperHalfPlane := by + rw [hβeq t ht] + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] + using hw.imag_positive T (α t) (hαD t ht) + have hβi : Function.Injective β := by + intro s t he + by_cases hs0 : s = 0 + · have ht0 : t = 0 := by + by_contra ht0 + have hh : 0 < (β t).im := hβH t (pos_iff_ne_zero.mpr ht0) + rw [← he, hs0, hβ0, Complex.zero_im] at hh + exact (lt_irrefl 0) hh + exact hs0.trans ht0.symm + · by_cases ht0 : t = 0 + · have hh : 0 < (β s).im := hβH s (pos_iff_ne_zero.mpr hs0) + rw [he, ht0, hβ0, Complex.zero_im] at hh + exact False.elim ((lt_irrefl 0) hh) + · apply hiα + rw [hβeq s (pos_iff_ne_zero.mpr hs0), hβeq t (pos_iff_ne_zero.mpr ht0)] at he + have hg : G T (α s) = G T (α t) := sub_left_inj.mp he + calc + α s = F T (G T (α s)) := + (hw.2.2.2.2.2.2.2.1 T _ (hαD s (pos_iff_ne_zero.mpr hs0))).symm + _ = F T (G T (α t)) := congrArg (F T) hg + _ = α t := hw.2.2.2.2.2.2.2.1 T _ (hαD t (pos_iff_ne_zero.mpr ht0)) + have hβescape : Tendsto β atTop (cocompact ℂ) := by + apply (hw.centered_chart_escapes T α hαD hescape).congr' + filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with t ht + exact (hβeq t ht).symm + exact ⟨β, hβc, hβi, hβ0, hβH, hβescape, hβeq⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexAngularEnergy.lean b/lean/OAI/Probability/SLE/Geometry/ComplexAngularEnergy.lean index a4a5c9ff8527d487a603dde2745527312f5cbe03..3f40b0da6d984a9a03ad6537b13f6116dcfea94f 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexAngularEnergy.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexAngularEnergy.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Geometry.SmallImageSemicircle - -namespace OAI - -/-! Complex-valued angular sections have small diameter when their -one-dimensional derivative energy is small. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma complex_angular_oscillation {f : ℝ → ℂ} {x y : ℝ} - (hx : x∈Ioo 0 Real.pi) (hy : y∈Ioo 0 Real.pi) - (hf : ∀ θ∈uIcc x y, DifferentiableAt ℝ f θ) - (hd : MemLp (deriv f) 2 (volume.restrict (Ioo 0 Real.pi))) : - ‖f y-f x‖ ≤ Real.sqrt Real.pi * - Real.sqrt (∫ θ in Ioo 0 Real.pi, ‖deriv f θ‖^2) := by - have hsub : uIoc x y⊆Ioo 0 Real.pi := by - intro θ hθ - exact ordConnected_Ioo.uIcc_subset hx hy (uIoc_subset_uIcc hθ) - have hi : IntegrableOn (fun θ => ‖deriv f θ‖) (Ioo 0 Real.pi) := - (hd.integrable (by norm_num)).norm - have hi' : IntegrableOn (deriv f) (Ioo 0 Real.pi) := hd.integrable (by norm_num) - have hid : IntervalIntegrable (deriv f) volume x y := - intervalIntegrable_iff.mpr (hi'.mono_set hsub) - calc - ‖f y-f x‖ = ‖∫ θ in x..y, deriv f θ‖ := - congrArg norm (intervalIntegral.integral_deriv_eq_sub hf hid).symm - _ ≤ ∫ θ in uIoc x y, ‖deriv f θ‖ := intervalIntegral.norm_integral_le_integral_norm_uIoc - _ ≤ ∫ θ in Ioo 0 Real.pi, ‖deriv f θ‖ := - setIntegral_mono_set hi (ae_of_all _ (fun _ => norm_nonneg _)) hsub.eventuallySubset - _ ≤ _ := by - simpa only [abs_norm] using SmallImageSemicircle.integral_abs_le_angular_energy hd.norm - -lemma holomorphic_angular_section_absolutelyContinuous - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f {z | 0 f ((v:ℂ)+(θ:ℂ)*Complex.I)) x y := by - have hc : ContDiffOn ℝ 1 (fun θ : ℝ => f ((v:ℂ)+(θ:ℂ)*Complex.I)) (uIcc x y) := by - intro θ hθ - have hθ' := ordConnected_Ioo.uIcc_subset hx hy hθ - have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 (t:ℂ)) θ := - Complex.ofRealCLM.contDiff.contDiffAt - exact (ha.comp θ (contDiffAt_const.add (hcast.mul contDiffAt_const))).contDiffWithinAt - obtain ⟨K,hK⟩ := hc.exists_lipschitzOnWith (by norm_num) (convex_uIcc x y) isCompact_uIcc - exact hK.absolutelyContinuousOnInterval - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.SmallImageSemicircle + +namespace OAI + +/-! Complex-valued angular sections have small diameter when their +one-dimensional derivative energy is small. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma complex_angular_oscillation {f : ℝ → ℂ} {x y : ℝ} + (hx : x∈Ioo 0 Real.pi) (hy : y∈Ioo 0 Real.pi) + (hf : ∀ θ∈uIcc x y, DifferentiableAt ℝ f θ) + (hd : MemLp (deriv f) 2 (volume.restrict (Ioo 0 Real.pi))) : + ‖f y-f x‖ ≤ Real.sqrt Real.pi * + Real.sqrt (∫ θ in Ioo 0 Real.pi, ‖deriv f θ‖^2) := by + have hsub : uIoc x y⊆Ioo 0 Real.pi := by + intro θ hθ + exact ordConnected_Ioo.uIcc_subset hx hy (uIoc_subset_uIcc hθ) + have hi : IntegrableOn (fun θ => ‖deriv f θ‖) (Ioo 0 Real.pi) := + (hd.integrable (by norm_num)).norm + have hi' : IntegrableOn (deriv f) (Ioo 0 Real.pi) := hd.integrable (by norm_num) + have hid : IntervalIntegrable (deriv f) volume x y := + intervalIntegrable_iff.mpr (hi'.mono_set hsub) + calc + ‖f y-f x‖ = ‖∫ θ in x..y, deriv f θ‖ := + congrArg norm (intervalIntegral.integral_deriv_eq_sub hf hid).symm + _ ≤ ∫ θ in uIoc x y, ‖deriv f θ‖ := intervalIntegral.norm_integral_le_integral_norm_uIoc + _ ≤ ∫ θ in Ioo 0 Real.pi, ‖deriv f θ‖ := + setIntegral_mono_set hi (ae_of_all _ (fun _ => norm_nonneg _)) hsub.eventuallySubset + _ ≤ _ := by + simpa only [abs_norm] using SmallImageSemicircle.integral_abs_le_angular_energy hd.norm + +lemma holomorphic_angular_section_absolutelyContinuous + {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f {z | 0 f ((v:ℂ)+(θ:ℂ)*Complex.I)) x y := by + have hc : ContDiffOn ℝ 1 (fun θ : ℝ => f ((v:ℂ)+(θ:ℂ)*Complex.I)) (uIcc x y) := by + intro θ hθ + have hθ' := ordConnected_Ioo.uIcc_subset hx hy hθ + have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 (t:ℂ)) θ := + Complex.ofRealCLM.contDiff.contDiffAt + exact (ha.comp θ (contDiffAt_const.add (hcast.mul contDiffAt_const))).contDiffWithinAt + obtain ⟨K,hK⟩ := hc.exists_lipschitzOnWith (by norm_num) (convex_uIcc x y) isCompact_uIcc + exact hK.absolutelyContinuousOnInterval + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexDirichletSlices.lean b/lean/OAI/Probability/SLE/Geometry/ComplexDirichletSlices.lean index 392fe98bf8041bf3250b3aac17aeccca1f09ca17..0796bf98bf5055fdb0b8d9af72caf882c04313f9 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexDirichletSlices.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexDirichletSlices.lean @@ -1,58 +1,58 @@ -import OAI.Probability.SLE.Geometry.ComplexAngularEnergy - -namespace OAI - -/-! Finite holomorphic energy gives square-integrable derivatives and -low-energy angular sections in each unit logarithmic strip. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma deriv_memLp_two_of_energy {f : ℂ → ℂ} {S : Set ℂ} (hS : IsOpen S) - (hf : DifferentiableOn ℂ f S) - (he : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞) : - MemLp (deriv f) 2 (volume.restrict S) := by - have hc : ContinuousOn (deriv f) S := (hf.deriv hS).continuousOn - have hm : AEStronglyMeasurable (deriv f) (volume.restrict S) := - hc.aestronglyMeasurable hS.measurableSet - apply (memLp_two_iff_integrable_sq_norm hm).mpr - exact ⟨hm.norm.pow 2, (hasFiniteIntegral_iff_ofReal - (ae_of_all _ (fun _ => sq_nonneg _))).mpr he⟩ - -lemma exists_complex_energy_section (g : ℝ × ℝ → ℂ) (q : ℝ) - (hg : MemLp g 2 ((volume.restrict (Ioo q (q+1))).prod - (volume.restrict (Ioo 0 Real.pi)))) : - ∃ v∈Ioo q (q+1), MemLp (fun θ => g (v,θ)) 2 (volume.restrict (Ioo 0 Real.pi)) ∧ - (∫ θ in Ioo 0 Real.pi, ‖g (v,θ)‖^2) ≤ - ∫ p, ‖g p‖^2 ∂((volume.restrict (Ioo q (q+1))).prod (volume.restrict (Ioo 0 Real.pi))) := by - let J := Ioo q (q+1) - let K := Ioo (0:ℝ) Real.pi - have hi := (memLp_two_iff_integrable_sq_norm hg.aestronglyMeasurable).mp hg - have hgood : ∀ᵐ v ∂volume.restrict J, - v∈J ∧ MemLp (fun θ => g (v,θ)) 2 (volume.restrict K) := by - filter_upwards [ae_restrict_mem measurableSet_Ioo,hi.prod_right_ae, - hg.aestronglyMeasurable.prodMk_left] with v hv hvi hvm - exact ⟨hv,(memLp_two_iff_integrable_sq_norm hvm).mpr hvi⟩ - have : IsProbabilityMeasure (volume.restrict J) := by - constructor - simp [J,Real.volume_Ioo] - let energy := fun v => ∫ θ in K, ‖g (v,θ)‖^2 - have hei : Integrable energy (volume.restrict J) := hi.integral_prod_left - let Good := fun v => v∈J ∧ MemLp (fun θ => g (v,θ)) 2 (volume.restrict K) - have hnull : (volume.restrict J) {v | ¬Good v}=0 := by - apply measure_eq_zero_iff_ae_notMem.mpr - filter_upwards [hgood] with v hv - exact fun h => h hv - obtain ⟨v,hv,hve⟩ := exists_notMem_null_le_integral hei hnull - have hv' : Good v := not_not.mp hv - refine ⟨v,hv'.1,hv'.2,?_⟩ - rw [integral_prod _ hi] - exact hve - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexAngularEnergy + +namespace OAI + +/-! Finite holomorphic energy gives square-integrable derivatives and +low-energy angular sections in each unit logarithmic strip. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma deriv_memLp_two_of_energy {f : ℂ → ℂ} {S : Set ℂ} (hS : IsOpen S) + (hf : DifferentiableOn ℂ f S) + (he : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞) : + MemLp (deriv f) 2 (volume.restrict S) := by + have hc : ContinuousOn (deriv f) S := (hf.deriv hS).continuousOn + have hm : AEStronglyMeasurable (deriv f) (volume.restrict S) := + hc.aestronglyMeasurable hS.measurableSet + apply (memLp_two_iff_integrable_sq_norm hm).mpr + exact ⟨hm.norm.pow 2, (hasFiniteIntegral_iff_ofReal + (ae_of_all _ (fun _ => sq_nonneg _))).mpr he⟩ + +lemma exists_complex_energy_section (g : ℝ × ℝ → ℂ) (q : ℝ) + (hg : MemLp g 2 ((volume.restrict (Ioo q (q+1))).prod + (volume.restrict (Ioo 0 Real.pi)))) : + ∃ v∈Ioo q (q+1), MemLp (fun θ => g (v,θ)) 2 (volume.restrict (Ioo 0 Real.pi)) ∧ + (∫ θ in Ioo 0 Real.pi, ‖g (v,θ)‖^2) ≤ + ∫ p, ‖g p‖^2 ∂((volume.restrict (Ioo q (q+1))).prod (volume.restrict (Ioo 0 Real.pi))) := by + let J := Ioo q (q+1) + let K := Ioo (0:ℝ) Real.pi + have hi := (memLp_two_iff_integrable_sq_norm hg.aestronglyMeasurable).mp hg + have hgood : ∀ᵐ v ∂volume.restrict J, + v∈J ∧ MemLp (fun θ => g (v,θ)) 2 (volume.restrict K) := by + filter_upwards [ae_restrict_mem measurableSet_Ioo,hi.prod_right_ae, + hg.aestronglyMeasurable.prodMk_left] with v hv hvi hvm + exact ⟨hv,(memLp_two_iff_integrable_sq_norm hvm).mpr hvi⟩ + have : IsProbabilityMeasure (volume.restrict J) := by + constructor + simp [J,Real.volume_Ioo] + let energy := fun v => ∫ θ in K, ‖g (v,θ)‖^2 + have hei : Integrable energy (volume.restrict J) := hi.integral_prod_left + let Good := fun v => v∈J ∧ MemLp (fun θ => g (v,θ)) 2 (volume.restrict K) + have hnull : (volume.restrict J) {v | ¬Good v}=0 := by + apply measure_eq_zero_iff_ae_notMem.mpr + filter_upwards [hgood] with v hv + exact fun h => h hv + obtain ⟨v,hv,hve⟩ := exists_notMem_null_le_integral hei hnull + have hv' : Good v := not_not.mp hv + refine ⟨v,hv'.1,hv'.2,?_⟩ + rw [integral_prod _ hi] + exact hve + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexDissipation.lean b/lean/OAI/Probability/SLE/Geometry/ComplexDissipation.lean index 103612e825a17782f374c3bc11e1c63b2c2263e3..cc14c663b97b42507fb48c0c92331572e27cf7d9 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexDissipation.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexDissipation.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Geometry.EndpointMonotonicity - -namespace OAI - -/-! A complex linear equation with nonpositive real coefficient contracts, -including an initial endpoint where the coefficient may be singular. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma complex_normSq_hasDerivWithinAt {S : Set ℝ} {t : ℝ} - (f : ℝ → ℂ) (v : ℂ) (hf : HasDerivWithinAt f v S t) : - HasDerivWithinAt (fun s => Complex.normSq (f s)) - (2*((f t).re*v.re+(f t).im*v.im)) S t := by - have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hf - have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hf - change HasDerivWithinAt (fun s => (f s).re) v.re S t at hr - change HasDerivWithinAt (fun s => (f s).im) v.im S t at hi - convert! (hr.pow 2).add (hi.pow 2) using 1 - · funext s - simp only [Complex.normSq_apply,pow_two,Pi.add_apply,Pi.mul_apply] - · ring - -theorem complex_linear_endpoint_contraction {T : ℝ} (hT : 0 ≤ T) - (f c : ℝ → ℂ) (hc : ContinuousOn f (Icc 0 T)) - (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (c t*f t) (Ici t) t) - (hb : ∀ t ∈ Ioc 0 T, (c t).re ≤ 0) : - ∀ t ∈ Icc 0 T, ‖f t‖ ≤ ‖f 0‖ := by - have hsq := le_initial_of_deriv_nonpos_on_Ioc hT - (fun t => Complex.normSq (f t)) - (fun t => 2*(c t).re*Complex.normSq (f t)) - (Complex.continuous_normSq.comp_continuousOn hc) (fun t ht => ?_) (fun t ht => ?_) - · intro t ht - have hh := hsq t ht - rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hh - nlinarith [norm_nonneg (f t),norm_nonneg (f 0)] - · convert! complex_normSq_hasDerivWithinAt f (c t*f t) (hd t ht) using 1 - simp only [Complex.mul_re,Complex.mul_im,Complex.normSq_apply] - ring - · exact mul_nonpos_of_nonpos_of_nonneg - (mul_nonpos_of_nonneg_of_nonpos (by norm_num) (hb t ht)) (Complex.normSq_nonneg _) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.EndpointMonotonicity + +namespace OAI + +/-! A complex linear equation with nonpositive real coefficient contracts, +including an initial endpoint where the coefficient may be singular. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma complex_normSq_hasDerivWithinAt {S : Set ℝ} {t : ℝ} + (f : ℝ → ℂ) (v : ℂ) (hf : HasDerivWithinAt f v S t) : + HasDerivWithinAt (fun s => Complex.normSq (f s)) + (2*((f t).re*v.re+(f t).im*v.im)) S t := by + have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hf + have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hf + change HasDerivWithinAt (fun s => (f s).re) v.re S t at hr + change HasDerivWithinAt (fun s => (f s).im) v.im S t at hi + convert! (hr.pow 2).add (hi.pow 2) using 1 + · funext s + simp only [Complex.normSq_apply,pow_two,Pi.add_apply,Pi.mul_apply] + · ring + +theorem complex_linear_endpoint_contraction {T : ℝ} (hT : 0 ≤ T) + (f c : ℝ → ℂ) (hc : ContinuousOn f (Icc 0 T)) + (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (c t*f t) (Ici t) t) + (hb : ∀ t ∈ Ioc 0 T, (c t).re ≤ 0) : + ∀ t ∈ Icc 0 T, ‖f t‖ ≤ ‖f 0‖ := by + have hsq := le_initial_of_deriv_nonpos_on_Ioc hT + (fun t => Complex.normSq (f t)) + (fun t => 2*(c t).re*Complex.normSq (f t)) + (Complex.continuous_normSq.comp_continuousOn hc) (fun t ht => ?_) (fun t ht => ?_) + · intro t ht + have hh := hsq t ht + rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hh + nlinarith [norm_nonneg (f t),norm_nonneg (f 0)] + · convert! complex_normSq_hasDerivWithinAt f (c t*f t) (hd t ht) using 1 + simp only [Complex.mul_re,Complex.mul_im,Complex.normSq_apply] + ring + · exact mul_nonpos_of_nonpos_of_nonneg + (mul_nonpos_of_nonneg_of_nonpos (by norm_num) (hb t ht)) (Complex.normSq_nonneg _) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexEndpointCluster.lean b/lean/OAI/Probability/SLE/Geometry/ComplexEndpointCluster.lean index e66953d8b0e459949d5ec225751e1d0811e86332..2a58a54fd2ea914759f3fb0f11a2a25d84ce8e9f 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexEndpointCluster.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexEndpointCluster.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Geometry.EndpointClusterLimit -import Mathlib.Analysis.Complex.Basic - -namespace OAI - -/-! Lifting a real cluster value when the imaginary coordinate tends to -zero. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology - -namespace SLEExactGauge - -lemma complex_cluster_of_real_cluster {α : Type*} {l : Filter α} {f : α → ℂ} {a : ℝ} - (him : Tendsto (fun x => (f x).im) l (𝓝 0)) - (ha : MapClusterPt a l (fun x => (f x).re)) : MapClusterPt (a:ℂ) l f := by - apply (Metric.nhds_basis_ball.mapClusterPt_iff_frequently).mpr - intro ε hε - have hre := ha.frequently (ball_mem_nhds a (half_pos hε)) - have himsmall : ∀ᶠ x in l, |(f x).im|<ε/2 := by - have hh : Tendsto (fun x => |(f x).im|) l (𝓝 0) := by - simpa only [norm_zero,Real.norm_eq_abs] using him.norm - exact hh.eventually (gt_mem_nhds (half_pos hε)) - apply (hre.and_eventually himsmall).mono - intro x hx - rw [mem_ball,dist_eq_norm] - have hr : |(f x).re-a|<ε/2 := by - simpa only [mem_ball,Real.dist_eq] using hx.1 - calc - ‖f x-(a:ℂ)‖ ≤ |(f x).re-a|+|(f x).im| := by - simpa only [sub_re,ofReal_re,sub_im,ofReal_im,sub_zero] using - norm_le_abs_re_add_abs_im (f x-(a:ℂ)) - _ < ε := by linarith [hx.2] - -lemma cluster_value_eq_limit {α X : Type*} [TopologicalSpace X] [T2Space X] - {l : Filter α} {f : α → X} {a b : X} - (ha : MapClusterPt a l f) (hb : Tendsto f l (𝓝 b)) : a=b := by - let : NeBot (𝓝 a ⊓ map f l) := ha - have h1 : Tendsto id (𝓝 a ⊓ map f l) (𝓝 a) := inf_le_left - have h2 : Tendsto id (𝓝 a ⊓ map f l) (𝓝 b) := inf_le_right.trans hb - exact tendsto_nhds_unique h1 h2 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.EndpointClusterLimit +import Mathlib.Analysis.Complex.Basic + +namespace OAI + +/-! Lifting a real cluster value when the imaginary coordinate tends to +zero. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology + +namespace SLEExactGauge + +lemma complex_cluster_of_real_cluster {α : Type*} {l : Filter α} {f : α → ℂ} {a : ℝ} + (him : Tendsto (fun x => (f x).im) l (𝓝 0)) + (ha : MapClusterPt a l (fun x => (f x).re)) : MapClusterPt (a:ℂ) l f := by + apply (Metric.nhds_basis_ball.mapClusterPt_iff_frequently).mpr + intro ε hε + have hre := ha.frequently (ball_mem_nhds a (half_pos hε)) + have himsmall : ∀ᶠ x in l, |(f x).im|<ε/2 := by + have hh : Tendsto (fun x => |(f x).im|) l (𝓝 0) := by + simpa only [norm_zero,Real.norm_eq_abs] using him.norm + exact hh.eventually (gt_mem_nhds (half_pos hε)) + apply (hre.and_eventually himsmall).mono + intro x hx + rw [mem_ball,dist_eq_norm] + have hr : |(f x).re-a|<ε/2 := by + simpa only [mem_ball,Real.dist_eq] using hx.1 + calc + ‖f x-(a:ℂ)‖ ≤ |(f x).re-a|+|(f x).im| := by + simpa only [sub_re,ofReal_re,sub_im,ofReal_im,sub_zero] using + norm_le_abs_re_add_abs_im (f x-(a:ℂ)) + _ < ε := by linarith [hx.2] + +lemma cluster_value_eq_limit {α X : Type*} [TopologicalSpace X] [T2Space X] + {l : Filter α} {f : α → X} {a b : X} + (ha : MapClusterPt a l f) (hb : Tendsto f l (𝓝 b)) : a=b := by + let : NeBot (𝓝 a ⊓ map f l) := ha + have h1 : Tendsto id (𝓝 a ⊓ map f l) (𝓝 a) := inf_le_left + have h2 : Tendsto id (𝓝 a ⊓ map f l) (𝓝 b) := inf_le_right.trans hb + exact tendsto_nhds_unique h1 h2 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexEndpointLimit.lean b/lean/OAI/Probability/SLE/Geometry/ComplexEndpointLimit.lean index 96f6f9ad545a0c1748b9bcba01842b8d4d0e69cd..96e257e9eb0a9387c81d6b26c5b201fe2d6c004c 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexEndpointLimit.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexEndpointLimit.lean @@ -1,59 +1,59 @@ -import OAI.Probability.SLE.Geometry.ComplexEndpointCluster - -namespace OAI - -/-! A bounded upper-half-plane path has a real endpoint limit when its -continuous image has a limit and the corresponding boundary fiber has no -interval. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology - -namespace SLEExactGauge - -lemma real_cluster_in_boundary_fiber {α : Type*} {l : Filter α} - {f : α → ℂ} {B : ℂ → ℂ} {a : ℝ} {b : ℂ} - (hB : ContinuousOn B {z : ℂ | 0≤z.im}) - (hmem : ∀ᶠ x in l, 0≤(f x).im) - (him : Tendsto (fun x => (f x).im) l (𝓝 0)) - (himage : Tendsto (fun x => B (f x)) l (𝓝 b)) - (ha : MapClusterPt a l (fun x => (f x).re)) : B (a:ℂ)=b := by - have hca := complex_cluster_of_real_cluster him ha - have hle : 𝓝 (a:ℂ) ⊓ map f l ≤ 𝓝[{z : ℂ | 0≤z.im}] (a:ℂ) := - le_inf inf_le_left (inf_le_right.trans (tendsto_principal.mpr hmem)) - have hcont : Tendsto B (𝓝[{z : ℂ | 0≤z.im}] (a:ℂ)) (𝓝 (B (a:ℂ))) := hB _ (by simp) - have hc := MapClusterPt.tendsto_comp' (hcont.mono_left hle) hca - exact cluster_value_eq_limit hc himage - -theorem exists_complex_endpoint_limit {f : ℝ → ℂ} {B : ℂ → ℂ} {b : ℂ} {D M : ℝ} - (hD : 0] (0:ℝ), ‖f t‖≤M) - (hmem : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), 0≤(f t).im) - (him : Tendsto (fun t => (f t).im) (𝓝[>] (0:ℝ)) (𝓝 0)) - (hB : ContinuousOn B {z : ℂ | 0≤z.im}) - (himage : Tendsto (fun t => B (f t)) (𝓝[>] (0:ℝ)) (𝓝 b)) - (hno : ∀ a d : ℝ, a] (0:ℝ)) (𝓝 (a:ℂ)) ∧ B (a:ℂ)=b := by - have hrbound : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), (f t).re∈Icc (-M) M := by - filter_upwards [hbound] with t ht - exact abs_le.mp ((abs_re_le_norm (f t)).trans ht) - have hnoCluster : ∀ a d : ℝ, a] (0:ℝ)) (fun t => (f t).re) := by - intro a d had hc - apply hno a d had - intro c hcd - exact real_cluster_in_boundary_fiber hB hmem him himage (hc c hcd) - obtain ⟨a,_,ha⟩ := exists_endpoint_limit_of_no_cluster_interval hD - (continuous_re.comp_continuousOn hf) isCompact_Icc hrbound hnoCluster - have hcomplex : Tendsto f (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by - have hh := (continuous_ofReal.continuousAt.tendsto.comp ha).add - ((continuous_ofReal.continuousAt.tendsto.comp him).mul_const I) - simpa only [Function.comp_apply,ofReal_zero,zero_mul,add_zero,re_add_im] using hh - refine ⟨a,hcomplex,?_⟩ - exact real_cluster_in_boundary_fiber hB hmem him himage ha.mapClusterPt - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexEndpointCluster + +namespace OAI + +/-! A bounded upper-half-plane path has a real endpoint limit when its +continuous image has a limit and the corresponding boundary fiber has no +interval. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology + +namespace SLEExactGauge + +lemma real_cluster_in_boundary_fiber {α : Type*} {l : Filter α} + {f : α → ℂ} {B : ℂ → ℂ} {a : ℝ} {b : ℂ} + (hB : ContinuousOn B {z : ℂ | 0≤z.im}) + (hmem : ∀ᶠ x in l, 0≤(f x).im) + (him : Tendsto (fun x => (f x).im) l (𝓝 0)) + (himage : Tendsto (fun x => B (f x)) l (𝓝 b)) + (ha : MapClusterPt a l (fun x => (f x).re)) : B (a:ℂ)=b := by + have hca := complex_cluster_of_real_cluster him ha + have hle : 𝓝 (a:ℂ) ⊓ map f l ≤ 𝓝[{z : ℂ | 0≤z.im}] (a:ℂ) := + le_inf inf_le_left (inf_le_right.trans (tendsto_principal.mpr hmem)) + have hcont : Tendsto B (𝓝[{z : ℂ | 0≤z.im}] (a:ℂ)) (𝓝 (B (a:ℂ))) := hB _ (by simp) + have hc := MapClusterPt.tendsto_comp' (hcont.mono_left hle) hca + exact cluster_value_eq_limit hc himage + +theorem exists_complex_endpoint_limit {f : ℝ → ℂ} {B : ℂ → ℂ} {b : ℂ} {D M : ℝ} + (hD : 0] (0:ℝ), ‖f t‖≤M) + (hmem : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), 0≤(f t).im) + (him : Tendsto (fun t => (f t).im) (𝓝[>] (0:ℝ)) (𝓝 0)) + (hB : ContinuousOn B {z : ℂ | 0≤z.im}) + (himage : Tendsto (fun t => B (f t)) (𝓝[>] (0:ℝ)) (𝓝 b)) + (hno : ∀ a d : ℝ, a] (0:ℝ)) (𝓝 (a:ℂ)) ∧ B (a:ℂ)=b := by + have hrbound : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), (f t).re∈Icc (-M) M := by + filter_upwards [hbound] with t ht + exact abs_le.mp ((abs_re_le_norm (f t)).trans ht) + have hnoCluster : ∀ a d : ℝ, a] (0:ℝ)) (fun t => (f t).re) := by + intro a d had hc + apply hno a d had + intro c hcd + exact real_cluster_in_boundary_fiber hB hmem him himage (hc c hcd) + obtain ⟨a,_,ha⟩ := exists_endpoint_limit_of_no_cluster_interval hD + (continuous_re.comp_continuousOn hf) isCompact_Icc hrbound hnoCluster + have hcomplex : Tendsto f (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by + have hh := (continuous_ofReal.continuousAt.tendsto.comp ha).add + ((continuous_ofReal.continuousAt.tendsto.comp him).mul_const I) + simpa only [Function.comp_apply,ofReal_zero,zero_mul,add_zero,re_add_im] using hh + refine ⟨a,hcomplex,?_⟩ + exact real_cluster_in_boundary_fiber hB hmem him himage ha.mapClusterPt + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexHarmonicGenerator.lean b/lean/OAI/Probability/SLE/Geometry/ComplexHarmonicGenerator.lean index 3cc345356a6eea5d38f6907118912f3e0577468c..d786862f2e154a27d6ec1f70d21dd62ce709d756 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexHarmonicGenerator.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexHarmonicGenerator.lean @@ -1,62 +1,62 @@ -import OAI.Probability.SLE.Stochastic.PlanarHarmonicLocalization -import OAI.Probability.SLE.Geometry.ConformalLaplacian - -namespace OAI - -/-! The planar Brownian generator of a complex-plane harmonic function is -one half of its real Laplacian. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Complex InnerProductSpace -open scoped Topology NNReal ENNReal Laplacian - -namespace SLEExactGauge - -lemma second_derivative_constant_curve {u : ℂ → ℝ} {ψ : ℝ → ℂ} {a : ℝ} {v : ℂ} - (hu : ContDiffAt ℝ 2 u (ψ a)) (hψ : ∀ r, HasDerivAt ψ v r) : - iteratedDeriv 2 (fun r => u (ψ r)) a = fderiv ℝ (fderiv ℝ u) (ψ a) v v := by - have hu1 : ContDiffAt ℝ 1 u (ψ a) := hu.of_le (by norm_num) - have hevent : ∀ᶠ z in 𝓝 (ψ a), ContDiffAt ℝ 1 u z := hu1.eventually (by norm_num) - have he : deriv (fun r => u (ψ r)) =ᶠ[𝓝 a] (fun r => fderiv ℝ u (ψ r) v) := by - filter_upwards [(hψ a).continuousAt.tendsto.eventually hevent] with r hr - exact ((hr.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt r (hψ r)).deriv - have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt - have hc := (hd.comp_hasDerivAt a (hψ a)).clm_apply (hasDerivAt_const a v) - have hv : deriv (fun r => fderiv ℝ u (ψ r) v) a=fderiv ℝ (fderiv ℝ u) (ψ a) v v := by - simpa only [Function.comp_apply,map_zero,add_zero] using hc.deriv - rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,he.deriv_eq,hv] - -lemma planarGenerator_complex_eq (u : ℂ → ℝ) (z : ℝ×ℝ) - (hu : ContDiffAt ℝ 2 u (Complex.equivRealProdCLM.symm z)) : - planarGenerator (u ∘ Complex.equivRealProdCLM.symm) z= - (1/2:ℝ)*(Δ u) (Complex.equivRealProdCLM.symm z) := by - have hx : ∀ r : ℝ, HasDerivAt (fun x : ℝ => (x:ℂ)+(z.2:ℂ)*I) 1 r := by - intro r - simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using - (Complex.ofRealCLM.hasDerivAt (x := r)).add_const ((z.2:ℂ)*I) - have hy : ∀ r : ℝ, HasDerivAt (fun y : ℝ => (z.1:ℂ)+(y:ℂ)*I) I r := by - intro r - simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul] using - ((Complex.ofRealCLM.hasDerivAt (x := r)).mul_const I).const_add (z.1:ℂ) - have hu' : ContDiffAt ℝ 2 u ((z.1:ℂ)+(z.2:ℂ)*I) := by - simpa only [Complex.equivRealProdCLM_symm_apply] using hu - have hxr := second_derivative_constant_curve (ψ := fun x : ℝ => (x:ℂ)+(z.2:ℂ)*I) - (a := z.1) hu' hx - have hyr := second_derivative_constant_curve (ψ := fun y : ℝ => (z.1:ℂ)+(y:ℂ)*I) - (a := z.2) hu' hy - simp only [Function.comp_apply,Complex.equivRealProdCLM_symm_apply,planarGenerator] - rw [hxr,hyr] - congr 1 - simp only [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply, - Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one] - -lemma planarGenerator_harmonicAt (u : ℂ → ℝ) (z : ℝ×ℝ) - (hu : HarmonicAt u (Complex.equivRealProdCLM.symm z)) : - planarGenerator (u ∘ Complex.equivRealProdCLM.symm) z=0 := by - rw [planarGenerator_complex_eq u z hu.1] - simp only [hu.2.self_of_nhds,Pi.zero_apply,mul_zero] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.PlanarHarmonicLocalization +import OAI.Probability.SLE.Geometry.ConformalLaplacian + +namespace OAI + +/-! The planar Brownian generator of a complex-plane harmonic function is +one half of its real Laplacian. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Complex InnerProductSpace +open scoped Topology NNReal ENNReal Laplacian + +namespace SLEExactGauge + +lemma second_derivative_constant_curve {u : ℂ → ℝ} {ψ : ℝ → ℂ} {a : ℝ} {v : ℂ} + (hu : ContDiffAt ℝ 2 u (ψ a)) (hψ : ∀ r, HasDerivAt ψ v r) : + iteratedDeriv 2 (fun r => u (ψ r)) a = fderiv ℝ (fderiv ℝ u) (ψ a) v v := by + have hu1 : ContDiffAt ℝ 1 u (ψ a) := hu.of_le (by norm_num) + have hevent : ∀ᶠ z in 𝓝 (ψ a), ContDiffAt ℝ 1 u z := hu1.eventually (by norm_num) + have he : deriv (fun r => u (ψ r)) =ᶠ[𝓝 a] (fun r => fderiv ℝ u (ψ r) v) := by + filter_upwards [(hψ a).continuousAt.tendsto.eventually hevent] with r hr + exact ((hr.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt r (hψ r)).deriv + have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt + have hc := (hd.comp_hasDerivAt a (hψ a)).clm_apply (hasDerivAt_const a v) + have hv : deriv (fun r => fderiv ℝ u (ψ r) v) a=fderiv ℝ (fderiv ℝ u) (ψ a) v v := by + simpa only [Function.comp_apply,map_zero,add_zero] using hc.deriv + rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,he.deriv_eq,hv] + +lemma planarGenerator_complex_eq (u : ℂ → ℝ) (z : ℝ×ℝ) + (hu : ContDiffAt ℝ 2 u (Complex.equivRealProdCLM.symm z)) : + planarGenerator (u ∘ Complex.equivRealProdCLM.symm) z= + (1/2:ℝ)*(Δ u) (Complex.equivRealProdCLM.symm z) := by + have hx : ∀ r : ℝ, HasDerivAt (fun x : ℝ => (x:ℂ)+(z.2:ℂ)*I) 1 r := by + intro r + simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using + (Complex.ofRealCLM.hasDerivAt (x := r)).add_const ((z.2:ℂ)*I) + have hy : ∀ r : ℝ, HasDerivAt (fun y : ℝ => (z.1:ℂ)+(y:ℂ)*I) I r := by + intro r + simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul] using + ((Complex.ofRealCLM.hasDerivAt (x := r)).mul_const I).const_add (z.1:ℂ) + have hu' : ContDiffAt ℝ 2 u ((z.1:ℂ)+(z.2:ℂ)*I) := by + simpa only [Complex.equivRealProdCLM_symm_apply] using hu + have hxr := second_derivative_constant_curve (ψ := fun x : ℝ => (x:ℂ)+(z.2:ℂ)*I) + (a := z.1) hu' hx + have hyr := second_derivative_constant_curve (ψ := fun y : ℝ => (z.1:ℂ)+(y:ℂ)*I) + (a := z.2) hu' hy + simp only [Function.comp_apply,Complex.equivRealProdCLM_symm_apply,planarGenerator] + rw [hxr,hyr] + congr 1 + simp only [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply, + Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one] + +lemma planarGenerator_harmonicAt (u : ℂ → ℝ) (z : ℝ×ℝ) + (hu : HarmonicAt u (Complex.equivRealProdCLM.symm z)) : + planarGenerator (u ∘ Complex.equivRealProdCLM.symm) z=0 := by + rw [planarGenerator_complex_eq u z hu.1] + simp only [hu.2.self_of_nhds,Pi.zero_apply,mul_zero] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexJordanExtension.lean b/lean/OAI/Probability/SLE/Geometry/ComplexJordanExtension.lean index 0a6b25cacd58c8757f2d8aa7141534004eeb9946..5256f900f2c86af3621b8ab16518abec144efa7a 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexJordanExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexJordanExtension.lean @@ -1,61 +1,61 @@ -import Schoenflies.JordanSchoenflies - -namespace OAI - -/-! Complex-coordinate application of the relative Schönflies theorem. -The prescribed boundary homeomorphism is retained pointwise. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -def complexPlaneCoordinates : ℂ ≃ₜ Schoenflies.Plane where - toFun z := !₂[z.re,z.im] - invFun v := ⟨v 0,v 1⟩ - left_inv _ := rfl - right_inv v := by - ext i - fin_cases i <;> rfl - continuous_toFun := by - apply (PiLp.continuous_toLp _ _).comp - apply continuous_pi - intro i - fin_cases i - · exact Complex.continuous_re - · exact Complex.continuous_im - continuous_invFun := by - have hc : Continuous (fun v : Schoenflies.Plane => - (v 0 : ℂ)+(v 1 : ℂ)*Complex.I) := by fun_prop - convert hc using 1 - funext v - apply Complex.ext <;> simp - -lemma complexPlaneCoordinates_isArcBetween - (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) : - Schoenflies.IsArcBetween (complexPlaneCoordinates '' (p '' Icc 0 1)) - (complexPlaneCoordinates (p 0)) (complexPlaneCoordinates (p 1)) := by - refine ⟨complexPlaneCoordinates ∘ p, - complexPlaneCoordinates.continuous.comp_continuousOn hc, - (complexPlaneCoordinates.injective.injOn).comp hi (mapsTo_image p _),?_,rfl,rfl⟩ - rw [image_image] - rfl - -theorem complex_jordan_extension {C D : Set ℂ} - (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) - (hD : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' D)) - (e : C ≃ₜ D) : - ∃ H : ℂ ≃ₜ ℂ, ∀ z : C, H z=e z := by - let ec := ((complexPlaneCoordinates.image C).symm.trans e).trans - (complexPlaneCoordinates.image D) - obtain ⟨F,hF⟩ := Schoenflies.jordan_schoenflies_of_homeomorph hC hD ec - refine ⟨(complexPlaneCoordinates.trans F).trans complexPlaneCoordinates.symm,?_⟩ - intro z - apply complexPlaneCoordinates.injective - have hh := hF ((complexPlaneCoordinates.image C) z) - simpa [ec] using hh - -end SLEExactGauge - -end - -end OAI +import Schoenflies.JordanSchoenflies + +namespace OAI + +/-! Complex-coordinate application of the relative Schönflies theorem. +The prescribed boundary homeomorphism is retained pointwise. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +def complexPlaneCoordinates : ℂ ≃ₜ Schoenflies.Plane where + toFun z := !₂[z.re,z.im] + invFun v := ⟨v 0,v 1⟩ + left_inv _ := rfl + right_inv v := by + ext i + fin_cases i <;> rfl + continuous_toFun := by + apply (PiLp.continuous_toLp _ _).comp + apply continuous_pi + intro i + fin_cases i + · exact Complex.continuous_re + · exact Complex.continuous_im + continuous_invFun := by + have hc : Continuous (fun v : Schoenflies.Plane => + (v 0 : ℂ)+(v 1 : ℂ)*Complex.I) := by fun_prop + convert hc using 1 + funext v + apply Complex.ext <;> simp + +lemma complexPlaneCoordinates_isArcBetween + (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) : + Schoenflies.IsArcBetween (complexPlaneCoordinates '' (p '' Icc 0 1)) + (complexPlaneCoordinates (p 0)) (complexPlaneCoordinates (p 1)) := by + refine ⟨complexPlaneCoordinates ∘ p, + complexPlaneCoordinates.continuous.comp_continuousOn hc, + (complexPlaneCoordinates.injective.injOn).comp hi (mapsTo_image p _),?_,rfl,rfl⟩ + rw [image_image] + rfl + +theorem complex_jordan_extension {C D : Set ℂ} + (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) + (hD : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' D)) + (e : C ≃ₜ D) : + ∃ H : ℂ ≃ₜ ℂ, ∀ z : C, H z=e z := by + let ec := ((complexPlaneCoordinates.image C).symm.trans e).trans + (complexPlaneCoordinates.image D) + obtain ⟨F,hF⟩ := Schoenflies.jordan_schoenflies_of_homeomorph hC hD ec + refine ⟨(complexPlaneCoordinates.trans F).trans complexPlaneCoordinates.symm,?_⟩ + intro z + apply complexPlaneCoordinates.injective + have hh := hF ((complexPlaneCoordinates.image C) z) + simpa [ec] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexJordanRegions.lean b/lean/OAI/Probability/SLE/Geometry/ComplexJordanRegions.lean index bdfcd338ec622e108140f167f7ecbe6d491e8ab3..9517b87a621d2ca3fc16e90d8704fdf7e52ba330 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexJordanRegions.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexJordanRegions.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Geometry.SchoenfliesInsideTransport - -namespace OAI - -/-! The bounded Jordan region in the complex coordinate chart. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -def complexInside (C : Set ℂ) : Set ℂ := - complexPlaneCoordinates ⁻¹' Schoenflies.inside (complexPlaneCoordinates '' C) - -theorem complexInside_image (C : Set ℂ) : - complexPlaneCoordinates '' complexInside C = - Schoenflies.inside (complexPlaneCoordinates '' C) := - image_preimage_eq _ complexPlaneCoordinates.surjective - -theorem complexInside_transport (h : ℂ ≃ₜ ℂ) (C : Set ℂ) : - h '' complexInside C=complexInside (h '' C) := by - let H := (complexPlaneCoordinates.symm.trans h).trans complexPlaneCoordinates - have he : H '' (complexPlaneCoordinates '' C)=complexPlaneCoordinates '' (h '' C) := by - ext x - simp [H] - apply complexPlaneCoordinates.injective.image_injective - rw [complexInside_image] - have hm : complexPlaneCoordinates '' (h '' complexInside C)= - H '' (complexPlaneCoordinates '' complexInside C) := by - ext x - simp [H] - rw [hm,complexInside_image,plane_homeomorph_inside,he] - -theorem complexInside_eq_open {C U : Set ℂ} - (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) - (hU : IsOpen U) (hconn : IsPreconnected U) (hne : U.Nonempty) - (hb : Bornology.IsBounded U) (hfront : frontier U=C) : complexInside C=U := by - apply complexPlaneCoordinates.injective.image_injective - rw [complexInside_image] - apply separating_inside_eq_open (Schoenflies.jordan_curve_theorem hC) - (complexPlaneCoordinates.isOpen_image.mpr hU) - (hconn.image _ complexPlaneCoordinates.continuous.continuousOn) - (hne.image _) ?_ ?_ - · exact (hb.isCompact_closure.image complexPlaneCoordinates.continuous).isBounded.subset - (image_mono subset_closure) - · rw [← complexPlaneCoordinates.image_frontier,hfront] - -theorem complexInside_closure {C : Set ℂ} - (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) : - closure (complexInside C)=complexInside C ∪ C := by - apply complexPlaneCoordinates.injective.image_injective - rw [complexPlaneCoordinates.image_closure,complexInside_image,image_union, - complexInside_image] - exact (Schoenflies.IsRegionOf.inside _).closure_eq - (Schoenflies.jordan_curve_theorem hC) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.SchoenfliesInsideTransport + +namespace OAI + +/-! The bounded Jordan region in the complex coordinate chart. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +def complexInside (C : Set ℂ) : Set ℂ := + complexPlaneCoordinates ⁻¹' Schoenflies.inside (complexPlaneCoordinates '' C) + +theorem complexInside_image (C : Set ℂ) : + complexPlaneCoordinates '' complexInside C = + Schoenflies.inside (complexPlaneCoordinates '' C) := + image_preimage_eq _ complexPlaneCoordinates.surjective + +theorem complexInside_transport (h : ℂ ≃ₜ ℂ) (C : Set ℂ) : + h '' complexInside C=complexInside (h '' C) := by + let H := (complexPlaneCoordinates.symm.trans h).trans complexPlaneCoordinates + have he : H '' (complexPlaneCoordinates '' C)=complexPlaneCoordinates '' (h '' C) := by + ext x + simp [H] + apply complexPlaneCoordinates.injective.image_injective + rw [complexInside_image] + have hm : complexPlaneCoordinates '' (h '' complexInside C)= + H '' (complexPlaneCoordinates '' complexInside C) := by + ext x + simp [H] + rw [hm,complexInside_image,plane_homeomorph_inside,he] + +theorem complexInside_eq_open {C U : Set ℂ} + (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) + (hU : IsOpen U) (hconn : IsPreconnected U) (hne : U.Nonempty) + (hb : Bornology.IsBounded U) (hfront : frontier U=C) : complexInside C=U := by + apply complexPlaneCoordinates.injective.image_injective + rw [complexInside_image] + apply separating_inside_eq_open (Schoenflies.jordan_curve_theorem hC) + (complexPlaneCoordinates.isOpen_image.mpr hU) + (hconn.image _ complexPlaneCoordinates.continuous.continuousOn) + (hne.image _) ?_ ?_ + · exact (hb.isCompact_closure.image complexPlaneCoordinates.continuous).isBounded.subset + (image_mono subset_closure) + · rw [← complexPlaneCoordinates.image_frontier,hfront] + +theorem complexInside_closure {C : Set ℂ} + (hC : Schoenflies.IsJordanCurve (complexPlaneCoordinates '' C)) : + closure (complexInside C)=complexInside C ∪ C := by + apply complexPlaneCoordinates.injective.image_injective + rw [complexPlaneCoordinates.image_closure,complexInside_image,image_union, + complexInside_image] + exact (Schoenflies.IsRegionOf.inside _).closure_eq + (Schoenflies.jordan_curve_theorem hC) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ComplexLipschitzExtension.lean b/lean/OAI/Probability/SLE/Geometry/ComplexLipschitzExtension.lean index 62e27cafb5cb439806a74de9efeee22e7f0a1c75..4e1a4e6f203609472a616efe69864cb6ce029968 100644 --- a/lean/OAI/Probability/SLE/Geometry/ComplexLipschitzExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ComplexLipschitzExtension.lean @@ -1,42 +1,42 @@ -import Mathlib - -namespace OAI - -/-! A continuous extension of a complex Lipschitz map from a time set. -The coordinatewise real extensions suffice for the raw flow endpoint. -/ -noncomputable section -open Set Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma exists_continuous_complex_extension {S : Set ℝ} {K : ℝ≥0} - (f : ℝ → ℂ) (hf : LipschitzOnWith K f S) : - ∃ g : ℝ → ℂ, Continuous g ∧ EqOn g f S := by - have hre : LipschitzOnWith K (fun t => (f t).re) S := by - apply LipschitzOnWith.of_dist_le_mul - intro x hx y hy - calc - dist (f x).re (f y).re = |(f x-f y).re| := by simp [Real.dist_eq] - _ ≤ ‖f x-f y‖ := Complex.abs_re_le_norm _ - _ ≤ K*dist x y := by simpa only [dist_eq_norm] using hf.dist_le_mul x hx y hy - have him : LipschitzOnWith K (fun t => (f t).im) S := by - apply LipschitzOnWith.of_dist_le_mul - intro x hx y hy - calc - dist (f x).im (f y).im = |(f x-f y).im| := by simp [Real.dist_eq] - _ ≤ ‖f x-f y‖ := Complex.abs_im_le_norm _ - _ ≤ K*dist x y := by simpa only [dist_eq_norm] using hf.dist_le_mul x hx y hy - obtain ⟨r,hr,her⟩ := hre.extend_real - obtain ⟨i,hi,hei⟩ := him.extend_real - let g (t : ℝ) : ℂ := (r t : ℂ)+(i t : ℂ)*I - refine ⟨g,(Complex.continuous_ofReal.comp hr.continuous).add - ((Complex.continuous_ofReal.comp hi.continuous).mul continuous_const),?_⟩ - intro t ht - apply Complex.ext <;> simp [g,← her ht,← hei ht] - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! A continuous extension of a complex Lipschitz map from a time set. +The coordinatewise real extensions suffice for the raw flow endpoint. -/ +noncomputable section +open Set Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma exists_continuous_complex_extension {S : Set ℝ} {K : ℝ≥0} + (f : ℝ → ℂ) (hf : LipschitzOnWith K f S) : + ∃ g : ℝ → ℂ, Continuous g ∧ EqOn g f S := by + have hre : LipschitzOnWith K (fun t => (f t).re) S := by + apply LipschitzOnWith.of_dist_le_mul + intro x hx y hy + calc + dist (f x).re (f y).re = |(f x-f y).re| := by simp [Real.dist_eq] + _ ≤ ‖f x-f y‖ := Complex.abs_re_le_norm _ + _ ≤ K*dist x y := by simpa only [dist_eq_norm] using hf.dist_le_mul x hx y hy + have him : LipschitzOnWith K (fun t => (f t).im) S := by + apply LipschitzOnWith.of_dist_le_mul + intro x hx y hy + calc + dist (f x).im (f y).im = |(f x-f y).im| := by simp [Real.dist_eq] + _ ≤ ‖f x-f y‖ := Complex.abs_im_le_norm _ + _ ≤ K*dist x y := by simpa only [dist_eq_norm] using hf.dist_le_mul x hx y hy + obtain ⟨r,hr,her⟩ := hre.extend_real + obtain ⟨i,hi,hei⟩ := him.extend_real + let g (t : ℝ) : ℂ := (r t : ℂ)+(i t : ℂ)*I + refine ⟨g,(Complex.continuous_ofReal.comp hr.continuous).add + ((Complex.continuous_ofReal.comp hi.continuous).mul continuous_const),?_⟩ + intro t ht + apply Complex.ext <;> simp [g,← her ht,← hei ht] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConcatenatedSlitGeometry.lean b/lean/OAI/Probability/SLE/Geometry/ConcatenatedSlitGeometry.lean index c378d26eaf27b58e3f0157437bb9169906c9d8f8..7c620994778151989ed7efb5ff6428afa0f457bd 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConcatenatedSlitGeometry.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConcatenatedSlitGeometry.lean @@ -1,104 +1,104 @@ -import OAI.Probability.SLE.Geometry.FiniteSimpleSlit -import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension - -namespace OAI - -/-! Geometric concatenation of an old simple slit and a fresh simple slit -through its inverse chart. The resulting curve has no extra filled components. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -def concatenateSlit (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) - (t : ℝ≥0) : ℂ := if t ≤ T then β t else F ((u:ℂ)+η (t-T)) - -lemma continuous_concatenateSlit - (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) - (hβ : Continuous β) (hη : Continuous η) (hη0 : η 0=0) - (hηH : ∀ t, 0 ≤ (η t).im) (hF : ContinuousOn F closedHalfPlane) - (hF0 : F (u:ℂ)=β T) : Continuous (concatenateSlit T β η F u) := by - have hc : Continuous (fun t : ℝ≥0 => F ((u:ℂ)+η (t-T))) := by - apply hF.comp_continuous (continuous_const.add (hη.comp (continuous_id.sub continuous_const))) - intro t - change 0 ≤ ((u:ℂ)+η (t-T)).im - simpa only [closedHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using hηH (t-T) - change Continuous ((Iic T).piecewise β (fun t => F ((u:ℂ)+η (t-T)))) - apply Continuous.piecewise _ hβ hc - intro t ht - have he : t=T := by simpa only [frontier_Iic,mem_singleton_iff] using ht - simp only [he,tsub_self,hη0,add_zero,hF0] - -lemma concatenateSlit_injective - (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) - (hβ : InjOn β (Icc 0 T)) (hη : Function.Injective η) - (hηH : ∀ t, 0 < t → η t ∈ upperHalfPlane) - (hF : InjOn F upperHalfPlane) - (himage : MapsTo F upperHalfPlane (survivingDomain β T)) : - Function.Injective (concatenateSlit T β η F u) := by - have hnew (t : ℝ≥0) (ht : T < t) : (u:ℂ)+η (t-T) ∈ upperHalfPlane := by - simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using - hηH (t-T) (tsub_pos_iff_lt.mpr ht) - intro s t he - by_cases hs : s ≤ T - · by_cases ht : t ≤ T - · exact hβ ⟨zero_le,hs⟩ ⟨zero_le,ht⟩ (by simpa only [concatenateSlit,ite_eq_left hs,ite_eq_left ht] using he) - · have him := himage (hnew t (lt_of_not_ge ht)) - apply False.elim - apply him.1.2 - refine ⟨s,⟨zero_le,hs⟩,?_⟩ - simpa only [concatenateSlit,ite_eq_left hs,ite_eq_right ht] using he - · by_cases ht : t ≤ T - · have him := himage (hnew s (lt_of_not_ge hs)) - apply False.elim - apply him.1.2 - refine ⟨t,⟨zero_le,ht⟩,?_⟩ - simpa only [concatenateSlit,ite_eq_right hs,ite_eq_left ht] using he.symm - · have hst := hF (hnew s (lt_of_not_ge hs)) (hnew t (lt_of_not_ge ht)) - (by simpa only [concatenateSlit,ite_eq_right hs,ite_eq_right ht] using he) - have heη := hη (add_left_cancel hst) - exact (tsub_left_inj (le_of_not_ge hs) (le_of_not_ge ht)).mp heη - -lemma concatenateSlit_positive - (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) - (hβ : ∀ t, 0 < t → β t ∈ upperHalfPlane) - (hη : ∀ t, 0 < t → η t ∈ upperHalfPlane) - (hF : MapsTo F upperHalfPlane upperHalfPlane) - (t : ℝ≥0) (ht : 0 < t) : concatenateSlit T β η F u t ∈ upperHalfPlane := by - by_cases h : t ≤ T - · simpa only [concatenateSlit,ite_eq_left h] using hβ t ht - · rw [concatenateSlit,ite_eq_right h] - apply hF - simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using - hη (t-T) (tsub_pos_iff_lt.mpr (lt_of_not_ge h)) - -theorem concatenateSlit_free_complement - (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) - (hβc : Continuous β) (hηc : Continuous η) (hβ0 : β 0=0) (hη0 : η 0=0) - (hβi : InjOn β (Icc 0 T)) (hηi : Function.Injective η) - (hβH : ∀ t, 0 < t → β t ∈ upperHalfPlane) - (hηH : ∀ t, 0 < t → η t ∈ upperHalfPlane) - (hFc : ContinuousOn F closedHalfPlane) (hF0 : F (u:ℂ)=β T) - (hFi : InjOn F upperHalfPlane) - (hFmap : MapsTo F upperHalfPlane (survivingDomain β T)) - (S : ℝ≥0) (hS : 0 < S) : - upperHalfPlane \ ((concatenateSlit T β η F u) '' Icc 0 S) ⊆ - survivingDomain (concatenateSlit T β η F u) S := by - have hηclosed (t : ℝ≥0) : 0 ≤ (η t).im := by - by_cases ht : t=0 - · simp [ht,hη0] - · exact (hηH t (pos_iff_ne_zero.mpr ht)).le - apply finite_simple_slit_free _ - (continuous_concatenateSlit T β η F u hβc hηc hη0 hηclosed hFc hF0) - (by simp only [concatenateSlit,ite_eq_left (show (0:ℝ≥0) ≤ T from zero_le),hβ0]) S hS - (concatenateSlit_injective T β η F u hβi hηi hηH hFi hFmap).injOn - intro t ht - exact concatenateSlit_positive T β η F u hβH hηH - (fun z hz => (hFmap hz).1.1) t ht.1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteSimpleSlit +import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension + +namespace OAI + +/-! Geometric concatenation of an old simple slit and a fresh simple slit +through its inverse chart. The resulting curve has no extra filled components. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +def concatenateSlit (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) + (t : ℝ≥0) : ℂ := if t ≤ T then β t else F ((u:ℂ)+η (t-T)) + +lemma continuous_concatenateSlit + (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) + (hβ : Continuous β) (hη : Continuous η) (hη0 : η 0=0) + (hηH : ∀ t, 0 ≤ (η t).im) (hF : ContinuousOn F closedHalfPlane) + (hF0 : F (u:ℂ)=β T) : Continuous (concatenateSlit T β η F u) := by + have hc : Continuous (fun t : ℝ≥0 => F ((u:ℂ)+η (t-T))) := by + apply hF.comp_continuous (continuous_const.add (hη.comp (continuous_id.sub continuous_const))) + intro t + change 0 ≤ ((u:ℂ)+η (t-T)).im + simpa only [closedHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using hηH (t-T) + change Continuous ((Iic T).piecewise β (fun t => F ((u:ℂ)+η (t-T)))) + apply Continuous.piecewise _ hβ hc + intro t ht + have he : t=T := by simpa only [frontier_Iic,mem_singleton_iff] using ht + simp only [he,tsub_self,hη0,add_zero,hF0] + +lemma concatenateSlit_injective + (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) + (hβ : InjOn β (Icc 0 T)) (hη : Function.Injective η) + (hηH : ∀ t, 0 < t → η t ∈ upperHalfPlane) + (hF : InjOn F upperHalfPlane) + (himage : MapsTo F upperHalfPlane (survivingDomain β T)) : + Function.Injective (concatenateSlit T β η F u) := by + have hnew (t : ℝ≥0) (ht : T < t) : (u:ℂ)+η (t-T) ∈ upperHalfPlane := by + simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using + hηH (t-T) (tsub_pos_iff_lt.mpr ht) + intro s t he + by_cases hs : s ≤ T + · by_cases ht : t ≤ T + · exact hβ ⟨zero_le,hs⟩ ⟨zero_le,ht⟩ (by simpa only [concatenateSlit,ite_eq_left hs,ite_eq_left ht] using he) + · have him := himage (hnew t (lt_of_not_ge ht)) + apply False.elim + apply him.1.2 + refine ⟨s,⟨zero_le,hs⟩,?_⟩ + simpa only [concatenateSlit,ite_eq_left hs,ite_eq_right ht] using he + · by_cases ht : t ≤ T + · have him := himage (hnew s (lt_of_not_ge hs)) + apply False.elim + apply him.1.2 + refine ⟨t,⟨zero_le,ht⟩,?_⟩ + simpa only [concatenateSlit,ite_eq_right hs,ite_eq_left ht] using he.symm + · have hst := hF (hnew s (lt_of_not_ge hs)) (hnew t (lt_of_not_ge ht)) + (by simpa only [concatenateSlit,ite_eq_right hs,ite_eq_right ht] using he) + have heη := hη (add_left_cancel hst) + exact (tsub_left_inj (le_of_not_ge hs) (le_of_not_ge ht)).mp heη + +lemma concatenateSlit_positive + (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) + (hβ : ∀ t, 0 < t → β t ∈ upperHalfPlane) + (hη : ∀ t, 0 < t → η t ∈ upperHalfPlane) + (hF : MapsTo F upperHalfPlane upperHalfPlane) + (t : ℝ≥0) (ht : 0 < t) : concatenateSlit T β η F u t ∈ upperHalfPlane := by + by_cases h : t ≤ T + · simpa only [concatenateSlit,ite_eq_left h] using hβ t ht + · rw [concatenateSlit,ite_eq_right h] + apply hF + simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,zero_add] using + hη (t-T) (tsub_pos_iff_lt.mpr (lt_of_not_ge h)) + +theorem concatenateSlit_free_complement + (T : ℝ≥0) (β η : ℝ≥0 → ℂ) (F : ℂ → ℂ) (u : ℝ) + (hβc : Continuous β) (hηc : Continuous η) (hβ0 : β 0=0) (hη0 : η 0=0) + (hβi : InjOn β (Icc 0 T)) (hηi : Function.Injective η) + (hβH : ∀ t, 0 < t → β t ∈ upperHalfPlane) + (hηH : ∀ t, 0 < t → η t ∈ upperHalfPlane) + (hFc : ContinuousOn F closedHalfPlane) (hF0 : F (u:ℂ)=β T) + (hFi : InjOn F upperHalfPlane) + (hFmap : MapsTo F upperHalfPlane (survivingDomain β T)) + (S : ℝ≥0) (hS : 0 < S) : + upperHalfPlane \ ((concatenateSlit T β η F u) '' Icc 0 S) ⊆ + survivingDomain (concatenateSlit T β η F u) S := by + have hηclosed (t : ℝ≥0) : 0 ≤ (η t).im := by + by_cases ht : t=0 + · simp [ht,hη0] + · exact (hηH t (pos_iff_ne_zero.mpr ht)).le + apply finite_simple_slit_free _ + (continuous_concatenateSlit T β η F u hβc hηc hη0 hηclosed hFc hF0) + (by simp only [concatenateSlit,ite_eq_left (show (0:ℝ≥0) ≤ T from zero_le),hβ0]) S hS + (concatenateSlit_injective T β η F u hβi hηi hηH hFi hFmap).injOn + intro t ht + exact concatenateSlit_positive T β η F u hβH hηH + (fun z hz => (hFmap hz).1.1) t ht.1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConditionalStopExtension.lean b/lean/OAI/Probability/SLE/Geometry/ConditionalStopExtension.lean index ddcd743282b194a36a9fa2b75d86e2318301d64e..523252aaa72808603b3a946d30fdfeee8daa11d9 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConditionalStopExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConditionalStopExtension.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Gauge.ExtendedExperimentStop - -namespace OAI - -/-! A conditional estimate at finite stopping times extends unchanged to -a stopping time finite almost surely on its admissible stopped-past event. -The geometric state is evaluated only on that event. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem conditional_event_bound_at_extended_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] (Q : Measure Ω) - (ℱ : Filtration ℝ≥0 mΩ) (p : ℝ≥0∞) (S : Set Ω) - (state : Ω → ℝ≥0 → Prop) - (hfiniteStep : ∀ (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) - (E : Set Ω), MeasurableSet[hτ.measurableSpace] E → - (∀ ω ∈ E, state ω (τ ω)) → p * Q E ≤ Q (E ∩ S)) - (σ : Ω → WithTop ℝ≥0) (hσ : IsStoppingTime ℱ σ) - (E : Set Ω) (hE : MeasurableSet[hσ.measurableSpace] E) - (hfinite : ∀ᵐ ω ∂Q, ω ∈ E → σ ω ≠ ⊤) - (hstate : ∀ ω ∈ E, state ω ((σ ω).untopD 0)) : - p * Q E ≤ Q (E ∩ S) := by - let Eₙ (n : ℕ) := E ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} - have hmono : Monotone Eₙ := by - intro n k hnk ω hω - refine ⟨hω.1, ?_⟩ - change σ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0) - exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk)) - have hcover : E =ᵐ[Q] ⋃ n, Eₙ n := by - filter_upwards [hfinite] with ω hω - apply propext - constructor - · intro hωE - lift σ ω to ℝ≥0 using hω hωE with t ht - obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) - exact mem_iUnion.mpr ⟨n, hωE, by - change σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) - rw [← ht] - exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)⟩ - · intro hωE - obtain ⟨n, hn⟩ := mem_iUnion.mp hωE - exact hn.1 - have hlocal (n : ℕ) : p * Q (Eₙ n) ≤ Q (E ∩ S) := by - have h := hfiniteStep (cutoffStop σ n) (cutoffStop_stopping ℱ σ hσ n) (Eₙ n) - (cutoffStop_success_measurable ℱ σ hσ n E hE) (fun ω hω => ?_) - · exact h.trans (measure_mono (inter_subset_inter inter_subset_left subset_rfl)) - · rw [cutoffStop_eq_on_success σ n ω hω.2] - exact hstate ω hω.1 - calc - _ = p * Q (⋃ n, Eₙ n) := by rw [measure_congr hcover] - _ = ⨆ n, p * Q (Eₙ n) := by rw [hmono.measure_iUnion, ENNReal.mul_iSup] - _ ≤ _ := iSup_le hlocal - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ExtendedExperimentStop + +namespace OAI + +/-! A conditional estimate at finite stopping times extends unchanged to +a stopping time finite almost surely on its admissible stopped-past event. +The geometric state is evaluated only on that event. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem conditional_event_bound_at_extended_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] (Q : Measure Ω) + (ℱ : Filtration ℝ≥0 mΩ) (p : ℝ≥0∞) (S : Set Ω) + (state : Ω → ℝ≥0 → Prop) + (hfiniteStep : ∀ (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) + (E : Set Ω), MeasurableSet[hτ.measurableSpace] E → + (∀ ω ∈ E, state ω (τ ω)) → p * Q E ≤ Q (E ∩ S)) + (σ : Ω → WithTop ℝ≥0) (hσ : IsStoppingTime ℱ σ) + (E : Set Ω) (hE : MeasurableSet[hσ.measurableSpace] E) + (hfinite : ∀ᵐ ω ∂Q, ω ∈ E → σ ω ≠ ⊤) + (hstate : ∀ ω ∈ E, state ω ((σ ω).untopD 0)) : + p * Q E ≤ Q (E ∩ S) := by + let Eₙ (n : ℕ) := E ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} + have hmono : Monotone Eₙ := by + intro n k hnk ω hω + refine ⟨hω.1, ?_⟩ + change σ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0) + exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk)) + have hcover : E =ᵐ[Q] ⋃ n, Eₙ n := by + filter_upwards [hfinite] with ω hω + apply propext + constructor + · intro hωE + lift σ ω to ℝ≥0 using hω hωE with t ht + obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) + exact mem_iUnion.mpr ⟨n, hωE, by + change σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) + rw [← ht] + exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)⟩ + · intro hωE + obtain ⟨n, hn⟩ := mem_iUnion.mp hωE + exact hn.1 + have hlocal (n : ℕ) : p * Q (Eₙ n) ≤ Q (E ∩ S) := by + have h := hfiniteStep (cutoffStop σ n) (cutoffStop_stopping ℱ σ hσ n) (Eₙ n) + (cutoffStop_success_measurable ℱ σ hσ n E hE) (fun ω hω => ?_) + · exact h.trans (measure_mono (inter_subset_inter inter_subset_left subset_rfl)) + · rw [cutoffStop_eq_on_success σ n ω hω.2] + exact hstate ω hω.1 + calc + _ = p * Q (⋃ n, Eₙ n) := by rw [measure_congr hcover] + _ = ⨆ n, p * Q (Eₙ n) := by rw [hmono.measure_iUnion, ENNReal.mul_iSup] + _ ≤ _ := iSup_le hlocal + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConformalDiskExit.lean b/lean/OAI/Probability/SLE/Geometry/ConformalDiskExit.lean index 9881525d4471afd5f82f0592fe25124874072475..89bcd7ffd22c2928432d14f77d5007d80bdb1507 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConformalDiskExit.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConformalDiskExit.lean @@ -1,62 +1,62 @@ -import OAI.Probability.SLE.Geometry.HolomorphicExitLimit - -namespace OAI - -/-! At a finite physical exit the limit of a conformal disk chart is on -the unit circle. Only continuity of the inverse inside the disk is used. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Target - -lemma disk_exit_limit_norm {D : Set ℂ} {F H : ℂ → ℂ} - (hF : MapsTo F D (ball 0 1)) (hH : ContinuousOn H (ball 0 1)) - (hHD : MapsTo H (ball 0 1) D) (hinv : ∀ z∈D, H (F z)=z) - (ζ : ℝ≥0 → ℂ) (hζ : Continuous ζ) (τ : ℝ≥0) (hτ : 0 < τ) - (hpath : ∀ t<τ, ζ t∈D) (hexit : ζ τ∉D) (c : ℂ) - (hc : Tendsto (fun t => F (ζ t)) (𝓝[<] τ) (𝓝 c)) : ‖c‖=1 := by - let : NeBot (𝓝[<] τ) := nhdsLT_neBot_of_exists_lt ⟨0,hτ⟩ - have hmem : ∀ᶠ t in 𝓝[<] τ, ζ t∈D := - (show ∀ᶠ t in 𝓝[<] τ, t<τ from self_mem_nhdsWithin).mono hpath - have hle : ‖c‖ ≤ 1 := le_of_tendsto hc.norm (hmem.mono fun t ht => - (mem_ball_zero_iff.mp (hF ht)).le) - apply le_antisymm hle - apply not_lt.mp - intro hlt - have hcb : c∈ball (0:ℂ) 1 := mem_ball_zero_iff.mpr hlt - have hHlim := ((hH c hcb).continuousAt (isOpen_ball.mem_nhds hcb)).tendsto.comp hc - have hsame : Tendsto ζ (𝓝[<] τ) (𝓝 (H c)) := - hHlim.congr' (hmem.mono fun t ht => hinv (ζ t) ht) - have hid : H c=ζ τ := tendsto_nhds_unique hsame - (hζ.continuousAt.tendsto.mono_left nhdsWithin_le_nhds) - exact hexit (hid ▸ hHD hcb) - -theorem conformal_disk_finite_exit_limit - (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) - (w : ℂ) (hw : w∈D) {F H : ℂ → ℂ} - (hFhol : DifferentiableOn ℂ F D) (hF : MapsTo F D (ball 0 1)) - (hH : ContinuousOn H (ball 0 1)) (hHD : MapsTo H (ball 0 1) D) - (hinv : ∀ z∈D, H (F z)=z) (T : ℝ≥0) : - ∀ᵐ b ∂W.prod W, ∀ τ : ℝ≥0, 0 < τ → τ ≤ T → - (∀ t<τ, w+Complex.equivRealProdCLM.symm (planarWiener t b)∈D) → - w+Complex.equivRealProdCLM.symm (planarWiener τ b)∉D → - ∃ c : ℂ, ‖c‖=1 ∧ - Tendsto (fun t => F (w+Complex.equivRealProdCLM.symm (planarWiener t b))) - (𝓝[<] τ) (𝓝 c) := by - have hb := holomorphic_domain_exit_limit W hW hD w hw hFhol T 1 - (fun z hz => (mem_ball_zero_iff.mp (hF hz)).le) - filter_upwards [hb] with b hb - intro τ hτ hτT hp hx - obtain ⟨c,hc⟩ := hb τ hτT hp hx - refine ⟨c,?_,hc⟩ - exact disk_exit_limit_norm hF hH hHD hinv _ - (continuous_const.add (Complex.equivRealProdCLM.symm.continuous.comp (planarWiener_continuous b))) - τ hτ hp hx c hc - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HolomorphicExitLimit + +namespace OAI + +/-! At a finite physical exit the limit of a conformal disk chart is on +the unit circle. Only continuity of the inverse inside the disk is used. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target + +lemma disk_exit_limit_norm {D : Set ℂ} {F H : ℂ → ℂ} + (hF : MapsTo F D (ball 0 1)) (hH : ContinuousOn H (ball 0 1)) + (hHD : MapsTo H (ball 0 1) D) (hinv : ∀ z∈D, H (F z)=z) + (ζ : ℝ≥0 → ℂ) (hζ : Continuous ζ) (τ : ℝ≥0) (hτ : 0 < τ) + (hpath : ∀ t<τ, ζ t∈D) (hexit : ζ τ∉D) (c : ℂ) + (hc : Tendsto (fun t => F (ζ t)) (𝓝[<] τ) (𝓝 c)) : ‖c‖=1 := by + let : NeBot (𝓝[<] τ) := nhdsLT_neBot_of_exists_lt ⟨0,hτ⟩ + have hmem : ∀ᶠ t in 𝓝[<] τ, ζ t∈D := + (show ∀ᶠ t in 𝓝[<] τ, t<τ from self_mem_nhdsWithin).mono hpath + have hle : ‖c‖ ≤ 1 := le_of_tendsto hc.norm (hmem.mono fun t ht => + (mem_ball_zero_iff.mp (hF ht)).le) + apply le_antisymm hle + apply not_lt.mp + intro hlt + have hcb : c∈ball (0:ℂ) 1 := mem_ball_zero_iff.mpr hlt + have hHlim := ((hH c hcb).continuousAt (isOpen_ball.mem_nhds hcb)).tendsto.comp hc + have hsame : Tendsto ζ (𝓝[<] τ) (𝓝 (H c)) := + hHlim.congr' (hmem.mono fun t ht => hinv (ζ t) ht) + have hid : H c=ζ τ := tendsto_nhds_unique hsame + (hζ.continuousAt.tendsto.mono_left nhdsWithin_le_nhds) + exact hexit (hid ▸ hHD hcb) + +theorem conformal_disk_finite_exit_limit + (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) + (w : ℂ) (hw : w∈D) {F H : ℂ → ℂ} + (hFhol : DifferentiableOn ℂ F D) (hF : MapsTo F D (ball 0 1)) + (hH : ContinuousOn H (ball 0 1)) (hHD : MapsTo H (ball 0 1) D) + (hinv : ∀ z∈D, H (F z)=z) (T : ℝ≥0) : + ∀ᵐ b ∂W.prod W, ∀ τ : ℝ≥0, 0 < τ → τ ≤ T → + (∀ t<τ, w+Complex.equivRealProdCLM.symm (planarWiener t b)∈D) → + w+Complex.equivRealProdCLM.symm (planarWiener τ b)∉D → + ∃ c : ℂ, ‖c‖=1 ∧ + Tendsto (fun t => F (w+Complex.equivRealProdCLM.symm (planarWiener t b))) + (𝓝[<] τ) (𝓝 c) := by + have hb := holomorphic_domain_exit_limit W hW hD w hw hFhol T 1 + (fun z hz => (mem_ball_zero_iff.mp (hF hz)).le) + filter_upwards [hb] with b hb + intro τ hτ hτT hp hx + obtain ⟨c,hc⟩ := hb τ hτT hp hx + refine ⟨c,?_,hc⟩ + exact disk_exit_limit_norm hF hH hHD hinv _ + (continuous_const.add (Complex.equivRealProdCLM.symm.continuous.comp (planarWiener_continuous b))) + τ hτ hp hx c hc + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConformalLaplacian.lean b/lean/OAI/Probability/SLE/Geometry/ConformalLaplacian.lean index 46dab310903ebd7db5a248a8b1f6b6a5e50bda56..7e3983c5257a71aa01304021d34ee88ed833e53a 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConformalLaplacian.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConformalLaplacian.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.RadiusClocks -import Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions - -namespace OAI - -/-! The real Laplacian transforms by the squared complex derivative. -This is the change-of-variables identity used for the exterior area proof. -/ -noncomputable section -open Set Filter Metric Function Complex InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma complex_composition_hessian {f : ℂ → ℂ} {u : ℂ → ℝ} {z : ℂ} - (hf : ContDiffAt ℝ 2 f z) (hu : ContDiffAt ℝ 2 u (f z)) (v : ℂ) : - fderiv ℝ (fderiv ℝ (u ∘ f)) z v v = - fderiv ℝ (fderiv ℝ u) (f z) (fderiv ℝ f z v) (fderiv ℝ f z v)+ - fderiv ℝ u (f z) (fderiv ℝ (fderiv ℝ f) z v v) := by - have hf1 : DifferentiableAt ℝ f z := hf.differentiableAt (by norm_num) - have hf2 : DifferentiableAt ℝ (fderiv ℝ f) z := (hf.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) - have hu2 : DifferentiableAt ℝ (fderiv ℝ u) (f z) := (hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) - have hc := (hu2.hasFDerivAt.comp z hf1.hasFDerivAt).clm_comp hf2.hasFDerivAt - have he : fderiv ℝ (u ∘ f) =ᶠ[𝓝 z] - (fun a => (fderiv ℝ u (f a)).comp (fderiv ℝ f a)) := by - filter_upwards [hf.eventually (by norm_num), - hf1.continuousAt.tendsto.eventually (hu.eventually (by norm_num))] with a hfa hua - exact fderiv_comp a (hua.differentiableAt (by norm_num)) (hfa.differentiableAt (by norm_num)) - have hd := hc.congr_of_eventuallyEq he - rw [hd.fderiv] - simp only [add_apply,ContinuousLinearMap.comp_apply, - ContinuousLinearMap.compL_apply,ContinuousLinearMap.flip_apply,Function.comp_apply] - ring - -lemma complex_fderiv_apply {f : ℂ → ℂ} {z : ℂ} (hf : DifferentiableAt ℂ f z) (v : ℂ) : - fderiv ℝ f z v=v*deriv f z := by - rw [hf.hasDerivAt.hasFDerivAt.restrictScalars ℝ |>.fderiv] - simp only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul] - -lemma laplacian_holomorphic_comp {f : ℂ → ℂ} {u : ℂ → ℝ} {z : ℂ} - (hf : AnalyticAt ℂ f z) (hu : ContDiffAt ℝ 2 u (f z)) : - (Δ (u ∘ f)) z=‖deriv f z‖^2*(Δ u) (f z) := by - have hfR : ContDiffAt ℝ 2 f z := hf.contDiffAt.restrict_scalars ℝ - have he0 : fderiv ℝ (fderiv ℝ f) z 1 1+fderiv ℝ (fderiv ℝ f) z I I=0 := by - have hh := hf.harmonicAt.2.self_of_nhds - simpa only [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply, - Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one,Pi.zero_apply] using hh - have hh := complex_hessian_rotation (u := u) (f z) (deriv f z) - rw [laplacian_eq_iteratedFDeriv_complexPlane] - simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one, - Matrix.cons_val_fin_one] - rw [complex_composition_hessian hfR hu 1,complex_composition_hessian hfR hu I, - complex_fderiv_apply hf.differentiableAt,complex_fderiv_apply hf.differentiableAt,one_mul] - calc - _ = (fderiv ℝ (fderiv ℝ u) (f z) (deriv f z) (deriv f z)+ - fderiv ℝ (fderiv ℝ u) (f z) (deriv f z*I) (deriv f z*I))+ - fderiv ℝ u (f z) (fderiv ℝ (fderiv ℝ f) z 1 1+fderiv ℝ (fderiv ℝ f) z I I) := by - rw [map_add] - simp only [mul_comm I] - ring - _ = _ := by rw [he0,map_zero,add_zero,hh] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.RadiusClocks +import Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions + +namespace OAI + +/-! The real Laplacian transforms by the squared complex derivative. +This is the change-of-variables identity used for the exterior area proof. -/ +noncomputable section +open Set Filter Metric Function Complex InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma complex_composition_hessian {f : ℂ → ℂ} {u : ℂ → ℝ} {z : ℂ} + (hf : ContDiffAt ℝ 2 f z) (hu : ContDiffAt ℝ 2 u (f z)) (v : ℂ) : + fderiv ℝ (fderiv ℝ (u ∘ f)) z v v = + fderiv ℝ (fderiv ℝ u) (f z) (fderiv ℝ f z v) (fderiv ℝ f z v)+ + fderiv ℝ u (f z) (fderiv ℝ (fderiv ℝ f) z v v) := by + have hf1 : DifferentiableAt ℝ f z := hf.differentiableAt (by norm_num) + have hf2 : DifferentiableAt ℝ (fderiv ℝ f) z := (hf.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) + have hu2 : DifferentiableAt ℝ (fderiv ℝ u) (f z) := (hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) + have hc := (hu2.hasFDerivAt.comp z hf1.hasFDerivAt).clm_comp hf2.hasFDerivAt + have he : fderiv ℝ (u ∘ f) =ᶠ[𝓝 z] + (fun a => (fderiv ℝ u (f a)).comp (fderiv ℝ f a)) := by + filter_upwards [hf.eventually (by norm_num), + hf1.continuousAt.tendsto.eventually (hu.eventually (by norm_num))] with a hfa hua + exact fderiv_comp a (hua.differentiableAt (by norm_num)) (hfa.differentiableAt (by norm_num)) + have hd := hc.congr_of_eventuallyEq he + rw [hd.fderiv] + simp only [add_apply,ContinuousLinearMap.comp_apply, + ContinuousLinearMap.compL_apply,ContinuousLinearMap.flip_apply,Function.comp_apply] + ring + +lemma complex_fderiv_apply {f : ℂ → ℂ} {z : ℂ} (hf : DifferentiableAt ℂ f z) (v : ℂ) : + fderiv ℝ f z v=v*deriv f z := by + rw [hf.hasDerivAt.hasFDerivAt.restrictScalars ℝ |>.fderiv] + simp only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul] + +lemma laplacian_holomorphic_comp {f : ℂ → ℂ} {u : ℂ → ℝ} {z : ℂ} + (hf : AnalyticAt ℂ f z) (hu : ContDiffAt ℝ 2 u (f z)) : + (Δ (u ∘ f)) z=‖deriv f z‖^2*(Δ u) (f z) := by + have hfR : ContDiffAt ℝ 2 f z := hf.contDiffAt.restrict_scalars ℝ + have he0 : fderiv ℝ (fderiv ℝ f) z 1 1+fderiv ℝ (fderiv ℝ f) z I I=0 := by + have hh := hf.harmonicAt.2.self_of_nhds + simpa only [laplacian_eq_iteratedFDeriv_complexPlane,iteratedFDeriv_two_apply, + Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one,Pi.zero_apply] using hh + have hh := complex_hessian_rotation (u := u) (f z) (deriv f z) + rw [laplacian_eq_iteratedFDeriv_complexPlane] + simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one, + Matrix.cons_val_fin_one] + rw [complex_composition_hessian hfR hu 1,complex_composition_hessian hfR hu I, + complex_fderiv_apply hf.differentiableAt,complex_fderiv_apply hf.differentiableAt,one_mul] + calc + _ = (fderiv ℝ (fderiv ℝ u) (f z) (deriv f z) (deriv f z)+ + fderiv ℝ (fderiv ℝ u) (f z) (deriv f z*I) (deriv f z*I))+ + fderiv ℝ u (f z) (fderiv ℝ (fderiv ℝ f) z 1 1+fderiv ℝ (fderiv ℝ f) z I I) := by + rw [map_add] + simp only [mul_comm I] + ring + _ = _ := by rw [he0,map_zero,add_zero,hh] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConformalLaplacianIntegral.lean b/lean/OAI/Probability/SLE/Geometry/ConformalLaplacianIntegral.lean index d723467ccf28b060167fd30293fc27272ca7468b..44881806c9c6db5777f94717b8946a1ca3e4ff46 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConformalLaplacianIntegral.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConformalLaplacianIntegral.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Geometry.ExteriorLaplacianTest -import OAI.Probability.SLE.Gauge.ConformalSausageTransfer - -namespace OAI - -/-! Change of variables for the Laplacian of a smooth real test under -an injective holomorphic map. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma integral_laplacian_holomorphic_image {D : Set ℂ} (hD : IsOpen D) - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {u : ℂ → ℝ} (hu : ContDiff ℝ 2 u) : - (∫ z in f '' D, (Δ u) z)=(∫ z in D, (Δ (u ∘ f)) z) := by - have hd (z : ℂ) (hz : z∈D) : HasDerivAt f (deriv f z) z := - (hf.differentiableAt (hD.mem_nhds hz)).hasDerivAt - rw [integral_image_eq_integral_abs_det_fderiv_smul volume hD.measurableSet - (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi] - apply setIntegral_congr_fun hD.measurableSet - intro z hz - dsimp only - rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),smul_eq_mul] - exact (laplacian_holomorphic_comp (hf.analyticAt (hD.mem_nhds hz)) hu.contDiffAt).symm - -lemma integrableOn_laplacian_holomorphic_comp {D : Set ℂ} (hD : IsOpen D) - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {u : ℂ → ℝ} (hu : ContDiff ℝ 2 u) (hs : HasCompactSupport u) : - IntegrableOn (Δ (u ∘ f)) D := by - have hd (z : ℂ) (hz : z∈D) : HasDerivAt f (deriv f z) z := - (hf.differentiableAt (hD.mem_nhds hz)).hasDerivAt - have hc : Continuous (Δ u) := - continuous_iff_continuousAt.mpr (fun z => plane_laplacian_continuousAt (hu.contDiffAt (x := z))) - have hsu := hasCompactSupport_laplacian hs - have hint := (integrableOn_image_iff_integrableOn_abs_det_fderiv_smul volume hD.measurableSet - (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi (Δ u)).mp - ((hc.integrable_of_hasCompactSupport hsu).integrableOn) - apply hint.congr_fun _ hD.measurableSet - intro z hz - dsimp only - rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),smul_eq_mul] - exact (laplacian_holomorphic_comp (hf.analyticAt (hD.mem_nhds hz)) hu.contDiffAt).symm - -lemma exterior_compact_test_laplacian_nonpos {D : Set ℂ} (hD : IsOpen D) - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - (hK : IsCompact (f '' D)ᶜ) : - ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ - (∀ z∈(f '' D)ᶜ, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ - IntegrableOn (Δ (u ∘ f)) D ∧ (∫ z in D, (Δ (u ∘ f)) z)≤0 := by - obtain ⟨u,hu,hs,he,hn⟩ := compact_quadratic_test_complement_integral _ hK - refine ⟨u,hu,hs,he,integrableOn_laplacian_holomorphic_comp hD hf hi hu hs,?_⟩ - rw [←integral_laplacian_holomorphic_image hD hf hi hu] - simpa only [compl_compl] using hn - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorLaplacianTest +import OAI.Probability.SLE.Gauge.ConformalSausageTransfer + +namespace OAI + +/-! Change of variables for the Laplacian of a smooth real test under +an injective holomorphic map. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma integral_laplacian_holomorphic_image {D : Set ℂ} (hD : IsOpen D) + {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {u : ℂ → ℝ} (hu : ContDiff ℝ 2 u) : + (∫ z in f '' D, (Δ u) z)=(∫ z in D, (Δ (u ∘ f)) z) := by + have hd (z : ℂ) (hz : z∈D) : HasDerivAt f (deriv f z) z := + (hf.differentiableAt (hD.mem_nhds hz)).hasDerivAt + rw [integral_image_eq_integral_abs_det_fderiv_smul volume hD.measurableSet + (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi] + apply setIntegral_congr_fun hD.measurableSet + intro z hz + dsimp only + rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),smul_eq_mul] + exact (laplacian_holomorphic_comp (hf.analyticAt (hD.mem_nhds hz)) hu.contDiffAt).symm + +lemma integrableOn_laplacian_holomorphic_comp {D : Set ℂ} (hD : IsOpen D) + {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {u : ℂ → ℝ} (hu : ContDiff ℝ 2 u) (hs : HasCompactSupport u) : + IntegrableOn (Δ (u ∘ f)) D := by + have hd (z : ℂ) (hz : z∈D) : HasDerivAt f (deriv f z) z := + (hf.differentiableAt (hD.mem_nhds hz)).hasDerivAt + have hc : Continuous (Δ u) := + continuous_iff_continuousAt.mpr (fun z => plane_laplacian_continuousAt (hu.contDiffAt (x := z))) + have hsu := hasCompactSupport_laplacian hs + have hint := (integrableOn_image_iff_integrableOn_abs_det_fderiv_smul volume hD.measurableSet + (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi (Δ u)).mp + ((hc.integrable_of_hasCompactSupport hsu).integrableOn) + apply hint.congr_fun _ hD.measurableSet + intro z hz + dsimp only + rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),smul_eq_mul] + exact (laplacian_holomorphic_comp (hf.analyticAt (hD.mem_nhds hz)) hu.contDiffAt).symm + +lemma exterior_compact_test_laplacian_nonpos {D : Set ℂ} (hD : IsOpen D) + {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + (hK : IsCompact (f '' D)ᶜ) : + ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ + (∀ z∈(f '' D)ᶜ, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ + IntegrableOn (Δ (u ∘ f)) D ∧ (∫ z in D, (Δ (u ∘ f)) z)≤0 := by + obtain ⟨u,hu,hs,he,hn⟩ := compact_quadratic_test_complement_integral _ hK + refine ⟨u,hu,hs,he,integrableOn_laplacian_holomorphic_comp hD hf hi hu hs,?_⟩ + rw [←integral_laplacian_holomorphic_image hD hf hi hu] + simpa only [compl_compl] using hn + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConformalRestartTransport.lean b/lean/OAI/Probability/SLE/Geometry/ConformalRestartTransport.lean index 89265a7f2ea9adebca35a7c207cdcd365c9c2500..436a2870b9603f1870886a4ad2a8c1fea73986a2 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConformalRestartTransport.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConformalRestartTransport.lean @@ -1,95 +1,95 @@ -import OAI.Probability.SLE.Mass.CanonicalImageMass -import OAI.Probability.SLE.Loewner.StoppedInversePaths - -namespace OAI - -/-! Fixed conformal changes of coordinates preserve restart increments. -Composition combines the two density weights before pushing forward. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge - -local instance conformalRestartLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -lemma weightedImageMass_comp - {X : Type*} [MeasurableSpace X] - (J : upperHalfPlane → X) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) - (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) - (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) - (η : Measure upperHalfPlane) : - weightedImageMass J w (weightedImageMass F v η) = - weightedImageMass (J ∘ F) (fun z => v z*w (F z)) η := by - classical - apply Measure.ext - intro A hA - rw [weightedImageMass_apply J hJ w _ A hA, - weightedImageMass_lintegral F hF v hv η ((J ⁻¹' A).indicator w) - (hw.indicator (hJ hA)), - weightedImageMass_apply (J ∘ F) (hJ.comp hF) _ η A hA] - apply lintegral_congr - intro z - change v z * (if J (F z) ∈ A then w (F z) else 0) = - (if J (F z) ∈ A then v z*w (F z) else 0) - split_ifs <;> simp - -lemma derivative_dimension_weight_comp - (κ : ℝ) (J F : ℂ → ℂ) (z : ℂ) - (hJ : DifferentiableAt ℂ J (F z)) (hF : DifferentiableAt ℂ F z) : - ENNReal.ofReal (‖deriv F z‖ ^ dimension κ) * - ENNReal.ofReal (‖deriv J (F z)‖ ^ dimension κ) = - ENNReal.ofReal (‖deriv (J ∘ F) z‖ ^ dimension κ) := by - rw [deriv_comp z hJ hF,norm_mul,Real.mul_rpow (norm_nonneg _) (norm_nonneg _), - ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _)] - exact mul_comm _ _ - -namespace Multipoint - -variable {D : Set ℂ} - -lemma conformalMass_sub - (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) - {μ ν : Measure upperHalfPlane} [IsFiniteMeasureOnCompacts ν] (hle : μ ≤ ν) : - conformalMass J w (ν-μ) = conformalMass J w ν-conformalMass J w μ := by - let : IsFiniteMeasureOnCompacts μ := - ⟨fun K hK => lt_of_le_of_lt (hle K) hK.measure_lt_top⟩ - let : IsLocallyFiniteMeasure (μ.withDensity (fun z => (w z : ℝ≥0∞))) := - IsLocallyFiniteMeasure.withDensity_coe w.continuous - let : SigmaFinite (conformalMass J w μ) := J.measurableEmbedding.sigmaFinite_map - apply (Measure.add_left_inj (conformalMass J w μ) _ _).mp - calc - conformalMass J w (ν-μ)+conformalMass J w μ = conformalMass J w ν := by - rw [← conformalMass_add,sigmaFinite_sub_add_cancel_of_le hle] - _ = (conformalMass J w ν-conformalMass J w μ)+conformalMass J w μ := - (sigmaFinite_sub_add_cancel_of_le (conformalMass_mono J w hle)).symm - -lemma conformalMass_weightedImageMass - (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) - (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) - (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) (η : Measure upperHalfPlane) : - conformalMass J w (weightedImageMass F v η) = - weightedImageMass (J ∘ F) (fun z => v z*(w (F z) : ℝ≥0∞)) η := - weightedImageMass_comp J J.continuous.measurable _ - w.continuous.measurable.coe_nnreal_ennreal F hF v hv η - -lemma conformalMass_increment_of_restart - (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) - {μ ν : Measure upperHalfPlane} [IsFiniteMeasureOnCompacts ν] (hle : μ ≤ ν) - (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) - (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) (η : Measure upperHalfPlane) - (hrestart : ν-μ = weightedImageMass F v η) : - conformalMass J w ν-conformalMass J w μ = - weightedImageMass (J ∘ F) (fun z => v z*(w (F z) : ℝ≥0∞)) η := by - rw [← conformalMass_sub J w hle,hrestart] - exact conformalMass_weightedImageMass J w F hF v hv η - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalImageMass +import OAI.Probability.SLE.Loewner.StoppedInversePaths + +namespace OAI + +/-! Fixed conformal changes of coordinates preserve restart increments. +Composition combines the two density weights before pushing forward. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge + +local instance conformalRestartLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +lemma weightedImageMass_comp + {X : Type*} [MeasurableSpace X] + (J : upperHalfPlane → X) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) + (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) + (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) + (η : Measure upperHalfPlane) : + weightedImageMass J w (weightedImageMass F v η) = + weightedImageMass (J ∘ F) (fun z => v z*w (F z)) η := by + classical + apply Measure.ext + intro A hA + rw [weightedImageMass_apply J hJ w _ A hA, + weightedImageMass_lintegral F hF v hv η ((J ⁻¹' A).indicator w) + (hw.indicator (hJ hA)), + weightedImageMass_apply (J ∘ F) (hJ.comp hF) _ η A hA] + apply lintegral_congr + intro z + change v z * (if J (F z) ∈ A then w (F z) else 0) = + (if J (F z) ∈ A then v z*w (F z) else 0) + split_ifs <;> simp + +lemma derivative_dimension_weight_comp + (κ : ℝ) (J F : ℂ → ℂ) (z : ℂ) + (hJ : DifferentiableAt ℂ J (F z)) (hF : DifferentiableAt ℂ F z) : + ENNReal.ofReal (‖deriv F z‖ ^ dimension κ) * + ENNReal.ofReal (‖deriv J (F z)‖ ^ dimension κ) = + ENNReal.ofReal (‖deriv (J ∘ F) z‖ ^ dimension κ) := by + rw [deriv_comp z hJ hF,norm_mul,Real.mul_rpow (norm_nonneg _) (norm_nonneg _), + ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _)] + exact mul_comm _ _ + +namespace Multipoint + +variable {D : Set ℂ} + +lemma conformalMass_sub + (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) + {μ ν : Measure upperHalfPlane} [IsFiniteMeasureOnCompacts ν] (hle : μ ≤ ν) : + conformalMass J w (ν-μ) = conformalMass J w ν-conformalMass J w μ := by + let : IsFiniteMeasureOnCompacts μ := + ⟨fun K hK => lt_of_le_of_lt (hle K) hK.measure_lt_top⟩ + let : IsLocallyFiniteMeasure (μ.withDensity (fun z => (w z : ℝ≥0∞))) := + IsLocallyFiniteMeasure.withDensity_coe w.continuous + let : SigmaFinite (conformalMass J w μ) := J.measurableEmbedding.sigmaFinite_map + apply (Measure.add_left_inj (conformalMass J w μ) _ _).mp + calc + conformalMass J w (ν-μ)+conformalMass J w μ = conformalMass J w ν := by + rw [← conformalMass_add,sigmaFinite_sub_add_cancel_of_le hle] + _ = (conformalMass J w ν-conformalMass J w μ)+conformalMass J w μ := + (sigmaFinite_sub_add_cancel_of_le (conformalMass_mono J w hle)).symm + +lemma conformalMass_weightedImageMass + (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) + (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) + (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) (η : Measure upperHalfPlane) : + conformalMass J w (weightedImageMass F v η) = + weightedImageMass (J ∘ F) (fun z => v z*(w (F z) : ℝ≥0∞)) η := + weightedImageMass_comp J J.continuous.measurable _ + w.continuous.measurable.coe_nnreal_ennreal F hF v hv η + +lemma conformalMass_increment_of_restart + (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) + {μ ν : Measure upperHalfPlane} [IsFiniteMeasureOnCompacts ν] (hle : μ ≤ ν) + (F : upperHalfPlane → upperHalfPlane) (hF : Measurable F) + (v : upperHalfPlane → ℝ≥0∞) (hv : Measurable v) (η : Measure upperHalfPlane) + (hrestart : ν-μ = weightedImageMass F v η) : + conformalMass J w ν-conformalMass J w μ = + weightedImageMass (J ∘ F) (fun z => v z*(w (F z) : ℝ≥0∞)) η := by + rw [← conformalMass_sub J w hle,hrestart] + exact conformalMass_weightedImageMass J w F hF v hv η + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ConjugateInversionChart.lean b/lean/OAI/Probability/SLE/Geometry/ConjugateInversionChart.lean index 321c3f600e476a46a3cb4447ea24f41370348217..71c516d5578146318207d4a57e790f444176b4b1 100644 --- a/lean/OAI/Probability/SLE/Geometry/ConjugateInversionChart.lean +++ b/lean/OAI/Probability/SLE/Geometry/ConjugateInversionChart.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Geometry.ReflectedInvertedUpperConnected -import OAI.Probability.SLE.Geometry.ExteriorJoukowski - -namespace OAI - -/-! Conjugate inversion identifies the upper half of the punctured disk -with the upper exterior disk. -/ -noncomputable section -open Set Complex Function Metric -open scoped ComplexConjugate - -namespace SLEExactGauge - -@[simp] lemma norm_conjugateInversion (z : ℂ) : ‖conjugateInversion z‖=‖z‖⁻¹ := by - simp only [conjugateInversion,norm_inv,norm_conj] - -lemma conjugateInversion_injective : Injective conjugateInversion := - Function.LeftInverse.injective conjugateInversion_involutive - -lemma conjugateInversion_disk_upper_bijOn : - BijOn conjugateInversion (ball (0:ℂ) 1 ∩ upperHalfPlane) exteriorUpperDisk := by - have hn : ∀ z ∈ upperHalfPlane, 0 < ‖z‖ := by - intro z hz - apply norm_pos_iff.mpr - intro he - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hz - refine ⟨?_,conjugateInversion_injective.injOn,?_⟩ - · intro z hz - refine ⟨?_,conjugateInversion_upper hz.2⟩ - rw [norm_conjugateInversion,inv_eq_one_div,lt_div_iff₀ (hn z hz.2)] - simpa only [one_mul,mem_ball,dist_zero_right] using hz.1 - · intro z hz - refine ⟨conjugateInversion z,⟨?_,conjugateInversion_upper hz.2⟩, - conjugateInversion_involutive z⟩ - rw [mem_ball,dist_zero_right,norm_conjugateInversion,inv_eq_one_div, - div_lt_iff₀ (hn z hz.2)] - simpa only [one_mul] using hz.1 - -lemma conjugateInversion_slit_bijOn - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - BijOn conjugateInversion (upperHalfPlane \ range p) - (invertedExterior (reflectedArcSet p) ∩ upperHalfPlane) := by - rw [reflected_inverted_upper_image p h0 hH] - exact conjugateInversion_injective.injOn.bijOn_image - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ReflectedInvertedUpperConnected +import OAI.Probability.SLE.Geometry.ExteriorJoukowski + +namespace OAI + +/-! Conjugate inversion identifies the upper half of the punctured disk +with the upper exterior disk. -/ +noncomputable section +open Set Complex Function Metric +open scoped ComplexConjugate + +namespace SLEExactGauge + +@[simp] lemma norm_conjugateInversion (z : ℂ) : ‖conjugateInversion z‖=‖z‖⁻¹ := by + simp only [conjugateInversion,norm_inv,norm_conj] + +lemma conjugateInversion_injective : Injective conjugateInversion := + Function.LeftInverse.injective conjugateInversion_involutive + +lemma conjugateInversion_disk_upper_bijOn : + BijOn conjugateInversion (ball (0:ℂ) 1 ∩ upperHalfPlane) exteriorUpperDisk := by + have hn : ∀ z ∈ upperHalfPlane, 0 < ‖z‖ := by + intro z hz + apply norm_pos_iff.mpr + intro he + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hz + refine ⟨?_,conjugateInversion_injective.injOn,?_⟩ + · intro z hz + refine ⟨?_,conjugateInversion_upper hz.2⟩ + rw [norm_conjugateInversion,inv_eq_one_div,lt_div_iff₀ (hn z hz.2)] + simpa only [one_mul,mem_ball,dist_zero_right] using hz.1 + · intro z hz + refine ⟨conjugateInversion z,⟨?_,conjugateInversion_upper hz.2⟩, + conjugateInversion_involutive z⟩ + rw [mem_ball,dist_zero_right,norm_conjugateInversion,inv_eq_one_div, + div_lt_iff₀ (hn z hz.2)] + simpa only [one_mul] using hz.1 + +lemma conjugateInversion_slit_bijOn + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + BijOn conjugateInversion (upperHalfPlane \ range p) + (invertedExterior (reflectedArcSet p) ∩ upperHalfPlane) := by + rw [reflected_inverted_upper_image p h0 hH] + exact conjugateInversion_injective.injOn.bijOn_image + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ContinuousRangeCompletion.lean b/lean/OAI/Probability/SLE/Geometry/ContinuousRangeCompletion.lean index a802b9fc187f3586b596b9aeac9aeb9ccb595ef2..bf8a972834f0b7b691aad60d44c9e9df8c6c92c6 100644 --- a/lean/OAI/Probability/SLE/Geometry/ContinuousRangeCompletion.lean +++ b/lean/OAI/Probability/SLE/Geometry/ContinuousRangeCompletion.lean @@ -1,114 +1,114 @@ -import Mathlib - -namespace OAI - -/-! Continuous images of Baire space are measurable in the completion of a -finite measure. This supplies the uncountable-time projection step for SLE. -/ -noncomputable section -open Set Filter MeasureTheory Function PiNat Metric -open scoped Topology ENNReal - -namespace SLEExactGauge - -def prefixCylinder (s : List ℕ) : Set (ℕ → ℕ) := - {b | res b s.length=s} - -lemma prefixCylinder_nil : prefixCylinder []=univ := by - ext b - simp [prefixCylinder] - -lemma prefixCylinder_children (s : List ℕ) : - prefixCylinder s=⋃ k : ℕ, prefixCylinder (k::s) := by - ext b - simp only [prefixCylinder,mem_ofPred_eq,mem_iUnion,List.length_cons,res_succ,List.cons.injEq] - constructor - · intro h - exact ⟨b s.length,rfl,h⟩ - · rintro ⟨k,_,h⟩ - exact h - -lemma prefixCylinder_res (b : ℕ → ℕ) (n : ℕ) : - prefixCylinder (res b n)=cylinder b n := by - rw [prefixCylinder,res_length,cylinder_eq_res] - -theorem nullMeasurableSet_continuous_baire_range - {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] - (μ : Measure X) [IsFiniteMeasure μ] - (f : (ℕ → ℕ) → X) (hf : Continuous f) : - NullMeasurableSet (range f) μ := by - classical - let A (s : List ℕ) := f '' prefixCylinder s - let E (s : List ℕ) := toMeasurable μ (A s) ∩ closure (A s) - have hE (s : List ℕ) : MeasurableSet (E s) := - (measurableSet_toMeasurable μ (A s)).inter isClosed_closure.measurableSet - have hAE (s : List ℕ) : A s ⊆ E s := - fun _ h => ⟨subset_toMeasurable μ (A s) h,subset_closure h⟩ - have hAchildren (s : List ℕ) : A s=⋃ k, A (k::s) := by - dsimp only [A] - rw [prefixCylinder_children,image_iUnion] - have hzero (s : List ℕ) : μ (E s \ ⋃ k, E (k::s))=0 := by - have hU : MeasurableSet (⋃ k, E (k::s)) := MeasurableSet.iUnion fun k => hE (k::s) - have hsub : A s ⊆ ⋃ k, E (k::s) := by - rw [hAchildren] - exact iUnion_mono fun k => hAE (k::s) - have hempty : A s ∩ (⋃ k, E (k::s))ᶜ=∅ := by - exact eq_empty_iff_forall_notMem.mpr (fun x hx => hx.2 (hsub hx.1)) - apply measure_mono_null (show E s \ ⋃ k, E (k::s) ⊆ - toMeasurable μ (A s) ∩ (⋃ k, E (k::s))ᶜ from fun _ h => ⟨h.1.1,h.2⟩) - rw [μ.measure_toMeasurable_inter hU.compl (measure_ne_top μ (A s)),hempty,measure_empty] - let N := ⋃ s : List ℕ, E s \ ⋃ k, E (k::s) - have hN : μ N=0 := measure_iUnion_null hzero - have hcover : range f ⊆ E [] := by - have h := hAE [] - simpa only [A,prefixCylinder_nil,image_univ] using h - have hinside (x : X) (hx : x ∈ E []) (hxN : x ∉ N) : x ∈ range f := by - have hchild (s : List ℕ) (hs : x ∈ E s) : ∃ k, x ∈ E (k::s) := by - by_contra h - apply hxN - exact mem_iUnion.mpr ⟨s,hs,by simpa only [mem_iUnion,not_exists] using h⟩ - let S := {s : List ℕ // x ∈ E s} - let F (s : S) : S := ⟨(hchild s.1 s.2).choose::s.1,(hchild s.1 s.2).choose_spec⟩ - let p (n : ℕ) : S := F^[n] ⟨[],hx⟩ - have hp (n : ℕ) : p (n+1)=F (p n) := by simp only [p,iterate_succ',comp_apply] - let b (n : ℕ) := (hchild (p n).1 (p n).2).choose - have hres (n : ℕ) : (p n).1=res b n := by - induction n with - | zero => rfl - | succ n ih => - rw [hp] - change b n::(p n).1=res b (n+1) - rw [res_succ,ih] - have hclosure (n : ℕ) : x ∈ closure (f '' cylinder b n) := by - have h := (p n).2.2 - simpa only [hres,A,prefixCylinder_res] using h - have heq : x=f b := by - apply eq_of_forall_dist_le - intro ε hε - obtain ⟨v,⟨c,n,rfl⟩,hb,hv⟩ := - (isTopologicalBasis_cylinders (fun _ : ℕ => ℕ)).exists_subset_of_mem_open - (show b ∈ f ⁻¹' ball (f b) ε from mem_ball_self hε) - (isOpen_ball.preimage hf) - have hsub : f '' cylinder b n ⊆ closedBall (f b) ε := by - rw [← mem_cylinder_iff_eq.mp hb] at hv - exact (image_subset_iff.mpr hv).trans ball_subset_closedBall - have hc := closure_minimal hsub isClosed_closedBall (hclosure n) - exact hc - exact ⟨b,heq.symm⟩ - apply (hE []).nullMeasurableSet.congr - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hN] with x hxN - exact propext ⟨fun hx => hinside x hx hxN,fun hx => hcover hx⟩ - -theorem nullMeasurableSet_of_analytic_finite - {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] - (μ : Measure X) [IsFiniteMeasure μ] {s : Set X} (hs : AnalyticSet s) : - NullMeasurableSet s μ := by - rw [AnalyticSet] at hs - rcases hs with rfl | ⟨f,hf,rfl⟩ - · exact MeasurableSet.empty.nullMeasurableSet - · exact nullMeasurableSet_continuous_baire_range μ f hf - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! Continuous images of Baire space are measurable in the completion of a +finite measure. This supplies the uncountable-time projection step for SLE. -/ +noncomputable section +open Set Filter MeasureTheory Function PiNat Metric +open scoped Topology ENNReal + +namespace SLEExactGauge + +def prefixCylinder (s : List ℕ) : Set (ℕ → ℕ) := + {b | res b s.length=s} + +lemma prefixCylinder_nil : prefixCylinder []=univ := by + ext b + simp [prefixCylinder] + +lemma prefixCylinder_children (s : List ℕ) : + prefixCylinder s=⋃ k : ℕ, prefixCylinder (k::s) := by + ext b + simp only [prefixCylinder,mem_ofPred_eq,mem_iUnion,List.length_cons,res_succ,List.cons.injEq] + constructor + · intro h + exact ⟨b s.length,rfl,h⟩ + · rintro ⟨k,_,h⟩ + exact h + +lemma prefixCylinder_res (b : ℕ → ℕ) (n : ℕ) : + prefixCylinder (res b n)=cylinder b n := by + rw [prefixCylinder,res_length,cylinder_eq_res] + +theorem nullMeasurableSet_continuous_baire_range + {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] + (μ : Measure X) [IsFiniteMeasure μ] + (f : (ℕ → ℕ) → X) (hf : Continuous f) : + NullMeasurableSet (range f) μ := by + classical + let A (s : List ℕ) := f '' prefixCylinder s + let E (s : List ℕ) := toMeasurable μ (A s) ∩ closure (A s) + have hE (s : List ℕ) : MeasurableSet (E s) := + (measurableSet_toMeasurable μ (A s)).inter isClosed_closure.measurableSet + have hAE (s : List ℕ) : A s ⊆ E s := + fun _ h => ⟨subset_toMeasurable μ (A s) h,subset_closure h⟩ + have hAchildren (s : List ℕ) : A s=⋃ k, A (k::s) := by + dsimp only [A] + rw [prefixCylinder_children,image_iUnion] + have hzero (s : List ℕ) : μ (E s \ ⋃ k, E (k::s))=0 := by + have hU : MeasurableSet (⋃ k, E (k::s)) := MeasurableSet.iUnion fun k => hE (k::s) + have hsub : A s ⊆ ⋃ k, E (k::s) := by + rw [hAchildren] + exact iUnion_mono fun k => hAE (k::s) + have hempty : A s ∩ (⋃ k, E (k::s))ᶜ=∅ := by + exact eq_empty_iff_forall_notMem.mpr (fun x hx => hx.2 (hsub hx.1)) + apply measure_mono_null (show E s \ ⋃ k, E (k::s) ⊆ + toMeasurable μ (A s) ∩ (⋃ k, E (k::s))ᶜ from fun _ h => ⟨h.1.1,h.2⟩) + rw [μ.measure_toMeasurable_inter hU.compl (measure_ne_top μ (A s)),hempty,measure_empty] + let N := ⋃ s : List ℕ, E s \ ⋃ k, E (k::s) + have hN : μ N=0 := measure_iUnion_null hzero + have hcover : range f ⊆ E [] := by + have h := hAE [] + simpa only [A,prefixCylinder_nil,image_univ] using h + have hinside (x : X) (hx : x ∈ E []) (hxN : x ∉ N) : x ∈ range f := by + have hchild (s : List ℕ) (hs : x ∈ E s) : ∃ k, x ∈ E (k::s) := by + by_contra h + apply hxN + exact mem_iUnion.mpr ⟨s,hs,by simpa only [mem_iUnion,not_exists] using h⟩ + let S := {s : List ℕ // x ∈ E s} + let F (s : S) : S := ⟨(hchild s.1 s.2).choose::s.1,(hchild s.1 s.2).choose_spec⟩ + let p (n : ℕ) : S := F^[n] ⟨[],hx⟩ + have hp (n : ℕ) : p (n+1)=F (p n) := by simp only [p,iterate_succ',comp_apply] + let b (n : ℕ) := (hchild (p n).1 (p n).2).choose + have hres (n : ℕ) : (p n).1=res b n := by + induction n with + | zero => rfl + | succ n ih => + rw [hp] + change b n::(p n).1=res b (n+1) + rw [res_succ,ih] + have hclosure (n : ℕ) : x ∈ closure (f '' cylinder b n) := by + have h := (p n).2.2 + simpa only [hres,A,prefixCylinder_res] using h + have heq : x=f b := by + apply eq_of_forall_dist_le + intro ε hε + obtain ⟨v,⟨c,n,rfl⟩,hb,hv⟩ := + (isTopologicalBasis_cylinders (fun _ : ℕ => ℕ)).exists_subset_of_mem_open + (show b ∈ f ⁻¹' ball (f b) ε from mem_ball_self hε) + (isOpen_ball.preimage hf) + have hsub : f '' cylinder b n ⊆ closedBall (f b) ε := by + rw [← mem_cylinder_iff_eq.mp hb] at hv + exact (image_subset_iff.mpr hv).trans ball_subset_closedBall + have hc := closure_minimal hsub isClosed_closedBall (hclosure n) + exact hc + exact ⟨b,heq.symm⟩ + apply (hE []).nullMeasurableSet.congr + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hN] with x hxN + exact propext ⟨fun hx => hinside x hx hxN,fun hx => hcover hx⟩ + +theorem nullMeasurableSet_of_analytic_finite + {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] + (μ : Measure X) [IsFiniteMeasure μ] {s : Set X} (hs : AnalyticSet s) : + NullMeasurableSet s μ := by + rw [AnalyticSet] at hs + rcases hs with rfl | ⟨f,hf,rfl⟩ + · exact MeasurableSet.empty.nullMeasurableSet + · exact nullMeasurableSet_continuous_baire_range μ f hf + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CountableInteriorIntervals.lean b/lean/OAI/Probability/SLE/Geometry/CountableInteriorIntervals.lean index dede935f7f327972c32451eb649b7b8deb242605..820b4199d88f758bb5c180dba9b2a615ce7c5a31 100644 --- a/lean/OAI/Probability/SLE/Geometry/CountableInteriorIntervals.lean +++ b/lean/OAI/Probability/SLE/Geometry/CountableInteriorIntervals.lean @@ -1,95 +1,95 @@ -import OAI.Probability.SLE.Gauge.CoverEvents -import Mathlib.Topology.Compactness.SigmaCompact - -namespace OAI - -/-! The deterministic countable certificates for the retained middle-strip -intervals. An actual open interior visit contains a nondegenerate rational -time interval, whose image lies in one fixed countable compact exhaustion. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem exists_countable_compact_exhaustion_open (O : Set ℂ) (hO : IsOpen O) : - ∃ K : ℕ → Set ℂ, (∀ n, IsCompact (K n)) ∧ (∀ n, K n ⊆ O) ∧ - (∀ S : Set ℂ, IsCompact S → S ⊆ O → ∃ n, S ⊆ K n) := by - let : LocallyCompactSpace O := hO.locallyCompactSpace - let L := CompactExhaustion.choice O - refine ⟨fun n => Subtype.val '' L n, ?_, ?_, ?_⟩ - · intro n - exact (L.isCompact n).image continuous_subtype_val - · rintro n z ⟨w, _, rfl⟩ - exact w.property - · intro S hS hSO - have hpre : IsCompact ((Subtype.val : O → ℂ) ⁻¹' S) := - Topology.IsEmbedding.subtypeVal.isInducing.isCompact_preimage' - hS (by simpa only [Subtype.range_coe] using hSO) - obtain ⟨n, hn⟩ := L.exists_superset_of_isCompact hpre - refine ⟨n, ?_⟩ - intro z hz - exact ⟨⟨z, hSO hz⟩, hn hz, rfl⟩ - -theorem exists_rational_interval_in_open_visit - (f : ℝ → ℂ) (hf : Continuous f) (O : Set ℂ) (hO : IsOpen O) - (a b : ℝ) (hvisit : ∃ u ∈ Ioo a b, f u ∈ O) : - ∃ s t : ℚ, a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < b ∧ - MapsTo f (Icc (s : ℝ) t) O := by - have hU : IsOpen (Ioo a b ∩ f ⁻¹' O) := isOpen_Ioo.inter (hO.preimage hf) - have hUn : (Ioo a b ∩ f ⁻¹' O).Nonempty := by - obtain ⟨u, hu, hfu⟩ := hvisit - exact ⟨u, hu, hfu⟩ - obtain ⟨l, r, hlr, hsub⟩ := hU.exists_Ioo_subset hUn - obtain ⟨s, hls, hsr⟩ := exists_rat_btwn hlr - obtain ⟨t, hst, htr⟩ := exists_rat_btwn hsr - refine ⟨s, t, (hsub ⟨hls, hsr⟩).1.1, hst, - (hsub ⟨hls.trans hst, htr⟩).1.2, ?_⟩ - intro u hu - exact (hsub ⟨hls.trans_le hu.1, hu.2.trans_lt htr⟩).2 - -/-- One countable compact family captures every open visit, at rational -times strictly between the entry and the later crossing. -/ -theorem rational_interval_compact_certificate - (O : Set ℂ) (hO : IsOpen O) : - ∃ K : ℕ → Set ℂ, (∀ n, IsCompact (K n)) ∧ (∀ n, K n ⊆ O) ∧ - ∀ (f : ℝ → ℂ), Continuous f → ∀ a b : ℝ, - (∃ u ∈ Ioo a b, f u ∈ O) → - ∃ (s t : ℚ) (n : ℕ), a < (s : ℝ) ∧ (s : ℝ) < t ∧ - (t : ℝ) < b ∧ f '' Icc (s : ℝ) t ⊆ K n := by - obtain ⟨K, hK, hKO, hcap⟩ := exists_countable_compact_exhaustion_open O hO - refine ⟨K, hK, hKO, ?_⟩ - intro f hf a b hvisit - obtain ⟨s, t, has, hst, htb, hmap⟩ := - exists_rational_interval_in_open_visit f hf O hO a b hvisit - obtain ⟨n, hn⟩ := hcap (f '' Icc (s : ℝ) t) - (isCompact_Icc.image hf) hmap.image_subset - exact ⟨s, t, n, has, hst, htb, hn⟩ - -local instance compactIntervalPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance compactIntervalPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem isClosed_interval_compact_certificate (s t : ℝ≥0) - {K : Set ℂ} (hK : IsClosed K) : - IsClosed {f : C(ℝ≥0, ℂ) | MapsTo f (Icc s t) K} := by - have he : {f : C(ℝ≥0, ℂ) | MapsTo f (Icc s t) K} = - ⋂ u ∈ Icc s t, (fun f : C(ℝ≥0, ℂ) => f u) ⁻¹' K := by - ext f - simp only [mem_ofPred_eq, MapsTo, mem_iInter, mem_preimage] - rw [he] - exact isClosed_iInter (fun u => isClosed_iInter (fun _ => - hK.preimage (continuous_eval_const u))) - -theorem measurableSet_interval_compact_certificate - {Ω : Type*} [MeasurableSpace Ω] - (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) - (s t : ℝ≥0) {K : Set ℂ} (hK : IsCompact K) : - MeasurableSet {ω | MapsTo (f ω) (Icc s t) K} := - hf (isClosed_interval_compact_certificate s t hK.isClosed).measurableSet - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.CoverEvents +import Mathlib.Topology.Compactness.SigmaCompact + +namespace OAI + +/-! The deterministic countable certificates for the retained middle-strip +intervals. An actual open interior visit contains a nondegenerate rational +time interval, whose image lies in one fixed countable compact exhaustion. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem exists_countable_compact_exhaustion_open (O : Set ℂ) (hO : IsOpen O) : + ∃ K : ℕ → Set ℂ, (∀ n, IsCompact (K n)) ∧ (∀ n, K n ⊆ O) ∧ + (∀ S : Set ℂ, IsCompact S → S ⊆ O → ∃ n, S ⊆ K n) := by + let : LocallyCompactSpace O := hO.locallyCompactSpace + let L := CompactExhaustion.choice O + refine ⟨fun n => Subtype.val '' L n, ?_, ?_, ?_⟩ + · intro n + exact (L.isCompact n).image continuous_subtype_val + · rintro n z ⟨w, _, rfl⟩ + exact w.property + · intro S hS hSO + have hpre : IsCompact ((Subtype.val : O → ℂ) ⁻¹' S) := + Topology.IsEmbedding.subtypeVal.isInducing.isCompact_preimage' + hS (by simpa only [Subtype.range_coe] using hSO) + obtain ⟨n, hn⟩ := L.exists_superset_of_isCompact hpre + refine ⟨n, ?_⟩ + intro z hz + exact ⟨⟨z, hSO hz⟩, hn hz, rfl⟩ + +theorem exists_rational_interval_in_open_visit + (f : ℝ → ℂ) (hf : Continuous f) (O : Set ℂ) (hO : IsOpen O) + (a b : ℝ) (hvisit : ∃ u ∈ Ioo a b, f u ∈ O) : + ∃ s t : ℚ, a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < b ∧ + MapsTo f (Icc (s : ℝ) t) O := by + have hU : IsOpen (Ioo a b ∩ f ⁻¹' O) := isOpen_Ioo.inter (hO.preimage hf) + have hUn : (Ioo a b ∩ f ⁻¹' O).Nonempty := by + obtain ⟨u, hu, hfu⟩ := hvisit + exact ⟨u, hu, hfu⟩ + obtain ⟨l, r, hlr, hsub⟩ := hU.exists_Ioo_subset hUn + obtain ⟨s, hls, hsr⟩ := exists_rat_btwn hlr + obtain ⟨t, hst, htr⟩ := exists_rat_btwn hsr + refine ⟨s, t, (hsub ⟨hls, hsr⟩).1.1, hst, + (hsub ⟨hls.trans hst, htr⟩).1.2, ?_⟩ + intro u hu + exact (hsub ⟨hls.trans_le hu.1, hu.2.trans_lt htr⟩).2 + +/-- One countable compact family captures every open visit, at rational +times strictly between the entry and the later crossing. -/ +theorem rational_interval_compact_certificate + (O : Set ℂ) (hO : IsOpen O) : + ∃ K : ℕ → Set ℂ, (∀ n, IsCompact (K n)) ∧ (∀ n, K n ⊆ O) ∧ + ∀ (f : ℝ → ℂ), Continuous f → ∀ a b : ℝ, + (∃ u ∈ Ioo a b, f u ∈ O) → + ∃ (s t : ℚ) (n : ℕ), a < (s : ℝ) ∧ (s : ℝ) < t ∧ + (t : ℝ) < b ∧ f '' Icc (s : ℝ) t ⊆ K n := by + obtain ⟨K, hK, hKO, hcap⟩ := exists_countable_compact_exhaustion_open O hO + refine ⟨K, hK, hKO, ?_⟩ + intro f hf a b hvisit + obtain ⟨s, t, has, hst, htb, hmap⟩ := + exists_rational_interval_in_open_visit f hf O hO a b hvisit + obtain ⟨n, hn⟩ := hcap (f '' Icc (s : ℝ) t) + (isCompact_Icc.image hf) hmap.image_subset + exact ⟨s, t, n, has, hst, htb, hn⟩ + +local instance compactIntervalPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance compactIntervalPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem isClosed_interval_compact_certificate (s t : ℝ≥0) + {K : Set ℂ} (hK : IsClosed K) : + IsClosed {f : C(ℝ≥0, ℂ) | MapsTo f (Icc s t) K} := by + have he : {f : C(ℝ≥0, ℂ) | MapsTo f (Icc s t) K} = + ⋂ u ∈ Icc s t, (fun f : C(ℝ≥0, ℂ) => f u) ⁻¹' K := by + ext f + simp only [mem_ofPred_eq, MapsTo, mem_iInter, mem_preimage] + rw [he] + exact isClosed_iInter (fun u => isClosed_iInter (fun _ => + hK.preimage (continuous_eval_const u))) + +theorem measurableSet_interval_compact_certificate + {Ω : Type*} [MeasurableSpace Ω] + (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) + (s t : ℝ≥0) {K : Set ℂ} (hK : IsCompact K) : + MeasurableSet {ω | MapsTo (f ω) (Icc s t) K} := + hf (isClosed_interval_compact_certificate s t hK.isClosed).measurableSet + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CountablePastBound.lean b/lean/OAI/Probability/SLE/Geometry/CountablePastBound.lean index d191d685fcc197fb3140775e097c472339da5df9..82534111995fecdc2457932e0afd4276fc608d78 100644 --- a/lean/OAI/Probability/SLE/Geometry/CountablePastBound.lean +++ b/lean/OAI/Probability/SLE/Geometry/CountablePastBound.lean @@ -1,106 +1,106 @@ -import OAI.Probability.SLE.Gauge.StrictPastTilted -import OAI.Probability.SLE.Gauge.ActualPastThreshold - -namespace OAI - -/-! Passing finite centered-law test bounds to countable detected mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint - -lemma range_image_event_union {X : Type*} (q : ℕ → ℝ≥0) (A : ℝ≥0 → Set X) : - (⋃ n, ⋃ t∈(Finset.range n).image q, A t) = ⋃ n, A (q n) := by - ext x - constructor - · intro hx - obtain ⟨n,hx⟩ := mem_iUnion.mp hx - obtain ⟨t,ht,hxt⟩ := mem_iUnion₂.mp hx - obtain ⟨k,_,rfl⟩ := Finset.mem_image.mp ht - exact mem_iUnion.mpr ⟨k,hxt⟩ - · intro hx - obtain ⟨n,hn⟩ := mem_iUnion.mp hx - exact mem_iUnion.mpr ⟨n+1,mem_iUnion₂.mpr ⟨q n, - Finset.mem_image.mpr ⟨n,Finset.mem_range.mpr (Nat.lt_succ_self n),rfl⟩,hn⟩⟩ - -namespace CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - -include C hB hc h0 hκ hκ8 F hw - -lemma countable_strict_past_le_of_finite_bound - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (J : ℕ → Finset ℝ≥0) (hJ : Monotone J) - (E : ℝ≥0 → Set Ω) - (hE : ∀ t, MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) - (K : Set upperHalfPlane) (hK : IsCompact K) (A : ℝ≥0∞) - (hbound : ∀ j, ∀ᶠ n in atTop, - (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, - ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) - (finiteStrictPastEvent (J j) E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) ≤ A) : - (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ n, ⋃ t∈J n, - if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P) ≤ A := by - rw [C.countable_strict_past_expectation hB hc h0 hκ hκ8 F hw J hJ E hE K hK] - apply iSup_le - intro j - rw [← C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw (J j) E - (fun t _ => hE t) K hK] - exact le_of_tendsto (C.finite_strict_past_tilted_limit hB hBm hc h0 hκ hκ8 γ G F hw u hu hmono hlim - (J j) E (fun t _ => hE t) K hK) (hbound j) - -lemma bad_cell_mass_le_of_finite_bound - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q) - (K : Set upperHalfPlane) (hK : IsCompact K) (H : ℝ) (hH : 0 < H) (A : ℝ≥0∞) - (hbound : ∀ j, ∀ᶠ n in atTop, - (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, - ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) - (finiteStrictPastEvent ((Finset.range j).image q) - (fun t => {ω | H < (C.process t ω K).toReal}) - (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) ≤ A) : - (∫⁻ ω, {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K}.indicator - (fun ω => C.rational.terminal ω K) ω ∂P) ≤ 2*A := by - have hE (t : ℝ≥0) : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] - {ω | H < (C.process t ω K).toReal} := - measurableSet_lt measurable_const - (((Measure.measurable_coe hK.measurableSet).comp (C.progressive.adapted t)).ennreal_toReal) - let E (t : ℝ≥0) : Set Ω := {ω | H < (C.process t ω K).toReal} - let J (n : ℕ) := (Finset.range n).image q - let D (ω : Ω) := C.rational.terminal ω (K ∩ ⋃ n, - if H < (C.process (q n) ω K).toReal then - {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)}ᶜ else ∅) - have hJ : Monotone J := fun n m hnm => Finset.image_subset_image (Finset.range_mono hnm) - have hD : (∫⁻ ω, D ω ∂P) ≤ A := by - have hh := C.countable_strict_past_le_of_finite_bound hB hc h0 hκ hκ8 F hw - u hu hmono hlim J hJ E hE K hK A hbound - simpa only [D, E, J, mem_ofPred_eq, range_image_event_union] using hh - calc - _ ≤ ∫⁻ ω, 2*D ω ∂P := by - apply lintegral_mono_ae - filter_upwards [C.past_threshold_detects_half_terminal hB hc h0 hκ hκ8 F hw q hq K hK H hH] with ω hω - by_cases hb : ENNReal.ofReal (2*H) < C.rational.terminal ω K - · rw [indicator_of_mem (show ω ∈ {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K} from hb)] - exact hω hb - · rw [indicator_of_notMem (show ω ∉ {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K} from hb)] - exact bot_le - _ = 2*(∫⁻ ω, D ω ∂P) := lintegral_const_mul' _ _ (by norm_num) - _ ≤ 2*A := mul_le_mul' le_rfl hD - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.StrictPastTilted +import OAI.Probability.SLE.Gauge.ActualPastThreshold + +namespace OAI + +/-! Passing finite centered-law test bounds to countable detected mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint + +lemma range_image_event_union {X : Type*} (q : ℕ → ℝ≥0) (A : ℝ≥0 → Set X) : + (⋃ n, ⋃ t∈(Finset.range n).image q, A t) = ⋃ n, A (q n) := by + ext x + constructor + · intro hx + obtain ⟨n,hx⟩ := mem_iUnion.mp hx + obtain ⟨t,ht,hxt⟩ := mem_iUnion₂.mp hx + obtain ⟨k,_,rfl⟩ := Finset.mem_image.mp ht + exact mem_iUnion.mpr ⟨k,hxt⟩ + · intro hx + obtain ⟨n,hn⟩ := mem_iUnion.mp hx + exact mem_iUnion.mpr ⟨n+1,mem_iUnion₂.mpr ⟨q n, + Finset.mem_image.mpr ⟨n,Finset.mem_range.mpr (Nat.lt_succ_self n),rfl⟩,hn⟩⟩ + +namespace CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + +include C hB hc h0 hκ hκ8 F hw + +lemma countable_strict_past_le_of_finite_bound + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (J : ℕ → Finset ℝ≥0) (hJ : Monotone J) + (E : ℝ≥0 → Set Ω) + (hE : ∀ t, MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] (E t)) + (K : Set upperHalfPlane) (hK : IsCompact K) (A : ℝ≥0∞) + (hbound : ∀ j, ∀ᶠ n in atTop, + (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, + ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) + (finiteStrictPastEvent (J j) E (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) ≤ A) : + (∫⁻ ω, C.rational.terminal ω (K ∩ ⋃ n, ⋃ t∈J n, + if ω∈E t then {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}ᶜ else ∅) ∂P) ≤ A := by + rw [C.countable_strict_past_expectation hB hc h0 hκ hκ8 F hw J hJ E hE K hK] + apply iSup_le + intro j + rw [← C.finite_strict_past_expectation hB hc h0 hκ hκ8 F hw (J j) E + (fun t _ => hE t) K hK] + exact le_of_tendsto (C.finite_strict_past_tilted_limit hB hBm hc h0 hκ hκ8 γ G F hw u hu hmono hlim + (J j) E (fun t _ => hE t) K hK) (hbound j) + +lemma bad_cell_mass_le_of_finite_bound + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q) + (K : Set upperHalfPlane) (hK : IsCompact K) (H : ℝ) (hH : 0 < H) (A : ℝ≥0∞) + (hbound : ∀ j, ∀ᶠ n in atTop, + (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' K, + ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z (u n) + (finiteStrictPastEvent ((Finset.range j).image q) + (fun t => {ω | H < (C.process t ω K).toReal}) + (fun ω => radiusLevelTime (γ ω) (G ω) z (u n))) ∂volume) ≤ A) : + (∫⁻ ω, {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K}.indicator + (fun ω => C.rational.terminal ω K) ω ∂P) ≤ 2*A := by + have hE (t : ℝ≥0) : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] + {ω | H < (C.process t ω K).toReal} := + measurableSet_lt measurable_const + (((Measure.measurable_coe hK.measurableSet).comp (C.progressive.adapted t)).ennreal_toReal) + let E (t : ℝ≥0) : Set Ω := {ω | H < (C.process t ω K).toReal} + let J (n : ℕ) := (Finset.range n).image q + let D (ω : Ω) := C.rational.terminal ω (K ∩ ⋃ n, + if H < (C.process (q n) ω K).toReal then + {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)}ᶜ else ∅) + have hJ : Monotone J := fun n m hnm => Finset.image_subset_image (Finset.range_mono hnm) + have hD : (∫⁻ ω, D ω ∂P) ≤ A := by + have hh := C.countable_strict_past_le_of_finite_bound hB hc h0 hκ hκ8 F hw + u hu hmono hlim J hJ E hE K hK A hbound + simpa only [D, E, J, mem_ofPred_eq, range_image_event_union] using hh + calc + _ ≤ ∫⁻ ω, 2*D ω ∂P := by + apply lintegral_mono_ae + filter_upwards [C.past_threshold_detects_half_terminal hB hc h0 hκ hκ8 F hw q hq K hK H hH] with ω hω + by_cases hb : ENNReal.ofReal (2*H) < C.rational.terminal ω K + · rw [indicator_of_mem (show ω ∈ {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K} from hb)] + exact hω hb + · rw [indicator_of_notMem (show ω ∉ {ω | ENNReal.ofReal (2*H) < C.rational.terminal ω K} from hb)] + exact bot_le + _ = 2*(∫⁻ ω, D ω ∂P) := lintegral_const_mul' _ _ (by norm_num) + _ ≤ 2*A := mul_le_mul' le_rfl hD + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CountableRemainderKernels.lean b/lean/OAI/Probability/SLE/Geometry/CountableRemainderKernels.lean index d911f5435c267d46241cc3505ddda7496b49c913..6315ea274afdf58d92fa9c362b04e7a9e5ad2947 100644 --- a/lean/OAI/Probability/SLE/Geometry/CountableRemainderKernels.lean +++ b/lean/OAI/Probability/SLE/Geometry/CountableRemainderKernels.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Mass.CountableLevelKernel -import OAI.Probability.SLE.Geometry.RemainderKernelCancellation - -namespace OAI - -/-! Positive measurable remaining masses, with the exact Green kernel, -for the actual common countable-time Radon construction. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem exists_countable_radon_remainder_kernels - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (t : ℕ → ℝ≥0) : - ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - ∃ ν η : ℕ → Ω → Measure upperHalfPlane, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Measurable (ν j)) ∧ - (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ - (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n - (levelMassTimes t j) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ - (∀ j, Measurable (η j)) ∧ - (∀ j ω, (η j ω).Regular ∧ IsFiniteMeasureOnCompacts (η j ω)) ∧ - (∀ᵐ ω ∂P, ∀ j, η j ω + ν (j+1) ω = ν 0 ω) ∧ - P.bind (ν 0) = initialGreenMeasure κ ∧ - ∀ j (E : Set Ω), - MeasurableSet[Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) (t j)] E → - (P.restrict E).bind (η j) = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (t j)) := by - obtain ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,hmean,hkernel⟩ := - exists_countable_level_radon_kernels hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth t - have horder : ∀ᵐ ω ∂P, ∀ j, ν (j+1) ω ≤ ν 0 ω := by - filter_upwards [hlim] with ω hω - intro j - let (i : ℕ) : (ν i ω).Regular := (hν i ω).1 - let (n i : ℕ) : IsFiniteMeasureOnCompacts - (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n (levelMassTimes t i) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n _ - apply radon_limits_mono _ _ (ν (j+1) ω) (ν 0 ω) (hω (j+1)) (hω 0) - intro n - simpa only [levelMassTimes_zero,levelMassTimes_succ] using - convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω - (le_top : (t j : ℝ≥0∞) ≤ ∞) - have hex (j : ℕ) := exists_measurable_halfPlane_difference P (ν 0) (ν (j+1)) - (hνm 0) (hνm (j+1)) (fun ω => (hν 0 ω).1) (fun ω => (hν (j+1) ω).1) - (horder.mono (fun _ hω => hω j)) - choose η hηm hηr hηsum using hex - refine ⟨s,w,ν,η,hw₀,hw₁,hνm,hν,hlim,hηm,hηr,ae_all_iff.mpr hηsum,hmean,?_⟩ - intro j E hE - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hbound : (P.restrict E).bind (ν (j+1)) ≤ initialGreenMeasure κ := by - rw [← hmean] - exact restricted_bind_le_of_ae_le P (ν (j+1)) (ν 0) (hνm (j+1)) (hνm 0) - (horder.mono (fun _ hω => hω j)) E - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν (j+1))) := - ⟨fun K hK => lt_of_le_of_lt (hbound K) hK.measure_lt_top⟩ - exact bind_remainder_eq_of_sum (P.restrict E) (ν (j+1)) (ν 0) (η j) - (hνm (j+1)) (hνm 0) (hηm j) _ (ae_restrict_of_ae (hηsum j)) (hkernel j E hE) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CountableLevelKernel +import OAI.Probability.SLE.Geometry.RemainderKernelCancellation + +namespace OAI + +/-! Positive measurable remaining masses, with the exact Green kernel, +for the actual common countable-time Radon construction. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem exists_countable_radon_remainder_kernels + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (t : ℕ → ℝ≥0) : + ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + ∃ ν η : ℕ → Ω → Measure upperHalfPlane, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Measurable (ν j)) ∧ + (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ + (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n + (levelMassTimes t j) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ + (∀ j, Measurable (η j)) ∧ + (∀ j ω, (η j ω).Regular ∧ IsFiniteMeasureOnCompacts (η j ω)) ∧ + (∀ᵐ ω ∂P, ∀ j, η j ω + ν (j+1) ω = ν 0 ω) ∧ + P.bind (ν 0) = initialGreenMeasure κ ∧ + ∀ j (E : Set Ω), + MeasurableSet[Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) (t j)] E → + (P.restrict E).bind (η j) = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (t j)) := by + obtain ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,hmean,hkernel⟩ := + exists_countable_level_radon_kernels hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth t + have horder : ∀ᵐ ω ∂P, ∀ j, ν (j+1) ω ≤ ν 0 ω := by + filter_upwards [hlim] with ω hω + intro j + let (i : ℕ) : (ν i ω).Regular := (hν i ω).1 + let (n i : ℕ) : IsFiniteMeasureOnCompacts + (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n (levelMassTimes t i) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n _ + apply radon_limits_mono _ _ (ν (j+1) ω) (ν 0 ω) (hω (j+1)) (hω 0) + intro n + simpa only [levelMassTimes_zero,levelMassTimes_succ] using + convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω + (le_top : (t j : ℝ≥0∞) ≤ ∞) + have hex (j : ℕ) := exists_measurable_halfPlane_difference P (ν 0) (ν (j+1)) + (hνm 0) (hνm (j+1)) (fun ω => (hν 0 ω).1) (fun ω => (hν (j+1) ω).1) + (horder.mono (fun _ hω => hω j)) + choose η hηm hηr hηsum using hex + refine ⟨s,w,ν,η,hw₀,hw₁,hνm,hν,hlim,hηm,hηr,ae_all_iff.mpr hηsum,hmean,?_⟩ + intro j E hE + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hbound : (P.restrict E).bind (ν (j+1)) ≤ initialGreenMeasure κ := by + rw [← hmean] + exact restricted_bind_le_of_ae_le P (ν (j+1)) (ν 0) (hνm (j+1)) (hνm 0) + (horder.mono (fun _ hω => hω j)) E + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν (j+1))) := + ⟨fun K hK => lt_of_le_of_lt (hbound K) hK.measure_lt_top⟩ + exact bind_remainder_eq_of_sum (P.restrict E) (ν (j+1)) (ν 0) (η j) + (hνm (j+1)) (hνm 0) (hηm j) _ (ae_restrict_of_ae (hηsum j)) (hkernel j E hE) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CrosscutExitSides.lean b/lean/OAI/Probability/SLE/Geometry/CrosscutExitSides.lean index 98895de2890d973b2ad694a54f1f6c7f770cc3f2..afa09d337acaddc90b2d4aba470f010ca1d6a250 100644 --- a/lean/OAI/Probability/SLE/Geometry/CrosscutExitSides.lean +++ b/lean/OAI/Probability/SLE/Geometry/CrosscutExitSides.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Geometry.FirstDomainExit -import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening -import OAI.Probability.SLE.Geometry.PublishedConformalExitLaw - -namespace OAI - -/-! A nonzero real map-out limit has the sign selected before the first -domain exit. This turns geometric guide avoidance into an actual exit-arc -event, once the published conformal Brownian exit law is supplied. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.CrosscutSides -open RibbonCoordinates Target - -theorem real_exit_positive_of_straightened_prefix - (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (g : ℝ → ℂ) {a τ x : ℝ} (haτ : a < τ) (hx : x ≠ 0) - (hprefix : ∀ t ∈ Ico a τ, 0 < (h (g t)).re) - (hlim : Tendsto g (𝓝[<] τ) (𝓝 (x : ℂ))) : 0 < x := by - have hh := straightening_preserves_real_exit_limit h hreal g x hlim - have he : ∀ᶠ t in 𝓝[<] τ, 0 ≤ (h (g t)).re := by - filter_upwards [Ioo_mem_nhdsLT haτ] with t ht - exact (hprefix t ⟨ht.1.le, ht.2⟩).le - exact lt_of_le_of_ne (ge_of_tendsto hh he) (Ne.symm hx) - -theorem real_exit_negative_of_straightened_prefix - (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (g : ℝ → ℂ) {a τ x : ℝ} (haτ : a < τ) (hx : x ≠ 0) - (hprefix : ∀ t ∈ Ico a τ, (h (g t)).re < 0) - (hlim : Tendsto g (𝓝[<] τ) (𝓝 (x : ℂ))) : x < 0 := by - have hh := straightening_preserves_real_exit_limit h hreal g x hlim - have he : ∀ᶠ t in 𝓝[<] τ, (h (g t)).re ≤ 0 := by - filter_upwards [Ioo_mem_nhdsLT haτ] with t ht - exact (hprefix t ⟨ht.1.le, ht.2⟩).le - exact lt_of_le_of_ne (le_of_tendsto hh he) hx - -/-- The scalar straightened side coordinate has a fixed sign until exit -because its zero set is precisely the complete guide. -/ -theorem positive_first_exit_belongs_to_arc - (D : Set ℂ) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) - (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (hg : ContinuousOn g D) (ζ : ℝ → ℂ) (hc : Continuous ζ) - {τ : ℝ} (hτ : UnitWindowFirstExit D ζ τ) - (haτ : (1 / 4 : ℝ) < τ) - (hn : ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (ζ t))).re ≠ 0) - (hstart : 0 < (h (g (ζ (1 / 4)))).re) - (x : ℝ) (hx : x ≠ 0) - (hlim : Tendsto (fun t => g (ζ t)) (𝓝[<] τ) (𝓝 (x : ℂ))) : 0 < x := by - have hj : ContinuousOn (fun z => (h (g z)).re) D := - Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) - have hin : MapsTo ζ (Ico (1 / 4 : ℝ) τ) D := - hτ.2.2.mono_left (Ico_subset_Ico_left (by norm_num)) - exact real_exit_positive_of_straightened_prefix h hreal (fun t => g (ζ t)) haτ hx - (exit_prefix_positive (fun z => (h (g z)).re) hj ζ hc.continuousOn hin hn haτ hstart) - hlim - -theorem negative_first_exit_belongs_to_arc - (D : Set ℂ) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) - (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (hg : ContinuousOn g D) (ζ : ℝ → ℂ) (hc : Continuous ζ) - {τ : ℝ} (hτ : UnitWindowFirstExit D ζ τ) - (haτ : (1 / 4 : ℝ) < τ) - (hn : ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (ζ t))).re ≠ 0) - (hstart : (h (g (ζ (1 / 4)))).re < 0) - (x : ℝ) (hx : x ≠ 0) - (hlim : Tendsto (fun t => g (ζ t)) (𝓝[<] τ) (𝓝 (x : ℂ))) : x < 0 := by - have hj : ContinuousOn (fun z => (h (g z)).re) D := - Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) - have hin : MapsTo ζ (Ico (1 / 4 : ℝ) τ) D := - hτ.2.2.mono_left (Ico_subset_Ico_left (by norm_num)) - exact real_exit_negative_of_straightened_prefix h hreal (fun t => g (ζ t)) haτ hx - (exit_prefix_negative (fun z => (h (g z)).re) hj ζ hc.continuousOn hin hn haτ hstart) - hlim - -end SLEExactGauge.CrosscutSides - -end - -end OAI +import OAI.Probability.SLE.Geometry.FirstDomainExit +import OAI.Probability.SLE.Geometry.PublishedCrosscutStraightening +import OAI.Probability.SLE.Geometry.PublishedConformalExitLaw + +namespace OAI + +/-! A nonzero real map-out limit has the sign selected before the first +domain exit. This turns geometric guide avoidance into an actual exit-arc +event, once the published conformal Brownian exit law is supplied. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.CrosscutSides +open RibbonCoordinates Target + +theorem real_exit_positive_of_straightened_prefix + (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (g : ℝ → ℂ) {a τ x : ℝ} (haτ : a < τ) (hx : x ≠ 0) + (hprefix : ∀ t ∈ Ico a τ, 0 < (h (g t)).re) + (hlim : Tendsto g (𝓝[<] τ) (𝓝 (x : ℂ))) : 0 < x := by + have hh := straightening_preserves_real_exit_limit h hreal g x hlim + have he : ∀ᶠ t in 𝓝[<] τ, 0 ≤ (h (g t)).re := by + filter_upwards [Ioo_mem_nhdsLT haτ] with t ht + exact (hprefix t ⟨ht.1.le, ht.2⟩).le + exact lt_of_le_of_ne (ge_of_tendsto hh he) (Ne.symm hx) + +theorem real_exit_negative_of_straightened_prefix + (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (g : ℝ → ℂ) {a τ x : ℝ} (haτ : a < τ) (hx : x ≠ 0) + (hprefix : ∀ t ∈ Ico a τ, (h (g t)).re < 0) + (hlim : Tendsto g (𝓝[<] τ) (𝓝 (x : ℂ))) : x < 0 := by + have hh := straightening_preserves_real_exit_limit h hreal g x hlim + have he : ∀ᶠ t in 𝓝[<] τ, (h (g t)).re ≤ 0 := by + filter_upwards [Ioo_mem_nhdsLT haτ] with t ht + exact (hprefix t ⟨ht.1.le, ht.2⟩).le + exact lt_of_le_of_ne (le_of_tendsto hh he) hx + +/-- The scalar straightened side coordinate has a fixed sign until exit +because its zero set is precisely the complete guide. -/ +theorem positive_first_exit_belongs_to_arc + (D : Set ℂ) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) + (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (hg : ContinuousOn g D) (ζ : ℝ → ℂ) (hc : Continuous ζ) + {τ : ℝ} (hτ : UnitWindowFirstExit D ζ τ) + (haτ : (1 / 4 : ℝ) < τ) + (hn : ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (ζ t))).re ≠ 0) + (hstart : 0 < (h (g (ζ (1 / 4)))).re) + (x : ℝ) (hx : x ≠ 0) + (hlim : Tendsto (fun t => g (ζ t)) (𝓝[<] τ) (𝓝 (x : ℂ))) : 0 < x := by + have hj : ContinuousOn (fun z => (h (g z)).re) D := + Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) + have hin : MapsTo ζ (Ico (1 / 4 : ℝ) τ) D := + hτ.2.2.mono_left (Ico_subset_Ico_left (by norm_num)) + exact real_exit_positive_of_straightened_prefix h hreal (fun t => g (ζ t)) haτ hx + (exit_prefix_positive (fun z => (h (g z)).re) hj ζ hc.continuousOn hin hn haτ hstart) + hlim + +theorem negative_first_exit_belongs_to_arc + (D : Set ℂ) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) + (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (hg : ContinuousOn g D) (ζ : ℝ → ℂ) (hc : Continuous ζ) + {τ : ℝ} (hτ : UnitWindowFirstExit D ζ τ) + (haτ : (1 / 4 : ℝ) < τ) + (hn : ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (ζ t))).re ≠ 0) + (hstart : (h (g (ζ (1 / 4)))).re < 0) + (x : ℝ) (hx : x ≠ 0) + (hlim : Tendsto (fun t => g (ζ t)) (𝓝[<] τ) (𝓝 (x : ℂ))) : x < 0 := by + have hj : ContinuousOn (fun z => (h (g z)).re) D := + Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) + have hin : MapsTo ζ (Ico (1 / 4 : ℝ) τ) D := + hτ.2.2.mono_left (Ico_subset_Ico_left (by norm_num)) + exact real_exit_negative_of_straightened_prefix h hreal (fun t => g (ζ t)) haτ hx + (exit_prefix_negative (fun z => (h (g z)).re) hj ζ hc.continuousOn hin hn haτ hstart) + hlim + +end SLEExactGauge.CrosscutSides + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CrosscutParameterInverse.lean b/lean/OAI/Probability/SLE/Geometry/CrosscutParameterInverse.lean index 43b2fc8eaeb54807d6d6e19aa8ab7eb68dc0102a..a9e63cf0782347bc44f10aa307ad91ddce809db3 100644 --- a/lean/OAI/Probability/SLE/Geometry/CrosscutParameterInverse.lean +++ b/lean/OAI/Probability/SLE/Geometry/CrosscutParameterInverse.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Geometry.CompactifiedCrosscut -import OAI.Probability.SLE.Geometry.DiskUpperStraightening -import OAI.Probability.SLE.Geometry.HalfPlaneCayleyInverse - -namespace OAI - -/-! The compactification parameter exactly matches the prescribed vertical ray. -/ -noncomputable section -open Set Complex -open scoped NNReal - -namespace SLEExactGauge - -def rayDiskParameter (t : ℝ≥0) : ℝ := (t:ℝ)/(1+(t:ℝ)) - -lemma rayDiskParameter_mem (t : ℝ≥0) : rayDiskParameter t ∈ Ico 0 1 := by - have hd : 0 < 1+(t:ℝ) := by positivity - refine ⟨div_nonneg t.property hd.le,?_⟩ - exact (div_lt_one hd).mpr (by linarith) - -lemma crosscutParameter_rayDiskParameter (t : ℝ≥0) : - crosscutParameter (rayDiskParameter t)=t := by - apply NNReal.coe_injective - have hs := rayDiskParameter_mem t - have hval : (crosscutParameter (rayDiskParameter t) : ℝ) = - rayDiskParameter t / (1-rayDiskParameter t) := - Real.coe_toNNReal _ (div_nonneg hs.1 (sub_pos.mpr hs.2).le) - rw [hval] - dsimp [rayDiskParameter] - have hd : (1+(t:ℝ)) ≠ 0 := by positivity - field_simp [hd] - ring - -lemma compactifiedCrosscut_rayDiskParameter (β : ℝ≥0 → ℂ) (t : ℝ≥0) : - compactifiedCrosscut β (rayDiskParameter t)=halfPlaneCayley (β t) := by - rw [compactifiedCrosscut_eq β (rayDiskParameter_mem t).2, - crosscutParameter_rayDiskParameter] - -lemma diskDiameter_rayDiskParameter (t : ℝ≥0) : - diskDiameter (rayDiskParameter t)=halfPlaneCayley ((t:ℂ)*I) := by - have hd : (1+(t:ℂ)) ≠ 0 := by - exact_mod_cast (show (1+(t:ℝ)) ≠ 0 by positivity) - have hi : (t:ℂ)*I+I ≠ 0 := by - have hh : (t:ℂ)*I+I=(1+(t:ℂ))*I := by ring - rw [hh] - exact mul_ne_zero hd Complex.I_ne_zero - have hr : 2*((t:ℝ)/(1+(t:ℝ)))-1=((t:ℝ)-1)/(1+(t:ℝ)) := by - have h : (1+(t:ℝ)) ≠ 0 := by positivity - field_simp - ring - unfold diskDiameter rayDiskParameter halfPlaneCayley - rw [hr] - push_cast - apply (div_eq_div_iff hd hi).mpr - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactifiedCrosscut +import OAI.Probability.SLE.Geometry.DiskUpperStraightening +import OAI.Probability.SLE.Geometry.HalfPlaneCayleyInverse + +namespace OAI + +/-! The compactification parameter exactly matches the prescribed vertical ray. -/ +noncomputable section +open Set Complex +open scoped NNReal + +namespace SLEExactGauge + +def rayDiskParameter (t : ℝ≥0) : ℝ := (t:ℝ)/(1+(t:ℝ)) + +lemma rayDiskParameter_mem (t : ℝ≥0) : rayDiskParameter t ∈ Ico 0 1 := by + have hd : 0 < 1+(t:ℝ) := by positivity + refine ⟨div_nonneg t.property hd.le,?_⟩ + exact (div_lt_one hd).mpr (by linarith) + +lemma crosscutParameter_rayDiskParameter (t : ℝ≥0) : + crosscutParameter (rayDiskParameter t)=t := by + apply NNReal.coe_injective + have hs := rayDiskParameter_mem t + have hval : (crosscutParameter (rayDiskParameter t) : ℝ) = + rayDiskParameter t / (1-rayDiskParameter t) := + Real.coe_toNNReal _ (div_nonneg hs.1 (sub_pos.mpr hs.2).le) + rw [hval] + dsimp [rayDiskParameter] + have hd : (1+(t:ℝ)) ≠ 0 := by positivity + field_simp [hd] + ring + +lemma compactifiedCrosscut_rayDiskParameter (β : ℝ≥0 → ℂ) (t : ℝ≥0) : + compactifiedCrosscut β (rayDiskParameter t)=halfPlaneCayley (β t) := by + rw [compactifiedCrosscut_eq β (rayDiskParameter_mem t).2, + crosscutParameter_rayDiskParameter] + +lemma diskDiameter_rayDiskParameter (t : ℝ≥0) : + diskDiameter (rayDiskParameter t)=halfPlaneCayley ((t:ℂ)*I) := by + have hd : (1+(t:ℂ)) ≠ 0 := by + exact_mod_cast (show (1+(t:ℝ)) ≠ 0 by positivity) + have hi : (t:ℂ)*I+I ≠ 0 := by + have hh : (t:ℂ)*I+I=(1+(t:ℂ))*I := by ring + rw [hh] + exact mul_ne_zero hd Complex.I_ne_zero + have hr : 2*((t:ℝ)/(1+(t:ℝ)))-1=((t:ℝ)-1)/(1+(t:ℝ)) := by + have h : (1+(t:ℝ)) ≠ 0 := by positivity + field_simp + ring + unfold diskDiameter rayDiskParameter halfPlaneCayley + rw [hr] + push_cast + apply (div_eq_div_iff hd hi).mpr + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/CurveJoin.lean b/lean/OAI/Probability/SLE/Geometry/CurveJoin.lean index 0030a8a5ace6224bd5dd74dcdb6ffb7999c73fda..4656fac75bdc8394e9fbd53d6c0769465e16e067 100644 --- a/lean/OAI/Probability/SLE/Geometry/CurveJoin.lean +++ b/lean/OAI/Probability/SLE/Geometry/CurveJoin.lean @@ -1,104 +1,104 @@ -import OAI.Probability.SLE.Gauge.NormalRibbonPatch - -namespace OAI - -/-! Joining the straight and circular pieces of the serpentine at matching -positions and tangents. The joins preserve the actual derivative and the -common tangent Lipschitz constant. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge.Serpentine - -def curveJoin (c : ℝ) (f g : ℝ → ℂ) (t : ℝ) : ℂ := if t ≤ c then f t else g t - -@[simp] lemma curveJoin_of_le {c t : ℝ} (h : t ≤ c) (f g : ℝ → ℂ) : - curveJoin c f g t = f t := by simp only [curveJoin, ite_eq_left h] - -@[simp] lemma curveJoin_of_gt {c t : ℝ} (h : c < t) (f g : ℝ → ℂ) : - curveJoin c f g t = g t := by simp only [curveJoin, ite_eq_right (not_le_of_gt h)] - -theorem curveJoin_continuous (c : ℝ) (f g : ℝ → ℂ) (hf : Continuous f) - (hg : Continuous g) (he : f c = g c) : Continuous (curveJoin c f g) := by - classical - change Continuous ((Iic c).piecewise f g) - apply Continuous.piecewise ?_ hf hg - intro t ht - have ht' : t = c := by simpa only [frontier_Iic, mem_singleton_iff] using ht - simpa only [ht'] using he - -lemma curveJoin_lipschitz {c : ℝ} {f g : ℝ → ℂ} {K : ℝ≥0} - (hf : LipschitzWith K f) (hg : LipschitzWith K g) (he : f c = g c) : - LipschitzWith K (curveJoin c f g) := by - apply lipschitzWith_iff_dist_le_mul.mpr - intro u v - by_cases hu : u ≤ c - · by_cases hv : v ≤ c - · simpa only [curveJoin_of_le hu, curveJoin_of_le hv] using hf.dist_le_mul u v - · have hcv : c < v := lt_of_not_ge hv - rw [curveJoin_of_le hu, curveJoin_of_gt hcv] - calc - dist (f u) (g v) ≤ dist (f u) (f c) + dist (g c) (g v) := by - simpa only [he] using dist_triangle (f u) (f c) (g v) - _ ≤ K * dist u c + K * dist c v := add_le_add (hf.dist_le_mul _ _) (hg.dist_le_mul _ _) - _ = K * dist u v := by - rw [Real.dist_eq, Real.dist_eq, Real.dist_eq, - abs_of_nonpos (sub_nonpos.mpr hu), abs_of_neg (sub_neg.mpr hcv), - abs_of_neg (sub_neg.mpr (hu.trans_lt hcv))] - ring - · have hcu : c < u := lt_of_not_ge hu - by_cases hv : v ≤ c - · rw [curveJoin_of_gt hcu, curveJoin_of_le hv] - calc - dist (g u) (f v) ≤ dist (g u) (g c) + dist (f c) (f v) := by - simpa only [he] using dist_triangle (g u) (g c) (f v) - _ ≤ K * dist u c + K * dist c v := add_le_add (hg.dist_le_mul _ _) (hf.dist_le_mul _ _) - _ = K * dist u v := by - rw [Real.dist_eq, Real.dist_eq, Real.dist_eq, - abs_of_pos (sub_pos.mpr hcu), abs_of_nonneg (sub_nonneg.mpr hv), - abs_of_pos (sub_pos.mpr (hv.trans_lt hcu))] - ring - · simpa only [curveJoin_of_gt hcu, curveJoin_of_gt (lt_of_not_ge hv)] using hg.dist_le_mul u v - -theorem curveJoin_hasDerivAt (c : ℝ) (f g df dg : ℝ → ℂ) - (hf : ∀ t, HasDerivAt f (df t) t) (hg : ∀ t, HasDerivAt g (dg t) t) - (he : f c = g c) (hd : df c = dg c) (t : ℝ) : - HasDerivAt (curveJoin c f g) (curveJoin c df dg t) t := by - by_cases hlt : t < c - · have heq : curveJoin c f g =ᶠ[𝓝 t] f := by - filter_upwards [eventually_lt_nhds hlt] with u hu - exact curveJoin_of_le hu.le f g - rw [curveJoin_of_le hlt.le] - exact (hf t).congr_of_eventuallyEq heq - · by_cases hgt : c < t - · have heq : curveJoin c f g =ᶠ[𝓝 t] g := by - filter_upwards [eventually_gt_nhds hgt] with u hu - exact curveJoin_of_gt hu f g - rw [curveJoin_of_gt hgt] - exact (hg t).congr_of_eventuallyEq heq - · have ht : t = c := le_antisymm (le_of_not_gt hgt) (le_of_not_gt hlt) - subst t - have hl : HasDerivWithinAt (curveJoin c f g) (df c) (Iic c) c := by - apply (hf c).hasDerivWithinAt.congr - · intro u hu - exact curveJoin_of_le hu f g - · exact curveJoin_of_le le_rfl f g - have hr : HasDerivWithinAt (curveJoin c f g) (df c) (Ici c) c := by - rw [hd] - apply (hg c).hasDerivWithinAt.congr - · intro u hu - by_cases heq : u = c - · simpa only [heq, curveJoin_of_le le_rfl] using he - · exact curveJoin_of_gt (lt_of_le_of_ne hu (Ne.symm heq)) f g - · simpa only [curveJoin_of_le le_rfl] using he - have hh := hl.union hr - rw [Iic_union_Ici] at hh - simpa only [curveJoin_of_le le_rfl] using hh.hasDerivAt (by simp) - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Gauge.NormalRibbonPatch + +namespace OAI + +/-! Joining the straight and circular pieces of the serpentine at matching +positions and tangents. The joins preserve the actual derivative and the +common tangent Lipschitz constant. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge.Serpentine + +def curveJoin (c : ℝ) (f g : ℝ → ℂ) (t : ℝ) : ℂ := if t ≤ c then f t else g t + +@[simp] lemma curveJoin_of_le {c t : ℝ} (h : t ≤ c) (f g : ℝ → ℂ) : + curveJoin c f g t = f t := by simp only [curveJoin, ite_eq_left h] + +@[simp] lemma curveJoin_of_gt {c t : ℝ} (h : c < t) (f g : ℝ → ℂ) : + curveJoin c f g t = g t := by simp only [curveJoin, ite_eq_right (not_le_of_gt h)] + +theorem curveJoin_continuous (c : ℝ) (f g : ℝ → ℂ) (hf : Continuous f) + (hg : Continuous g) (he : f c = g c) : Continuous (curveJoin c f g) := by + classical + change Continuous ((Iic c).piecewise f g) + apply Continuous.piecewise ?_ hf hg + intro t ht + have ht' : t = c := by simpa only [frontier_Iic, mem_singleton_iff] using ht + simpa only [ht'] using he + +lemma curveJoin_lipschitz {c : ℝ} {f g : ℝ → ℂ} {K : ℝ≥0} + (hf : LipschitzWith K f) (hg : LipschitzWith K g) (he : f c = g c) : + LipschitzWith K (curveJoin c f g) := by + apply lipschitzWith_iff_dist_le_mul.mpr + intro u v + by_cases hu : u ≤ c + · by_cases hv : v ≤ c + · simpa only [curveJoin_of_le hu, curveJoin_of_le hv] using hf.dist_le_mul u v + · have hcv : c < v := lt_of_not_ge hv + rw [curveJoin_of_le hu, curveJoin_of_gt hcv] + calc + dist (f u) (g v) ≤ dist (f u) (f c) + dist (g c) (g v) := by + simpa only [he] using dist_triangle (f u) (f c) (g v) + _ ≤ K * dist u c + K * dist c v := add_le_add (hf.dist_le_mul _ _) (hg.dist_le_mul _ _) + _ = K * dist u v := by + rw [Real.dist_eq, Real.dist_eq, Real.dist_eq, + abs_of_nonpos (sub_nonpos.mpr hu), abs_of_neg (sub_neg.mpr hcv), + abs_of_neg (sub_neg.mpr (hu.trans_lt hcv))] + ring + · have hcu : c < u := lt_of_not_ge hu + by_cases hv : v ≤ c + · rw [curveJoin_of_gt hcu, curveJoin_of_le hv] + calc + dist (g u) (f v) ≤ dist (g u) (g c) + dist (f c) (f v) := by + simpa only [he] using dist_triangle (g u) (g c) (f v) + _ ≤ K * dist u c + K * dist c v := add_le_add (hg.dist_le_mul _ _) (hf.dist_le_mul _ _) + _ = K * dist u v := by + rw [Real.dist_eq, Real.dist_eq, Real.dist_eq, + abs_of_pos (sub_pos.mpr hcu), abs_of_nonneg (sub_nonneg.mpr hv), + abs_of_pos (sub_pos.mpr (hv.trans_lt hcu))] + ring + · simpa only [curveJoin_of_gt hcu, curveJoin_of_gt (lt_of_not_ge hv)] using hg.dist_le_mul u v + +theorem curveJoin_hasDerivAt (c : ℝ) (f g df dg : ℝ → ℂ) + (hf : ∀ t, HasDerivAt f (df t) t) (hg : ∀ t, HasDerivAt g (dg t) t) + (he : f c = g c) (hd : df c = dg c) (t : ℝ) : + HasDerivAt (curveJoin c f g) (curveJoin c df dg t) t := by + by_cases hlt : t < c + · have heq : curveJoin c f g =ᶠ[𝓝 t] f := by + filter_upwards [eventually_lt_nhds hlt] with u hu + exact curveJoin_of_le hu.le f g + rw [curveJoin_of_le hlt.le] + exact (hf t).congr_of_eventuallyEq heq + · by_cases hgt : c < t + · have heq : curveJoin c f g =ᶠ[𝓝 t] g := by + filter_upwards [eventually_gt_nhds hgt] with u hu + exact curveJoin_of_gt hu f g + rw [curveJoin_of_gt hgt] + exact (hg t).congr_of_eventuallyEq heq + · have ht : t = c := le_antisymm (le_of_not_gt hgt) (le_of_not_gt hlt) + subst t + have hl : HasDerivWithinAt (curveJoin c f g) (df c) (Iic c) c := by + apply (hf c).hasDerivWithinAt.congr + · intro u hu + exact curveJoin_of_le hu f g + · exact curveJoin_of_le le_rfl f g + have hr : HasDerivWithinAt (curveJoin c f g) (df c) (Ici c) c := by + rw [hd] + apply (hg c).hasDerivWithinAt.congr + · intro u hu + by_cases heq : u = c + · simpa only [heq, curveJoin_of_le le_rfl] using he + · exact curveJoin_of_gt (lt_of_le_of_ne hu (Ne.symm heq)) f g + · simpa only [curveJoin_of_le le_rfl] using he + have hh := hl.union hr + rw [Iic_union_Ici] at hh + simpa only [curveJoin_of_le le_rfl] using hh.hasDerivAt (by simp) + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiagonalComplexLimit.lean b/lean/OAI/Probability/SLE/Geometry/DiagonalComplexLimit.lean index e25a9a56c2ef2ebf9790d8fd23d3e3b1ee84b935..22a6a1bfc2935fafbae8a97b8d32bb971fe5b3ec 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiagonalComplexLimit.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiagonalComplexLimit.lean @@ -1,31 +1,31 @@ -import OAI.Probability.SLE.Stochastic.DrivingEndpointUniqueness - -namespace OAI - -/-! A diagonal choice for the two limits used at a slit tip. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma diagonal_complex_limit {f : ℕ → ℕ → ℂ} {g : ℕ → ℂ} {b : ℂ} - (hf : ∀ k, Tendsto (fun n => f n k) atTop (𝓝 (g k))) - (hg : Tendsto g atTop (𝓝 b)) : - ∃ φ : ℕ → ℕ, (∀ n, n≤φ n) ∧ Tendsto (fun n => f (φ n) n) atTop (𝓝 b) := by - have hN (k : ℕ) : ∃ N : ℕ, ∀ n≥N, dist (f n k) (g k)<1/((k:ℝ)+1) := by - exact eventually_atTop.mp ((Metric.tendsto_nhds.mp (hf k)) _ (by positivity)) - choose N hN using hN - let φ := fun n => max n (N n) - refine ⟨φ,fun n => le_max_left _ _,Metric.tendsto_nhds.mpr ?_⟩ - intro ε hε - filter_upwards [(Metric.tendsto_nhds.mp hg) (ε/2) (half_pos hε), - tendsto_one_div_add_atTop_nhds_zero_nat.eventually (gt_mem_nhds (half_pos hε))] with n hn he - have hd := hN n (φ n) (le_max_right _ _) - exact (dist_triangle _ (g n) b).trans_lt (by linarith) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.DrivingEndpointUniqueness + +namespace OAI + +/-! A diagonal choice for the two limits used at a slit tip. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma diagonal_complex_limit {f : ℕ → ℕ → ℂ} {g : ℕ → ℂ} {b : ℂ} + (hf : ∀ k, Tendsto (fun n => f n k) atTop (𝓝 (g k))) + (hg : Tendsto g atTop (𝓝 b)) : + ∃ φ : ℕ → ℕ, (∀ n, n≤φ n) ∧ Tendsto (fun n => f (φ n) n) atTop (𝓝 b) := by + have hN (k : ℕ) : ∃ N : ℕ, ∀ n≥N, dist (f n k) (g k)<1/((k:ℝ)+1) := by + exact eventually_atTop.mp ((Metric.tendsto_nhds.mp (hf k)) _ (by positivity)) + choose N hN using hN + let φ := fun n => max n (N n) + refine ⟨φ,fun n => le_max_left _ _,Metric.tendsto_nhds.mpr ?_⟩ + intro ε hε + filter_upwards [(Metric.tendsto_nhds.mp hg) (ε/2) (half_pos hε), + tendsto_one_div_add_atTop_nhds_zero_nat.eventually (gt_mem_nhds (half_pos hε))] with n hn he + have hd := hN n (φ n) (le_max_right _ _) + exact (dist_triangle _ (g n) b).trans_lt (by linarith) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskBoundaryRegion.lean b/lean/OAI/Probability/SLE/Geometry/DiskBoundaryRegion.lean index b7e081a88b32247a7b51bb9a206fe18225e7012e..9de989c0ba86b57efefee10ef403e959ecf72e94 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskBoundaryRegion.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskBoundaryRegion.lean @@ -1,128 +1,128 @@ -import OAI.Probability.SLE.Geometry.HalfDiskGeometry - -namespace OAI - -/-! The full circle and its two half-disk Jordan regions. -/ -noncomputable section -open Set Complex Metric -open scoped ComplexConjugate - -namespace SLEExactGauge - -lemma diskLowerArc_continuous : Continuous diskLowerArc := - Complex.continuous_conj.comp diskUpperArc_continuous - -lemma diskLowerArc_injOn : InjOn diskLowerArc (Icc 0 1) := by - intro s hs t ht he - exact diskUpperArc_injOn hs ht (Complex.conjCLE.injective he) - -lemma diskLowerArc_zero : diskLowerArc 0 = -1 := by - simp [diskLowerArc,diskUpperArc_zero] - -lemma diskLowerArc_one : diskLowerArc 1 = 1 := by - simp [diskLowerArc,diskUpperArc_one] - -theorem disk_circle_jordan : - Schoenflies.IsJordanCurve (complexPlaneCoordinates '' {z : ℂ | ‖z‖=1}) := by - rw [← disk_semicircle_union,image_union] - have hA := complexPlaneCoordinates_isArcBetween diskUpperArc - diskUpperArc_continuous.continuousOn diskUpperArc_injOn - have hB := complexPlaneCoordinates_isArcBetween diskLowerArc - diskLowerArc_continuous.continuousOn diskLowerArc_injOn - rw [diskLowerArc_zero,diskLowerArc_one,← diskUpperArc_zero,← diskUpperArc_one] at hB - apply Schoenflies.isJordanCurve_union hA hB - rintro _ ⟨a,ha,rfl⟩ ⟨b,hb,he⟩ - have hab := complexPlaneCoordinates.injective he - have hz : a ∈ ({-1,1}:Set ℂ) := by - rw [← disk_semicircle_inter] - exact ⟨ha,hab ▸ hb⟩ - rcases hz with hz|hz - · left - rw [hz,diskUpperArc_zero] - · right - rw [hz,diskUpperArc_one] - -theorem complexInside_unitDisk : - complexInside {z : ℂ | ‖z‖=1} = ball (0:ℂ) 1 := by - apply complexInside_eq_open disk_circle_jordan isOpen_ball - (convex_ball _ _).isPreconnected (nonempty_ball.mpr (by norm_num)) - isBounded_ball - rw [frontier_ball _ (by norm_num : (1:ℝ)≠0)] - ext z - simp - -def openLowerDisk : Set ℂ := {z | ‖z‖ < 1 ∧ z.im < 0} -def closedLowerDisk : Set ℂ := {z | ‖z‖ ≤ 1 ∧ z.im ≤ 0} - -lemma conj_image_openUpperDisk : conj '' openUpperDisk=openLowerDisk := by - ext z - constructor - · rintro ⟨w,hw,rfl⟩ - simpa [openUpperDisk,openLowerDisk] using hw - · intro hz - refine ⟨conj z,?_,Complex.conj_conj z⟩ - simpa [openUpperDisk,openLowerDisk] using hz - -lemma conj_image_closedUpperDisk : conj '' closedUpperDisk=closedLowerDisk := by - ext z - constructor - · rintro ⟨w,hw,rfl⟩ - simpa [closedUpperDisk,closedLowerDisk] using hw - · intro hz - refine ⟨conj z,?_,Complex.conj_conj z⟩ - simpa [closedUpperDisk,closedLowerDisk] using hz - -lemma openLowerDisk_closure : closure openLowerDisk=closedLowerDisk := by - have hh := Complex.conjCLE.toHomeomorph.image_closure openUpperDisk - change conj '' closure openUpperDisk=closure (conj '' openUpperDisk) at hh - rw [openUpperDisk_closure,conj_image_openUpperDisk,conj_image_closedUpperDisk] at hh - exact hh.symm - -lemma conj_image_diskDiameter : conj '' (diskDiameter '' Icc 0 1)=diskDiameter '' Icc 0 1 := by - ext z - constructor - · rintro ⟨_,⟨s,hs,rfl⟩,rfl⟩ - simpa only [diskDiameter,Complex.conj_ofReal] using mem_image_of_mem diskDiameter hs - · rintro ⟨s,hs,rfl⟩ - refine ⟨diskDiameter s,mem_image_of_mem _ hs,?_⟩ - exact Complex.conj_ofReal _ - -theorem complexInside_lowerDisk : - complexInside (diskDiameter '' Icc 0 1 ∪ diskLowerArc '' Icc 0 1)=openLowerDisk := by - have he : conj '' (diskUpperArc '' Icc 0 1)=diskLowerArc '' Icc 0 1 := by - rw [image_image] - rfl - have hh := complexInside_transport Complex.conjCLE.toHomeomorph - (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1) - change conj '' complexInside (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)= - complexInside (conj '' (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)) at hh - rw [image_union,conj_image_diskDiameter,he,complexInside_upperDisk, - conj_image_openUpperDisk] at hh - exact hh.symm - -lemma closed_halfDisks_union : closedUpperDisk ∪ closedLowerDisk=closedBall (0:ℂ) 1 := by - ext z - simp only [mem_union,closedUpperDisk,closedLowerDisk,mem_ofPred,mem_closedBall,dist_zero_right] - constructor - · rintro (h|h) <;> exact h.1 - · intro hz - rcases le_total 0 z.im with hi|hi - · exact Or.inl ⟨hz,hi⟩ - · exact Or.inr ⟨hz,hi⟩ - -lemma closed_halfDisks_inter : - closedUpperDisk ∩ closedLowerDisk=diskDiameter '' Icc 0 1 := by - rw [diskDiameter_image] - ext z - simp only [closedUpperDisk,closedLowerDisk,mem_inter_iff,mem_ofPred] - constructor - · rintro ⟨hp,hm⟩ - exact ⟨hp.1,le_antisymm hm.2 hp.2⟩ - · rintro ⟨hn,hi⟩ - exact ⟨⟨hn,hi.ge⟩,⟨hn,hi.le⟩⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfDiskGeometry + +namespace OAI + +/-! The full circle and its two half-disk Jordan regions. -/ +noncomputable section +open Set Complex Metric +open scoped ComplexConjugate + +namespace SLEExactGauge + +lemma diskLowerArc_continuous : Continuous diskLowerArc := + Complex.continuous_conj.comp diskUpperArc_continuous + +lemma diskLowerArc_injOn : InjOn diskLowerArc (Icc 0 1) := by + intro s hs t ht he + exact diskUpperArc_injOn hs ht (Complex.conjCLE.injective he) + +lemma diskLowerArc_zero : diskLowerArc 0 = -1 := by + simp [diskLowerArc,diskUpperArc_zero] + +lemma diskLowerArc_one : diskLowerArc 1 = 1 := by + simp [diskLowerArc,diskUpperArc_one] + +theorem disk_circle_jordan : + Schoenflies.IsJordanCurve (complexPlaneCoordinates '' {z : ℂ | ‖z‖=1}) := by + rw [← disk_semicircle_union,image_union] + have hA := complexPlaneCoordinates_isArcBetween diskUpperArc + diskUpperArc_continuous.continuousOn diskUpperArc_injOn + have hB := complexPlaneCoordinates_isArcBetween diskLowerArc + diskLowerArc_continuous.continuousOn diskLowerArc_injOn + rw [diskLowerArc_zero,diskLowerArc_one,← diskUpperArc_zero,← diskUpperArc_one] at hB + apply Schoenflies.isJordanCurve_union hA hB + rintro _ ⟨a,ha,rfl⟩ ⟨b,hb,he⟩ + have hab := complexPlaneCoordinates.injective he + have hz : a ∈ ({-1,1}:Set ℂ) := by + rw [← disk_semicircle_inter] + exact ⟨ha,hab ▸ hb⟩ + rcases hz with hz|hz + · left + rw [hz,diskUpperArc_zero] + · right + rw [hz,diskUpperArc_one] + +theorem complexInside_unitDisk : + complexInside {z : ℂ | ‖z‖=1} = ball (0:ℂ) 1 := by + apply complexInside_eq_open disk_circle_jordan isOpen_ball + (convex_ball _ _).isPreconnected (nonempty_ball.mpr (by norm_num)) + isBounded_ball + rw [frontier_ball _ (by norm_num : (1:ℝ)≠0)] + ext z + simp + +def openLowerDisk : Set ℂ := {z | ‖z‖ < 1 ∧ z.im < 0} +def closedLowerDisk : Set ℂ := {z | ‖z‖ ≤ 1 ∧ z.im ≤ 0} + +lemma conj_image_openUpperDisk : conj '' openUpperDisk=openLowerDisk := by + ext z + constructor + · rintro ⟨w,hw,rfl⟩ + simpa [openUpperDisk,openLowerDisk] using hw + · intro hz + refine ⟨conj z,?_,Complex.conj_conj z⟩ + simpa [openUpperDisk,openLowerDisk] using hz + +lemma conj_image_closedUpperDisk : conj '' closedUpperDisk=closedLowerDisk := by + ext z + constructor + · rintro ⟨w,hw,rfl⟩ + simpa [closedUpperDisk,closedLowerDisk] using hw + · intro hz + refine ⟨conj z,?_,Complex.conj_conj z⟩ + simpa [closedUpperDisk,closedLowerDisk] using hz + +lemma openLowerDisk_closure : closure openLowerDisk=closedLowerDisk := by + have hh := Complex.conjCLE.toHomeomorph.image_closure openUpperDisk + change conj '' closure openUpperDisk=closure (conj '' openUpperDisk) at hh + rw [openUpperDisk_closure,conj_image_openUpperDisk,conj_image_closedUpperDisk] at hh + exact hh.symm + +lemma conj_image_diskDiameter : conj '' (diskDiameter '' Icc 0 1)=diskDiameter '' Icc 0 1 := by + ext z + constructor + · rintro ⟨_,⟨s,hs,rfl⟩,rfl⟩ + simpa only [diskDiameter,Complex.conj_ofReal] using mem_image_of_mem diskDiameter hs + · rintro ⟨s,hs,rfl⟩ + refine ⟨diskDiameter s,mem_image_of_mem _ hs,?_⟩ + exact Complex.conj_ofReal _ + +theorem complexInside_lowerDisk : + complexInside (diskDiameter '' Icc 0 1 ∪ diskLowerArc '' Icc 0 1)=openLowerDisk := by + have he : conj '' (diskUpperArc '' Icc 0 1)=diskLowerArc '' Icc 0 1 := by + rw [image_image] + rfl + have hh := complexInside_transport Complex.conjCLE.toHomeomorph + (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1) + change conj '' complexInside (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)= + complexInside (conj '' (diskDiameter '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)) at hh + rw [image_union,conj_image_diskDiameter,he,complexInside_upperDisk, + conj_image_openUpperDisk] at hh + exact hh.symm + +lemma closed_halfDisks_union : closedUpperDisk ∪ closedLowerDisk=closedBall (0:ℂ) 1 := by + ext z + simp only [mem_union,closedUpperDisk,closedLowerDisk,mem_ofPred,mem_closedBall,dist_zero_right] + constructor + · rintro (h|h) <;> exact h.1 + · intro hz + rcases le_total 0 z.im with hi|hi + · exact Or.inl ⟨hz,hi⟩ + · exact Or.inr ⟨hz,hi⟩ + +lemma closed_halfDisks_inter : + closedUpperDisk ∩ closedLowerDisk=diskDiameter '' Icc 0 1 := by + rw [diskDiameter_image] + ext z + simp only [closedUpperDisk,closedLowerDisk,mem_inter_iff,mem_ofPred] + constructor + · rintro ⟨hp,hm⟩ + exact ⟨hp.1,le_antisymm hm.2 hp.2⟩ + · rintro ⟨hn,hi⟩ + exact ⟨⟨hn,hi.ge⟩,⟨hn,hi.le⟩⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskCrosscutBoundary.lean b/lean/OAI/Probability/SLE/Geometry/DiskCrosscutBoundary.lean index d30c014ed3cbaa2608586dac47c46b574e6d9051..87f4f79727547a91a307a9ddc42a1ce9e70ee6f6 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskCrosscutBoundary.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskCrosscutBoundary.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Geometry.ComplexJordanExtension -import OAI.Probability.SLE.Geometry.ArcBoundaryReplacement - -namespace OAI - -/-! The upper semicircle supplies the fixed boundary arc for the first -relative Schoenflies extension of a compactified half-plane crosscut. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -def diskUpperArc (s : ℝ) : ℂ := - (-Real.cos (Real.pi*s) : ℂ)+(Real.sin (Real.pi*s) : ℂ)*I - -lemma diskUpperArc_continuous : Continuous diskUpperArc := by - unfold diskUpperArc - fun_prop - -lemma diskUpperArc_zero : diskUpperArc 0 = -1 := by simp [diskUpperArc] -lemma diskUpperArc_one : diskUpperArc 1 = 1 := by simp [diskUpperArc] - -lemma diskUpperArc_re (s : ℝ) : (diskUpperArc s).re = -Real.cos (Real.pi*s) := by - simp only [diskUpperArc,Complex.add_re,Complex.neg_re,Complex.ofReal_re, - Complex.mul_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,zero_mul, - sub_zero,add_zero] - -lemma diskUpperArc_im (s : ℝ) : (diskUpperArc s).im = Real.sin (Real.pi*s) := by - simp only [diskUpperArc,Complex.add_im,Complex.neg_im,Complex.ofReal_im,neg_zero, - Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero, - add_zero,zero_add] - -lemma diskUpperArc_norm (s : ℝ) : ‖diskUpperArc s‖=1 := by - have hn : Complex.normSq (diskUpperArc s)=1 := by - rw [Complex.normSq_apply,diskUpperArc_re,diskUpperArc_im] - nlinarith [Real.sin_sq_add_cos_sq (Real.pi*s)] - rw [Complex.normSq_eq_norm_sq] at hn - nlinarith [norm_nonneg (diskUpperArc s)] - -lemma diskUpperArc_injOn : InjOn diskUpperArc (Icc 0 1) := by - intro s hs t ht he - have hr : -Real.cos (Real.pi*s) = -Real.cos (Real.pi*t) := by - simpa only [diskUpperArc_re] using congrArg Complex.re he - have heq : Real.pi*s=Real.pi*t := Real.strictAntiOn_cos.injOn - ⟨mul_nonneg Real.pi_pos.le hs.1,mul_le_of_le_one_right Real.pi_pos.le hs.2⟩ - ⟨mul_nonneg Real.pi_pos.le ht.1,mul_le_of_le_one_right Real.pi_pos.le ht.2⟩ (by linarith) - exact mul_left_cancel₀ Real.pi_ne_zero heq - -lemma disk_crosscut_meets_upper_at_endpoints - (p : ℝ → ℂ) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) - (s : ℝ) (hs : s ∈ Icc 0 1) (hmem : p s ∈ diskUpperArc '' Icc 0 1) : - s=0 ∨ s=1 := by - by_contra hn - have hs0 : 0 < s := lt_of_le_of_ne hs.1 (by intro he; exact hn (Or.inl he.symm)) - have hs1 : s < 1 := lt_of_le_of_ne hs.2 (by intro he; exact hn (Or.inr he)) - obtain ⟨t,_ht,he⟩ := hmem - have hh := hp s ⟨hs0,hs1⟩ - rw [← he,diskUpperArc_norm] at hh - exact lt_irrefl _ hh - -theorem disk_crosscut_upper_jordan - (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : - Schoenflies.IsJordanCurve - (complexPlaneCoordinates '' (p '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)) := by - have hpArc := complexPlaneCoordinates_isArcBetween p hc hi - have haArc := complexPlaneCoordinates_isArcBetween diskUpperArc - diskUpperArc_continuous.continuousOn diskUpperArc_injOn - rw [diskUpperArc_zero,diskUpperArc_one,← h0,← h1] at haArc - rw [image_union] - apply Schoenflies.isJordanCurve_union hpArc haArc - rintro _ ⟨_,⟨s,hs,rfl⟩,rfl⟩ ⟨_,⟨t,ht,rfl⟩,he⟩ - have het : diskUpperArc t=p s := complexPlaneCoordinates.injective he - rcases disk_crosscut_meets_upper_at_endpoints p hp s hs ⟨t,ht,het⟩ with rfl | rfl - · exact Or.inl rfl - · exact Or.inr rfl - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexJordanExtension +import OAI.Probability.SLE.Geometry.ArcBoundaryReplacement + +namespace OAI + +/-! The upper semicircle supplies the fixed boundary arc for the first +relative Schoenflies extension of a compactified half-plane crosscut. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +def diskUpperArc (s : ℝ) : ℂ := + (-Real.cos (Real.pi*s) : ℂ)+(Real.sin (Real.pi*s) : ℂ)*I + +lemma diskUpperArc_continuous : Continuous diskUpperArc := by + unfold diskUpperArc + fun_prop + +lemma diskUpperArc_zero : diskUpperArc 0 = -1 := by simp [diskUpperArc] +lemma diskUpperArc_one : diskUpperArc 1 = 1 := by simp [diskUpperArc] + +lemma diskUpperArc_re (s : ℝ) : (diskUpperArc s).re = -Real.cos (Real.pi*s) := by + simp only [diskUpperArc,Complex.add_re,Complex.neg_re,Complex.ofReal_re, + Complex.mul_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,zero_mul, + sub_zero,add_zero] + +lemma diskUpperArc_im (s : ℝ) : (diskUpperArc s).im = Real.sin (Real.pi*s) := by + simp only [diskUpperArc,Complex.add_im,Complex.neg_im,Complex.ofReal_im,neg_zero, + Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero, + add_zero,zero_add] + +lemma diskUpperArc_norm (s : ℝ) : ‖diskUpperArc s‖=1 := by + have hn : Complex.normSq (diskUpperArc s)=1 := by + rw [Complex.normSq_apply,diskUpperArc_re,diskUpperArc_im] + nlinarith [Real.sin_sq_add_cos_sq (Real.pi*s)] + rw [Complex.normSq_eq_norm_sq] at hn + nlinarith [norm_nonneg (diskUpperArc s)] + +lemma diskUpperArc_injOn : InjOn diskUpperArc (Icc 0 1) := by + intro s hs t ht he + have hr : -Real.cos (Real.pi*s) = -Real.cos (Real.pi*t) := by + simpa only [diskUpperArc_re] using congrArg Complex.re he + have heq : Real.pi*s=Real.pi*t := Real.strictAntiOn_cos.injOn + ⟨mul_nonneg Real.pi_pos.le hs.1,mul_le_of_le_one_right Real.pi_pos.le hs.2⟩ + ⟨mul_nonneg Real.pi_pos.le ht.1,mul_le_of_le_one_right Real.pi_pos.le ht.2⟩ (by linarith) + exact mul_left_cancel₀ Real.pi_ne_zero heq + +lemma disk_crosscut_meets_upper_at_endpoints + (p : ℝ → ℂ) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) + (s : ℝ) (hs : s ∈ Icc 0 1) (hmem : p s ∈ diskUpperArc '' Icc 0 1) : + s=0 ∨ s=1 := by + by_contra hn + have hs0 : 0 < s := lt_of_le_of_ne hs.1 (by intro he; exact hn (Or.inl he.symm)) + have hs1 : s < 1 := lt_of_le_of_ne hs.2 (by intro he; exact hn (Or.inr he)) + obtain ⟨t,_ht,he⟩ := hmem + have hh := hp s ⟨hs0,hs1⟩ + rw [← he,diskUpperArc_norm] at hh + exact lt_irrefl _ hh + +theorem disk_crosscut_upper_jordan + (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : + Schoenflies.IsJordanCurve + (complexPlaneCoordinates '' (p '' Icc 0 1 ∪ diskUpperArc '' Icc 0 1)) := by + have hpArc := complexPlaneCoordinates_isArcBetween p hc hi + have haArc := complexPlaneCoordinates_isArcBetween diskUpperArc + diskUpperArc_continuous.continuousOn diskUpperArc_injOn + rw [diskUpperArc_zero,diskUpperArc_one,← h0,← h1] at haArc + rw [image_union] + apply Schoenflies.isJordanCurve_union hpArc haArc + rintro _ ⟨_,⟨s,hs,rfl⟩,rfl⟩ ⟨_,⟨t,ht,rfl⟩,he⟩ + have het : diskUpperArc t=p s := complexPlaneCoordinates.injective he + rcases disk_crosscut_meets_upper_at_endpoints p hp s hs ⟨t,ht,het⟩ with rfl | rfl + · exact Or.inl rfl + · exact Or.inr rfl + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskCrosscutClosedRegions.lean b/lean/OAI/Probability/SLE/Geometry/DiskCrosscutClosedRegions.lean index 76abab8e76638a7c857d8867b8fecb4423f98698..7419103739d5a76f1c9e299e90e944e6c02a9a74 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskCrosscutClosedRegions.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskCrosscutClosedRegions.lean @@ -1,140 +1,140 @@ -import OAI.Probability.SLE.Geometry.DiskCrosscutPartition - -namespace OAI - -/-! The closures of the two disk crosscut regions cover the closed disk -and meet exactly along the crosscut. -/ -noncomputable section -open Set Complex Metric -open scoped ComplexConjugate - -namespace SLEExactGauge - -theorem disk_crosscut_lower_jordan - (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : - Schoenflies.IsJordanCurve - (complexPlaneCoordinates '' (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1)) := by - have hpArc := complexPlaneCoordinates_isArcBetween p hc hi - have haArc := complexPlaneCoordinates_isArcBetween diskLowerArc - diskLowerArc_continuous.continuousOn diskLowerArc_injOn - rw [diskLowerArc_zero,diskLowerArc_one,← h0,← h1] at haArc - rw [image_union] - apply Schoenflies.isJordanCurve_union haArc hpArc - rintro _ ⟨_,⟨t,ht,rfl⟩,rfl⟩ ⟨_,⟨s,hs,rfl⟩,he⟩ - have heq : p s=diskLowerArc t := complexPlaneCoordinates.injective he - have hend : s=0 ∨ s=1 := by - by_contra hn - have hs0 : 0 c '' S) disk_semicircle_union - · rw [← image_inter c.injective] - rw [disk_semicircle_inter,image_pair] - have hPin : c '' P \ {c (-1),c 1} ⊆ Schoenflies.inside (c '' C) := by - rintro z ⟨⟨_,⟨s,hs,rfl⟩,rfl⟩,hne⟩ - have hs0 : s ≠ 0 := by - intro he - apply hne - simp [he,h0] - have hs1 : s ≠ 1 := by - intro he - apply hne - simp [he,h1] - have hball : p s ∈ ball (0:ℂ) 1 := by - simpa only [mem_ball,dist_zero_right] using - hp s ⟨lt_of_le_of_ne hs.1 (Ne.symm hs0),lt_of_le_of_ne hs.2 hs1⟩ - have hin : p s ∈ complexInside C := by rwa [complexInside_unitDisk] - exact hin - have hP : Schoenflies.IsArcBetween (c '' P) (c (-1)) (c 1) := by - simpa only [h0,h1] using complexPlaneCoordinates_isArcBetween p hc hi - obtain ⟨he,hd⟩ := plane_crosscut_partition_of_straightening disk_circle_jordan hP hcut - (mem_image_of_mem c (by simp)) (mem_image_of_mem c (by simp)) hPin G hpoly - dsimp [c,C,P,A,B] at he hd - refine ⟨?_,?_⟩ - · rw [← complexInside_unitDisk] - apply complexPlaneCoordinates.injective.image_injective - simpa only [image_sdiff complexPlaneCoordinates.injective,complexInside_image,image_union] using he - · rw [Set.disjoint_left] at hd ⊢ - intro z hz hz' - apply hd (a := complexPlaneCoordinates z) - · have hh : complexPlaneCoordinates z ∈ Schoenflies.inside - (complexPlaneCoordinates '' (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1)) := hz - simpa only [image_union] using hh - · have hh : complexPlaneCoordinates z ∈ Schoenflies.inside - (complexPlaneCoordinates '' (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1)) := hz' - simpa only [image_union] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskBoundaryRegion +import OAI.Probability.SLE.Geometry.TransportCrosscutPartition + +namespace OAI + +/-! The two bounded regions of a compact disk crosscut exhaust its complement. -/ +noncomputable section +open Set Complex Metric + +namespace SLEExactGauge + +lemma diskDiameter_plane_segment : + complexPlaneCoordinates '' (diskDiameter '' Icc 0 1)= + segment ℝ (complexPlaneCoordinates (-1)) (complexPlaneCoordinates 1) := by + rw [image_image,segment_eq_image_lineMap] + congr 1 + funext s + rw [AffineMap.lineMap_apply_module] + change !₂[(diskDiameter s).re,(diskDiameter s).im]= + (1-s) • !₂[(-1:ℂ).re,(-1:ℂ).im] + s • !₂[(1:ℂ).re,(1:ℂ).im] + ext i + fin_cases i + · norm_num [diskDiameter] + ring + · norm_num [diskDiameter] + +theorem disk_crosscut_partition + (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : + ball (0:ℂ) 1 \ (p '' Icc 0 1) = + complexInside (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1) ∪ + complexInside (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1) ∧ + Disjoint (complexInside (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1)) + (complexInside (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1)) := by + let c := complexPlaneCoordinates + let C : Set ℂ := {z | ‖z‖=1} + let P := p '' Icc 0 1 + let A := diskUpperArc '' Icc 0 1 + let B := diskLowerArc '' Icc 0 1 + obtain ⟨H,hHp,hHA⟩ := disk_crosscut_upper_straightening p hc hi h0 h1 hp + have hHP : H '' P=diskDiameter '' Icc 0 1 := by + ext z + constructor + · rintro ⟨_,⟨s,hs,rfl⟩,rfl⟩ + exact ⟨s,hs,(hHp s hs).symm⟩ + · rintro ⟨s,hs,rfl⟩ + exact ⟨p s,mem_image_of_mem p hs,hHp s hs⟩ + let G : Schoenflies.Plane ≃ₜ Schoenflies.Plane := (c.symm.trans H).trans c + have hGP : G '' (c '' P)=c '' (H '' P) := by + ext z + simp [G] + have hpoly : Schoenflies.IsPolygonal (G '' (c '' P)) := by + rw [hGP,hHP,diskDiameter_plane_segment] + exact Schoenflies.isPolygonal_segment _ _ + have hcut : Schoenflies.IsCutPair (c '' C) (c (-1)) (c 1) (c '' A) (c '' B) := by + refine ⟨?_,?_,?_,?_⟩ + · simpa only [diskUpperArc_zero,diskUpperArc_one] using + complexPlaneCoordinates_isArcBetween diskUpperArc + diskUpperArc_continuous.continuousOn diskUpperArc_injOn + · simpa only [diskLowerArc_zero,diskLowerArc_one] using + complexPlaneCoordinates_isArcBetween diskLowerArc + diskLowerArc_continuous.continuousOn diskLowerArc_injOn + · rw [← image_union] + exact congrArg (fun S => c '' S) disk_semicircle_union + · rw [← image_inter c.injective] + rw [disk_semicircle_inter,image_pair] + have hPin : c '' P \ {c (-1),c 1} ⊆ Schoenflies.inside (c '' C) := by + rintro z ⟨⟨_,⟨s,hs,rfl⟩,rfl⟩,hne⟩ + have hs0 : s ≠ 0 := by + intro he + apply hne + simp [he,h0] + have hs1 : s ≠ 1 := by + intro he + apply hne + simp [he,h1] + have hball : p s ∈ ball (0:ℂ) 1 := by + simpa only [mem_ball,dist_zero_right] using + hp s ⟨lt_of_le_of_ne hs.1 (Ne.symm hs0),lt_of_le_of_ne hs.2 hs1⟩ + have hin : p s ∈ complexInside C := by rwa [complexInside_unitDisk] + exact hin + have hP : Schoenflies.IsArcBetween (c '' P) (c (-1)) (c 1) := by + simpa only [h0,h1] using complexPlaneCoordinates_isArcBetween p hc hi + obtain ⟨he,hd⟩ := plane_crosscut_partition_of_straightening disk_circle_jordan hP hcut + (mem_image_of_mem c (by simp)) (mem_image_of_mem c (by simp)) hPin G hpoly + dsimp [c,C,P,A,B] at he hd + refine ⟨?_,?_⟩ + · rw [← complexInside_unitDisk] + apply complexPlaneCoordinates.injective.image_injective + simpa only [image_sdiff complexPlaneCoordinates.injective,complexInside_image,image_union] using he + · rw [Set.disjoint_left] at hd ⊢ + intro z hz hz' + apply hd (a := complexPlaneCoordinates z) + · have hh : complexPlaneCoordinates z ∈ Schoenflies.inside + (complexPlaneCoordinates '' (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1)) := hz + simpa only [image_union] using hh + · have hh : complexPlaneCoordinates z ∈ Schoenflies.inside + (complexPlaneCoordinates '' (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1)) := hz' + simpa only [image_union] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskGrowthFromSecondCoefficient.lean b/lean/OAI/Probability/SLE/Geometry/DiskGrowthFromSecondCoefficient.lean index e092c626d4fd5d22c04df9bdba54e7f4dc0ed6dc..442e5f0c618d7e142548ab60ab178ef0fe32eb8d 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskGrowthFromSecondCoefficient.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskGrowthFromSecondCoefficient.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Loewner.UnivalentRadialDerivative -import OAI.Probability.SLE.Multipoint.MultipointGeometry - -namespace OAI - -/-! Integrating the sharp second-coefficient estimate gives the growth -bounds for normalized univalent disk maps. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -theorem diskGrowth_of_secondDerivative - (hsecond : ∀ f : ℂ → ℂ, DifferentiableOn ℂ f (ball 0 1) → - InjOn f (ball 0 1) → ‖deriv (deriv f) 0‖≤4*‖deriv f 0‖) : - PublishedDiskUnivalentGrowth := by - intro f hf hi z hz - by_cases hz0 : z=0 - · simp [hz0] - let r := ‖z‖ - have hrp : 0 - univalent_preSchwarzian_of_second_derivative hsecond hf hi (a := a) ha - let v (t : ℝ) := f ((t:ℂ)*u) - let v' (t : ℝ) := u*deriv f ((t:ℂ)*u) - let g (t : ℝ) := ‖deriv f 0‖*(1+t)/(1-t)^3 - let G (t : ℝ) := ‖deriv f 0‖*t/(1-t)^2 - have hat (t : ℝ) (ht : t∈Icc 0 r) : (t:ℂ)*u∈ball 0 1 := by - rw [mem_ball_zero_iff,norm_mul,hu,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] - exact ht.2.trans_lt hr1 - have hv' (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt v (v' t) t := by - have hd := ((hf.differentiableAt (isOpen_ball.mem_nhds (hat t ht))).hasDerivAt.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t - ((Complex.ofRealCLM.hasDerivAt (x := t)).mul_const u) - simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, - smul_eq_mul,Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul,Function.comp_def,v,v'] using hd - have hv'c : ContinuousOn v' (Icc 0 r) := by - apply continuousOn_const.mul - intro t ht - have houter : ContinuousAt (deriv f) ((t:ℂ)*u) := - (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv.continuousAt - exact (ContinuousAt.comp (g := deriv f) (f := fun s : ℝ => (s:ℂ)*u) - houter (by fun_prop)).continuousWithinAt - have hgc : ContinuousOn g (Icc 0 r) := by - apply ContinuousOn.div (by fun_prop) (by fun_prop) - intro t ht - exact pow_ne_zero _ (by linarith [ht.2,hr1]) - have hG (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt G (g t) t := by - have hm : 1-t≠0 := by linarith [ht.2,hr1] - have h := ((hasDerivAt_id t).const_mul ‖deriv f 0‖).div - (((hasDerivAt_id t).const_sub 1).pow 2) (pow_ne_zero _ hm) - change HasDerivAt G _ t at h - convert h using 1 - dsimp [g] - field_simp - ring - have hvint : ∫ t in (0:ℝ)..r, v' t=f z-f 0 := by - have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) - (fun t ht => hv' t (by simpa only [uIcc_of_le hrp.le] using ht)) - (hv'c.intervalIntegrable_of_Icc hrp.le) - simpa only [v,hru,Complex.ofReal_zero,zero_mul] using h - have hgint : ∫ t in (0:ℝ)..r, g t=G r := by - have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) - (fun t ht => hG t (by simpa only [uIcc_of_le hrp.le] using ht)) - (hgc.intervalIntegrable_of_Icc hrp.le) - simpa [G] using h - calc - ‖f z-f 0‖ = ‖∫ t in (0:ℝ)..r, v' t‖ := congrArg norm hvint.symm - _ ≤ ∫ t in (0:ℝ)..r, ‖v' t‖ := intervalIntegral.norm_integral_le_integral_norm hrp.le - _ ≤ ∫ t in (0:ℝ)..r, g t := by - apply intervalIntegral.integral_mono_on hrp.le - (hv'c.norm.intervalIntegrable_of_Icc hrp.le) (hgc.intervalIntegrable_of_Icc hrp.le) - intro t ht - dsimp only [v'] - rw [norm_mul,hu,one_mul] - exact univalent_radial_derivative_bound hf hi hpre u hu t ⟨ht.1,ht.2.trans_lt hr1⟩ - _ = ‖deriv f 0‖*‖z‖/(1-‖z‖)^2 := hgint - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentRadialDerivative +import OAI.Probability.SLE.Multipoint.MultipointGeometry + +namespace OAI + +/-! Integrating the sharp second-coefficient estimate gives the growth +bounds for normalized univalent disk maps. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +theorem diskGrowth_of_secondDerivative + (hsecond : ∀ f : ℂ → ℂ, DifferentiableOn ℂ f (ball 0 1) → + InjOn f (ball 0 1) → ‖deriv (deriv f) 0‖≤4*‖deriv f 0‖) : + PublishedDiskUnivalentGrowth := by + intro f hf hi z hz + by_cases hz0 : z=0 + · simp [hz0] + let r := ‖z‖ + have hrp : 0 + univalent_preSchwarzian_of_second_derivative hsecond hf hi (a := a) ha + let v (t : ℝ) := f ((t:ℂ)*u) + let v' (t : ℝ) := u*deriv f ((t:ℂ)*u) + let g (t : ℝ) := ‖deriv f 0‖*(1+t)/(1-t)^3 + let G (t : ℝ) := ‖deriv f 0‖*t/(1-t)^2 + have hat (t : ℝ) (ht : t∈Icc 0 r) : (t:ℂ)*u∈ball 0 1 := by + rw [mem_ball_zero_iff,norm_mul,hu,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] + exact ht.2.trans_lt hr1 + have hv' (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt v (v' t) t := by + have hd := ((hf.differentiableAt (isOpen_ball.mem_nhds (hat t ht))).hasDerivAt.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t + ((Complex.ofRealCLM.hasDerivAt (x := t)).mul_const u) + simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, + smul_eq_mul,Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul,Function.comp_def,v,v'] using hd + have hv'c : ContinuousOn v' (Icc 0 r) := by + apply continuousOn_const.mul + intro t ht + have houter : ContinuousAt (deriv f) ((t:ℂ)*u) := + (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv.continuousAt + exact (ContinuousAt.comp (g := deriv f) (f := fun s : ℝ => (s:ℂ)*u) + houter (by fun_prop)).continuousWithinAt + have hgc : ContinuousOn g (Icc 0 r) := by + apply ContinuousOn.div (by fun_prop) (by fun_prop) + intro t ht + exact pow_ne_zero _ (by linarith [ht.2,hr1]) + have hG (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt G (g t) t := by + have hm : 1-t≠0 := by linarith [ht.2,hr1] + have h := ((hasDerivAt_id t).const_mul ‖deriv f 0‖).div + (((hasDerivAt_id t).const_sub 1).pow 2) (pow_ne_zero _ hm) + change HasDerivAt G _ t at h + convert h using 1 + dsimp [g] + field_simp + ring + have hvint : ∫ t in (0:ℝ)..r, v' t=f z-f 0 := by + have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) + (fun t ht => hv' t (by simpa only [uIcc_of_le hrp.le] using ht)) + (hv'c.intervalIntegrable_of_Icc hrp.le) + simpa only [v,hru,Complex.ofReal_zero,zero_mul] using h + have hgint : ∫ t in (0:ℝ)..r, g t=G r := by + have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) + (fun t ht => hG t (by simpa only [uIcc_of_le hrp.le] using ht)) + (hgc.intervalIntegrable_of_Icc hrp.le) + simpa [G] using h + calc + ‖f z-f 0‖ = ‖∫ t in (0:ℝ)..r, v' t‖ := congrArg norm hvint.symm + _ ≤ ∫ t in (0:ℝ)..r, ‖v' t‖ := intervalIntegral.norm_integral_le_integral_norm hrp.le + _ ≤ ∫ t in (0:ℝ)..r, g t := by + apply intervalIntegral.integral_mono_on hrp.le + (hv'c.norm.intervalIntegrable_of_Icc hrp.le) (hgc.intervalIntegrable_of_Icc hrp.le) + intro t ht + dsimp only [v'] + rw [norm_mul,hu,one_mul] + exact univalent_radial_derivative_bound hf hi hpre u hu t ⟨ht.1,ht.2.trans_lt hr1⟩ + _ = ‖deriv f 0‖*‖z‖/(1-‖z‖)^2 := hgint + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskKernelRemainder.lean b/lean/OAI/Probability/SLE/Geometry/DiskKernelRemainder.lean index 27a87f6065763b8909b91b0c93f0ae493259b274..83fca4dae09c7742b27700efa294ad3cff8e906c 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskKernelRemainder.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskKernelRemainder.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Geometry.PositiveRealDiskEstimate - -namespace OAI - -/-! A quantitative first-order kernel estimate for the inverted -positive-real function of a small slit. -/ -noncomputable section -open Set Filter Function Metric Complex -open scoped Topology - -namespace SLEExactGauge - -lemma disk_kernel_remainder {v w : ℂ} {m : ℝ} (hm : 0 h) t z hz - have hr : ‖deriv (physicalDiskChart id F G t z) 0‖ = fullRadius G t z := by - simpa [physicalConformalRadius] using hder - have hn : ‖w‖ ≤ 1/4 := by simpa only [mem_closedBall,dist_zero_right] using hw' - simpa only [hr] using univalent_quarter_derivative_bounds hgrowth hd hi hn - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskRecentering +import OAI.Probability.SLE.Geometry.PhysicalCellDisk + +namespace OAI + +/-! Two-sided derivative comparison on the fixed quarter disk. -/ + +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma univalent_quarter_derivative_lower (hgrowth : PublishedDiskUnivalentGrowth) + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) {w : ℂ} (hw : ‖w‖ ≤ 1/4) : + ‖deriv f 0‖ / 16 ≤ ‖deriv f w‖ := by + have hw1 : ‖w‖ < 1 := by linarith + let g := f ∘ diskRecenter w + have hgd : DifferentiableOn ℂ g (ball 0 1) := + hd.comp (diskRecenter_differentiable hw1) (diskRecenter_maps hw1) + have hgi : InjOn g (ball 0 1) := hi.comp (diskRecenter_injOn hw1) (diskRecenter_maps hw1) + have hnorm (z : ℂ) (hz : z ∈ ball 0 1) : + ‖deriv g z‖ = ‖deriv f (diskRecenter w z)‖ * ‖deriv (diskRecenter w) z‖ := by + have hf := hd.differentiableAt (isOpen_ball.mem_nhds (diskRecenter_maps hw1 hz)) + have hg := (diskRecenter_hasDerivAt hw1 (mem_ball_zero_iff.mp hz)).differentiableAt + change ‖deriv (f ∘ diskRecenter w) z‖ = _ + rw [deriv_comp z hf hg,norm_mul] + have hzero := hnorm 0 (mem_ball_self (by norm_num : (0:ℝ)<1)) + simp only [diskRecenter_zero,diskRecenter_deriv_zero hw1] at hzero + have hneg := hnorm (-w) (mem_ball_zero_iff.mpr (by simpa only [norm_neg] using hw1)) + simp only [diskRecenter_neg,diskRecenter_deriv_neg hw1] at hneg + have hh := univalent_quarter_derivative_upper hgrowth hgd hgi + (show ‖-w‖ ≤ 1/4 by simpa only [norm_neg] using hw) + rw [hzero,hneg] at hh + let δ : ℝ := 1-‖w‖^2 + have hδ : 0 < δ := (diskRecenter_det hw1).2 + have hδ1 : δ ≤ 1 := by dsimp [δ]; nlinarith [sq_nonneg ‖w‖] + change ‖deriv f 0‖ * δ⁻¹ ≤ 16 * (‖deriv f w‖ * δ) at hh + have hbound : ‖deriv f 0‖ ≤ 16*‖deriv f w‖*δ^2 := by + calc + _ = (‖deriv f 0‖*δ⁻¹)*δ := by field_simp + _ ≤ (16*(‖deriv f w‖*δ))*δ := mul_le_mul_of_nonneg_right hh hδ.le + _ = _ := by ring + have hsq : δ^2 ≤ 1 := by nlinarith + have hlast := hbound.trans (mul_le_of_le_one_right + (mul_nonneg (by norm_num : (0:ℝ)≤16) (norm_nonneg _)) hsq) + linarith + +lemma univalent_quarter_derivative_bounds (hgrowth : PublishedDiskUnivalentGrowth) + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) {w : ℂ} (hw : ‖w‖ ≤ 1/4) : + ‖deriv f 0‖ / 16 ≤ ‖deriv f w‖ ∧ ‖deriv f w‖ ≤ 16*‖deriv f 0‖ := + ⟨univalent_quarter_derivative_lower hgrowth hd hi hw, + univalent_quarter_derivative_upper hgrowth hd hi hw⟩ + +lemma IsCapacityTwoWitness.quarter_disk_derivative_bounds + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) + (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) + (w : ℂ) (hw' : w ∈ closedBall 0 (1/4)) : + fullRadius G t z / 16 ≤ ‖deriv (physicalDiskChart id F G t z) w‖ ∧ + ‖deriv (physicalDiskChart id F G t z) w‖ ≤ 16*fullRadius G t z := by + obtain ⟨hd,hi,_,hder,_⟩ := hw.physical_disk_chart id differentiableOn_id + (fun _ _ _ _ h => h) t z hz + have hr : ‖deriv (physicalDiskChart id F G t z) 0‖ = fullRadius G t z := by + simpa [physicalConformalRadius] using hder + have hn : ‖w‖ ≤ 1/4 := by simpa only [mem_closedBall,dist_zero_right] using hw' + simpa only [hr] using univalent_quarter_derivative_bounds hgrowth hd hi hn + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskQuarterGeometry.lean b/lean/OAI/Probability/SLE/Geometry/DiskQuarterGeometry.lean index db1a25aad321d52f175c575b3dd7a75deff2b895..314d0ad4c50050bb5655d119c69f5b40af666423 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskQuarterGeometry.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskQuarterGeometry.lean @@ -1,99 +1,99 @@ -import OAI.Probability.SLE.Multipoint.MultipointGeometry -import OAI.Probability.SLE.AngularLaws - -namespace OAI - -/-! Compact disk-chart bounds used to compare physical cell mass with -canonical disk mass. Coarse universal constants suffice for the gauge bounds. -/ - -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma univalent_ball_subset_image - {f : ℂ → ℂ} {r : ℝ} (hr : 0 < r) - (hd : DifferentiableOn ℂ f (ball 0 r)) (hi : InjOn f (ball 0 r)) : - ball (f 0) (r * ‖deriv f 0‖ / (128 * Real.pi)) ⊆ f '' ball 0 r := by - intro z hz - by_contra hn - have h := univalent_omitted_value_bound_ball hr hd hi z hn - have hp : 0 < 128 * Real.pi := mul_pos (by norm_num) Real.pi_pos - have hz' : dist z (f 0) < r * ‖deriv f 0‖ / (128 * Real.pi) := hz - have hh := (lt_div_iff₀ hp).mp hz' - rw [dist_eq_norm, norm_sub_rev] at hh - nlinarith - -lemma univalent_half_growth_bound (hgrowth : PublishedDiskUnivalentGrowth) - {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) - (hi : InjOn f (ball 0 1)) {z : ℂ} (hz : ‖z‖ ≤ 1/2) : - ‖f z - f 0‖ ≤ 2 * ‖deriv f 0‖ := by - have hz1 : ‖z‖ < 1 := by linarith - have hh := hgrowth f hd hi z (mem_ball_zero_iff.mpr hz1) - apply hh.trans - have hp : 0 < (1-‖z‖)^2 := sq_pos_of_pos (by linarith) - apply (div_le_iff₀ hp).mpr - have hsq : (1/4:ℝ) ≤ (1-‖z‖)^2 := by nlinarith [norm_nonneg z] - have hprod := mul_le_mul_of_nonneg_left hsq (norm_nonneg (deriv f 0)) - have hnorm := mul_le_mul_of_nonneg_left hz (norm_nonneg (deriv f 0)) - nlinarith - -lemma univalent_quarter_derivative_upper (hgrowth : PublishedDiskUnivalentGrowth) - {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) - (hi : InjOn f (ball 0 1)) {z : ℂ} (hz : ‖z‖ ≤ 1/4) : - ‖deriv f z‖ ≤ 16 * ‖deriv f 0‖ := by - have hsub : ball z (1/4) ⊆ ball (0:ℂ) 1 := by - intro w hw - have hdist : ‖w-z‖ < 1/4 := by simpa only [mem_ball,dist_eq_norm] using hw - have hnorm : ‖w‖ ≤ ‖w-z‖+‖z‖ := by simpa only [norm_sub_rev, add_comm] using norm_le_insert z w - apply mem_ball_zero_iff.mpr - linarith - have hmaps : MapsTo f (ball z (1/4)) (closedBall (f z) (4*‖deriv f 0‖)) := by - intro w hw - have hdist : ‖w-z‖ < 1/4 := by simpa only [mem_ball,dist_eq_norm] using hw - have hnorm : ‖w‖ ≤ ‖w-z‖+‖z‖ := by simpa only [norm_sub_rev, add_comm] using norm_le_insert z w - have hw2 : ‖w‖ ≤ 1/2 := by linarith - have hz2 : ‖z‖ ≤ 1/2 := by linarith - have hb₁ := univalent_half_growth_bound hgrowth hd hi hw2 - have hb₂ := univalent_half_growth_bound hgrowth hd hi hz2 - have htri : ‖f w-f z‖ ≤ ‖f w-f 0‖+‖f z-f 0‖ := by - simpa only [norm_sub_rev] using norm_sub_le_norm_sub_add_norm_sub (f w) (f 0) (f z) - change dist (f w) (f z) ≤ 4*‖deriv f 0‖ - rw [dist_eq_norm] - linarith - have hh := Complex.norm_deriv_le_div_of_mapsTo_ball (hd.mono hsub) hmaps (by norm_num : (0:ℝ)<1/4) - norm_num at hh ⊢ - linarith - -lemma univalent_quarter_closed_image_contains - {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) - (hi : InjOn f (ball 0 1)) (hder : 0 < ‖deriv f 0‖) : - closedBall (f 0) (‖deriv f 0‖ / (1024 * Real.pi)) ⊆ - f '' closedBall 0 (1/4) := by - have hsub : ball (0:ℂ) (1/4) ⊆ ball 0 1 := ball_subset_ball (by norm_num) - have h := univalent_ball_subset_image (by norm_num : (0:ℝ)<1/4) (hd.mono hsub) (hi.mono hsub) - intro z hz - apply (image_mono ball_subset_closedBall) - apply h - have hp : 0 < Real.pi := Real.pi_pos - have hstrict : ‖deriv f 0‖ / (1024 * Real.pi) < (1/4)*‖deriv f 0‖ / (128 * Real.pi) := by - apply (div_lt_div_iff₀ (by positivity) (by positivity)).mpr - nlinarith - exact lt_of_le_of_lt hz hstrict - -lemma univalent_quarter_closed_image_bound (hgrowth : PublishedDiskUnivalentGrowth) - {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) - (hi : InjOn f (ball 0 1)) : - f '' closedBall 0 (1/4) ⊆ closedBall (f 0) (2*‖deriv f 0‖) := by - rintro z ⟨w,hw,rfl⟩ - have hw' : ‖w‖ ≤ 1/2 := by - have hh : ‖w‖ ≤ 1/4 := by simpa only [mem_closedBall,dist_zero_right] using hw - linarith - simpa only [mem_closedBall,dist_eq_norm] using univalent_half_growth_bound hgrowth hd hi hw' - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Multipoint.MultipointGeometry +import OAI.Probability.SLE.AngularLaws + +namespace OAI + +/-! Compact disk-chart bounds used to compare physical cell mass with +canonical disk mass. Coarse universal constants suffice for the gauge bounds. -/ + +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma univalent_ball_subset_image + {f : ℂ → ℂ} {r : ℝ} (hr : 0 < r) + (hd : DifferentiableOn ℂ f (ball 0 r)) (hi : InjOn f (ball 0 r)) : + ball (f 0) (r * ‖deriv f 0‖ / (128 * Real.pi)) ⊆ f '' ball 0 r := by + intro z hz + by_contra hn + have h := univalent_omitted_value_bound_ball hr hd hi z hn + have hp : 0 < 128 * Real.pi := mul_pos (by norm_num) Real.pi_pos + have hz' : dist z (f 0) < r * ‖deriv f 0‖ / (128 * Real.pi) := hz + have hh := (lt_div_iff₀ hp).mp hz' + rw [dist_eq_norm, norm_sub_rev] at hh + nlinarith + +lemma univalent_half_growth_bound (hgrowth : PublishedDiskUnivalentGrowth) + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) {z : ℂ} (hz : ‖z‖ ≤ 1/2) : + ‖f z - f 0‖ ≤ 2 * ‖deriv f 0‖ := by + have hz1 : ‖z‖ < 1 := by linarith + have hh := hgrowth f hd hi z (mem_ball_zero_iff.mpr hz1) + apply hh.trans + have hp : 0 < (1-‖z‖)^2 := sq_pos_of_pos (by linarith) + apply (div_le_iff₀ hp).mpr + have hsq : (1/4:ℝ) ≤ (1-‖z‖)^2 := by nlinarith [norm_nonneg z] + have hprod := mul_le_mul_of_nonneg_left hsq (norm_nonneg (deriv f 0)) + have hnorm := mul_le_mul_of_nonneg_left hz (norm_nonneg (deriv f 0)) + nlinarith + +lemma univalent_quarter_derivative_upper (hgrowth : PublishedDiskUnivalentGrowth) + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) {z : ℂ} (hz : ‖z‖ ≤ 1/4) : + ‖deriv f z‖ ≤ 16 * ‖deriv f 0‖ := by + have hsub : ball z (1/4) ⊆ ball (0:ℂ) 1 := by + intro w hw + have hdist : ‖w-z‖ < 1/4 := by simpa only [mem_ball,dist_eq_norm] using hw + have hnorm : ‖w‖ ≤ ‖w-z‖+‖z‖ := by simpa only [norm_sub_rev, add_comm] using norm_le_insert z w + apply mem_ball_zero_iff.mpr + linarith + have hmaps : MapsTo f (ball z (1/4)) (closedBall (f z) (4*‖deriv f 0‖)) := by + intro w hw + have hdist : ‖w-z‖ < 1/4 := by simpa only [mem_ball,dist_eq_norm] using hw + have hnorm : ‖w‖ ≤ ‖w-z‖+‖z‖ := by simpa only [norm_sub_rev, add_comm] using norm_le_insert z w + have hw2 : ‖w‖ ≤ 1/2 := by linarith + have hz2 : ‖z‖ ≤ 1/2 := by linarith + have hb₁ := univalent_half_growth_bound hgrowth hd hi hw2 + have hb₂ := univalent_half_growth_bound hgrowth hd hi hz2 + have htri : ‖f w-f z‖ ≤ ‖f w-f 0‖+‖f z-f 0‖ := by + simpa only [norm_sub_rev] using norm_sub_le_norm_sub_add_norm_sub (f w) (f 0) (f z) + change dist (f w) (f z) ≤ 4*‖deriv f 0‖ + rw [dist_eq_norm] + linarith + have hh := Complex.norm_deriv_le_div_of_mapsTo_ball (hd.mono hsub) hmaps (by norm_num : (0:ℝ)<1/4) + norm_num at hh ⊢ + linarith + +lemma univalent_quarter_closed_image_contains + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) (hder : 0 < ‖deriv f 0‖) : + closedBall (f 0) (‖deriv f 0‖ / (1024 * Real.pi)) ⊆ + f '' closedBall 0 (1/4) := by + have hsub : ball (0:ℂ) (1/4) ⊆ ball 0 1 := ball_subset_ball (by norm_num) + have h := univalent_ball_subset_image (by norm_num : (0:ℝ)<1/4) (hd.mono hsub) (hi.mono hsub) + intro z hz + apply (image_mono ball_subset_closedBall) + apply h + have hp : 0 < Real.pi := Real.pi_pos + have hstrict : ‖deriv f 0‖ / (1024 * Real.pi) < (1/4)*‖deriv f 0‖ / (128 * Real.pi) := by + apply (div_lt_div_iff₀ (by positivity) (by positivity)).mpr + nlinarith + exact lt_of_le_of_lt hz hstrict + +lemma univalent_quarter_closed_image_bound (hgrowth : PublishedDiskUnivalentGrowth) + {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) + (hi : InjOn f (ball 0 1)) : + f '' closedBall 0 (1/4) ⊆ closedBall (f 0) (2*‖deriv f 0‖) := by + rintro z ⟨w,hw,rfl⟩ + have hw' : ‖w‖ ≤ 1/2 := by + have hh : ‖w‖ ≤ 1/4 := by simpa only [mem_closedBall,dist_zero_right] using hw + linarith + simpa only [mem_closedBall,dist_eq_norm] using univalent_half_growth_bound hgrowth hd hi hw' + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskRecentering.lean b/lean/OAI/Probability/SLE/Geometry/DiskRecentering.lean index 421a1e514d7ad39dd78451ce2b38b6fe12817693..afd0437fba3a7a51db7914527b6792cb4d163864 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskRecentering.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskRecentering.lean @@ -1,99 +1,99 @@ -import OAI.Probability.SLE.Geometry.DiskQuarterGeometry - -namespace OAI - -/-! The elementary disk automorphism needed to compare derivatives on the -quarter disk. -/ - -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -def diskRecenter (w z : ℂ) : ℂ := (w+z)/(1+star w*z) - -lemma diskRecenter_den_ne {w z : ℂ} (hw : ‖w‖ < 1) (hz : ‖z‖ < 1) : - 1+star w*z ≠ 0 := by - intro h - have hh : star w*z = -1 := by linear_combination h - have hn := congrArg norm hh - simp only [norm_mul,norm_star,norm_neg,norm_one] at hn - nlinarith [norm_nonneg w,norm_nonneg z] - -lemma diskRecenter_det {w : ℂ} (hw : ‖w‖ < 1) : - 1-star w*w = ((1-‖w‖^2:ℝ):ℂ) ∧ 0 < 1-‖w‖^2 := by - constructor - · simp only [Complex.star_def,Complex.conj_mul',Complex.ofReal_sub,Complex.ofReal_one,Complex.ofReal_pow] - · nlinarith [norm_nonneg w] - -lemma diskRecenter_maps {w : ℂ} (hw : ‖w‖ < 1) : - MapsTo (diskRecenter w) (ball 0 1) (ball 0 1) := by - intro z hz - have hz' : ‖z‖ < 1 := mem_ball_zero_iff.mp hz - have hd := diskRecenter_den_ne hw hz' - have hg : Complex.normSq (1+star w*z)-Complex.normSq (w+z) = - (1-Complex.normSq w)*(1-Complex.normSq z) := by - simp only [Complex.normSq_apply,Complex.add_re,Complex.add_im,Complex.one_re,Complex.one_im, - Complex.mul_re,Complex.mul_im,Complex.star_def,Complex.conj_re,Complex.conj_im] - ring - have hp : 0 < (1-Complex.normSq w)*(1-Complex.normSq z) := by - rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] - exact mul_pos (diskRecenter_det hw).2 (diskRecenter_det hz').2 - rw [mem_ball_zero_iff,diskRecenter,norm_div] - apply (div_lt_one (norm_pos_iff.mpr hd)).mpr - simp only [Complex.normSq_eq_norm_sq] at hg hp - nlinarith [norm_nonneg (w+z),norm_nonneg (1+star w*z)] - -lemma diskRecenter_injOn {w : ℂ} (hw : ‖w‖ < 1) : - InjOn (diskRecenter w) (ball 0 1) := by - intro x hx y hy hxy - have hx' := diskRecenter_den_ne hw (mem_ball_zero_iff.mp hx) - have hy' := diskRecenter_den_ne hw (mem_ball_zero_iff.mp hy) - have hh := (div_eq_div_iff hx' hy').mp hxy - have he : (1-star w*w)*(x-y)=0 := by linear_combination hh - have hn : 1-star w*w ≠ 0 := by - rw [(diskRecenter_det hw).1] - exact_mod_cast (diskRecenter_det hw).2.ne' - exact sub_eq_zero.mp ((mul_eq_zero.mp he).resolve_left hn) - -lemma diskRecenter_hasDerivAt {w z : ℂ} (hw : ‖w‖ < 1) (hz : ‖z‖ < 1) : - HasDerivAt (diskRecenter w) ((1-star w*w)/(1+star w*z)^2) z := by - have hh := ((hasDerivAt_id z).const_add w).div - (((hasDerivAt_id z).const_mul (star w)).const_add 1) (diskRecenter_den_ne hw hz) - convert hh using 1 <;> first | rfl | (dsimp; ring) - -lemma diskRecenter_differentiable {w : ℂ} (hw : ‖w‖ < 1) : - DifferentiableOn ℂ (diskRecenter w) (ball 0 1) := by - intro z hz - exact (diskRecenter_hasDerivAt hw (mem_ball_zero_iff.mp hz)).differentiableAt.differentiableWithinAt - -@[simp] lemma diskRecenter_zero (w : ℂ) : diskRecenter w 0=w := by simp [diskRecenter] -@[simp] lemma diskRecenter_neg (w : ℂ) : diskRecenter w (-w)=0 := by simp [diskRecenter] - -lemma diskRecenter_deriv_zero {w : ℂ} (hw : ‖w‖ < 1) : - ‖deriv (diskRecenter w) 0‖=1-‖w‖^2 := by - rw [(diskRecenter_hasDerivAt hw (by simp)).deriv] - simp only [mul_zero,add_zero,one_pow,div_one,(diskRecenter_det hw).1] - exact Complex.norm_of_nonneg (diskRecenter_det hw).2.le - -lemma diskRecenter_deriv_neg {w : ℂ} (hw : ‖w‖ < 1) : - ‖deriv (diskRecenter w) (-w)‖=(1-‖w‖^2)⁻¹ := by - have hz : ‖-w‖ < 1 := by simpa only [norm_neg] using hw - rw [(diskRecenter_hasDerivAt hw hz).deriv] - have he : 1+star w*(-w)=1-star w*w := by ring - rw [he] - have hn : 1-star w*w ≠ 0 := by - rw [(diskRecenter_det hw).1] - exact_mod_cast (diskRecenter_det hw).2.ne' - have hquot : (1-star w*w)/(1-star w*w)^2=(1-star w*w)⁻¹ := by - field_simp [hn] - rw [hquot] - simp only [norm_inv,(diskRecenter_det hw).1, - Complex.norm_of_nonneg (diskRecenter_det hw).2.le] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskQuarterGeometry + +namespace OAI + +/-! The elementary disk automorphism needed to compare derivatives on the +quarter disk. -/ + +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +def diskRecenter (w z : ℂ) : ℂ := (w+z)/(1+star w*z) + +lemma diskRecenter_den_ne {w z : ℂ} (hw : ‖w‖ < 1) (hz : ‖z‖ < 1) : + 1+star w*z ≠ 0 := by + intro h + have hh : star w*z = -1 := by linear_combination h + have hn := congrArg norm hh + simp only [norm_mul,norm_star,norm_neg,norm_one] at hn + nlinarith [norm_nonneg w,norm_nonneg z] + +lemma diskRecenter_det {w : ℂ} (hw : ‖w‖ < 1) : + 1-star w*w = ((1-‖w‖^2:ℝ):ℂ) ∧ 0 < 1-‖w‖^2 := by + constructor + · simp only [Complex.star_def,Complex.conj_mul',Complex.ofReal_sub,Complex.ofReal_one,Complex.ofReal_pow] + · nlinarith [norm_nonneg w] + +lemma diskRecenter_maps {w : ℂ} (hw : ‖w‖ < 1) : + MapsTo (diskRecenter w) (ball 0 1) (ball 0 1) := by + intro z hz + have hz' : ‖z‖ < 1 := mem_ball_zero_iff.mp hz + have hd := diskRecenter_den_ne hw hz' + have hg : Complex.normSq (1+star w*z)-Complex.normSq (w+z) = + (1-Complex.normSq w)*(1-Complex.normSq z) := by + simp only [Complex.normSq_apply,Complex.add_re,Complex.add_im,Complex.one_re,Complex.one_im, + Complex.mul_re,Complex.mul_im,Complex.star_def,Complex.conj_re,Complex.conj_im] + ring + have hp : 0 < (1-Complex.normSq w)*(1-Complex.normSq z) := by + rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] + exact mul_pos (diskRecenter_det hw).2 (diskRecenter_det hz').2 + rw [mem_ball_zero_iff,diskRecenter,norm_div] + apply (div_lt_one (norm_pos_iff.mpr hd)).mpr + simp only [Complex.normSq_eq_norm_sq] at hg hp + nlinarith [norm_nonneg (w+z),norm_nonneg (1+star w*z)] + +lemma diskRecenter_injOn {w : ℂ} (hw : ‖w‖ < 1) : + InjOn (diskRecenter w) (ball 0 1) := by + intro x hx y hy hxy + have hx' := diskRecenter_den_ne hw (mem_ball_zero_iff.mp hx) + have hy' := diskRecenter_den_ne hw (mem_ball_zero_iff.mp hy) + have hh := (div_eq_div_iff hx' hy').mp hxy + have he : (1-star w*w)*(x-y)=0 := by linear_combination hh + have hn : 1-star w*w ≠ 0 := by + rw [(diskRecenter_det hw).1] + exact_mod_cast (diskRecenter_det hw).2.ne' + exact sub_eq_zero.mp ((mul_eq_zero.mp he).resolve_left hn) + +lemma diskRecenter_hasDerivAt {w z : ℂ} (hw : ‖w‖ < 1) (hz : ‖z‖ < 1) : + HasDerivAt (diskRecenter w) ((1-star w*w)/(1+star w*z)^2) z := by + have hh := ((hasDerivAt_id z).const_add w).div + (((hasDerivAt_id z).const_mul (star w)).const_add 1) (diskRecenter_den_ne hw hz) + convert hh using 1 <;> first | rfl | (dsimp; ring) + +lemma diskRecenter_differentiable {w : ℂ} (hw : ‖w‖ < 1) : + DifferentiableOn ℂ (diskRecenter w) (ball 0 1) := by + intro z hz + exact (diskRecenter_hasDerivAt hw (mem_ball_zero_iff.mp hz)).differentiableAt.differentiableWithinAt + +@[simp] lemma diskRecenter_zero (w : ℂ) : diskRecenter w 0=w := by simp [diskRecenter] +@[simp] lemma diskRecenter_neg (w : ℂ) : diskRecenter w (-w)=0 := by simp [diskRecenter] + +lemma diskRecenter_deriv_zero {w : ℂ} (hw : ‖w‖ < 1) : + ‖deriv (diskRecenter w) 0‖=1-‖w‖^2 := by + rw [(diskRecenter_hasDerivAt hw (by simp)).deriv] + simp only [mul_zero,add_zero,one_pow,div_one,(diskRecenter_det hw).1] + exact Complex.norm_of_nonneg (diskRecenter_det hw).2.le + +lemma diskRecenter_deriv_neg {w : ℂ} (hw : ‖w‖ < 1) : + ‖deriv (diskRecenter w) (-w)‖=(1-‖w‖^2)⁻¹ := by + have hz : ‖-w‖ < 1 := by simpa only [norm_neg] using hw + rw [(diskRecenter_hasDerivAt hw hz).deriv] + have he : 1+star w*(-w)=1-star w*w := by ring + rw [he] + have hn : 1-star w*w ≠ 0 := by + rw [(diskRecenter_det hw).1] + exact_mod_cast (diskRecenter_det hw).2.ne' + have hquot : (1-star w*w)/(1-star w*w)^2=(1-star w*w)⁻¹ := by + field_simp [hn] + rw [hquot] + simp only [norm_inv,(diskRecenter_det hw).1, + Complex.norm_of_nonneg (diskRecenter_det hw).2.le] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskSemicircleGeometry.lean b/lean/OAI/Probability/SLE/Geometry/DiskSemicircleGeometry.lean index b36f2379184dd5f80ca0057e5eb5859d2bbee9d3..7d5072f38e76e2fea84e1d0b4becaf5d390706cd 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskSemicircleGeometry.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskSemicircleGeometry.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Geometry.DiskLowerStraightening - -namespace OAI - -/-! Concrete boundary sets used in the relative disk extension. -/ -noncomputable section -open Set Complex -open scoped ComplexConjugate - -namespace SLEExactGauge - -lemma diskUpperArc_im_nonneg {s : ℝ} (hs : s ∈ Icc 0 1) : - 0 ≤ (diskUpperArc s).im := by - rw [diskUpperArc_im] - exact Real.sin_nonneg_of_mem_Icc - ⟨mul_nonneg Real.pi_pos.le hs.1,mul_le_of_le_one_right Real.pi_pos.le hs.2⟩ - -theorem diskUpperArc_image : - diskUpperArc '' Icc 0 1 = {z : ℂ | ‖z‖=1 ∧ 0 ≤ z.im} := by - ext z - constructor - · rintro ⟨s,hs,rfl⟩ - exact ⟨diskUpperArc_norm s,diskUpperArc_im_nonneg hs⟩ - · rintro ⟨hn,hi⟩ - have hr : |z.re| ≤ 1 := by simpa only [hn] using abs_re_le_norm z - have ha := abs_le.mp hr - let s := Real.arccos (-z.re)/Real.pi - have hs : s ∈ Icc 0 1 := by - constructor - · exact div_nonneg (Real.arccos_nonneg _) Real.pi_pos.le - · exact (div_le_one Real.pi_pos).mpr (Real.arccos_le_pi _) - refine ⟨s,hs,?_⟩ - have hps : Real.pi*s=Real.arccos (-z.re) := by dsimp [s]; field_simp - have hc : Real.cos (Real.pi*s) = -z.re := by - rw [hps] - exact Real.cos_arccos (by linarith [ha.2]) (by linarith [ha.1]) - have hsin := diskUpperArc_im_nonneg hs - have hsq : z.re^2+z.im^2=1 := by - have he := Complex.normSq_eq_norm_sq z - rw [Complex.normSq_apply,hn] at he - nlinarith - apply Complex.ext - · rw [diskUpperArc_re] - rw [hc]; ring - · rw [diskUpperArc_im] - rw [diskUpperArc_im] at hsin - nlinarith [Real.sin_sq_add_cos_sq (Real.pi*s)] - -theorem diskLowerArc_image : - diskLowerArc '' Icc 0 1 = {z : ℂ | ‖z‖=1 ∧ z.im ≤ 0} := by - ext z - constructor - · rintro ⟨s,hs,rfl⟩ - have hh : diskUpperArc s ∈ diskUpperArc '' Icc 0 1 := mem_image_of_mem _ hs - rw [diskUpperArc_image] at hh - simpa [diskLowerArc] using hh - · rintro ⟨hn,hi⟩ - have hc : conj z ∈ diskUpperArc '' Icc 0 1 := by - rw [diskUpperArc_image] - simpa using And.intro hn hi - obtain ⟨s,hs,he⟩ := hc - refine ⟨s,hs,?_⟩ - dsimp [diskLowerArc] - rw [he,Complex.conj_conj] - -theorem disk_semicircle_union : - diskUpperArc '' Icc 0 1 ∪ diskLowerArc '' Icc 0 1 = {z : ℂ | ‖z‖=1} := by - rw [diskUpperArc_image,diskLowerArc_image] - ext z - simp only [mem_union,mem_ofPred] - constructor - · rintro (h|h) <;> exact h.1 - · intro h - rcases le_total 0 z.im with hi|hi - · exact Or.inl ⟨h,hi⟩ - · exact Or.inr ⟨h,hi⟩ - -theorem disk_semicircle_inter : - diskUpperArc '' Icc 0 1 ∩ diskLowerArc '' Icc 0 1 = ({-1,1} : Set ℂ) := by - rw [diskUpperArc_image,diskLowerArc_image] - ext z - constructor - · rintro ⟨⟨hn,hp⟩,⟨_,hm⟩⟩ - have hi : z.im=0 := le_antisymm hm hp - have hz : z=(z.re : ℂ) := Complex.ext rfl hi - rw [hz,Complex.norm_real,Real.norm_eq_abs] at hn - rcases (abs_eq (by norm_num : (0:ℝ) ≤ 1)).mp hn with he|he - · right - rw [hz,he] - norm_num - · left - rw [hz,he] - norm_num - · intro hz - rcases hz with hz|hz <;> simp_all - -theorem diskDiameter_image : - diskDiameter '' Icc 0 1 = {z : ℂ | ‖z‖ ≤ 1 ∧ z.im=0} := by - ext z - constructor - · rintro ⟨s,hs,rfl⟩ - change ‖((2*s-1:ℝ):ℂ)‖ ≤ 1 ∧ _ - rw [Complex.norm_real,Real.norm_eq_abs] - exact ⟨abs_le.mpr ⟨by linarith [hs.1],by linarith [hs.2]⟩,rfl⟩ - · rintro ⟨hn,hi⟩ - have hr := (abs_re_le_norm z).trans hn - have ha := abs_le.mp hr - refine ⟨(z.re+1)/2,⟨by linarith [ha.1],by linarith [ha.2]⟩,?_⟩ - apply Complex.ext - · change 2*((z.re+1)/2)-1=z.re - ring - · exact hi.symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskLowerStraightening + +namespace OAI + +/-! Concrete boundary sets used in the relative disk extension. -/ +noncomputable section +open Set Complex +open scoped ComplexConjugate + +namespace SLEExactGauge + +lemma diskUpperArc_im_nonneg {s : ℝ} (hs : s ∈ Icc 0 1) : + 0 ≤ (diskUpperArc s).im := by + rw [diskUpperArc_im] + exact Real.sin_nonneg_of_mem_Icc + ⟨mul_nonneg Real.pi_pos.le hs.1,mul_le_of_le_one_right Real.pi_pos.le hs.2⟩ + +theorem diskUpperArc_image : + diskUpperArc '' Icc 0 1 = {z : ℂ | ‖z‖=1 ∧ 0 ≤ z.im} := by + ext z + constructor + · rintro ⟨s,hs,rfl⟩ + exact ⟨diskUpperArc_norm s,diskUpperArc_im_nonneg hs⟩ + · rintro ⟨hn,hi⟩ + have hr : |z.re| ≤ 1 := by simpa only [hn] using abs_re_le_norm z + have ha := abs_le.mp hr + let s := Real.arccos (-z.re)/Real.pi + have hs : s ∈ Icc 0 1 := by + constructor + · exact div_nonneg (Real.arccos_nonneg _) Real.pi_pos.le + · exact (div_le_one Real.pi_pos).mpr (Real.arccos_le_pi _) + refine ⟨s,hs,?_⟩ + have hps : Real.pi*s=Real.arccos (-z.re) := by dsimp [s]; field_simp + have hc : Real.cos (Real.pi*s) = -z.re := by + rw [hps] + exact Real.cos_arccos (by linarith [ha.2]) (by linarith [ha.1]) + have hsin := diskUpperArc_im_nonneg hs + have hsq : z.re^2+z.im^2=1 := by + have he := Complex.normSq_eq_norm_sq z + rw [Complex.normSq_apply,hn] at he + nlinarith + apply Complex.ext + · rw [diskUpperArc_re] + rw [hc]; ring + · rw [diskUpperArc_im] + rw [diskUpperArc_im] at hsin + nlinarith [Real.sin_sq_add_cos_sq (Real.pi*s)] + +theorem diskLowerArc_image : + diskLowerArc '' Icc 0 1 = {z : ℂ | ‖z‖=1 ∧ z.im ≤ 0} := by + ext z + constructor + · rintro ⟨s,hs,rfl⟩ + have hh : diskUpperArc s ∈ diskUpperArc '' Icc 0 1 := mem_image_of_mem _ hs + rw [diskUpperArc_image] at hh + simpa [diskLowerArc] using hh + · rintro ⟨hn,hi⟩ + have hc : conj z ∈ diskUpperArc '' Icc 0 1 := by + rw [diskUpperArc_image] + simpa using And.intro hn hi + obtain ⟨s,hs,he⟩ := hc + refine ⟨s,hs,?_⟩ + dsimp [diskLowerArc] + rw [he,Complex.conj_conj] + +theorem disk_semicircle_union : + diskUpperArc '' Icc 0 1 ∪ diskLowerArc '' Icc 0 1 = {z : ℂ | ‖z‖=1} := by + rw [diskUpperArc_image,diskLowerArc_image] + ext z + simp only [mem_union,mem_ofPred] + constructor + · rintro (h|h) <;> exact h.1 + · intro h + rcases le_total 0 z.im with hi|hi + · exact Or.inl ⟨h,hi⟩ + · exact Or.inr ⟨h,hi⟩ + +theorem disk_semicircle_inter : + diskUpperArc '' Icc 0 1 ∩ diskLowerArc '' Icc 0 1 = ({-1,1} : Set ℂ) := by + rw [diskUpperArc_image,diskLowerArc_image] + ext z + constructor + · rintro ⟨⟨hn,hp⟩,⟨_,hm⟩⟩ + have hi : z.im=0 := le_antisymm hm hp + have hz : z=(z.re : ℂ) := Complex.ext rfl hi + rw [hz,Complex.norm_real,Real.norm_eq_abs] at hn + rcases (abs_eq (by norm_num : (0:ℝ) ≤ 1)).mp hn with he|he + · right + rw [hz,he] + norm_num + · left + rw [hz,he] + norm_num + · intro hz + rcases hz with hz|hz <;> simp_all + +theorem diskDiameter_image : + diskDiameter '' Icc 0 1 = {z : ℂ | ‖z‖ ≤ 1 ∧ z.im=0} := by + ext z + constructor + · rintro ⟨s,hs,rfl⟩ + change ‖((2*s-1:ℝ):ℂ)‖ ≤ 1 ∧ _ + rw [Complex.norm_real,Real.norm_eq_abs] + exact ⟨abs_le.mpr ⟨by linarith [hs.1],by linarith [hs.2]⟩,rfl⟩ + · rintro ⟨hn,hi⟩ + have hr := (abs_re_le_norm z).trans hn + have ha := abs_le.mp hr + refine ⟨(z.re+1)/2,⟨by linarith [ha.1],by linarith [ha.2]⟩,?_⟩ + apply Complex.ext + · change 2*((z.re+1)/2)-1=z.re + ring + · exact hi.symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskSideRegionImages.lean b/lean/OAI/Probability/SLE/Geometry/DiskSideRegionImages.lean index 67fff132c20cd9f8eaf3663cec4c87701df02107..002aad7a643078fad7bbff5f2a89f95fb8136d6f 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskSideRegionImages.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskSideRegionImages.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Geometry.DiskCrosscutClosedRegions - -namespace OAI - -/-! Each prescribed side extension transports the corresponding closed region. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -lemma image_interval_of_agree (H : ℂ ≃ₜ ℂ) (p q : ℝ → ℂ) - (h : ∀ s ∈ Icc 0 1, H (p s)=q s) : H '' (p '' Icc 0 1)=q '' Icc 0 1 := by - ext z - constructor - · rintro ⟨_,⟨s,hs,rfl⟩,rfl⟩ - exact ⟨s,hs,(h s hs).symm⟩ - · rintro ⟨s,hs,rfl⟩ - exact ⟨p s,mem_image_of_mem p hs,h s hs⟩ - -lemma image_eq_self_of_fixed (H : ℂ ≃ₜ ℂ) (A : Set ℂ) - (h : ∀ z ∈ A, H z=z) : H '' A=A := by - ext z - constructor - · rintro ⟨w,hw,rfl⟩ - rwa [h w hw] - · intro hz - exact ⟨z,hz,h z hz⟩ - -theorem disk_upper_region_image (p : ℝ → ℂ) (H : ℂ ≃ₜ ℂ) - (hP : ∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) - (hA : ∀ z ∈ diskUpperArc '' Icc 0 1, H z=z) : - H '' closure (complexInside (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1))= - closedUpperDisk := by - rw [H.image_closure,complexInside_transport,image_union, - image_interval_of_agree H p diskDiameter hP, - image_eq_self_of_fixed H _ hA,union_comm,complexInside_upperDisk,openUpperDisk_closure] - -theorem disk_lower_region_image (p : ℝ → ℂ) (H : ℂ ≃ₜ ℂ) - (hP : ∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) - (hA : ∀ z ∈ diskLowerArc '' Icc 0 1, H z=z) : - H '' closure (complexInside (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1))= - closedLowerDisk := by - rw [H.image_closure,complexInside_transport,image_union, - image_interval_of_agree H p diskDiameter hP, - image_eq_self_of_fixed H _ hA,union_comm,complexInside_lowerDisk,openLowerDisk_closure] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskCrosscutClosedRegions + +namespace OAI + +/-! Each prescribed side extension transports the corresponding closed region. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +lemma image_interval_of_agree (H : ℂ ≃ₜ ℂ) (p q : ℝ → ℂ) + (h : ∀ s ∈ Icc 0 1, H (p s)=q s) : H '' (p '' Icc 0 1)=q '' Icc 0 1 := by + ext z + constructor + · rintro ⟨_,⟨s,hs,rfl⟩,rfl⟩ + exact ⟨s,hs,(h s hs).symm⟩ + · rintro ⟨s,hs,rfl⟩ + exact ⟨p s,mem_image_of_mem p hs,h s hs⟩ + +lemma image_eq_self_of_fixed (H : ℂ ≃ₜ ℂ) (A : Set ℂ) + (h : ∀ z ∈ A, H z=z) : H '' A=A := by + ext z + constructor + · rintro ⟨w,hw,rfl⟩ + rwa [h w hw] + · intro hz + exact ⟨z,hz,h z hz⟩ + +theorem disk_upper_region_image (p : ℝ → ℂ) (H : ℂ ≃ₜ ℂ) + (hP : ∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) + (hA : ∀ z ∈ diskUpperArc '' Icc 0 1, H z=z) : + H '' closure (complexInside (diskUpperArc '' Icc 0 1 ∪ p '' Icc 0 1))= + closedUpperDisk := by + rw [H.image_closure,complexInside_transport,image_union, + image_interval_of_agree H p diskDiameter hP, + image_eq_self_of_fixed H _ hA,union_comm,complexInside_upperDisk,openUpperDisk_closure] + +theorem disk_lower_region_image (p : ℝ → ℂ) (H : ℂ ≃ₜ ℂ) + (hP : ∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) + (hA : ∀ z ∈ diskLowerArc '' Icc 0 1, H z=z) : + H '' closure (complexInside (diskLowerArc '' Icc 0 1 ∪ p '' Icc 0 1))= + closedLowerDisk := by + rw [H.image_closure,complexInside_transport,image_union, + image_interval_of_agree H p diskDiameter hP, + image_eq_self_of_fixed H _ hA,union_comm,complexInside_lowerDisk,openLowerDisk_closure] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskTwoSideGlue.lean b/lean/OAI/Probability/SLE/Geometry/DiskTwoSideGlue.lean index e50243fcdf95c437607a7f40e9bec5c9a4e05381..14701716808b160e169d2be7a8f48119f90a85df 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskTwoSideGlue.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskTwoSideGlue.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Geometry.DiskBoundaryRegion -import OAI.Probability.SLE.Geometry.ClosedComplexPair - -namespace OAI - -/-! Assemble two compatible side maps into a homeomorphism of the closed disk. -/ -noncomputable section -open Set Complex Metric -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge - -lemma closedLowerDisk_isClosed : IsClosed closedLowerDisk := - (isClosed_le continuous_norm continuous_const).inter - (isClosed_le Complex.continuous_im continuous_const) - -def closedDiskTwoSideHomeomorph {A B : Set ℂ} - (hA : IsClosed A) (hB : IsClosed B) - (hAB : A ∪ B=closedBall (0:ℂ) 1) - (H G : ℂ ≃ₜ ℂ) - (hH : H '' A=closedUpperDisk) (hG : G '' B=closedLowerDisk) - (he : ∀ z ∈ A ∩ B, H z=G z) - (hi : ∀ z ∈ closedUpperDisk ∩ closedLowerDisk, H.symm z=G.symm z) : - closedBall (0:ℂ) 1 ≃ₜ closedBall (0:ℂ) 1 := by - let e : A ≃ₜ closedUpperDisk := (H.image A).trans (Homeomorph.setCongr hH) - let d : B ≃ₜ closedLowerDisk := (G.image B).trans (Homeomorph.setCongr hG) - let F := closedComplexPairHomeomorph hA closedUpperDisk_isClosed hB - closedLowerDisk_isClosed e d (fun x hx => he x ⟨x.property,hx⟩) - (fun x hx => hi x ⟨x.property,hx⟩) - exact ((Homeomorph.setCongr hAB.symm).trans F).trans - (Homeomorph.setCongr closed_halfDisks_union) - -lemma closedDiskTwoSideHomeomorph_apply {A B : Set ℂ} - (hA : IsClosed A) (hB : IsClosed B) - (hAB : A ∪ B=closedBall (0:ℂ) 1) - (H G : ℂ ≃ₜ ℂ) - (hH : H '' A=closedUpperDisk) (hG : G '' B=closedLowerDisk) - (he : ∀ z ∈ A ∩ B, H z=G z) - (hi : ∀ z ∈ closedUpperDisk ∩ closedLowerDisk, H.symm z=G.symm z) - (z : closedBall (0:ℂ) 1) : - (closedDiskTwoSideHomeomorph hA hB hAB H G hH hG he hi z : ℂ)= - if z.val ∈ A then H z else G z := by - classical - change (if ha : z.val ∈ A then H z else - if hb : z.val ∈ B then G z else z.val)=_ - by_cases ha : z.val ∈ A - · rw [dite_eq_left ha,ite_eq_left ha] - · have hb : z.val ∈ B := ((hAB.symm ▸ z.property) : z.val ∈ A ∪ B).resolve_left ha - rw [dite_eq_right ha,dite_eq_left hb,ite_eq_right ha] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskBoundaryRegion +import OAI.Probability.SLE.Geometry.ClosedComplexPair + +namespace OAI + +/-! Assemble two compatible side maps into a homeomorphism of the closed disk. -/ +noncomputable section +open Set Complex Metric +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge + +lemma closedLowerDisk_isClosed : IsClosed closedLowerDisk := + (isClosed_le continuous_norm continuous_const).inter + (isClosed_le Complex.continuous_im continuous_const) + +def closedDiskTwoSideHomeomorph {A B : Set ℂ} + (hA : IsClosed A) (hB : IsClosed B) + (hAB : A ∪ B=closedBall (0:ℂ) 1) + (H G : ℂ ≃ₜ ℂ) + (hH : H '' A=closedUpperDisk) (hG : G '' B=closedLowerDisk) + (he : ∀ z ∈ A ∩ B, H z=G z) + (hi : ∀ z ∈ closedUpperDisk ∩ closedLowerDisk, H.symm z=G.symm z) : + closedBall (0:ℂ) 1 ≃ₜ closedBall (0:ℂ) 1 := by + let e : A ≃ₜ closedUpperDisk := (H.image A).trans (Homeomorph.setCongr hH) + let d : B ≃ₜ closedLowerDisk := (G.image B).trans (Homeomorph.setCongr hG) + let F := closedComplexPairHomeomorph hA closedUpperDisk_isClosed hB + closedLowerDisk_isClosed e d (fun x hx => he x ⟨x.property,hx⟩) + (fun x hx => hi x ⟨x.property,hx⟩) + exact ((Homeomorph.setCongr hAB.symm).trans F).trans + (Homeomorph.setCongr closed_halfDisks_union) + +lemma closedDiskTwoSideHomeomorph_apply {A B : Set ℂ} + (hA : IsClosed A) (hB : IsClosed B) + (hAB : A ∪ B=closedBall (0:ℂ) 1) + (H G : ℂ ≃ₜ ℂ) + (hH : H '' A=closedUpperDisk) (hG : G '' B=closedLowerDisk) + (he : ∀ z ∈ A ∩ B, H z=G z) + (hi : ∀ z ∈ closedUpperDisk ∩ closedLowerDisk, H.symm z=G.symm z) + (z : closedBall (0:ℂ) 1) : + (closedDiskTwoSideHomeomorph hA hB hAB H G hH hG he hi z : ℂ)= + if z.val ∈ A then H z else G z := by + classical + change (if ha : z.val ∈ A then H z else + if hb : z.val ∈ B then G z else z.val)=_ + by_cases ha : z.val ∈ A + · rw [dite_eq_left ha,ite_eq_left ha] + · have hb : z.val ∈ B := ((hAB.symm ▸ z.property) : z.val ∈ A ∪ B).resolve_left ha + rw [dite_eq_right ha,dite_eq_left hb,ite_eq_right ha] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/DiskUpperStraightening.lean b/lean/OAI/Probability/SLE/Geometry/DiskUpperStraightening.lean index ed61f85e3225be4d29c6d4da1b83a596c6033999..86e72b696d62d589bbbf8941218404c8ff6a1af1 100644 --- a/lean/OAI/Probability/SLE/Geometry/DiskUpperStraightening.lean +++ b/lean/OAI/Probability/SLE/Geometry/DiskUpperStraightening.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Geometry.DiskCrosscutBoundary - -namespace OAI - -/-! Straighten the disk crosscut with its exact parameter, while fixing -the upper semicircle. The second side is handled separately. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -def diskDiameter (s : ℝ) : ℂ := ((2*s-1 : ℝ) : ℂ) - -lemma diskDiameter_continuous : Continuous diskDiameter := by - unfold diskDiameter - fun_prop - -lemma diskDiameter_injOn : InjOn diskDiameter (Icc 0 1) := by - intro s _ t _ he - have hh := Complex.ofReal_inj.mp he - linarith - -lemma diskDiameter_interior (s : ℝ) (hs : s ∈ Ioo 0 1) : ‖diskDiameter s‖ < 1 := by - change ‖((2*s-1 : ℝ) : ℂ)‖ < 1 - rw [Complex.norm_real,Real.norm_eq_abs] - exact abs_lt.mpr ⟨by linarith [hs.1],by linarith [hs.2]⟩ - -theorem disk_crosscut_upper_straightening - (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) - (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : - ∃ H : ℂ ≃ₜ ℂ, - (∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) ∧ - ∀ z ∈ diskUpperArc '' Icc 0 1, H z=z := by - classical - let B : Set ℂ := diskUpperArc '' Icc 0 1 - have hB : IsClosed B := (isCompact_Icc.image diskUpperArc_continuous).isClosed - have hq0 : diskDiameter 0=p 0 := by simp [diskDiameter,h0] - have hq1 : diskDiameter 1=p 1 := by norm_num [diskDiameter,h1] - let e := fixedArcBoundaryReplacement p diskDiameter B hc hi - diskDiameter_continuous.continuousOn diskDiameter_injOn hB hq0 hq1 - (disk_crosscut_meets_upper_at_endpoints p hp) - (disk_crosscut_meets_upper_at_endpoints diskDiameter diskDiameter_interior) - obtain ⟨H,hH⟩ := complex_jordan_extension (disk_crosscut_upper_jordan p hc hi h0 h1 hp) - (disk_crosscut_upper_jordan diskDiameter diskDiameter_continuous.continuousOn - diskDiameter_injOn (by norm_num [diskDiameter]) (by norm_num [diskDiameter]) - diskDiameter_interior) e - refine ⟨H,?_,?_⟩ - · intro s hs - have hh := hH ⟨p s,Or.inl (mem_image_of_mem p hs)⟩ - apply hh.trans - change (if hx : p s ∈ p '' Icc 0 1 then - (arcParameterReplacement p diskDiameter hc hi diskDiameter_continuous.continuousOn - diskDiameter_injOn ⟨p s,hx⟩ : ℂ) else p s)=diskDiameter s - rw [dite_eq_left (mem_image_of_mem p hs)] - exact arcParameterReplacement_apply p diskDiameter hc hi - diskDiameter_continuous.continuousOn diskDiameter_injOn s hs - · intro z hz - have hh := hH ⟨z,Or.inr hz⟩ - apply hh.trans - change (if hx : z ∈ p '' Icc 0 1 then - (arcParameterReplacement p diskDiameter hc hi diskDiameter_continuous.continuousOn - diskDiameter_injOn ⟨z,hx⟩ : ℂ) else z)=z - split_ifs with hx - · obtain ⟨s,hs,rfl⟩ := hx - rw [arcParameterReplacement_apply p diskDiameter hc hi - diskDiameter_continuous.continuousOn diskDiameter_injOn s hs] - rcases disk_crosscut_meets_upper_at_endpoints p hp s hs hz with rfl | rfl - · exact hq0 - · exact hq1 - · rfl - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskCrosscutBoundary + +namespace OAI + +/-! Straighten the disk crosscut with its exact parameter, while fixing +the upper semicircle. The second side is handled separately. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +def diskDiameter (s : ℝ) : ℂ := ((2*s-1 : ℝ) : ℂ) + +lemma diskDiameter_continuous : Continuous diskDiameter := by + unfold diskDiameter + fun_prop + +lemma diskDiameter_injOn : InjOn diskDiameter (Icc 0 1) := by + intro s _ t _ he + have hh := Complex.ofReal_inj.mp he + linarith + +lemma diskDiameter_interior (s : ℝ) (hs : s ∈ Ioo 0 1) : ‖diskDiameter s‖ < 1 := by + change ‖((2*s-1 : ℝ) : ℂ)‖ < 1 + rw [Complex.norm_real,Real.norm_eq_abs] + exact abs_lt.mpr ⟨by linarith [hs.1],by linarith [hs.2]⟩ + +theorem disk_crosscut_upper_straightening + (p : ℝ → ℂ) (hc : ContinuousOn p (Icc 0 1)) (hi : InjOn p (Icc 0 1)) + (h0 : p 0=-1) (h1 : p 1=1) (hp : ∀ s ∈ Ioo 0 1, ‖p s‖ < 1) : + ∃ H : ℂ ≃ₜ ℂ, + (∀ s ∈ Icc 0 1, H (p s)=diskDiameter s) ∧ + ∀ z ∈ diskUpperArc '' Icc 0 1, H z=z := by + classical + let B : Set ℂ := diskUpperArc '' Icc 0 1 + have hB : IsClosed B := (isCompact_Icc.image diskUpperArc_continuous).isClosed + have hq0 : diskDiameter 0=p 0 := by simp [diskDiameter,h0] + have hq1 : diskDiameter 1=p 1 := by norm_num [diskDiameter,h1] + let e := fixedArcBoundaryReplacement p diskDiameter B hc hi + diskDiameter_continuous.continuousOn diskDiameter_injOn hB hq0 hq1 + (disk_crosscut_meets_upper_at_endpoints p hp) + (disk_crosscut_meets_upper_at_endpoints diskDiameter diskDiameter_interior) + obtain ⟨H,hH⟩ := complex_jordan_extension (disk_crosscut_upper_jordan p hc hi h0 h1 hp) + (disk_crosscut_upper_jordan diskDiameter diskDiameter_continuous.continuousOn + diskDiameter_injOn (by norm_num [diskDiameter]) (by norm_num [diskDiameter]) + diskDiameter_interior) e + refine ⟨H,?_,?_⟩ + · intro s hs + have hh := hH ⟨p s,Or.inl (mem_image_of_mem p hs)⟩ + apply hh.trans + change (if hx : p s ∈ p '' Icc 0 1 then + (arcParameterReplacement p diskDiameter hc hi diskDiameter_continuous.continuousOn + diskDiameter_injOn ⟨p s,hx⟩ : ℂ) else p s)=diskDiameter s + rw [dite_eq_left (mem_image_of_mem p hs)] + exact arcParameterReplacement_apply p diskDiameter hc hi + diskDiameter_continuous.continuousOn diskDiameter_injOn s hs + · intro z hz + have hh := hH ⟨z,Or.inr hz⟩ + apply hh.trans + change (if hx : z ∈ p '' Icc 0 1 then + (arcParameterReplacement p diskDiameter hc hi diskDiameter_continuous.continuousOn + diskDiameter_injOn ⟨z,hx⟩ : ℂ) else z)=z + split_ifs with hx + · obtain ⟨s,hs,rfl⟩ := hx + rw [arcParameterReplacement_apply p diskDiameter hc hi + diskDiameter_continuous.continuousOn diskDiameter_injOn s hs] + rcases disk_crosscut_meets_upper_at_endpoints p hp s hs hz with rfl | rfl + · exact hq0 + · exact hq1 + · rfl + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/EndpointClusterInterval.lean b/lean/OAI/Probability/SLE/Geometry/EndpointClusterInterval.lean index 4fd09531c13a09917351a6bd7ea74782d914cf52..2dcc2ed4e8e7318b44fb40352769bb88c5e7d2e8 100644 --- a/lean/OAI/Probability/SLE/Geometry/EndpointClusterInterval.lean +++ b/lean/OAI/Probability/SLE/Geometry/EndpointClusterInterval.lean @@ -1,52 +1,52 @@ -import Mathlib.Topology.Order.IntermediateValue -import Mathlib.Topology.MetricSpace.Pseudo.Defs -import Mathlib.Topology.ClusterPt -import Mathlib.Topology.Instances.Real.Lemmas - -namespace OAI - -/-! The real cluster values of a continuous path at a one-sided endpoint -contain every value between two of them. -/ -noncomputable section -open Set Filter Metric -open scoped Topology - -namespace SLEExactGauge - -lemma endpoint_intermediate_level {f : ℝ → ℝ} {D a b c : ℝ} - (hD : 0] (0:ℝ)) f) (hb : MapClusterPt b (𝓝[>] (0:ℝ)) f) - (hac : a] (0:ℝ), t∈Ioo 0 ε := by - filter_upwards [self_mem_nhdsWithin, - mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds hε)] with t ht htε - exact ⟨ht,htε⟩ - obtain ⟨u,hu,huε⟩ := ((ha.frequently (Iio_mem_nhds hac)).and_eventually hsmall).exists - obtain ⟨v,hv,hvε⟩ := ((hb.frequently (Ioi_mem_nhds hcb)).and_eventually hsmall).exists - have hcε : ContinuousOn f (Ioo 0 ε) := hC.mono (Ioo_subset_Ioo le_rfl (min_le_right δ D)) - obtain ⟨t,ht,he⟩ := isPreconnected_Ioo.intermediate_value huε hvε hcε ⟨hu.le,hv.le⟩ - exact ⟨t,⟨ht.1,ht.2.trans_le (min_le_left δ D)⟩,he⟩ - -theorem endpoint_cluster_interval {f : ℝ → ℝ} {D a b : ℝ} - (hD : 0] (0:ℝ)) f) (hb : MapClusterPt b (𝓝[>] (0:ℝ)) f) : - ∀ c∈Ioo a b, MapClusterPt c (𝓝[>] (0:ℝ)) f := by - intro c hc - apply clusterPt_iff_nonempty.mpr - intro W hW V hV - have hpre : f ⁻¹' V∈𝓝[>] (0:ℝ) := hV - obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhdsWithin_iff.mp hpre - obtain ⟨t,ht,he⟩ := endpoint_intermediate_level hD hC ha hb hc.1 hc.2 hδ - have htB : t∈ball (0:ℝ) δ := by - simpa only [mem_ball,Real.dist_eq,sub_zero,abs_of_pos ht.1] using ht.2 - have hfV : f t∈V := hsub ⟨htB,ht.1⟩ - exact ⟨c,mem_of_mem_nhds hW,he ▸ hfV⟩ - -end SLEExactGauge - -end - -end OAI +import Mathlib.Topology.Order.IntermediateValue +import Mathlib.Topology.MetricSpace.Pseudo.Defs +import Mathlib.Topology.ClusterPt +import Mathlib.Topology.Instances.Real.Lemmas + +namespace OAI + +/-! The real cluster values of a continuous path at a one-sided endpoint +contain every value between two of them. -/ +noncomputable section +open Set Filter Metric +open scoped Topology + +namespace SLEExactGauge + +lemma endpoint_intermediate_level {f : ℝ → ℝ} {D a b c : ℝ} + (hD : 0] (0:ℝ)) f) (hb : MapClusterPt b (𝓝[>] (0:ℝ)) f) + (hac : a] (0:ℝ), t∈Ioo 0 ε := by + filter_upwards [self_mem_nhdsWithin, + mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds hε)] with t ht htε + exact ⟨ht,htε⟩ + obtain ⟨u,hu,huε⟩ := ((ha.frequently (Iio_mem_nhds hac)).and_eventually hsmall).exists + obtain ⟨v,hv,hvε⟩ := ((hb.frequently (Ioi_mem_nhds hcb)).and_eventually hsmall).exists + have hcε : ContinuousOn f (Ioo 0 ε) := hC.mono (Ioo_subset_Ioo le_rfl (min_le_right δ D)) + obtain ⟨t,ht,he⟩ := isPreconnected_Ioo.intermediate_value huε hvε hcε ⟨hu.le,hv.le⟩ + exact ⟨t,⟨ht.1,ht.2.trans_le (min_le_left δ D)⟩,he⟩ + +theorem endpoint_cluster_interval {f : ℝ → ℝ} {D a b : ℝ} + (hD : 0] (0:ℝ)) f) (hb : MapClusterPt b (𝓝[>] (0:ℝ)) f) : + ∀ c∈Ioo a b, MapClusterPt c (𝓝[>] (0:ℝ)) f := by + intro c hc + apply clusterPt_iff_nonempty.mpr + intro W hW V hV + have hpre : f ⁻¹' V∈𝓝[>] (0:ℝ) := hV + obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhdsWithin_iff.mp hpre + obtain ⟨t,ht,he⟩ := endpoint_intermediate_level hD hC ha hb hc.1 hc.2 hδ + have htB : t∈ball (0:ℝ) δ := by + simpa only [mem_ball,Real.dist_eq,sub_zero,abs_of_pos ht.1] using ht.2 + have hfV : f t∈V := hsub ⟨htB,ht.1⟩ + exact ⟨c,mem_of_mem_nhds hW,he ▸ hfV⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/EndpointClusterLimit.lean b/lean/OAI/Probability/SLE/Geometry/EndpointClusterLimit.lean index 1f05ac457a1248a7590f97dd588c62ee68006212..cccd7f2ac28df9a4c711c3fd5d381ad34199ec4f 100644 --- a/lean/OAI/Probability/SLE/Geometry/EndpointClusterLimit.lean +++ b/lean/OAI/Probability/SLE/Geometry/EndpointClusterLimit.lean @@ -1,31 +1,31 @@ -import OAI.Probability.SLE.Geometry.EndpointClusterInterval -import Mathlib.Topology.Compactness.Compact - -namespace OAI - -/-! A bounded continuous real path has an endpoint limit when its cluster -values contain no nontrivial interval. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -theorem exists_endpoint_limit_of_no_cluster_interval {f : ℝ → ℝ} {D : ℝ} - (hD : 0] (0:ℝ), f t∈K) - (hno : ∀ a b : ℝ, a] (0:ℝ)) f) : - ∃ a∈K, Tendsto f (𝓝[>] (0:ℝ)) (𝓝 a) := by - obtain ⟨a,haK,ha⟩ := hK.exists_mapClusterPt (tendsto_principal.mpr hmem) - refine ⟨a,haK,hK.tendsto_nhds_of_unique_mapClusterPt hmem ?_⟩ - intro b _ hb - by_contra hba - rcases lt_or_gt_of_ne hba with hlt | hgt - · exact hno b a hlt (endpoint_cluster_interval hD hC hb ha) - · exact hno a b hgt (endpoint_cluster_interval hD hC ha hb) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.EndpointClusterInterval +import Mathlib.Topology.Compactness.Compact + +namespace OAI + +/-! A bounded continuous real path has an endpoint limit when its cluster +values contain no nontrivial interval. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +theorem exists_endpoint_limit_of_no_cluster_interval {f : ℝ → ℝ} {D : ℝ} + (hD : 0] (0:ℝ), f t∈K) + (hno : ∀ a b : ℝ, a] (0:ℝ)) f) : + ∃ a∈K, Tendsto f (𝓝[>] (0:ℝ)) (𝓝 a) := by + obtain ⟨a,haK,ha⟩ := hK.exists_mapClusterPt (tendsto_principal.mpr hmem) + refine ⟨a,haK,hK.tendsto_nhds_of_unique_mapClusterPt hmem ?_⟩ + intro b _ hb + by_contra hba + rcases lt_or_gt_of_ne hba with hlt | hgt + · exact hno b a hlt (endpoint_cluster_interval hD hC hb ha) + · exact hno a b hgt (endpoint_cluster_interval hD hC ha hb) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/EndpointDerivativeBound.lean b/lean/OAI/Probability/SLE/Geometry/EndpointDerivativeBound.lean index fcc79cb89981c78294f53160334f427a750fdeec..0668c1d6cb7c152f3234bcf88aeab6bd732f184f 100644 --- a/lean/OAI/Probability/SLE/Geometry/EndpointDerivativeBound.lean +++ b/lean/OAI/Probability/SLE/Geometry/EndpointDerivativeBound.lean @@ -1,52 +1,52 @@ -import Mathlib - -namespace OAI - -/-! A derivative bound on positive times extends to the endpoint when -the function is continuous there. Used for squared reverse coordinates. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -lemma norm_sub_initial_le_of_deriv_on_Ioc - {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] - {T C : ℝ} (hT : 0 ≤ T) (f f' : ℝ → E) - (hc : ContinuousOn f (Icc 0 T)) - (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (f' t) (Ici t) t) - (hb : ∀ t ∈ Ioc 0 T, ‖f' t‖ ≤ C) : - ∀ t ∈ Icc 0 T, ‖f t-f 0‖ ≤ C*t := by - intro t ht - by_cases ht0 : t=0 - · subst t; simp - have htp : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0) - let x (n : ℕ) : ℝ := t/((n : ℝ)+1) - have hx (n : ℕ) : 0 < x n := by dsimp [x]; positivity - have hxt (n : ℕ) : x n ≤ t := by - apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr - nlinarith [Nat.cast_nonneg (α := ℝ) n] - have hx0 : Tendsto x atTop (𝓝 (0 : ℝ)) := by - have h := tendsto_one_div_add_atTop_nhds_zero_nat.const_mul t - simpa only [x,mul_one_div,mul_zero] using h - have hxin : Tendsto x atTop (𝓝[Icc 0 T] 0) := tendsto_nhdsWithin_iff.mpr - ⟨hx0,Eventually.of_forall (fun n => ⟨(hx n).le,(hxt n).trans ht.2⟩)⟩ - have hfx := (hc 0 ⟨le_rfl,hT⟩).tendsto.comp hxin - have hleft : Tendsto (fun n => ‖f t-f (x n)‖) atTop (𝓝 ‖f t-f 0‖) := - (hfx.const_sub (f t)).norm - have hright : Tendsto (fun n => C*(t-x n)) atTop (𝓝 (C*t)) := by - simpa only [sub_zero] using (hx0.const_sub t).const_mul C - apply le_of_tendsto_of_tendsto hleft hright - apply Eventually.of_forall - intro n - have hsub : Icc (x n) t ⊆ Icc 0 T := Icc_subset_Icc (hx n).le ht.2 - apply norm_image_sub_le_of_norm_deriv_right_le_segment (hc.mono hsub) - (fun s hs => hd s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) - (fun s hs => hb s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) - t ⟨hxt n,le_rfl⟩ - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! A derivative bound on positive times extends to the endpoint when +the function is continuous there. Used for squared reverse coordinates. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +lemma norm_sub_initial_le_of_deriv_on_Ioc + {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] + {T C : ℝ} (hT : 0 ≤ T) (f f' : ℝ → E) + (hc : ContinuousOn f (Icc 0 T)) + (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (f' t) (Ici t) t) + (hb : ∀ t ∈ Ioc 0 T, ‖f' t‖ ≤ C) : + ∀ t ∈ Icc 0 T, ‖f t-f 0‖ ≤ C*t := by + intro t ht + by_cases ht0 : t=0 + · subst t; simp + have htp : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0) + let x (n : ℕ) : ℝ := t/((n : ℝ)+1) + have hx (n : ℕ) : 0 < x n := by dsimp [x]; positivity + have hxt (n : ℕ) : x n ≤ t := by + apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr + nlinarith [Nat.cast_nonneg (α := ℝ) n] + have hx0 : Tendsto x atTop (𝓝 (0 : ℝ)) := by + have h := tendsto_one_div_add_atTop_nhds_zero_nat.const_mul t + simpa only [x,mul_one_div,mul_zero] using h + have hxin : Tendsto x atTop (𝓝[Icc 0 T] 0) := tendsto_nhdsWithin_iff.mpr + ⟨hx0,Eventually.of_forall (fun n => ⟨(hx n).le,(hxt n).trans ht.2⟩)⟩ + have hfx := (hc 0 ⟨le_rfl,hT⟩).tendsto.comp hxin + have hleft : Tendsto (fun n => ‖f t-f (x n)‖) atTop (𝓝 ‖f t-f 0‖) := + (hfx.const_sub (f t)).norm + have hright : Tendsto (fun n => C*(t-x n)) atTop (𝓝 (C*t)) := by + simpa only [sub_zero] using (hx0.const_sub t).const_mul C + apply le_of_tendsto_of_tendsto hleft hright + apply Eventually.of_forall + intro n + have hsub : Icc (x n) t ⊆ Icc 0 T := Icc_subset_Icc (hx n).le ht.2 + apply norm_image_sub_le_of_norm_deriv_right_le_segment (hc.mono hsub) + (fun s hs => hd s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) + (fun s hs => hb s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) + t ⟨hxt n,le_rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/EndpointMonotonicity.lean b/lean/OAI/Probability/SLE/Geometry/EndpointMonotonicity.lean index e01eef2e2acb4f20b0a30a700efff75dabe602f7..bb63d446b71ef2e2e639ae51a471809cd3137f7b 100644 --- a/lean/OAI/Probability/SLE/Geometry/EndpointMonotonicity.lean +++ b/lean/OAI/Probability/SLE/Geometry/EndpointMonotonicity.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Geometry.EndpointDerivativeBound - -namespace OAI - -/-! One-sided derivative comparisons remain valid at a continuous endpoint, -even when the derivative at that endpoint is singular. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -lemma le_initial_of_deriv_nonpos_on_Ioc {T : ℝ} (hT : 0 ≤ T) - (f f' : ℝ → ℝ) (hc : ContinuousOn f (Icc 0 T)) - (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (f' t) (Ici t) t) - (hb : ∀ t ∈ Ioc 0 T, f' t ≤ 0) : - ∀ t ∈ Icc 0 T, f t ≤ f 0 := by - intro t ht - by_cases ht0 : t=0 - · subst t; exact le_rfl - have htp : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0) - let x (n : ℕ) : ℝ := t/((n : ℝ)+1) - have hx (n : ℕ) : 0 < x n := by dsimp [x]; positivity - have hxt (n : ℕ) : x n ≤ t := by - apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr - nlinarith [Nat.cast_nonneg (α := ℝ) n] - have hx0 : Tendsto x atTop (𝓝 (0 : ℝ)) := by - have h := tendsto_one_div_add_atTop_nhds_zero_nat.const_mul t - simpa only [x,mul_one_div,mul_zero] using h - have hxin : Tendsto x atTop (𝓝[Icc 0 T] 0) := tendsto_nhdsWithin_iff.mpr - ⟨hx0,Eventually.of_forall (fun n => ⟨(hx n).le,(hxt n).trans ht.2⟩)⟩ - apply ge_of_tendsto ((hc 0 ⟨le_rfl,hT⟩).tendsto.comp hxin) - apply Eventually.of_forall - intro n - have hsub : Icc (x n) t ⊆ Icc 0 T := Icc_subset_Icc (hx n).le ht.2 - exact image_le_of_deriv_right_le_deriv_boundary (hc.mono hsub) - (fun s hs => hd s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) - (B := fun _ => f (x n)) (B' := fun _ => 0) le_rfl continuousOn_const - (fun s _ => hasDerivWithinAt_const s (Ici s) (f (x n))) - (fun s hs => hb s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) ⟨hxt n,le_rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.EndpointDerivativeBound + +namespace OAI + +/-! One-sided derivative comparisons remain valid at a continuous endpoint, +even when the derivative at that endpoint is singular. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +lemma le_initial_of_deriv_nonpos_on_Ioc {T : ℝ} (hT : 0 ≤ T) + (f f' : ℝ → ℝ) (hc : ContinuousOn f (Icc 0 T)) + (hd : ∀ t ∈ Ioc 0 T, HasDerivWithinAt f (f' t) (Ici t) t) + (hb : ∀ t ∈ Ioc 0 T, f' t ≤ 0) : + ∀ t ∈ Icc 0 T, f t ≤ f 0 := by + intro t ht + by_cases ht0 : t=0 + · subst t; exact le_rfl + have htp : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm ht0) + let x (n : ℕ) : ℝ := t/((n : ℝ)+1) + have hx (n : ℕ) : 0 < x n := by dsimp [x]; positivity + have hxt (n : ℕ) : x n ≤ t := by + apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr + nlinarith [Nat.cast_nonneg (α := ℝ) n] + have hx0 : Tendsto x atTop (𝓝 (0 : ℝ)) := by + have h := tendsto_one_div_add_atTop_nhds_zero_nat.const_mul t + simpa only [x,mul_one_div,mul_zero] using h + have hxin : Tendsto x atTop (𝓝[Icc 0 T] 0) := tendsto_nhdsWithin_iff.mpr + ⟨hx0,Eventually.of_forall (fun n => ⟨(hx n).le,(hxt n).trans ht.2⟩)⟩ + apply ge_of_tendsto ((hc 0 ⟨le_rfl,hT⟩).tendsto.comp hxin) + apply Eventually.of_forall + intro n + have hsub : Icc (x n) t ⊆ Icc 0 T := Icc_subset_Icc (hx n).le ht.2 + exact image_le_of_deriv_right_le_deriv_boundary (hc.mono hsub) + (fun s hs => hd s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) + (B := fun _ => f (x n)) (B' := fun _ => 0) le_rfl continuousOn_const + (fun s _ => hasDerivWithinAt_const s (Ici s) (f (x n))) + (fun s hs => hb s ⟨(hx n).trans_le hs.1,hs.2.le.trans ht.2⟩) ⟨hxt n,le_rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExplicitInputDischarge.lean b/lean/OAI/Probability/SLE/Geometry/ExplicitInputDischarge.lean index 157d63241e41cbfbfce8de331a706509f6eb6b98..17a918e42031375208a891dec900c92a0d7aeeb0 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExplicitInputDischarge.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExplicitInputDischarge.lean @@ -1,36 +1,36 @@ -import OAI.Probability.SLE.Geometry.ActualExplicitTransfer -import OAI.Probability.SLE.Geometry.ActualWienerSupport - -namespace OAI - -/-! The explicit exact-gauge theorem using planar Wiener support -derived from the Brownian law. -/ -noncomputable section -open Set MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance explicitDischargeMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance explicitDischargeBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem ordinary_explicit_mainConclusion_of_remaining_inputs - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) - (hcompact : PublishedHalfPlaneUnivalentCompactness) - (hboundary : PublishedFiniteLoewnerBoundaryExtension) - (hsupport : PublishedGeometricTubeSupport P γ) - (hplanar : PublishedPlanarDiskOpenness) - (hstraight : PublishedHalfPlaneCrosscutStraightening) - (hexit : PublishedHalfPlaneConformalExitLaw) : - MainConclusion P (explicitGauge κ) γ := - ordinary_explicit_mainConclusion_of_published_inputs P κ hκ hκ8 γ hSLE - htrace hcompact hboundary hsupport hplanar hstraight hexit actual_planarWienerUnitSupport - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ActualExplicitTransfer +import OAI.Probability.SLE.Geometry.ActualWienerSupport + +namespace OAI + +/-! The explicit exact-gauge theorem using planar Wiener support +derived from the Brownian law. -/ +noncomputable section +open Set MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance explicitDischargeMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance explicitDischargeBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem ordinary_explicit_mainConclusion_of_remaining_inputs + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (htrace : PublishedTraceGeneration (ordinaryRegularDriverLaw hSLE) κ (fun t p => p t)) + (hcompact : PublishedHalfPlaneUnivalentCompactness) + (hboundary : PublishedFiniteLoewnerBoundaryExtension) + (hsupport : PublishedGeometricTubeSupport P γ) + (hplanar : PublishedPlanarDiskOpenness) + (hstraight : PublishedHalfPlaneCrosscutStraightening) + (hexit : PublishedHalfPlaneConformalExitLaw) : + MainConclusion P (explicitGauge κ) γ := + ordinary_explicit_mainConclusion_of_published_inputs P κ hκ hκ8 γ hSLE + htrace hcompact hboundary hsupport hplanar hstraight hexit actual_planarWienerUnitSupport + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExplicitPublishedMain.lean b/lean/OAI/Probability/SLE/Geometry/ExplicitPublishedMain.lean index 760e94982b831b289e620721e745b0d5137d1350..93350614c79194eee800da1269538967464ae80d 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExplicitPublishedMain.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExplicitPublishedMain.lean @@ -1,25 +1,25 @@ -import OAI.Probability.SLE.Gauge.GaugeCrosscutDischarge -import OAI.Probability.SLE.Geometry.ActualGeometricTubeSupport - -namespace OAI - -/-! The explicit exact Hausdorff gauge theorem for ordinary chordal SLE. -/ -noncomputable section -open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem ordinary_explicit_mainConclusion_unconditional - {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) (γ : Ω → ℝ≥0 → ℂ) - (hSLE : IsOrdinaryChordalSLE κ γ P) : - MainConclusion P (explicitGauge κ) γ := - ordinary_explicit_mainConclusion_of_support P κ hκ hκ8 γ hSLE - (ordinary_geometricTubeSupport P κ hκ γ hSLE) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.GaugeCrosscutDischarge +import OAI.Probability.SLE.Geometry.ActualGeometricTubeSupport + +namespace OAI + +/-! The explicit exact Hausdorff gauge theorem for ordinary chordal SLE. -/ +noncomputable section +open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem ordinary_explicit_mainConclusion_unconditional + {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) (γ : Ω → ℝ≥0 → ℂ) + (hSLE : IsOrdinaryChordalSLE κ γ P) : + MainConclusion P (explicitGauge κ) γ := + ordinary_explicit_mainConclusion_of_support P κ hκ hκ8 γ hSLE + (ordinary_geometricTubeSupport P κ hκ γ hSLE) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExtendedDiskAngle.lean b/lean/OAI/Probability/SLE/Geometry/ExtendedDiskAngle.lean index 53b633baf62e0e5ff144f29b04c06896ab70c779..c6d5da5592697a9c62eb523e88e92553047e8c13 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExtendedDiskAngle.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExtendedDiskAngle.lean @@ -1,69 +1,69 @@ -import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle -import OAI.Probability.SLE.Geometry.FiniteStoppedScalar - -namespace OAI - -/-! The full marked angle at a possibly infinite stopping time is -measurable in its stopped past, with value zero on the infinite event. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -def extendedMarkedAngle {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (σ : Ω → WithTop ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := - if σ ω=⊤ then 0 else - (stoppedMarkedCenter κ B γ G (fun ω => (σ ω).untopD 0) z ω).arg - -lemma stoppedMarkedCenter_time_congr {Ω : Type*} - (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (σ τ : Ω → ℝ≥0) (z : ℂ) (ω : Ω) (h : σ ω=τ ω) : - stoppedMarkedCenter κ B γ G σ z ω=stoppedMarkedCenter κ B γ G τ z ω := by - classical - exact congrArg (fun t => (if z∈survivingDomain (γ ω) t then G ω t z else 0)- - ((Real.sqrt κ*B t ω:ℝ):ℂ)) h - -lemma extendedMarkedAngle_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (σ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) σ) - (z : ℂ) : - Measurable[hσ.measurableSpace] (extendedMarkedAngle κ B γ G σ z) := by - let X := fun n : ℕ => (finiteStopEvent σ n).indicator (fun ω => - (stoppedMarkedCenter κ B γ G (cutoffStop σ n) z ω).arg) - have hX (n : ℕ) : Measurable[hσ.measurableSpace] (X n) := by - have hcut := cutoffStop_stopping _ σ hσ (n:ℝ≥0) - have hp := (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw - (cutoffStop σ n) hcut z).carg - exact (hp.mono (cutoffStop_past_le _ σ hσ n) le_rfl).indicator - (finiteStopEvent_measurable _ σ hσ n) - apply @measurable_of_tendsto_metrizable Ω ℝ hσ.measurableSpace _ _ _ _ X - (extendedMarkedAngle κ B γ G σ z) hX - apply tendsto_pi_nhds.mpr - intro ω - by_cases hs : σ ω=⊤ - · have he (n : ℕ) : X n ω=0 := by simp [X,finiteStopEvent,hs] - simpa only [he,extendedMarkedAngle,hs,ite_true] using - (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) - · have hf : σ ω<⊤ := lt_top_iff_ne_top.mpr hs - have hmem : ω∈⋃ n, finiteStopEvent σ n := by rw [iUnion_finiteStopEvent]; exact hf - obtain ⟨n,hn⟩ := mem_iUnion.mp hmem - apply tendsto_const_nhds.congr' - filter_upwards [eventually_ge_atTop n,cutoffStop_eventually_eq σ ω hf] with k hk he - dsimp only [X] - rw [indicator_of_mem (finiteStopEvent_mono σ hk hn), - stoppedMarkedCenter_time_congr κ B γ G (cutoffStop σ k) - (fun ω => (σ ω).untopD 0) z ω he] - exact ite_eq_right hs - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle +import OAI.Probability.SLE.Geometry.FiniteStoppedScalar + +namespace OAI + +/-! The full marked angle at a possibly infinite stopping time is +measurable in its stopped past, with value zero on the infinite event. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +def extendedMarkedAngle {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (σ : Ω → WithTop ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := + if σ ω=⊤ then 0 else + (stoppedMarkedCenter κ B γ G (fun ω => (σ ω).untopD 0) z ω).arg + +lemma stoppedMarkedCenter_time_congr {Ω : Type*} + (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (σ τ : Ω → ℝ≥0) (z : ℂ) (ω : Ω) (h : σ ω=τ ω) : + stoppedMarkedCenter κ B γ G σ z ω=stoppedMarkedCenter κ B γ G τ z ω := by + classical + exact congrArg (fun t => (if z∈survivingDomain (γ ω) t then G ω t z else 0)- + ((Real.sqrt κ*B t ω:ℝ):ℂ)) h + +lemma extendedMarkedAngle_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (σ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) σ) + (z : ℂ) : + Measurable[hσ.measurableSpace] (extendedMarkedAngle κ B γ G σ z) := by + let X := fun n : ℕ => (finiteStopEvent σ n).indicator (fun ω => + (stoppedMarkedCenter κ B γ G (cutoffStop σ n) z ω).arg) + have hX (n : ℕ) : Measurable[hσ.measurableSpace] (X n) := by + have hcut := cutoffStop_stopping _ σ hσ (n:ℝ≥0) + have hp := (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw + (cutoffStop σ n) hcut z).carg + exact (hp.mono (cutoffStop_past_le _ σ hσ n) le_rfl).indicator + (finiteStopEvent_measurable _ σ hσ n) + apply @measurable_of_tendsto_metrizable Ω ℝ hσ.measurableSpace _ _ _ _ X + (extendedMarkedAngle κ B γ G σ z) hX + apply tendsto_pi_nhds.mpr + intro ω + by_cases hs : σ ω=⊤ + · have he (n : ℕ) : X n ω=0 := by simp [X,finiteStopEvent,hs] + simpa only [he,extendedMarkedAngle,hs,ite_true] using + (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) + · have hf : σ ω<⊤ := lt_top_iff_ne_top.mpr hs + have hmem : ω∈⋃ n, finiteStopEvent σ n := by rw [iUnion_finiteStopEvent]; exact hf + obtain ⟨n,hn⟩ := mem_iUnion.mp hmem + apply tendsto_const_nhds.congr' + filter_upwards [eventually_ge_atTop n,cutoffStop_eventually_eq σ ω hf] with k hk he + dsimp only [X] + rw [indicator_of_mem (finiteStopEvent_mono σ hk hn), + stoppedMarkedCenter_time_congr κ B γ G (cutoffStop σ k) + (fun ω => (σ ω).untopD 0) z ω he] + exact ite_eq_right hs + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExtendedMarkedAngle.lean b/lean/OAI/Probability/SLE/Geometry/ExtendedMarkedAngle.lean index 5ed19ee48becdcb6c8f60569b0542f90d218aab3..bc906648334f81ae493410bc8db4bb0e651e283d 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExtendedMarkedAngle.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExtendedMarkedAngle.lean @@ -1,168 +1,168 @@ -import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular -import OAI.Probability.SLE.Angle.ActualAngleWarmup - -namespace OAI - -/-! The good-angle event at an actual radius level belongs to that literal -stopping time's past. Infinite stopping times are excluded explicitly. -Measurability follows by finite cutoff exhaustion of the stopped chart. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -/-- The stopped marked state at a finite extended stopping time. -/ -def extendedMarkedGood {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (σ : Ω → WithTop ℝ≥0) (z : ℂ) (s : ℝ) : Set Ω := - {ω | σ ω ≠ ⊤ ∧ ω ∈ stoppedMarkedGood κ B γ G (fun ω => (σ ω).untopD 0) z s} - -lemma stoppedMarkedGood_time_congr - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (τ₁ τ₂ : Ω → ℝ≥0) (z : ℂ) (s : ℝ) (ω : Ω) (he : τ₁ ω = τ₂ ω) : - ω ∈ stoppedMarkedGood κ B γ G τ₁ z s ↔ - ω ∈ stoppedMarkedGood κ B γ G τ₂ z s := by - classical - exact Iff.of_eq (congrArg (fun t : ℝ≥0 => z ∈ survivingDomain (γ ω) t ∧ - s ≤ (unitMarkedState ((if z ∈ survivingDomain (γ ω) t then G ω t z else 0) - - (Real.sqrt κ * B t ω : ℝ))).im) he) - -lemma extendedMarkedGood_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (σ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) σ) - (z : ℂ) (s : ℝ) : - MeasurableSet[hσ.measurableSpace] (extendedMarkedGood κ B γ G σ z s) := by - let E (n : ℕ) := stoppedMarkedGood κ B γ G (cutoffStop σ (n : ℝ≥0)) z s ∩ - {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} - have hE (n : ℕ) : MeasurableSet[hσ.measurableSpace] (E n) := by - have hcut := cutoffStop_stopping _ σ hσ (n : ℝ≥0) - have hpast := hcut.measurableSpace_mono hσ (fun ω => - (le_of_eq (cutoffStop_coe σ (n : ℝ≥0) ω)).trans (min_le_left _ _)) - exact (hpast _ (source_stoppedMarkedGood_measurable B hm hc κ γ G F hw - (cutoffStop σ (n : ℝ≥0)) hcut z s)).inter (hσ.measurableSet_le' (n : ℝ≥0)) - have he : extendedMarkedGood κ B γ G σ z s = ⋃ n, E n := by - ext ω - constructor - · rintro ⟨hf, hg⟩ - lift σ ω to ℝ≥0 using hf with t ht - obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) - have hle : σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) := by - rw [← ht] - exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) - apply mem_iUnion.mpr - refine ⟨n, ?_, hle⟩ - exact (stoppedMarkedGood_time_congr κ B γ G (cutoffStop σ (n : ℝ≥0)) - (fun ω => (σ ω).untopD 0) z s ω (cutoffStop_eq_on_success σ _ ω hle)).mpr hg - · intro hg - obtain ⟨n, hn, hle⟩ := mem_iUnion.mp hg - refine ⟨ne_top_of_le_ne_top WithTop.coe_ne_top hle, ?_⟩ - exact (stoppedMarkedGood_time_congr κ B γ G (cutoffStop σ (n : ℝ≥0)) - (fun ω => (σ ω).untopD 0) z s ω (cutoffStop_eq_on_success σ _ ω hle)).mp hn - rw [he] - exact MeasurableSet.iUnion hE - -lemma extendedMarkedGood_eq_actual_angle - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (σ : Ω → WithTop ℝ≥0) (z : ℂ) (s : ℝ) : - extendedMarkedGood κ B γ G σ z s = - {ω | σ ω ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) ∧ - s ≤ Real.sin (Complex.arg (G ω ((σ ω).untopD 0) z - - (Real.sqrt κ * B ((σ ω).untopD 0) ω : ℝ)))} := by - ext ω - change (σ ω ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) ∧ _) ↔ _ - simp only [mem_ofPred_eq] - by_cases hl : z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) - · rw [stoppedMarkedUnit_im_on_live B κ γ G (fun ω => (σ ω).untopD 0) z ω hl] - · simp only [hl, false_and, and_false] - -lemma canonical_good_angle_eq_extendedMarkedGood - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (u s : ℝ) (hu : 0 ≤ u) : - {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧ - s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} = - extendedMarkedGood κ B γ G - (fun ω => (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0)) z s := by - rw [canonical_good_angle_source_eq κ B hc γ G F hw z u s hu] - rw [extendedMarkedGood_eq_actual_angle κ B γ G - (fun ω => (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0)) z s] - ext ω - simp only [mem_ofPred_eq] - by_cases hf : radiusLevelTime (γ ω) (G ω) z u < ∞ - · obtain ⟨T, hT, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top (γ ω) (G ω) z u hf - have he := radiusLevelTime_eq_of_live_level ((hc ω).const_mul _) (hw ω) z u T hT hR - rw [he] - change ((T : ℝ≥0∞) < ∞ ∧ - s ≤ Real.sin (Complex.arg (G ω (T : ℝ≥0∞).toNNReal z - - (Real.sqrt κ * B (T : ℝ≥0∞).toNNReal ω : ℝ)))) ↔ - ((T : WithTop ℝ≥0) ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((T : WithTop ℝ≥0).untopD 0) ∧ - s ≤ Real.sin (Complex.arg (G ω ((T : WithTop ℝ≥0).untopD 0) z - - (Real.sqrt κ * B ((T : WithTop ℝ≥0).untopD 0) ω : ℝ)))) - simp only [ENNReal.coe_lt_top, ENNReal.toNNReal_coe, - WithTop.untopD_coe, hT, true_and] - constructor - · intro hs - exact ⟨WithTop.coe_ne_top, hs⟩ - · exact fun h => h.2 - · have he : radiusLevelTime (γ ω) (G ω) z u = ∞ := not_lt_top_iff.mp hf - simp only [he, lt_self_iff_false, false_and] - constructor - · exact False.elim - · intro h - exact h.1 rfl - -/-- The jointly measurable canonical angle event also belongs to the -literal actual radius stop's past. -/ -theorem canonical_good_angle_measurable_at_radius - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u s : ℝ) (hu : 0 ≤ u) : - MeasurableSet[(actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 - γ G F hw z hz u hu).measurableSpace] - {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧ - s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} := by - rw [canonical_good_angle_eq_extendedMarkedGood κ B hc γ G F hw z u s hu] - exact extendedMarkedGood_measurable B hm hc κ γ G F hw _ - (actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu) z s - -/-- The same stopped-past measurability for the literal geometric angle. -/ -theorem actual_good_angle_measurable_at_radius - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u s : ℝ) (hu : 0 ≤ u) : - MeasurableSet[(actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 - γ G F hw z hz u hu).measurableSpace] - {ω | radiusLevelTime (γ ω) (G ω) z u < ∞ ∧ - s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z - - (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω : ℝ)))} := by - rw [← canonical_good_angle_source_eq κ B hc γ G F hw z u s hu] - exact canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u s hu - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular +import OAI.Probability.SLE.Angle.ActualAngleWarmup + +namespace OAI + +/-! The good-angle event at an actual radius level belongs to that literal +stopping time's past. Infinite stopping times are excluded explicitly. +Measurability follows by finite cutoff exhaustion of the stopped chart. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +/-- The stopped marked state at a finite extended stopping time. -/ +def extendedMarkedGood {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (σ : Ω → WithTop ℝ≥0) (z : ℂ) (s : ℝ) : Set Ω := + {ω | σ ω ≠ ⊤ ∧ ω ∈ stoppedMarkedGood κ B γ G (fun ω => (σ ω).untopD 0) z s} + +lemma stoppedMarkedGood_time_congr + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (τ₁ τ₂ : Ω → ℝ≥0) (z : ℂ) (s : ℝ) (ω : Ω) (he : τ₁ ω = τ₂ ω) : + ω ∈ stoppedMarkedGood κ B γ G τ₁ z s ↔ + ω ∈ stoppedMarkedGood κ B γ G τ₂ z s := by + classical + exact Iff.of_eq (congrArg (fun t : ℝ≥0 => z ∈ survivingDomain (γ ω) t ∧ + s ≤ (unitMarkedState ((if z ∈ survivingDomain (γ ω) t then G ω t z else 0) - + (Real.sqrt κ * B t ω : ℝ))).im) he) + +lemma extendedMarkedGood_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (σ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) σ) + (z : ℂ) (s : ℝ) : + MeasurableSet[hσ.measurableSpace] (extendedMarkedGood κ B γ G σ z s) := by + let E (n : ℕ) := stoppedMarkedGood κ B γ G (cutoffStop σ (n : ℝ≥0)) z s ∩ + {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} + have hE (n : ℕ) : MeasurableSet[hσ.measurableSpace] (E n) := by + have hcut := cutoffStop_stopping _ σ hσ (n : ℝ≥0) + have hpast := hcut.measurableSpace_mono hσ (fun ω => + (le_of_eq (cutoffStop_coe σ (n : ℝ≥0) ω)).trans (min_le_left _ _)) + exact (hpast _ (source_stoppedMarkedGood_measurable B hm hc κ γ G F hw + (cutoffStop σ (n : ℝ≥0)) hcut z s)).inter (hσ.measurableSet_le' (n : ℝ≥0)) + have he : extendedMarkedGood κ B γ G σ z s = ⋃ n, E n := by + ext ω + constructor + · rintro ⟨hf, hg⟩ + lift σ ω to ℝ≥0 using hf with t ht + obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) + have hle : σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) := by + rw [← ht] + exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) + apply mem_iUnion.mpr + refine ⟨n, ?_, hle⟩ + exact (stoppedMarkedGood_time_congr κ B γ G (cutoffStop σ (n : ℝ≥0)) + (fun ω => (σ ω).untopD 0) z s ω (cutoffStop_eq_on_success σ _ ω hle)).mpr hg + · intro hg + obtain ⟨n, hn, hle⟩ := mem_iUnion.mp hg + refine ⟨ne_top_of_le_ne_top WithTop.coe_ne_top hle, ?_⟩ + exact (stoppedMarkedGood_time_congr κ B γ G (cutoffStop σ (n : ℝ≥0)) + (fun ω => (σ ω).untopD 0) z s ω (cutoffStop_eq_on_success σ _ ω hle)).mp hn + rw [he] + exact MeasurableSet.iUnion hE + +lemma extendedMarkedGood_eq_actual_angle + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (σ : Ω → WithTop ℝ≥0) (z : ℂ) (s : ℝ) : + extendedMarkedGood κ B γ G σ z s = + {ω | σ ω ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) ∧ + s ≤ Real.sin (Complex.arg (G ω ((σ ω).untopD 0) z - + (Real.sqrt κ * B ((σ ω).untopD 0) ω : ℝ)))} := by + ext ω + change (σ ω ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) ∧ _) ↔ _ + simp only [mem_ofPred_eq] + by_cases hl : z ∈ survivingDomain (γ ω) ((σ ω).untopD 0) + · rw [stoppedMarkedUnit_im_on_live B κ γ G (fun ω => (σ ω).untopD 0) z ω hl] + · simp only [hl, false_and, and_false] + +lemma canonical_good_angle_eq_extendedMarkedGood + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (u s : ℝ) (hu : 0 ≤ u) : + {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧ + s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} = + extendedMarkedGood κ B γ G + (fun ω => (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0)) z s := by + rw [canonical_good_angle_source_eq κ B hc γ G F hw z u s hu] + rw [extendedMarkedGood_eq_actual_angle κ B γ G + (fun ω => (radiusLevelTime (γ ω) (G ω) z u : WithTop ℝ≥0)) z s] + ext ω + simp only [mem_ofPred_eq] + by_cases hf : radiusLevelTime (γ ω) (G ω) z u < ∞ + · obtain ⟨T, hT, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top (γ ω) (G ω) z u hf + have he := radiusLevelTime_eq_of_live_level ((hc ω).const_mul _) (hw ω) z u T hT hR + rw [he] + change ((T : ℝ≥0∞) < ∞ ∧ + s ≤ Real.sin (Complex.arg (G ω (T : ℝ≥0∞).toNNReal z - + (Real.sqrt κ * B (T : ℝ≥0∞).toNNReal ω : ℝ)))) ↔ + ((T : WithTop ℝ≥0) ≠ ⊤ ∧ z ∈ survivingDomain (γ ω) ((T : WithTop ℝ≥0).untopD 0) ∧ + s ≤ Real.sin (Complex.arg (G ω ((T : WithTop ℝ≥0).untopD 0) z - + (Real.sqrt κ * B ((T : WithTop ℝ≥0).untopD 0) ω : ℝ)))) + simp only [ENNReal.coe_lt_top, ENNReal.toNNReal_coe, + WithTop.untopD_coe, hT, true_and] + constructor + · intro hs + exact ⟨WithTop.coe_ne_top, hs⟩ + · exact fun h => h.2 + · have he : radiusLevelTime (γ ω) (G ω) z u = ∞ := not_lt_top_iff.mp hf + simp only [he, lt_self_iff_false, false_and] + constructor + · exact False.elim + · intro h + exact h.1 rfl + +/-- The jointly measurable canonical angle event also belongs to the +literal actual radius stop's past. -/ +theorem canonical_good_angle_measurable_at_radius + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u s : ℝ) (hu : 0 ≤ u) : + MeasurableSet[(actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 + γ G F hw z hz u hu).measurableSpace] + {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧ + s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} := by + rw [canonical_good_angle_eq_extendedMarkedGood κ B hc γ G F hw z u s hu] + exact extendedMarkedGood_measurable B hm hc κ γ G F hw _ + (actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu) z s + +/-- The same stopped-past measurability for the literal geometric angle. -/ +theorem actual_good_angle_measurable_at_radius + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u s : ℝ) (hu : 0 ≤ u) : + MeasurableSet[(actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 + γ G F hw z hz u hu).measurableSpace] + {ω | radiusLevelTime (γ ω) (G ω) z u < ∞ ∧ + s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z - + (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω : ℝ)))} := by + rw [← canonical_good_angle_source_eq κ B hc γ G F hw z u s hu] + exact canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u s hu + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExtendedStopLocalization.lean b/lean/OAI/Probability/SLE/Geometry/ExtendedStopLocalization.lean index e727c863d4567db6e2f4af47aad4fa016dc3bf76..05ea63a651826d590dee4d3d974ab6809af647b9 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExtendedStopLocalization.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExtendedStopLocalization.lean @@ -1,123 +1,123 @@ -import OAI.Probability.SLE.Gauge.ExtendedExperimentStop - -namespace OAI - -/-! Finite localization of a possibly infinite stopping time. The value -`S.untopD 0` is used only on the event where `S` is finite; it is not asserted -to be a stopping time on the whole sample space. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -variable {Ω : Type*} - -def finiteStopEvent (S : Ω → WithTop ℝ≥0) (n : ℕ) : Set Ω := - {ω | S ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} - -lemma cutoffStop_le_horizon (S : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : - cutoffStop S H ω ≤ H := by - exact WithTop.coe_le_coe.mp ((cutoffStop_coe S H ω).trans_le (min_le_right _ _)) - -lemma cutoffStop_le_stop (S : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : - (cutoffStop S H ω : WithTop ℝ≥0) ≤ S ω := by - rw [cutoffStop_coe] - exact min_le_left _ _ - -lemma cutoffStop_monotone (S : Ω → WithTop ℝ≥0) (ω : Ω) : - Monotone (fun H : ℝ≥0 => cutoffStop S H ω) := by - intro H K hHK - apply WithTop.coe_le_coe.mp - rw [cutoffStop_coe,cutoffStop_coe] - exact min_le_min_left _ (WithTop.coe_le_coe.mpr hHK) - -lemma finiteStopEvent_mono (S : Ω → WithTop ℝ≥0) : Monotone (finiteStopEvent S) := by - intro n k hnk ω hω - change S ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) at hω - change S ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0) - exact hω.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk)) - -lemma iUnion_finiteStopEvent (S : Ω → WithTop ℝ≥0) : - (⋃ n, finiteStopEvent S n) = {ω | S ω < ⊤} := by - ext ω - simp only [mem_iUnion,finiteStopEvent,mem_ofPred_eq] - constructor - · rintro ⟨n,hn⟩ - exact hn.trans_lt (WithTop.coe_lt_top _) - · intro hω - lift S ω to ℝ≥0 using hω.ne with t ht - obtain ⟨n,hn⟩ := exists_nat_ge (t : ℝ) - refine ⟨n,?_⟩ - exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) - -lemma cutoffStop_eqOn_finiteStopEvent (S : Ω → WithTop ℝ≥0) (n : ℕ) : - EqOn (cutoffStop S n) (fun ω => (S ω).untopD 0) (finiteStopEvent S n) := by - intro ω hω - exact cutoffStop_eq_on_success S n ω hω - -lemma cutoffStop_eventually_eq (S : Ω → WithTop ℝ≥0) (ω : Ω) (hω : S ω < ⊤) : - ∀ᶠ n : ℕ in atTop, cutoffStop S n ω = (S ω).untopD 0 := by - have hmem : ω ∈ ⋃ n, finiteStopEvent S n := by - rw [iUnion_finiteStopEvent S] - exact hω - obtain ⟨n,hn⟩ := mem_iUnion.mp hmem - filter_upwards [eventually_ge_atTop n] with k hk - exact cutoffStop_eqOn_finiteStopEvent S k (finiteStopEvent_mono S hk hn) - -section Measurable - -variable [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (S : Ω → WithTop ℝ≥0) (hS : IsStoppingTime ℱ S) - -lemma finiteStopEvent_measurable (n : ℕ) : - MeasurableSet[hS.measurableSpace] (finiteStopEvent S n) := - hS.measurableSet_le' n - -lemma finiteStopEvent_measurable_cutoff (n : ℕ) : - MeasurableSet[(cutoffStop_stopping ℱ S hS n).measurableSpace] (finiteStopEvent S n) := by - simpa only [Set.univ_inter,finiteStopEvent] using - cutoffStop_success_measurable ℱ S hS n Set.univ MeasurableSet.univ - -lemma finiteStopEvent_inter_measurable_cutoff (n : ℕ) (E : Set Ω) - (hE : MeasurableSet[hS.measurableSpace] E) : - MeasurableSet[(cutoffStop_stopping ℱ S hS n).measurableSpace] (E ∩ finiteStopEvent S n) := - cutoffStop_success_measurable ℱ S hS n E hE - -lemma finiteStopEvent_all_measurable : - MeasurableSet[hS.measurableSpace] {ω | S ω < ⊤} := by - rw [← iUnion_finiteStopEvent S] - exact MeasurableSet.iUnion (finiteStopEvent_measurable ℱ S hS) - -lemma cutoffStop_past_le (n : ℕ) : - (cutoffStop_stopping ℱ S hS n).measurableSpace ≤ hS.measurableSpace := - IsStoppingTime.measurableSpace_mono _ hS (cutoffStop_le_stop S n) - -lemma ae_of_cutoffStop (P : Measure Ω) (Q : Ω → ℝ≥0 → Prop) - (hQ : ∀ n : ℕ, ∀ᵐ ω ∂P, Q ω (cutoffStop S n ω)) : - ∀ᵐ ω ∂P, S ω < ⊤ → Q ω ((S ω).untopD 0) := by - filter_upwards [ae_all_iff.mpr hQ] with ω hω hfinite - have hmem : ω ∈ ⋃ n, finiteStopEvent S n := by - rw [iUnion_finiteStopEvent S] - exact hfinite - obtain ⟨n,hn⟩ := mem_iUnion.mp hmem - simpa only [cutoffStop_eqOn_finiteStopEvent S n hn] using hω n - -end Measurable - -lemma sourceRestartDriverPath_eq_on_finiteStopEvent - (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) - (S : Ω → WithTop ℝ≥0) (n : ℕ) (ω : Ω) (hω : ω ∈ finiteStopEvent S n) : - sourceRestartDriverPath B hc (cutoffStop S n) ω = - sourceRestartDriverPath B hc (fun ω => (S ω).untopD 0) ω := by - ext t - change B (cutoffStop S n ω+t) ω-B (cutoffStop S n ω) ω = - B ((S ω).untopD 0+t) ω-B ((S ω).untopD 0) ω - rw [cutoffStop_eqOn_finiteStopEvent S n hω] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ExtendedExperimentStop + +namespace OAI + +/-! Finite localization of a possibly infinite stopping time. The value +`S.untopD 0` is used only on the event where `S` is finite; it is not asserted +to be a stopping time on the whole sample space. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +variable {Ω : Type*} + +def finiteStopEvent (S : Ω → WithTop ℝ≥0) (n : ℕ) : Set Ω := + {ω | S ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} + +lemma cutoffStop_le_horizon (S : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : + cutoffStop S H ω ≤ H := by + exact WithTop.coe_le_coe.mp ((cutoffStop_coe S H ω).trans_le (min_le_right _ _)) + +lemma cutoffStop_le_stop (S : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : + (cutoffStop S H ω : WithTop ℝ≥0) ≤ S ω := by + rw [cutoffStop_coe] + exact min_le_left _ _ + +lemma cutoffStop_monotone (S : Ω → WithTop ℝ≥0) (ω : Ω) : + Monotone (fun H : ℝ≥0 => cutoffStop S H ω) := by + intro H K hHK + apply WithTop.coe_le_coe.mp + rw [cutoffStop_coe,cutoffStop_coe] + exact min_le_min_left _ (WithTop.coe_le_coe.mpr hHK) + +lemma finiteStopEvent_mono (S : Ω → WithTop ℝ≥0) : Monotone (finiteStopEvent S) := by + intro n k hnk ω hω + change S ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) at hω + change S ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0) + exact hω.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk)) + +lemma iUnion_finiteStopEvent (S : Ω → WithTop ℝ≥0) : + (⋃ n, finiteStopEvent S n) = {ω | S ω < ⊤} := by + ext ω + simp only [mem_iUnion,finiteStopEvent,mem_ofPred_eq] + constructor + · rintro ⟨n,hn⟩ + exact hn.trans_lt (WithTop.coe_lt_top _) + · intro hω + lift S ω to ℝ≥0 using hω.ne with t ht + obtain ⟨n,hn⟩ := exists_nat_ge (t : ℝ) + refine ⟨n,?_⟩ + exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) + +lemma cutoffStop_eqOn_finiteStopEvent (S : Ω → WithTop ℝ≥0) (n : ℕ) : + EqOn (cutoffStop S n) (fun ω => (S ω).untopD 0) (finiteStopEvent S n) := by + intro ω hω + exact cutoffStop_eq_on_success S n ω hω + +lemma cutoffStop_eventually_eq (S : Ω → WithTop ℝ≥0) (ω : Ω) (hω : S ω < ⊤) : + ∀ᶠ n : ℕ in atTop, cutoffStop S n ω = (S ω).untopD 0 := by + have hmem : ω ∈ ⋃ n, finiteStopEvent S n := by + rw [iUnion_finiteStopEvent S] + exact hω + obtain ⟨n,hn⟩ := mem_iUnion.mp hmem + filter_upwards [eventually_ge_atTop n] with k hk + exact cutoffStop_eqOn_finiteStopEvent S k (finiteStopEvent_mono S hk hn) + +section Measurable + +variable [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (S : Ω → WithTop ℝ≥0) (hS : IsStoppingTime ℱ S) + +lemma finiteStopEvent_measurable (n : ℕ) : + MeasurableSet[hS.measurableSpace] (finiteStopEvent S n) := + hS.measurableSet_le' n + +lemma finiteStopEvent_measurable_cutoff (n : ℕ) : + MeasurableSet[(cutoffStop_stopping ℱ S hS n).measurableSpace] (finiteStopEvent S n) := by + simpa only [Set.univ_inter,finiteStopEvent] using + cutoffStop_success_measurable ℱ S hS n Set.univ MeasurableSet.univ + +lemma finiteStopEvent_inter_measurable_cutoff (n : ℕ) (E : Set Ω) + (hE : MeasurableSet[hS.measurableSpace] E) : + MeasurableSet[(cutoffStop_stopping ℱ S hS n).measurableSpace] (E ∩ finiteStopEvent S n) := + cutoffStop_success_measurable ℱ S hS n E hE + +lemma finiteStopEvent_all_measurable : + MeasurableSet[hS.measurableSpace] {ω | S ω < ⊤} := by + rw [← iUnion_finiteStopEvent S] + exact MeasurableSet.iUnion (finiteStopEvent_measurable ℱ S hS) + +lemma cutoffStop_past_le (n : ℕ) : + (cutoffStop_stopping ℱ S hS n).measurableSpace ≤ hS.measurableSpace := + IsStoppingTime.measurableSpace_mono _ hS (cutoffStop_le_stop S n) + +lemma ae_of_cutoffStop (P : Measure Ω) (Q : Ω → ℝ≥0 → Prop) + (hQ : ∀ n : ℕ, ∀ᵐ ω ∂P, Q ω (cutoffStop S n ω)) : + ∀ᵐ ω ∂P, S ω < ⊤ → Q ω ((S ω).untopD 0) := by + filter_upwards [ae_all_iff.mpr hQ] with ω hω hfinite + have hmem : ω ∈ ⋃ n, finiteStopEvent S n := by + rw [iUnion_finiteStopEvent S] + exact hfinite + obtain ⟨n,hn⟩ := mem_iUnion.mp hmem + simpa only [cutoffStop_eqOn_finiteStopEvent S n hn] using hω n + +end Measurable + +lemma sourceRestartDriverPath_eq_on_finiteStopEvent + (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) + (S : Ω → WithTop ℝ≥0) (n : ℕ) (ω : Ω) (hω : ω ∈ finiteStopEvent S n) : + sourceRestartDriverPath B hc (cutoffStop S n) ω = + sourceRestartDriverPath B hc (fun ω => (S ω).untopD 0) ω := by + ext t + change B (cutoffStop S n ω+t) ω-B (cutoffStop S n ω) ω = + B ((S ω).untopD 0+t) ω-B ((S ω).untopD 0) ω + rw [cutoffStop_eqOn_finiteStopEvent S n hω] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorCapacityRemainder.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorCapacityRemainder.lean index 8e17a0c68390007eb89c6d8373acb6478b09dd66..290bc272f6f6f9c7026ade969072632ce01f4883 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorCapacityRemainder.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorCapacityRemainder.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Geometry.ExteriorHerglotzPositive -import OAI.Probability.SLE.Geometry.DiskKernelRemainder - -namespace OAI - -/-! Capacity-relative remainder for a hydrodynamically normalized -reflected slit contained in a small disk. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem exterior_capacity_remainder {G : ℂ → ℂ} {R c : ℝ} - (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) - (hs : ∀ z : ℂ, R≤‖z‖ → z.im*(G z-z).im≤0) - {z : ℂ} (hz : 2*R≤‖z‖) : - ‖G z-z-(c:ℂ)/z‖≤10*c*R/‖z‖^2 := by - have hn : 0<‖z‖ := lt_of_lt_of_le (by linarith : 0<2*R) hz - have hz0 : z≠0 := norm_pos_iff.mp hn - have hR0 : (R:ℂ)≠0 := ofReal_ne_zero.mpr hR.ne' - let w : ℂ := (R:ℂ)/z - have hw0 : w≠0 := div_ne_zero hR0 hz0 - have hwn : ‖w‖=R/‖z‖ := by - simp only [w,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR] - have hw : ‖w‖≤1/2 := by rw [hwn]; apply (div_le_iff₀ hn).mpr; linarith - have hH := exteriorHerglotzTransform_differentiableOn hR hG hc - have hpos := exteriorHerglotzTransform_nonneg hR hG hc hs - have hb := positive_real_disk_kernel_remainder (div_pos hc0 hR) - (hH.mono ball_subset_closedBall) (by simp only [exteriorHerglotzTransform_zero,ofReal_div]) - (fun v hv => hpos v (ball_subset_closedBall hv)) hw - have hw2 : (1:ℂ)-w^2≠0 := by - have hlt : ‖w^2‖<1 := by rw [norm_pow]; nlinarith [norm_nonneg w] - intro he - have he' : w^2=1 := (sub_eq_zero.mp he).symm - rw [he',norm_one] at hlt - exact lt_irrefl _ hlt - have hinv : (R:ℂ)/w=z := by dsimp only [w]; field_simp - have he : w*exteriorHerglotzTransform G R c w/(1-w^2)-(c/R:ℝ)*w= - G z-z-(c:ℂ)/z := by - have hm : (c/R:ℝ)*w=(c:ℂ)/z := by - dsimp only [w] - rw [ofReal_div] - field_simp [hR0] - rw [hm,exteriorHerglotzTransform_eq_raw hw0] - unfold exteriorHerglotzRaw - rw [hinv] - congr 1 - field_simp [hw0,hw2] - rw [he,hwn] at hb - have hbR : 10*(c/R)*(R/‖z‖)^2=10*c*R/‖z‖^2 := by - field_simp [hR.ne',hn.ne'] - rwa [hbR] at hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorHerglotzPositive +import OAI.Probability.SLE.Geometry.DiskKernelRemainder + +namespace OAI + +/-! Capacity-relative remainder for a hydrodynamically normalized +reflected slit contained in a small disk. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +theorem exterior_capacity_remainder {G : ℂ → ℂ} {R c : ℝ} + (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) + (hs : ∀ z : ℂ, R≤‖z‖ → z.im*(G z-z).im≤0) + {z : ℂ} (hz : 2*R≤‖z‖) : + ‖G z-z-(c:ℂ)/z‖≤10*c*R/‖z‖^2 := by + have hn : 0<‖z‖ := lt_of_lt_of_le (by linarith : 0<2*R) hz + have hz0 : z≠0 := norm_pos_iff.mp hn + have hR0 : (R:ℂ)≠0 := ofReal_ne_zero.mpr hR.ne' + let w : ℂ := (R:ℂ)/z + have hw0 : w≠0 := div_ne_zero hR0 hz0 + have hwn : ‖w‖=R/‖z‖ := by + simp only [w,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR] + have hw : ‖w‖≤1/2 := by rw [hwn]; apply (div_le_iff₀ hn).mpr; linarith + have hH := exteriorHerglotzTransform_differentiableOn hR hG hc + have hpos := exteriorHerglotzTransform_nonneg hR hG hc hs + have hb := positive_real_disk_kernel_remainder (div_pos hc0 hR) + (hH.mono ball_subset_closedBall) (by simp only [exteriorHerglotzTransform_zero,ofReal_div]) + (fun v hv => hpos v (ball_subset_closedBall hv)) hw + have hw2 : (1:ℂ)-w^2≠0 := by + have hlt : ‖w^2‖<1 := by rw [norm_pow]; nlinarith [norm_nonneg w] + intro he + have he' : w^2=1 := (sub_eq_zero.mp he).symm + rw [he',norm_one] at hlt + exact lt_irrefl _ hlt + have hinv : (R:ℂ)/w=z := by dsimp only [w]; field_simp + have he : w*exteriorHerglotzTransform G R c w/(1-w^2)-(c/R:ℝ)*w= + G z-z-(c:ℂ)/z := by + have hm : (c/R:ℝ)*w=(c:ℂ)/z := by + dsimp only [w] + rw [ofReal_div] + field_simp [hR0] + rw [hm,exteriorHerglotzTransform_eq_raw hw0] + unfold exteriorHerglotzRaw + rw [hinv] + congr 1 + field_simp [hw0,hw2] + rw [he,hwn] at hb + have hbR : 10*(c/R)*(R/‖z‖)^2=10*c*R/‖z‖^2 := by + field_simp [hR.ne',hn.ne'] + rwa [hbR] at hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorCompactFlux.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorCompactFlux.lean index 44db8d9f7700516be59a70f67c07e1ac62ec7dd3..5a47c1507ddf9fb05b33c5bb733f96b3a27176b1 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorCompactFlux.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorCompactFlux.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Geometry.AnnulusPolarIntegral - -namespace OAI - -/-! A compactly supported test with nonpositive exterior Laplacian -integral has nonnegative inward boundary flux. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma exterior_compact_flux_nonneg {v : ℂ → ℝ} {a : ℝ} (ha : 0 hz.1) - intro z hz - apply hzero z - exact le_of_not_gt (fun h => hz.2 ⟨hz.1,h⟩) - have hrzero (θ : ℝ) : radialLaplacianFlux v (R,θ)=0 := by - have hn : R≤‖polarPoint R θ‖ := by - rw [polarPoint_norm,abs_of_pos (ha.trans_le haR)] - simp only [radialLaplacianFlux,fderiv_of_notMem_tsupport ℝ (hnot _ hn), - zero_apply,mul_zero] - have he := annulus_integral_laplacian_flux ha haR (fun z hz _ => hv z hz) - rw [hrestrict,he] at hneg - simp only [hrzero,intervalIntegral.integral_zero,zero_sub] at hneg - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.AnnulusPolarIntegral + +namespace OAI + +/-! A compactly supported test with nonpositive exterior Laplacian +integral has nonnegative inward boundary flux. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma exterior_compact_flux_nonneg {v : ℂ → ℝ} {a : ℝ} (ha : 0 hz.1) + intro z hz + apply hzero z + exact le_of_not_gt (fun h => hz.2 ⟨hz.1,h⟩) + have hrzero (θ : ℝ) : radialLaplacianFlux v (R,θ)=0 := by + have hn : R≤‖polarPoint R θ‖ := by + rw [polarPoint_norm,abs_of_pos (ha.trans_le haR)] + simp only [radialLaplacianFlux,fderiv_of_notMem_tsupport ℝ (hnot _ hn), + zero_apply,mul_zero] + have he := annulus_integral_laplacian_flux ha haR (fun z hz _ => hv z hz) + rw [hrestrict,he] at hneg + simp only [hrzero,intervalIntegral.integral_zero,zero_sub] at hneg + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorFirstCoefficient.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorFirstCoefficient.lean index d203acf6bffc461803e79479965301c23eec1fb9..79ee5f1d4ac3dbaf670ea7c7871de98763c28b55 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorFirstCoefficient.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorFirstCoefficient.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Geometry.HolomorphicAreaAverage -import OAI.Probability.SLE.Geometry.AnalyticFluxEnergy - -namespace OAI - -/-! The first inverse-radius coefficient of a univalent exterior map is -bounded by its nonnegative signed boundary area. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real -open scoped Topology Laplacian ComplexConjugate - -namespace SLEExactGauge - -lemma exterior_first_coefficient_le_one {f C : ℂ → ℂ} - (hf : DifferentiableOn ℂ f {z : ℂ | 1<‖z‖}) - (hC : DifferentiableOn ℂ C (ball 0 1)) - (he : ∀ z : ℂ, 1<‖z‖ → f z=z+C z⁻¹) - (hflux : ∀ r : ℝ, 1 ‖f z‖^2) (r,θ)) : - ‖deriv C 0‖≤1 := by - have hbound (r : ℝ) (hr : 1 z+C z⁻¹) := by - filter_upwards [(isOpen_lt continuous_const continuous_norm).mem_nhds - (show 1<‖z‖ by rwa [hzr])] with w hw - exact he w hw - have hh := (hd.congr_of_eventuallyEq heloc).deriv - convert! hh using 1 - ring - have harea := inverse_chart_area_average hr0 hCa hval hder - have hfp := hflux r hr - rw [holomorphic_flux_eq_area_average hfa,harea] at hfp - have hcp := analytic_flux_ge_first_coefficient hri hCa - rw [holomorphic_flux_eq_area_average (hCa.mono (by - rw [abs_of_pos hri]; exact sphere_subset_closedBall))] at hcp - have hs : r⁻¹^2*‖deriv C 0‖^2≤r^2 := by - apply (mul_le_mul_iff_right₀ (show 0<4*Real.pi by positivity)).mp - nlinarith only [hfp,hcp] - have hrnorm : r⁻¹*‖deriv C 0‖≤r := by - apply (sq_le_sq₀ (mul_nonneg hri.le (norm_nonneg _)) hr0.le).mp - nlinarith only [hs] - have hh : ‖deriv C 0‖≤r*r := by - apply (div_le_iff₀ hr0).mp - simpa only [div_eq_mul_inv,mul_comm] using hrnorm - simpa only [pow_two] using hh - let r (n : ℕ) := 1+1/((n:ℝ)+1) - have hr (n : ℕ) : 1 (r n)^2) atTop (𝓝 (1:ℝ)) := by simpa using ht.pow 2 - exact ge_of_tendsto ht2 (Eventually.of_forall (fun n => hbound (r n) (hr n))) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HolomorphicAreaAverage +import OAI.Probability.SLE.Geometry.AnalyticFluxEnergy + +namespace OAI + +/-! The first inverse-radius coefficient of a univalent exterior map is +bounded by its nonnegative signed boundary area. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real +open scoped Topology Laplacian ComplexConjugate + +namespace SLEExactGauge + +lemma exterior_first_coefficient_le_one {f C : ℂ → ℂ} + (hf : DifferentiableOn ℂ f {z : ℂ | 1<‖z‖}) + (hC : DifferentiableOn ℂ C (ball 0 1)) + (he : ∀ z : ℂ, 1<‖z‖ → f z=z+C z⁻¹) + (hflux : ∀ r : ℝ, 1 ‖f z‖^2) (r,θ)) : + ‖deriv C 0‖≤1 := by + have hbound (r : ℝ) (hr : 1 z+C z⁻¹) := by + filter_upwards [(isOpen_lt continuous_const continuous_norm).mem_nhds + (show 1<‖z‖ by rwa [hzr])] with w hw + exact he w hw + have hh := (hd.congr_of_eventuallyEq heloc).deriv + convert! hh using 1 + ring + have harea := inverse_chart_area_average hr0 hCa hval hder + have hfp := hflux r hr + rw [holomorphic_flux_eq_area_average hfa,harea] at hfp + have hcp := analytic_flux_ge_first_coefficient hri hCa + rw [holomorphic_flux_eq_area_average (hCa.mono (by + rw [abs_of_pos hri]; exact sphere_subset_closedBall))] at hcp + have hs : r⁻¹^2*‖deriv C 0‖^2≤r^2 := by + apply (mul_le_mul_iff_right₀ (show 0<4*Real.pi by positivity)).mp + nlinarith only [hfp,hcp] + have hrnorm : r⁻¹*‖deriv C 0‖≤r := by + apply (sq_le_sq₀ (mul_nonneg hri.le (norm_nonneg _)) hr0.le).mp + nlinarith only [hs] + have hh : ‖deriv C 0‖≤r*r := by + apply (div_le_iff₀ hr0).mp + simpa only [div_eq_mul_inv,mul_comm] using hrnorm + simpa only [pow_two] using hh + let r (n : ℕ) := 1+1/((n:ℝ)+1) + have hr (n : ℕ) : 1 (r n)^2) atTop (𝓝 (1:ℝ)) := by simpa using ht.pow 2 + exact ge_of_tendsto ht2 (Eventually.of_forall (fun n => hbound (r n) (hr n))) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzAnalytic.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzAnalytic.lean index 41690acdf7fbcc69bdc0149196ad43855fe96fba..e8e51d3343a4dc0ebfe1aef2eadf000bbbfce8b5 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzAnalytic.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzAnalytic.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Geometry.ExteriorHerglotzTransform - -namespace OAI - -/-! The hydrodynamic singularity of the inverted transform is removable. -Only analyticity outside the chosen circle is required. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma exteriorHerglotzTransform_eq_raw {G : ℂ → ℂ} {R c : ℝ} {w : ℂ} - (hw : w≠0) : exteriorHerglotzTransform G R c w=exteriorHerglotzRaw G R w := - Function.update_of_ne hw _ _ - -lemma exteriorHerglotzTransform_differentiableAt_ne_zero {G : ℂ → ℂ} {R c : ℝ} - (hR : 0 (R:ℂ)/v) w := - (differentiableAt_const (R:ℂ)).div differentiableAt_id hw - have hd : DifferentiableAt ℂ (exteriorHerglotzRaw G R) w := - ((((differentiableAt_const (1:ℂ)).sub (differentiableAt_id.pow 2)).div - differentiableAt_id hw).mul (((hG _ hz).comp w hi).sub hi)) - apply hd.congr_of_eventuallyEq - filter_upwards [isOpen_ne.mem_nhds hw] with v hv - exact exteriorHerglotzTransform_eq_raw hv - -theorem exteriorHerglotzTransform_differentiableOn {G : ℂ → ℂ} {R c : ℝ} - (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) : - DifferentiableOn ℂ (exteriorHerglotzTransform G R c) (closedBall 0 1) := by - have hzero : DifferentiableAt ℂ (exteriorHerglotzTransform G R c) 0 := by - apply (Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt ?_ - (exteriorHerglotzTransform_continuousAt_zero hR.ne' hc)).differentiableAt - have hb : ∀ᶠ w : ℂ in 𝓝[≠] (0:ℂ), w∈ball 0 1 := - Filter.Eventually.filter_mono inf_le_left (ball_mem_nhds (0:ℂ) zero_lt_one) - filter_upwards [self_mem_nhdsWithin,hb] with w hw hw1 - exact exteriorHerglotzTransform_differentiableAt_ne_zero hR hG hw - (mem_ball_zero_iff.mp hw1).le - intro w hw - rcases eq_or_ne w 0 with rfl|hne - · exact hzero.differentiableWithinAt - · exact (exteriorHerglotzTransform_differentiableAt_ne_zero hR hG hne - (mem_closedBall_zero_iff.mp hw)).differentiableWithinAt - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorHerglotzTransform + +namespace OAI + +/-! The hydrodynamic singularity of the inverted transform is removable. +Only analyticity outside the chosen circle is required. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma exteriorHerglotzTransform_eq_raw {G : ℂ → ℂ} {R c : ℝ} {w : ℂ} + (hw : w≠0) : exteriorHerglotzTransform G R c w=exteriorHerglotzRaw G R w := + Function.update_of_ne hw _ _ + +lemma exteriorHerglotzTransform_differentiableAt_ne_zero {G : ℂ → ℂ} {R c : ℝ} + (hR : 0 (R:ℂ)/v) w := + (differentiableAt_const (R:ℂ)).div differentiableAt_id hw + have hd : DifferentiableAt ℂ (exteriorHerglotzRaw G R) w := + ((((differentiableAt_const (1:ℂ)).sub (differentiableAt_id.pow 2)).div + differentiableAt_id hw).mul (((hG _ hz).comp w hi).sub hi)) + apply hd.congr_of_eventuallyEq + filter_upwards [isOpen_ne.mem_nhds hw] with v hv + exact exteriorHerglotzTransform_eq_raw hv + +theorem exteriorHerglotzTransform_differentiableOn {G : ℂ → ℂ} {R c : ℝ} + (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) : + DifferentiableOn ℂ (exteriorHerglotzTransform G R c) (closedBall 0 1) := by + have hzero : DifferentiableAt ℂ (exteriorHerglotzTransform G R c) 0 := by + apply (Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt ?_ + (exteriorHerglotzTransform_continuousAt_zero hR.ne' hc)).differentiableAt + have hb : ∀ᶠ w : ℂ in 𝓝[≠] (0:ℂ), w∈ball 0 1 := + Filter.Eventually.filter_mono inf_le_left (ball_mem_nhds (0:ℂ) zero_lt_one) + filter_upwards [self_mem_nhdsWithin,hb] with w hw hw1 + exact exteriorHerglotzTransform_differentiableAt_ne_zero hR hG hw + (mem_ball_zero_iff.mp hw1).le + intro w hw + rcases eq_or_ne w 0 with rfl|hne + · exact hzero.differentiableWithinAt + · exact (exteriorHerglotzTransform_differentiableAt_ne_zero hR hG hne + (mem_closedBall_zero_iff.mp hw)).differentiableWithinAt + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzPositive.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzPositive.lean index bf381b97c6a404f573d4b06fbf5e4ac07d504bd8..2938a5808380f76eecdea5af2567920d7079a105 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzPositive.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzPositive.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Geometry.ExteriorHerglotzAnalytic -import OAI.Probability.SLE.Geometry.PositiveRealDiskEstimate - -namespace OAI - -/-! The normalized height inequality becomes positivity of the real -part after inversion. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma exteriorHerglotzTransform_sphere_re {G : ℂ → ℂ} {R c : ℝ} - {w : ℂ} (hw : ‖w‖=1) : - (exteriorHerglotzTransform G R c w).re= - 2*w.im*(G ((R:ℂ)/w)-(R:ℂ)/w).im := by - have hw0 : w≠0 := by intro he; simp [he] at hw - have he : ((1:ℂ)-w^2)/w=conj w-w := by - rw [← Complex.inv_eq_conj hw] - field_simp [hw0] - rw [exteriorHerglotzTransform_eq_raw hw0] - simp only [exteriorHerglotzRaw,he,mul_re,sub_re,sub_im,conj_re,conj_im] - ring - -theorem exteriorHerglotzTransform_nonneg {G : ℂ → ℂ} {R c : ℝ} - (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) - (hs : ∀ z : ℂ, R≤‖z‖ → z.im*(G z-z).im≤0) : - ∀ w∈closedBall (0:ℂ) 1, 0≤(exteriorHerglotzTransform G R c w).re := by - apply real_nonneg_of_sphere_nonneg zero_lt_one - (exteriorHerglotzTransform_differentiableOn hR hG hc) - intro w hw - have hwn : ‖w‖=1 := by simpa only [mem_sphere,dist_zero_right] using hw - have hz : ‖(R:ℂ)/w‖=R := by - rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR,hwn,div_one] - have him : ((R:ℂ)/w).im= -R*w.im := by - rw [div_eq_mul_inv,Complex.inv_eq_conj hwn] - simp only [mul_im,ofReal_re,ofReal_im,conj_im,zero_mul,add_zero] - ring - have hh := hs ((R:ℂ)/w) hz.ge - rw [him] at hh - rw [exteriorHerglotzTransform_sphere_re hwn] - have hp : 0≤w.im*(G ((R:ℂ)/w)-(R:ℂ)/w).im := by nlinarith - nlinarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorHerglotzAnalytic +import OAI.Probability.SLE.Geometry.PositiveRealDiskEstimate + +namespace OAI + +/-! The normalized height inequality becomes positivity of the real +part after inversion. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma exteriorHerglotzTransform_sphere_re {G : ℂ → ℂ} {R c : ℝ} + {w : ℂ} (hw : ‖w‖=1) : + (exteriorHerglotzTransform G R c w).re= + 2*w.im*(G ((R:ℂ)/w)-(R:ℂ)/w).im := by + have hw0 : w≠0 := by intro he; simp [he] at hw + have he : ((1:ℂ)-w^2)/w=conj w-w := by + rw [← Complex.inv_eq_conj hw] + field_simp [hw0] + rw [exteriorHerglotzTransform_eq_raw hw0] + simp only [exteriorHerglotzRaw,he,mul_re,sub_re,sub_im,conj_re,conj_im] + ring + +theorem exteriorHerglotzTransform_nonneg {G : ℂ → ℂ} {R c : ℝ} + (hR : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) + (hs : ∀ z : ℂ, R≤‖z‖ → z.im*(G z-z).im≤0) : + ∀ w∈closedBall (0:ℂ) 1, 0≤(exteriorHerglotzTransform G R c w).re := by + apply real_nonneg_of_sphere_nonneg zero_lt_one + (exteriorHerglotzTransform_differentiableOn hR hG hc) + intro w hw + have hwn : ‖w‖=1 := by simpa only [mem_sphere,dist_zero_right] using hw + have hz : ‖(R:ℂ)/w‖=R := by + rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR,hwn,div_one] + have him : ((R:ℂ)/w).im= -R*w.im := by + rw [div_eq_mul_inv,Complex.inv_eq_conj hwn] + simp only [mul_im,ofReal_re,ofReal_im,conj_im,zero_mul,add_zero] + ring + have hh := hs ((R:ℂ)/w) hz.ge + rw [him] at hh + rw [exteriorHerglotzTransform_sphere_re hwn] + have hp : 0≤w.im*(G ((R:ℂ)/w)-(R:ℂ)/w).im := by nlinarith + nlinarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzTransform.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzTransform.lean index 4d8b42668016a5fcb7b0e99c6549c2d98e6429de..4ade881544d0c1d51896675ce7916ffdfc1d6b52 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzTransform.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorHerglotzTransform.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Loewner.ReflectedExteriorAnalytic -import Mathlib.Analysis.Complex.RemovableSingularity - -namespace OAI - -/-! Inversion converts the vanishing exterior hydrodynamic displacement -into a disk function whose value at zero is exactly capacity/radius. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -def exteriorHerglotzRaw (G : ℂ → ℂ) (R : ℝ) (w : ℂ) : ℂ := - (1-w^2)/w*(G ((R:ℂ)/w)-(R:ℂ)/w) - -noncomputable def exteriorHerglotzTransform (G : ℂ → ℂ) (R c : ℝ) : ℂ → ℂ := - Function.update (exteriorHerglotzRaw G R) 0 ((c:ℂ)/(R:ℂ)) - -@[simp] lemma exteriorHerglotzTransform_zero (G : ℂ → ℂ) (R c : ℝ) : - exteriorHerglotzTransform G R c 0=(c:ℂ)/(R:ℂ) := Function.update_self _ _ _ - -lemma exteriorHerglotzRaw_identity (G : ℂ → ℂ) {R : ℝ} (hR : R≠0) - {w : ℂ} (hw : w≠0) : - exteriorHerglotzRaw G R w=(1-w^2)/(R:ℂ)* - (((R:ℂ)/w)*(G ((R:ℂ)/w)-(R:ℂ)/w)) := by - unfold exteriorHerglotzRaw - field_simp [ofReal_ne_zero.mpr hR,hw] - -lemma scaled_inverse_tendsto_cocompact {R : ℝ} (hR : R≠0) : - Tendsto (fun w : ℂ => (R:ℂ)/w) (𝓝[≠] (0:ℂ)) (cocompact ℂ) := by - have hi : Tendsto (fun w : ℂ => w⁻¹) (𝓝[≠] (0:ℂ)) (cocompact ℂ) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_nhdsNE_zero : Tendsto (Inv.inv : ℂ → ℂ) (𝓝[≠] (0:ℂ)) (Bornology.cobounded ℂ)) - have hh := - (Homeomorph.mulLeft₀ (R:ℂ) (ofReal_ne_zero.mpr hR)).isClosedEmbedding.tendsto_cocompact.comp hi - change Tendsto (fun w : ℂ => (R:ℂ)*w⁻¹) (𝓝[≠] (0:ℂ)) (cocompact ℂ) at hh - simpa only [div_eq_mul_inv] using hh - -theorem exteriorHerglotzTransform_continuousAt_zero {G : ℂ → ℂ} {R c : ℝ} (hR : R≠0) - (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) : - ContinuousAt (exteriorHerglotzTransform G R c) 0 := by - apply continuousAt_update_same.mpr - have hsmall : Tendsto (fun w : ℂ => (1-w^2)/(R:ℂ)) (𝓝[≠] (0:ℂ)) (𝓝 (1/(R:ℂ))) := by - have hh : Continuous (fun w : ℂ => (1-w^2)/(R:ℂ)) := - (continuous_const.sub (continuous_id.pow 2)).div_const _ - have ht := (hh.tendsto 0).mono_left (show 𝓝[≠] (0:ℂ)≤𝓝 (0:ℂ) from inf_le_left) - simpa only [zero_pow two_ne_zero,sub_zero] using ht - have ht := hsmall.mul (hc.comp (scaled_inverse_tendsto_cocompact hR)) - have he : (1/(R:ℂ))*(c:ℂ)=(c:ℂ)/(R:ℂ) := by ring - rw [he] at ht - apply ht.congr' - filter_upwards [self_mem_nhdsWithin] with w hw - exact (exteriorHerglotzRaw_identity G hR hw).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReflectedExteriorAnalytic +import Mathlib.Analysis.Complex.RemovableSingularity + +namespace OAI + +/-! Inversion converts the vanishing exterior hydrodynamic displacement +into a disk function whose value at zero is exactly capacity/radius. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +def exteriorHerglotzRaw (G : ℂ → ℂ) (R : ℝ) (w : ℂ) : ℂ := + (1-w^2)/w*(G ((R:ℂ)/w)-(R:ℂ)/w) + +noncomputable def exteriorHerglotzTransform (G : ℂ → ℂ) (R c : ℝ) : ℂ → ℂ := + Function.update (exteriorHerglotzRaw G R) 0 ((c:ℂ)/(R:ℂ)) + +@[simp] lemma exteriorHerglotzTransform_zero (G : ℂ → ℂ) (R c : ℝ) : + exteriorHerglotzTransform G R c 0=(c:ℂ)/(R:ℂ) := Function.update_self _ _ _ + +lemma exteriorHerglotzRaw_identity (G : ℂ → ℂ) {R : ℝ} (hR : R≠0) + {w : ℂ} (hw : w≠0) : + exteriorHerglotzRaw G R w=(1-w^2)/(R:ℂ)* + (((R:ℂ)/w)*(G ((R:ℂ)/w)-(R:ℂ)/w)) := by + unfold exteriorHerglotzRaw + field_simp [ofReal_ne_zero.mpr hR,hw] + +lemma scaled_inverse_tendsto_cocompact {R : ℝ} (hR : R≠0) : + Tendsto (fun w : ℂ => (R:ℂ)/w) (𝓝[≠] (0:ℂ)) (cocompact ℂ) := by + have hi : Tendsto (fun w : ℂ => w⁻¹) (𝓝[≠] (0:ℂ)) (cocompact ℂ) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_nhdsNE_zero : Tendsto (Inv.inv : ℂ → ℂ) (𝓝[≠] (0:ℂ)) (Bornology.cobounded ℂ)) + have hh := + (Homeomorph.mulLeft₀ (R:ℂ) (ofReal_ne_zero.mpr hR)).isClosedEmbedding.tendsto_cocompact.comp hi + change Tendsto (fun w : ℂ => (R:ℂ)*w⁻¹) (𝓝[≠] (0:ℂ)) (cocompact ℂ) at hh + simpa only [div_eq_mul_inv] using hh + +theorem exteriorHerglotzTransform_continuousAt_zero {G : ℂ → ℂ} {R c : ℝ} (hR : R≠0) + (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) : + ContinuousAt (exteriorHerglotzTransform G R c) 0 := by + apply continuousAt_update_same.mpr + have hsmall : Tendsto (fun w : ℂ => (1-w^2)/(R:ℂ)) (𝓝[≠] (0:ℂ)) (𝓝 (1/(R:ℂ))) := by + have hh : Continuous (fun w : ℂ => (1-w^2)/(R:ℂ)) := + (continuous_const.sub (continuous_id.pow 2)).div_const _ + have ht := (hh.tendsto 0).mono_left (show 𝓝[≠] (0:ℂ)≤𝓝 (0:ℂ) from inf_le_left) + simpa only [zero_pow two_ne_zero,sub_zero] using ht + have ht := hsmall.mul (hc.comp (scaled_inverse_tendsto_cocompact hR)) + have he : (1/(R:ℂ))*(c:ℂ)=(c:ℂ)/(R:ℂ) := by ring + rw [he] at ht + apply ht.congr' + filter_upwards [self_mem_nhdsWithin] with w hw + exact (exteriorHerglotzRaw_identity G hR hw).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorJoukowski.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorJoukowski.lean index 17cfbfdc57be466e2c385d089131fe4d78267be1..72548a8af327016af4648e1330490aaf3837dccd 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorJoukowski.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorJoukowski.lean @@ -1,116 +1,116 @@ -import OAI.Probability.SLE.LoewnerCharts -import Mathlib.Analysis.Complex.Polynomial.Basic - -namespace OAI - -/-! The Joukowski chart from the upper half of the exterior disk to the -half-plane. This is the final elementary chart in exterior slit normalization. -/ -noncomputable section -open Set Complex - -namespace SLEExactGauge - -def exteriorUpperDisk : Set ℂ := {w | 1 < ‖w‖ ∧ 0 < w.im} - -def exteriorJoukowski (w : ℂ) : ℂ := w+w⁻¹ - -lemma exteriorJoukowski_im (w : ℂ) : - (exteriorJoukowski w).im=w.im-w.im/‖w‖^2 := by - simp only [exteriorJoukowski,add_im,inv_im,normSq_eq_norm_sq,neg_div,sub_eq_add_neg] - -lemma exteriorJoukowski_im_mul {w : ℂ} (hw : w ≠ 0) : - (exteriorJoukowski w).im*‖w‖^2=w.im*(‖w‖^2-1) := by - rw [exteriorJoukowski_im] - have hn : ‖w‖^2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.mpr hw) - field_simp - -lemma exteriorJoukowski_mapsTo : MapsTo exteriorJoukowski exteriorUpperDisk upperHalfPlane := by - intro w hw - change 1 < ‖w‖ ∧ 0 < w.im at hw - have hwn : 0 < ‖w‖ := zero_lt_one.trans hw.1 - have hw0 : w ≠ 0 := norm_ne_zero_iff.mp hwn.ne' - have hsq : 1 < ‖w‖^2 := by nlinarith - have he := exteriorJoukowski_im_mul hw0 - change 0 < (exteriorJoukowski w).im - nlinarith [mul_pos hw.2 (sub_pos.mpr hsq),sq_nonneg ‖w‖] - -lemma exteriorJoukowski_injOn : InjOn exteriorJoukowski {w : ℂ | 1 < ‖w‖} := by - intro u hu v hv he - change 1 < ‖u‖ at hu - change 1 < ‖v‖ at hv - have hu0 : u ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hu).ne' - have hv0 : v ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hv).ne' - have hf : (u-v)*(u*v-1)=0 := by - calc - (u-v)*(u*v-1)=(exteriorJoukowski u-exteriorJoukowski v)*(u*v) := by - dsimp only [exteriorJoukowski] - field_simp [hu0,hv0] - ring - _ = 0 := by rw [he,sub_self,zero_mul] - rcases mul_eq_zero.mp hf with h|h - · exact sub_eq_zero.mp h - · have hnorm : ‖u‖*‖v‖=1 := by - simpa only [norm_mul,norm_one] using congrArg norm (sub_eq_zero.mp h) - have hmul := mul_lt_mul_of_pos_left hv (zero_lt_one.trans hu) - have hp : 1 < ‖u‖*‖v‖ := hu.trans (by simpa only [mul_one] using hmul) - linarith - -lemma exteriorJoukowski_surjOn : SurjOn exteriorJoukowski exteriorUpperDisk upperHalfPlane := by - intro z hz - obtain ⟨s,hs⟩ := IsAlgClosed.exists_pow_nat_eq (z^2-4) (by decide : 0 < (2:ℕ)) - let u : ℂ := (z+s)/2 - let v : ℂ := (z-s)/2 - have hsum : u+v=z := by dsimp [u,v]; ring - have hprod : u*v=1 := by - dsimp only [u,v] - linear_combination -hs/4 - have hu0 : u ≠ 0 := by intro he; rw [he,zero_mul] at hprod; exact zero_ne_one hprod - have hv0 : v ≠ 0 := by intro he; rw [he,mul_zero] at hprod; exact zero_ne_one hprod - have huinv : u⁻¹=v := by - apply (mul_left_cancel₀ hu0) - rw [mul_inv_cancel₀ hu0,hprod] - have hvinv : v⁻¹=u := by rw [← huinv,inv_inv] - have huJ : exteriorJoukowski u=z := by rw [exteriorJoukowski,huinv,hsum] - have hvJ : exteriorJoukowski v=z := by rw [exteriorJoukowski,hvinv,add_comm,hsum] - have hunit (w : ℂ) (hw0 : w ≠ 0) (hwJ : exteriorJoukowski w=z) : ‖w‖ ≠ 1 := by - intro h - have he := exteriorJoukowski_im_mul hw0 - rw [hwJ,h] at he - norm_num at he - exact hz.ne' he - have hex : ∃ w : ℂ, 1 < ‖w‖ ∧ exteriorJoukowski w=z := by - by_cases hu : 1 < ‖u‖ - · exact ⟨u,hu,huJ⟩ - · have hvnorm : ‖u‖*‖v‖=1 := by - simpa only [norm_mul,norm_one] using congrArg norm hprod - have hult : ‖u‖ < 1 := lt_of_le_of_ne (le_of_not_gt hu) (hunit u hu0 huJ) - have hvn : 0 ≤ ‖v‖ := norm_nonneg _ - have hup : 0 < ‖u‖ := norm_pos_iff.mpr hu0 - have hvp : 0 < ‖v‖ := norm_pos_iff.mpr hv0 - have hh := mul_lt_mul_of_pos_right hult hvp - exact ⟨v,by nlinarith, hvJ⟩ - obtain ⟨w,hwn,hwJ⟩ := hex - have hw0 : w ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hwn).ne' - have hi : 0 < w.im := by - have he := exteriorJoukowski_im_mul hw0 - rw [hwJ] at he - have hsq : 1 < ‖w‖^2 := by nlinarith - have hh : 0 < z.im*‖w‖^2 := mul_pos hz (sq_pos_of_pos (zero_lt_one.trans hwn)) - nlinarith - exact ⟨w,⟨hwn,hi⟩,hwJ⟩ - -theorem exteriorJoukowski_bijOn : BijOn exteriorJoukowski exteriorUpperDisk upperHalfPlane := - ⟨exteriorJoukowski_mapsTo,exteriorJoukowski_injOn.mono (fun _ h => h.1), - exteriorJoukowski_surjOn⟩ - -theorem exteriorJoukowski_differentiableOn : - DifferentiableOn ℂ exteriorJoukowski exteriorUpperDisk := by - intro w hw - exact (differentiableAt_id.add (differentiableAt_id.inv - (norm_ne_zero_iff.mp (zero_lt_one.trans hw.1).ne'))).differentiableWithinAt - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerCharts +import Mathlib.Analysis.Complex.Polynomial.Basic + +namespace OAI + +/-! The Joukowski chart from the upper half of the exterior disk to the +half-plane. This is the final elementary chart in exterior slit normalization. -/ +noncomputable section +open Set Complex + +namespace SLEExactGauge + +def exteriorUpperDisk : Set ℂ := {w | 1 < ‖w‖ ∧ 0 < w.im} + +def exteriorJoukowski (w : ℂ) : ℂ := w+w⁻¹ + +lemma exteriorJoukowski_im (w : ℂ) : + (exteriorJoukowski w).im=w.im-w.im/‖w‖^2 := by + simp only [exteriorJoukowski,add_im,inv_im,normSq_eq_norm_sq,neg_div,sub_eq_add_neg] + +lemma exteriorJoukowski_im_mul {w : ℂ} (hw : w ≠ 0) : + (exteriorJoukowski w).im*‖w‖^2=w.im*(‖w‖^2-1) := by + rw [exteriorJoukowski_im] + have hn : ‖w‖^2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.mpr hw) + field_simp + +lemma exteriorJoukowski_mapsTo : MapsTo exteriorJoukowski exteriorUpperDisk upperHalfPlane := by + intro w hw + change 1 < ‖w‖ ∧ 0 < w.im at hw + have hwn : 0 < ‖w‖ := zero_lt_one.trans hw.1 + have hw0 : w ≠ 0 := norm_ne_zero_iff.mp hwn.ne' + have hsq : 1 < ‖w‖^2 := by nlinarith + have he := exteriorJoukowski_im_mul hw0 + change 0 < (exteriorJoukowski w).im + nlinarith [mul_pos hw.2 (sub_pos.mpr hsq),sq_nonneg ‖w‖] + +lemma exteriorJoukowski_injOn : InjOn exteriorJoukowski {w : ℂ | 1 < ‖w‖} := by + intro u hu v hv he + change 1 < ‖u‖ at hu + change 1 < ‖v‖ at hv + have hu0 : u ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hu).ne' + have hv0 : v ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hv).ne' + have hf : (u-v)*(u*v-1)=0 := by + calc + (u-v)*(u*v-1)=(exteriorJoukowski u-exteriorJoukowski v)*(u*v) := by + dsimp only [exteriorJoukowski] + field_simp [hu0,hv0] + ring + _ = 0 := by rw [he,sub_self,zero_mul] + rcases mul_eq_zero.mp hf with h|h + · exact sub_eq_zero.mp h + · have hnorm : ‖u‖*‖v‖=1 := by + simpa only [norm_mul,norm_one] using congrArg norm (sub_eq_zero.mp h) + have hmul := mul_lt_mul_of_pos_left hv (zero_lt_one.trans hu) + have hp : 1 < ‖u‖*‖v‖ := hu.trans (by simpa only [mul_one] using hmul) + linarith + +lemma exteriorJoukowski_surjOn : SurjOn exteriorJoukowski exteriorUpperDisk upperHalfPlane := by + intro z hz + obtain ⟨s,hs⟩ := IsAlgClosed.exists_pow_nat_eq (z^2-4) (by decide : 0 < (2:ℕ)) + let u : ℂ := (z+s)/2 + let v : ℂ := (z-s)/2 + have hsum : u+v=z := by dsimp [u,v]; ring + have hprod : u*v=1 := by + dsimp only [u,v] + linear_combination -hs/4 + have hu0 : u ≠ 0 := by intro he; rw [he,zero_mul] at hprod; exact zero_ne_one hprod + have hv0 : v ≠ 0 := by intro he; rw [he,mul_zero] at hprod; exact zero_ne_one hprod + have huinv : u⁻¹=v := by + apply (mul_left_cancel₀ hu0) + rw [mul_inv_cancel₀ hu0,hprod] + have hvinv : v⁻¹=u := by rw [← huinv,inv_inv] + have huJ : exteriorJoukowski u=z := by rw [exteriorJoukowski,huinv,hsum] + have hvJ : exteriorJoukowski v=z := by rw [exteriorJoukowski,hvinv,add_comm,hsum] + have hunit (w : ℂ) (hw0 : w ≠ 0) (hwJ : exteriorJoukowski w=z) : ‖w‖ ≠ 1 := by + intro h + have he := exteriorJoukowski_im_mul hw0 + rw [hwJ,h] at he + norm_num at he + exact hz.ne' he + have hex : ∃ w : ℂ, 1 < ‖w‖ ∧ exteriorJoukowski w=z := by + by_cases hu : 1 < ‖u‖ + · exact ⟨u,hu,huJ⟩ + · have hvnorm : ‖u‖*‖v‖=1 := by + simpa only [norm_mul,norm_one] using congrArg norm hprod + have hult : ‖u‖ < 1 := lt_of_le_of_ne (le_of_not_gt hu) (hunit u hu0 huJ) + have hvn : 0 ≤ ‖v‖ := norm_nonneg _ + have hup : 0 < ‖u‖ := norm_pos_iff.mpr hu0 + have hvp : 0 < ‖v‖ := norm_pos_iff.mpr hv0 + have hh := mul_lt_mul_of_pos_right hult hvp + exact ⟨v,by nlinarith, hvJ⟩ + obtain ⟨w,hwn,hwJ⟩ := hex + have hw0 : w ≠ 0 := norm_ne_zero_iff.mp (zero_lt_one.trans hwn).ne' + have hi : 0 < w.im := by + have he := exteriorJoukowski_im_mul hw0 + rw [hwJ] at he + have hsq : 1 < ‖w‖^2 := by nlinarith + have hh : 0 < z.im*‖w‖^2 := mul_pos hz (sq_pos_of_pos (zero_lt_one.trans hwn)) + nlinarith + exact ⟨w,⟨hwn,hi⟩,hwJ⟩ + +theorem exteriorJoukowski_bijOn : BijOn exteriorJoukowski exteriorUpperDisk upperHalfPlane := + ⟨exteriorJoukowski_mapsTo,exteriorJoukowski_injOn.mono (fun _ h => h.1), + exteriorJoukowski_surjOn⟩ + +theorem exteriorJoukowski_differentiableOn : + DifferentiableOn ℂ exteriorJoukowski exteriorUpperDisk := by + intro w hw + exact (differentiableAt_id.add (differentiableAt_id.inv + (norm_ne_zero_iff.mp (zero_lt_one.trans hw.1).ne'))).differentiableWithinAt + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorLaplacianTest.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorLaplacianTest.lean index ea986f1461ff053c367a9ea0a0bea3ba1f221224..442614b445906e122ff2781af9fdd9b4bd7fadd1 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorLaplacianTest.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorLaplacianTest.lean @@ -1,70 +1,70 @@ -import OAI.Probability.SLE.Geometry.CompactLaplacianIntegral -import OAI.Probability.SLE.Geometry.ConformalLaplacian - -namespace OAI - -/-! A compactly supported quadratic test near the omitted set of an -exterior conformal map. Its Laplacian has nonpositive integral over the -image domain. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory InnerProductSpace -open scoped Topology Laplacian - -namespace SLEExactGauge - -lemma complex_laplacian_norm_sq (z : ℂ) : (Δ (fun w : ℂ => ‖w‖^2)) z=4 := by - let L : ℂ →L[ℝ] (ℂ →L[ℝ] ℝ) := 2 • innerSL ℝ - have hd : fderiv ℝ (fun w : ℂ => ‖w‖^2)=L := fderiv_norm_sq - rw [laplacian_eq_iteratedFDeriv_complexPlane] - simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, - Matrix.cons_val_one,Matrix.cons_val_fin_one,hd,L.fderiv] - simp only [L,two_smul,add_apply] - change (inner ℝ (1:ℂ) 1+inner ℝ (1:ℂ) 1)+(inner ℝ Complex.I Complex.I+inner ℝ Complex.I Complex.I)=4 - simp only [real_inner_self_eq_norm_sq,norm_one,Complex.norm_I] - norm_num - -lemma exists_compact_quadratic_test (K : Set ℂ) (hK : IsCompact K) : - ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ - (∀ z∈K, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ - (∀ z∈K, (Δ u) z=4) := by - obtain ⟨R,hR,hRK⟩ := hK.isBounded.exists_pos_norm_lt - let b : ContDiffBump (0:ℂ) := ⟨R,R+1,hR,by linarith⟩ - let u (z : ℂ) := b z*‖z‖^2 - have hu : ContDiff ℝ 2 u := b.contDiff.mul (contDiff_id.norm_sq ℝ) - have hs : HasCompactSupport u := b.hasCompactSupport.mul_right - have he (z : ℂ) (hz : z∈K) : u =ᶠ[𝓝 z] (fun w => ‖w‖^2) := by - have hb := b.eventuallyEq_one_of_mem_ball (show z∈ball (0:ℂ) b.rIn by - simpa only [mem_ball_zero_iff] using hRK z hz) - filter_upwards [hb] with w hw - simp only [u,hw,Pi.one_apply,one_mul] - exact ⟨u,hu,hs,he,fun z hz => (laplacian_congr_nhds (he z hz)).self_of_nhds.trans (complex_laplacian_norm_sq z)⟩ - -lemma hasCompactSupport_laplacian {u : ℂ → ℝ} (hs : HasCompactSupport u) : - HasCompactSupport (Δ u) := by - have h := (hs.fderiv ℝ |>.fderiv ℝ).comp_left - (g := fun L : ℂ →L[ℝ] (ℂ →L[ℝ] ℝ) => L 1 1+L Complex.I Complex.I) (by simp) - simpa only [Function.comp_def,laplacian_eq_iteratedFDeriv_complexPlane, - iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, - Matrix.cons_val_one,Matrix.cons_val_fin_one] using h - -lemma compact_quadratic_test_complement_integral - (K : Set ℂ) (hK : IsCompact K) : - ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ - (∀ z∈K, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ - (∫ z in Kᶜ, (Δ u) z)≤0 := by - obtain ⟨u,hu,hs,he,hΔ⟩ := exists_compact_quadratic_test K hK - have hcont : Continuous (Δ u) := by - exact continuous_iff_continuousAt.mpr (fun z => plane_laplacian_continuousAt (hu.contDiffAt (x := z))) - have hsupport := hasCompactSupport_laplacian hs - have hi : Integrable (Δ u) := hcont.integrable_of_hasCompactSupport hsupport - have hzero := integral_laplacian_compact_eq_zero hu hs - have hsum := integral_add_compl hK.measurableSet hi - have hpos : 0≤∫ z in K, (Δ u) z := setIntegral_nonneg hK.measurableSet (fun z hz => by rw [hΔ z hz]; norm_num) - refine ⟨u,hu,hs,he,?_⟩ - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactLaplacianIntegral +import OAI.Probability.SLE.Geometry.ConformalLaplacian + +namespace OAI + +/-! A compactly supported quadratic test near the omitted set of an +exterior conformal map. Its Laplacian has nonpositive integral over the +image domain. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory InnerProductSpace +open scoped Topology Laplacian + +namespace SLEExactGauge + +lemma complex_laplacian_norm_sq (z : ℂ) : (Δ (fun w : ℂ => ‖w‖^2)) z=4 := by + let L : ℂ →L[ℝ] (ℂ →L[ℝ] ℝ) := 2 • innerSL ℝ + have hd : fderiv ℝ (fun w : ℂ => ‖w‖^2)=L := fderiv_norm_sq + rw [laplacian_eq_iteratedFDeriv_complexPlane] + simp only [iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, + Matrix.cons_val_one,Matrix.cons_val_fin_one,hd,L.fderiv] + simp only [L,two_smul,add_apply] + change (inner ℝ (1:ℂ) 1+inner ℝ (1:ℂ) 1)+(inner ℝ Complex.I Complex.I+inner ℝ Complex.I Complex.I)=4 + simp only [real_inner_self_eq_norm_sq,norm_one,Complex.norm_I] + norm_num + +lemma exists_compact_quadratic_test (K : Set ℂ) (hK : IsCompact K) : + ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ + (∀ z∈K, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ + (∀ z∈K, (Δ u) z=4) := by + obtain ⟨R,hR,hRK⟩ := hK.isBounded.exists_pos_norm_lt + let b : ContDiffBump (0:ℂ) := ⟨R,R+1,hR,by linarith⟩ + let u (z : ℂ) := b z*‖z‖^2 + have hu : ContDiff ℝ 2 u := b.contDiff.mul (contDiff_id.norm_sq ℝ) + have hs : HasCompactSupport u := b.hasCompactSupport.mul_right + have he (z : ℂ) (hz : z∈K) : u =ᶠ[𝓝 z] (fun w => ‖w‖^2) := by + have hb := b.eventuallyEq_one_of_mem_ball (show z∈ball (0:ℂ) b.rIn by + simpa only [mem_ball_zero_iff] using hRK z hz) + filter_upwards [hb] with w hw + simp only [u,hw,Pi.one_apply,one_mul] + exact ⟨u,hu,hs,he,fun z hz => (laplacian_congr_nhds (he z hz)).self_of_nhds.trans (complex_laplacian_norm_sq z)⟩ + +lemma hasCompactSupport_laplacian {u : ℂ → ℝ} (hs : HasCompactSupport u) : + HasCompactSupport (Δ u) := by + have h := (hs.fderiv ℝ |>.fderiv ℝ).comp_left + (g := fun L : ℂ →L[ℝ] (ℂ →L[ℝ] ℝ) => L 1 1+L Complex.I Complex.I) (by simp) + simpa only [Function.comp_def,laplacian_eq_iteratedFDeriv_complexPlane, + iteratedFDeriv_two_apply,Fin.isValue,Matrix.cons_val_zero, + Matrix.cons_val_one,Matrix.cons_val_fin_one] using h + +lemma compact_quadratic_test_complement_integral + (K : Set ℂ) (hK : IsCompact K) : + ∃ u : ℂ → ℝ, ContDiff ℝ 2 u ∧ HasCompactSupport u ∧ + (∀ z∈K, u =ᶠ[𝓝 z] (fun w => ‖w‖^2)) ∧ + (∫ z in Kᶜ, (Δ u) z)≤0 := by + obtain ⟨u,hu,hs,he,hΔ⟩ := exists_compact_quadratic_test K hK + have hcont : Continuous (Δ u) := by + exact continuous_iff_continuousAt.mpr (fun z => plane_laplacian_continuousAt (hu.contDiffAt (x := z))) + have hsupport := hasCompactSupport_laplacian hs + have hi : Integrable (Δ u) := hcont.integrable_of_hasCompactSupport hsupport + have hzero := integral_laplacian_compact_eq_zero hu hs + have hsum := integral_add_compl hK.measurableSet hi + have hpos : 0≤∫ z in K, (Δ u) z := setIntegral_nonneg hK.measurableSet (fun z hz => by rw [hΔ z hz]; norm_num) + refine ⟨u,hu,hs,he,?_⟩ + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/ExteriorLaurent.lean b/lean/OAI/Probability/SLE/Geometry/ExteriorLaurent.lean index b37275b87a970c0ffe314170c45f673474888e11..f249aaa78d3636b3848729c8ab185609c359fadf 100644 --- a/lean/OAI/Probability/SLE/Geometry/ExteriorLaurent.lean +++ b/lean/OAI/Probability/SLE/Geometry/ExteriorLaurent.lean @@ -1,91 +1,91 @@ -import OAI.Probability.SLE.Loewner.SlitExteriorChart -import OAI.Probability.SLE.Geometry.SymmetricSlope - -namespace OAI - -/-! Removing the pole of the exterior slit chart at infinity gives its -first two Laurent coefficients without choosing a power-series expansion. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -def slitInfinityFactor (f : ℂ → ℂ) (a : ℝ) (w : ℂ) : ℂ := - (a:ℂ) * ((dslope f 0 w)⁻¹+w^2*dslope f 0 w) - -lemma slitInfinityFactor_zero {f : ℂ → ℂ} {a : ℝ} - (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : slitInfinityFactor f a 0=1 := by - simp only [slitInfinityFactor,dslope_same,hd,zero_pow two_ne_zero,zero_mul,add_zero] - exact mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr ha) - -lemma slitInfinityFactor_analyticAt {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : - AnalyticAt ℂ (slitInfinityFactor f a) 0 := by - have hq := analyticAt_dslope_zero hf - have hq0 : dslope f 0 0 ≠ 0 := by - rw [dslope_same,hd] - exact Complex.ofReal_ne_zero.mpr ha - exact analyticAt_const.mul ((hq.inv hq0).add ((analyticAt_id.pow 2).mul hq)) - -lemma slitInfinityFactor_reflection {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) - {f : ℂ → ℂ} (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) : - ∀ z ∈ D, slitInfinityFactor f a (conj z)=conj (slitInfinityFactor f a z) := by - intro z hz - simp only [slitInfinityFactor,dslope_reflection hD hs z hz,map_mul,map_add,map_inv₀, - map_pow,conj_ofReal] - -lemma slitInfinityFactor_identity {f : ℂ → ℂ} (hf0 : f 0=0) (a : ℝ) - {z : ℂ} (hz : z ≠ 0) : - z*slitInfinityFactor f a z⁻¹=(a:ℂ)*exteriorJoukowski (slitExteriorMap f z) := by - have hq : dslope f 0 z⁻¹=z*f z⁻¹ := by - simp only [dslope_of_ne _ (inv_ne_zero hz),slope,vsub_eq_sub,sub_zero,hf0,inv_inv,smul_eq_mul] - simp only [slitInfinityFactor,hq,exteriorJoukowski,slitExteriorMap,inv_inv,mul_inv_rev] - field_simp - -lemma slitInfinityFactor_remainder {f : ℂ → ℂ} {a : ℝ} - (hf0 : f 0=0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) {z : ℂ} (hz : z ≠ 0) : - z*((a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- - deriv (slitInfinityFactor f a) 0-z)= - dslope (dslope (slitInfinityFactor f a) 0) 0 z⁻¹ := by - have h0 := slitInfinityFactor_zero ha hd - have h1 := sub_smul_dslope (slitInfinityFactor f a) (0:ℂ) z⁻¹ - have h2 := sub_smul_dslope (dslope (slitInfinityFactor f a) 0) (0:ℂ) z⁻¹ - simp only [sub_zero,smul_eq_mul,dslope_same,h0] at h1 h2 - have hmul : (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)= - z*slitInfinityFactor f a z⁻¹ := (slitInfinityFactor_identity hf0 a hz).symm - rw [hmul] - have hz1 : z*z⁻¹=1 := mul_inv_cancel₀ hz - calc - z*(z*slitInfinityFactor f a z⁻¹-deriv (slitInfinityFactor f a) 0-z) = - z*(z*(z⁻¹*dslope (slitInfinityFactor f a) 0 z⁻¹+1)- - deriv (slitInfinityFactor f a) 0-z) := by rw [h1]; ring - _ = z*(dslope (slitInfinityFactor f a) 0 z⁻¹-deriv (slitInfinityFactor f a) 0) := by - rw [mul_add,← mul_assoc,hz1,one_mul,mul_one] - ring - _ = z*(z⁻¹*dslope (dslope (slitInfinityFactor f a) 0) 0 z⁻¹) := by rw [h2] - _ = _ := by rw [← mul_assoc,hz1,one_mul] - -lemma slitExteriorMap_laurent_limit {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (hf0 : f 0=0) (ha : a ≠ 0) - (hd : deriv f 0=(a:ℂ)) : - Tendsto (fun z : ℂ => z*((a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- - deriv (slitInfinityFactor f a) 0-z)) (cocompact ℂ) - (𝓝 (deriv (dslope (slitInfinityFactor f a) 0) 0)) := by - have hH := slitInfinityFactor_analyticAt hf ha hd - have hq := analyticAt_dslope_zero hH - have hr := analyticAt_dslope_zero hq - have hinv : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - have ht := hr.continuousAt.tendsto.comp hinv - simp only [dslope_same] at ht - apply ht.congr' - filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz - exact (slitInfinityFactor_remainder hf0 ha hd hz).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitExteriorChart +import OAI.Probability.SLE.Geometry.SymmetricSlope + +namespace OAI + +/-! Removing the pole of the exterior slit chart at infinity gives its +first two Laurent coefficients without choosing a power-series expansion. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +def slitInfinityFactor (f : ℂ → ℂ) (a : ℝ) (w : ℂ) : ℂ := + (a:ℂ) * ((dslope f 0 w)⁻¹+w^2*dslope f 0 w) + +lemma slitInfinityFactor_zero {f : ℂ → ℂ} {a : ℝ} + (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : slitInfinityFactor f a 0=1 := by + simp only [slitInfinityFactor,dslope_same,hd,zero_pow two_ne_zero,zero_mul,add_zero] + exact mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr ha) + +lemma slitInfinityFactor_analyticAt {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : + AnalyticAt ℂ (slitInfinityFactor f a) 0 := by + have hq := analyticAt_dslope_zero hf + have hq0 : dslope f 0 0 ≠ 0 := by + rw [dslope_same,hd] + exact Complex.ofReal_ne_zero.mpr ha + exact analyticAt_const.mul ((hq.inv hq0).add ((analyticAt_id.pow 2).mul hq)) + +lemma slitInfinityFactor_reflection {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) + {f : ℂ → ℂ} (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) : + ∀ z ∈ D, slitInfinityFactor f a (conj z)=conj (slitInfinityFactor f a z) := by + intro z hz + simp only [slitInfinityFactor,dslope_reflection hD hs z hz,map_mul,map_add,map_inv₀, + map_pow,conj_ofReal] + +lemma slitInfinityFactor_identity {f : ℂ → ℂ} (hf0 : f 0=0) (a : ℝ) + {z : ℂ} (hz : z ≠ 0) : + z*slitInfinityFactor f a z⁻¹=(a:ℂ)*exteriorJoukowski (slitExteriorMap f z) := by + have hq : dslope f 0 z⁻¹=z*f z⁻¹ := by + simp only [dslope_of_ne _ (inv_ne_zero hz),slope,vsub_eq_sub,sub_zero,hf0,inv_inv,smul_eq_mul] + simp only [slitInfinityFactor,hq,exteriorJoukowski,slitExteriorMap,inv_inv,mul_inv_rev] + field_simp + +lemma slitInfinityFactor_remainder {f : ℂ → ℂ} {a : ℝ} + (hf0 : f 0=0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) {z : ℂ} (hz : z ≠ 0) : + z*((a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- + deriv (slitInfinityFactor f a) 0-z)= + dslope (dslope (slitInfinityFactor f a) 0) 0 z⁻¹ := by + have h0 := slitInfinityFactor_zero ha hd + have h1 := sub_smul_dslope (slitInfinityFactor f a) (0:ℂ) z⁻¹ + have h2 := sub_smul_dslope (dslope (slitInfinityFactor f a) 0) (0:ℂ) z⁻¹ + simp only [sub_zero,smul_eq_mul,dslope_same,h0] at h1 h2 + have hmul : (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)= + z*slitInfinityFactor f a z⁻¹ := (slitInfinityFactor_identity hf0 a hz).symm + rw [hmul] + have hz1 : z*z⁻¹=1 := mul_inv_cancel₀ hz + calc + z*(z*slitInfinityFactor f a z⁻¹-deriv (slitInfinityFactor f a) 0-z) = + z*(z*(z⁻¹*dslope (slitInfinityFactor f a) 0 z⁻¹+1)- + deriv (slitInfinityFactor f a) 0-z) := by rw [h1]; ring + _ = z*(dslope (slitInfinityFactor f a) 0 z⁻¹-deriv (slitInfinityFactor f a) 0) := by + rw [mul_add,← mul_assoc,hz1,one_mul,mul_one] + ring + _ = z*(z⁻¹*dslope (dslope (slitInfinityFactor f a) 0) 0 z⁻¹) := by rw [h2] + _ = _ := by rw [← mul_assoc,hz1,one_mul] + +lemma slitExteriorMap_laurent_limit {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (hf0 : f 0=0) (ha : a ≠ 0) + (hd : deriv f 0=(a:ℂ)) : + Tendsto (fun z : ℂ => z*((a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- + deriv (slitInfinityFactor f a) 0-z)) (cocompact ℂ) + (𝓝 (deriv (dslope (slitInfinityFactor f a) 0) 0)) := by + have hH := slitInfinityFactor_analyticAt hf ha hd + have hq := analyticAt_dslope_zero hH + have hr := analyticAt_dslope_zero hq + have hinv : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + have ht := hr.continuousAt.tendsto.comp hinv + simp only [dslope_same] at ht + apply ht.congr' + filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz + exact (slitInfinityFactor_remainder hf0 ha hd hz).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteBoundaryRestart.lean b/lean/OAI/Probability/SLE/Geometry/FiniteBoundaryRestart.lean index 6caa929bfa5e998e6abe5c2db680a25059dbab2c..c963ebae7bad9980fb3554884dc920c119dde5fb 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteBoundaryRestart.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteBoundaryRestart.lean @@ -1,141 +1,141 @@ -import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension -import OAI.Probability.SLE.Geometry.MeasurableBoundaryRestart - -namespace OAI - -/-! Finite-valued boundary charts introduce no measurable selection into -an actual stopped restart: their boundary evaluations are determined by the -interior map. The same past-measurable family realizes all future times. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance : LocallyCompactSpace closedHalfPlane := - isClosed_closedHalfPlane.locallyCompactSpace -local instance finiteRestartPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance finiteRestartPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ -local instance finiteRestartDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance finiteRestartDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -lemma closed_halfplane_map_ext {Y : Type*} [TopologicalSpace Y] [T2Space Y] - (f g : C(closedHalfPlane, Y)) - (heq : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f z = g z) : f = g := by - ext z - let zₙ (n : ℕ) : closedHalfPlane := - ⟨z + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I, by - change 0 ≤ (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, - mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] - exact add_nonneg z.2 (by positivity)⟩ - have hH (n : ℕ) : (zₙ n).1 ∈ upperHalfPlane := by - change 0 < (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im - simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, - mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] - exact add_pos_of_nonneg_of_pos z.2 (by positivity) - have hz : Tendsto zₙ atTop (𝓝 z) := by - apply tendsto_subtype_rng.mpr - simpa only [Function.comp_def, Complex.ofReal_zero, zero_mul, add_zero] using - tendsto_const_nhds.add (((Complex.continuous_ofReal.tendsto 0).comp - (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))).mul_const Complex.I) - exact tendsto_nhds_unique ((f.continuous.tendsto z).comp hz) - (((g.continuous.tendsto z).comp hz).congr (fun n => (heq (zₙ n) (hH n)).symm)) - -lemma finite_boundary_extension_family_measurable - {Ω : Type*} [MeasurableSpace Ω] - (f : Ω → ℂ → ℂ) (Fb : Ω → C(closedHalfPlane, ℂ)) - (hf : ∀ z ∈ upperHalfPlane, Measurable (fun ω => f ω z)) - (hext : ∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb ω z = f ω z) : - Measurable Fb := by - let Fc : Ω → C(closedHalfPlane, OnePoint ℂ) := fun ω => - ⟨fun z => (Fb ω z : OnePoint ℂ), OnePoint.continuous_coe.comp (Fb ω).continuous⟩ - apply ContinuousMap.measurable_iff_eval.mpr - intro z - have hc := boundary_extension_eval_measurable f Fc hf - (fun ω w hw => congrArg (fun v : ℂ => (v : OnePoint ℂ)) (hext ω w hw)) z - exact OnePoint.isOpenEmbedding_coe.measurableEmbedding.measurable_comp_iff.mp hc - -/-- Finite boundary evaluations are determined by the stopping-time past. -/ -theorem source_stopped_finiteBoundaryPath_measurable - (hpublished : PublishedFiniteLoewnerBoundaryExtension) - {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) : - ∃ Fb : Ω → C(closedHalfPlane, ℂ), - Measurable[hτ.measurableSpace] Fb ∧ - (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → - Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ - ∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω) := by - choose Fb hext htip using fun ω => - normalized_finite_capacity_inverse_boundary_extension hpublished - (continuous_const.mul (hc ω)) (hw ω) (τ ω) 1 zero_lt_one - have he (ω : Ω) (z : closedHalfPlane) (hz : z.1 ∈ upperHalfPlane) : - Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by - simpa only [Complex.ofReal_one, one_mul, add_comm, Pi.mul_apply] using hext ω z hz - refine ⟨Fb, ?_, he, htip⟩ - apply @finite_boundary_extension_family_measurable Ω hτ.measurableSpace - (fun ω z => F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) Fb _ he - intro z hz - exact (continuous_eval_const (⟨z, hz⟩ : upperHalfPlane)).measurable.comp - (source_stopped_inversePath_measurable B hm hc κ γ G F hw τ hτ) - -/-- The actual future trace is the finite boundary image of the fresh -trace, using a chart measurable in precisely the same stopped past. -/ -theorem source_measurable_finite_boundary_restart - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (hpublished : PublishedFiniteLoewnerBoundaryExtension) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) : - ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), ∃ Fb : Ω → C(closedHalfPlane, ℂ), - Measurable Φ ∧ - (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ - Measurable[hτ.measurableSpace] Fb ∧ - (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → - Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ - (∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω)) ∧ - (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → - (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) = - P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧ - ∀ᵐ ω ∂P, ∀ t : ℝ≥0, - γ ω (τ ω + t) = - finiteBoundaryPath (Fb ω) (Φ (sourceRestartDriverPath B hc τ ω)) t := by - obtain ⟨Φ, Fc, hΦm, hΦ, _, hFc, _, hlaw, hrestart⟩ := - source_measurable_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw - hpublished.to_onePoint τ hτ - obtain ⟨Fb, hFbm, hFb, hFb0⟩ := source_stopped_finiteBoundaryPath_measurable - hpublished B hm hc κ γ G F hw τ hτ - have he (ω : Ω) : - (⟨fun z => (Fb ω z : OnePoint ℂ), - OnePoint.continuous_coe.comp (Fb ω).continuous⟩ : - C(closedHalfPlane, OnePoint ℂ)) = Fc ω := by - apply closed_halfplane_map_ext - intro z hz - exact (congrArg (fun w : ℂ => (w : OnePoint ℂ)) (hFb ω z hz)).trans - (hFc ω z hz).symm - refine ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, hFb0, hlaw, ?_⟩ - filter_upwards [hrestart] with ω hω - intro t - apply OnePoint.coe_injective - rw [hω t] - exact (congrArg (fun f : C(closedHalfPlane, OnePoint ℂ) => - f (liftClosedPath (Φ (sourceRestartDriverPath B hc τ ω)) t)) (he ω)).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.PublishedFiniteBoundaryExtension +import OAI.Probability.SLE.Geometry.MeasurableBoundaryRestart + +namespace OAI + +/-! Finite-valued boundary charts introduce no measurable selection into +an actual stopped restart: their boundary evaluations are determined by the +interior map. The same past-measurable family realizes all future times. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance : LocallyCompactSpace closedHalfPlane := + isClosed_closedHalfPlane.locallyCompactSpace +local instance finiteRestartPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance finiteRestartPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ +local instance finiteRestartDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance finiteRestartDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +lemma closed_halfplane_map_ext {Y : Type*} [TopologicalSpace Y] [T2Space Y] + (f g : C(closedHalfPlane, Y)) + (heq : ∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane → f z = g z) : f = g := by + ext z + let zₙ (n : ℕ) : closedHalfPlane := + ⟨z + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I, by + change 0 ≤ (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, + mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] + exact add_nonneg z.2 (by positivity)⟩ + have hH (n : ℕ) : (zₙ n).1 ∈ upperHalfPlane := by + change 0 < (z.1 + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * Complex.I).im + simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, + mul_one, Complex.ofReal_im, Complex.I_re, mul_zero, add_zero] + exact add_pos_of_nonneg_of_pos z.2 (by positivity) + have hz : Tendsto zₙ atTop (𝓝 z) := by + apply tendsto_subtype_rng.mpr + simpa only [Function.comp_def, Complex.ofReal_zero, zero_mul, add_zero] using + tendsto_const_nhds.add (((Complex.continuous_ofReal.tendsto 0).comp + (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))).mul_const Complex.I) + exact tendsto_nhds_unique ((f.continuous.tendsto z).comp hz) + (((g.continuous.tendsto z).comp hz).congr (fun n => (heq (zₙ n) (hH n)).symm)) + +lemma finite_boundary_extension_family_measurable + {Ω : Type*} [MeasurableSpace Ω] + (f : Ω → ℂ → ℂ) (Fb : Ω → C(closedHalfPlane, ℂ)) + (hf : ∀ z ∈ upperHalfPlane, Measurable (fun ω => f ω z)) + (hext : ∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → Fb ω z = f ω z) : + Measurable Fb := by + let Fc : Ω → C(closedHalfPlane, OnePoint ℂ) := fun ω => + ⟨fun z => (Fb ω z : OnePoint ℂ), OnePoint.continuous_coe.comp (Fb ω).continuous⟩ + apply ContinuousMap.measurable_iff_eval.mpr + intro z + have hc := boundary_extension_eval_measurable f Fc hf + (fun ω w hw => congrArg (fun v : ℂ => (v : OnePoint ℂ)) (hext ω w hw)) z + exact OnePoint.isOpenEmbedding_coe.measurableEmbedding.measurable_comp_iff.mp hc + +/-- Finite boundary evaluations are determined by the stopping-time past. -/ +theorem source_stopped_finiteBoundaryPath_measurable + (hpublished : PublishedFiniteLoewnerBoundaryExtension) + {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) : + ∃ Fb : Ω → C(closedHalfPlane, ℂ), + Measurable[hτ.measurableSpace] Fb ∧ + (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → + Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ + ∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω) := by + choose Fb hext htip using fun ω => + normalized_finite_capacity_inverse_boundary_extension hpublished + (continuous_const.mul (hc ω)) (hw ω) (τ ω) 1 zero_lt_one + have he (ω : Ω) (z : closedHalfPlane) (hz : z.1 ∈ upperHalfPlane) : + Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by + simpa only [Complex.ofReal_one, one_mul, add_comm, Pi.mul_apply] using hext ω z hz + refine ⟨Fb, ?_, he, htip⟩ + apply @finite_boundary_extension_family_measurable Ω hτ.measurableSpace + (fun ω z => F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) Fb _ he + intro z hz + exact (continuous_eval_const (⟨z, hz⟩ : upperHalfPlane)).measurable.comp + (source_stopped_inversePath_measurable B hm hc κ γ G F hw τ hτ) + +/-- The actual future trace is the finite boundary image of the fresh +trace, using a chart measurable in precisely the same stopped past. -/ +theorem source_measurable_finite_boundary_restart + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (hpublished : PublishedFiniteLoewnerBoundaryExtension) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) : + ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), ∃ Fb : Ω → C(closedHalfPlane, ℂ), + Measurable Φ ∧ + (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧ + Measurable[hτ.measurableSpace] Fb ∧ + (∀ ω (z : closedHalfPlane), z.1 ∈ upperHalfPlane → + Fb ω z = F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ))) ∧ + (∀ ω, Fb ω ⟨0, by simp [closedHalfPlane]⟩ = γ ω (τ ω)) ∧ + (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → + (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) = + P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧ + ∀ᵐ ω ∂P, ∀ t : ℝ≥0, + γ ω (τ ω + t) = + finiteBoundaryPath (Fb ω) (Φ (sourceRestartDriverPath B hc τ ω)) t := by + obtain ⟨Φ, Fc, hΦm, hΦ, _, hFc, _, hlaw, hrestart⟩ := + source_measurable_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw + hpublished.to_onePoint τ hτ + obtain ⟨Fb, hFbm, hFb, hFb0⟩ := source_stopped_finiteBoundaryPath_measurable + hpublished B hm hc κ γ G F hw τ hτ + have he (ω : Ω) : + (⟨fun z => (Fb ω z : OnePoint ℂ), + OnePoint.continuous_coe.comp (Fb ω).continuous⟩ : + C(closedHalfPlane, OnePoint ℂ)) = Fc ω := by + apply closed_halfplane_map_ext + intro z hz + exact (congrArg (fun w : ℂ => (w : OnePoint ℂ)) (hFb ω z hz)).trans + (hFc ω z hz).symm + refine ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, hFb0, hlaw, ?_⟩ + filter_upwards [hrestart] with ω hω + intro t + apply OnePoint.coe_injective + rw [hω t] + exact (congrArg (fun f : C(closedHalfPlane, OnePoint ℂ) => + f (liftClosedPath (Φ (sourceRestartDriverPath B hc τ ω)) t)) (he ω)).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteCellTailKernel.lean b/lean/OAI/Probability/SLE/Geometry/FiniteCellTailKernel.lean index e6f973f378b196a8894820d46e87737c440dae5f..26c8a7b59a7efed189c8604f9c233ccefb56046e 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteCellTailKernel.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteCellTailKernel.lean @@ -1,212 +1,212 @@ -import OAI.Probability.SLE.Gauge.PrescribedDiskTestKernel -import OAI.Probability.SLE.Gauge.PrescribedDiskConditional - -namespace OAI - -/-! The upper physical cell comparison gives the reverse test inequality: -a large canonical cell mass forces a large normalized finite disk mass. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance cellKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance cellKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def CanonicalMass.finiteCellMassTest - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (S : Ω → WithTop ℝ≥0) - (K : Set ℂ) (b : ℝ≥0∞) : Set Ω := - finiteStoppedThreshold (fun t ω => C.process t ω (Subtype.val ⁻¹' K)) S b - -lemma CanonicalMass.finiteCellMassTest_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun t => - (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) S) - (K : Set ℂ) (hK : MeasurableSet K) (b : ℝ≥0∞) : - MeasurableSet[hS.measurableSpace] (C.finiteCellMassTest S K b) := by - apply finiteStoppedThreshold_measurable _ _ S hS b - intro t - exact (Measure.measurable_coe (hK.preimage measurable_subtype_coe)).comp (C.progressive t) - -lemma cell_ball_live_of_radius - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) - (t : ℝ≥0) (z : ℂ) (hz : z∈survivingDomain γ t) (r : ℝ) (hr : 0 B t ω)) - (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) - (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) - (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hgrowth : PublishedDiskUnivalentGrowth) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) - (hu : 0≤u) (hv : 0 radiusLevelTime (γ ω) (G ω) z (u+v)) K - (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) ≤ - ∫⁻ ω in A∩E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v - (stoppedMarkedCenter κ B γ G τ z ω).arg - ∂finiteCenteredLaw P κ B γ G z (u+v) := by - classical - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let ν := P.map (sourceDriverPath B hc) - let : IsProbabilityMeasure ν := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - let a := stoppedMarkedScale κ B γ G τ z - let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg - let Y := normalizedRestartDriverPath B hc τ a - have ham : Measurable[hτ.measurableSpace] a := - source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z - have ha : ∀ ω, 0 (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) A hA - have hpolar (ω : Ω) (hω : ω∈E) : - θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z= - (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by - have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω) - simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω), - stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2 - obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state - hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase - intro A hA - let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) - let q := fun ω => R.angleDiskTail ν κ b v (θ ω) - let test := C.finiteCellMassTest T K - (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) - let S := E∩test - let Z := fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) - (γ ω) (G ω) z (u+v)/initialGreen κ z) - let H : Set (Ω × C(ℝ≥0,ℝ)) := {x | x.1∈E ∧ b≤R.diskAtRadius κ (θ x.1) v x.2} - let Dens : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x => - M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2) - have hH : @MeasurableSet _ (hτ.measurableSpace.prod (borel _)) H := - (hE.preimage measurable_fst).inter (measurableSet_le measurable_const - (@RawCanonicalMass.diskAtRadius_measurable (Ω × C(ℝ≥0,ℝ)) - (hτ.measurableSpace.prod (borel _)) R κ _ _ (hθm.comp measurable_fst) - measurable_const Prod.snd measurable_snd)) - have hzm : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by - unfold Target.diskMapInv - fun_prop - have hDens : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ Dens := - (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp - (measurable_const.prodMk ((hzm.comp measurable_fst).prodMk measurable_snd))) - have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz - (u+v) (by linarith) - have htest : MeasurableSet test := hT.measurableSpace_le _ - (C.finiteCellMassTest_measurable T hT K hK _) - have hS : MeasurableSet S := (hτ.measurableSpace_le _ hE).inter htest - have hsec (ω : Ω) : (∫⁻ p, H.indicator Dens (ω,p) ∂ν)= - E.indicator (fun ω => M ω*q ω) ω := by - by_cases hω : ω∈E - · let V := {p | b≤R.diskAtRadius κ (θ ω) v p} - have hV : MeasurableSet V := measurableSet_le measurable_const - (R.diskAtRadius_measurable κ (fun _ => θ ω) (fun _ => v) - measurable_const measurable_const id measurable_id) - have hid : (fun p => H.indicator Dens (ω,p)) = - (fun p => M ω*V.indicator (fun p => canonicalFiniteDensity κ - (v,Target.diskMapInv (θ ω) 0,p)) p) := by - funext p - by_cases hp : p∈V - · rw [indicator_of_mem (show (ω,p)∈H from ⟨hω,hp⟩),indicator_of_mem hp] - · rw [indicator_of_notMem (show (ω,p)∉H from fun h => hp h.2), - indicator_of_notMem hp,mul_zero] - rw [hid,lintegral_const_mul' _ _ (hMf ω),lintegral_indicator hV, - ← withDensity_apply _ hV,indicator_of_mem hω] - rfl - · have hn (p : C(ℝ≥0,ℝ)) : (ω,p)∉H := fun h => hω h.1 - simp only [indicator_of_notMem hω] - simp_rw [indicator_of_notMem (hn _)] - exact lintegral_zero - have hden := normalized_restart_disk_density hB hm hc γ G F hw hpublished τ hτ a ham ha - have hcell := C.normalized_cell_bounds_at_radius hB hc hκ F hw R he D hD hDf hDp - hgrowth hpublished τ hτ a ham ha - have hactual : ∀ᵐ ω ∂P, S.indicator Z ω≤H.indicator Dens (ω,Y ω) := by - filter_upwards [hden,hcell,hMactual] with ω hd hb hMω - by_cases hSω : ω∈S - · obtain ⟨hωE,ht⟩ := hSω - have hpol := hpolar ω hωE - have hmass := (hb (θ ω) hpol.1 z (hlive ω hωE) r hr (hR ω hωE) - hpol.2 u hu (hbase ω hωE) v hv ht.1).2 K hK hKball (hKball.trans (cell_ball_live_of_radius (hw ω) hgrowth (τ ω) z (hlive ω hωE) r hr (hR ω hωE))) - have heval : extendedCanonicalMass C.process C.rational.terminal (T ω) ω= - C.process ((T ω).untopD 0) ω := by - change (if T ω<⊤ then C.process ((T ω).untopD 0) ω else C.rational.terminal ω)=_ - exact ite_eq_left ht.1 - rw [heval] at hmass - have hfpos : 0 C.process t ω (Subtype.val ⁻¹' K)) S b + +lemma CanonicalMass.finiteCellMassTest_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun t => + (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) S) + (K : Set ℂ) (hK : MeasurableSet K) (b : ℝ≥0∞) : + MeasurableSet[hS.measurableSpace] (C.finiteCellMassTest S K b) := by + apply finiteStoppedThreshold_measurable _ _ S hS b + intro t + exact (Measure.measurable_coe (hK.preimage measurable_subtype_coe)).comp (C.progressive t) + +lemma cell_ball_live_of_radius + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) + (t : ℝ≥0) (z : ℂ) (hz : z∈survivingDomain γ t) (r : ℝ) (hr : 0 B t ω)) + (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D) + (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D) + (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hgrowth : PublishedDiskUnivalentGrowth) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (z : ℂ) (hz : z∈upperHalfPlane) (u v r b : ℝ) + (hu : 0≤u) (hv : 0 radiusLevelTime (γ ω) (G ω) z (u+v)) K + (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) ≤ + ∫⁻ ω in A∩E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v + (stoppedMarkedCenter κ B γ G τ z ω).arg + ∂finiteCenteredLaw P κ B γ G z (u+v) := by + classical + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let ν := P.map (sourceDriverPath B hc) + let : IsProbabilityMeasure ν := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + let a := stoppedMarkedScale κ B γ G τ z + let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg + let Y := normalizedRestartDriverPath B hc τ a + have ham : Measurable[hτ.measurableSpace] a := + source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z + have ha : ∀ ω, 0 (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) A hA + have hpolar (ω : Ω) (hω : ω∈E) : + θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z= + (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by + have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω) + simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω), + stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2 + obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state + hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase + intro A hA + let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v) + let q := fun ω => R.angleDiskTail ν κ b v (θ ω) + let test := C.finiteCellMassTest T K + (ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*ENNReal.ofReal b) + let S := E∩test + let Z := fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) + (γ ω) (G ω) z (u+v)/initialGreen κ z) + let H : Set (Ω × C(ℝ≥0,ℝ)) := {x | x.1∈E ∧ b≤R.diskAtRadius κ (θ x.1) v x.2} + let Dens : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x => + M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2) + have hH : @MeasurableSet _ (hτ.measurableSpace.prod (borel _)) H := + (hE.preimage measurable_fst).inter (measurableSet_le measurable_const + (@RawCanonicalMass.diskAtRadius_measurable (Ω × C(ℝ≥0,ℝ)) + (hτ.measurableSpace.prod (borel _)) R κ _ _ (hθm.comp measurable_fst) + measurable_const Prod.snd measurable_snd)) + have hzm : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by + unfold Target.diskMapInv + fun_prop + have hDens : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ Dens := + (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp + (measurable_const.prodMk ((hzm.comp measurable_fst).prodMk measurable_snd))) + have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz + (u+v) (by linarith) + have htest : MeasurableSet test := hT.measurableSpace_le _ + (C.finiteCellMassTest_measurable T hT K hK _) + have hS : MeasurableSet S := (hτ.measurableSpace_le _ hE).inter htest + have hsec (ω : Ω) : (∫⁻ p, H.indicator Dens (ω,p) ∂ν)= + E.indicator (fun ω => M ω*q ω) ω := by + by_cases hω : ω∈E + · let V := {p | b≤R.diskAtRadius κ (θ ω) v p} + have hV : MeasurableSet V := measurableSet_le measurable_const + (R.diskAtRadius_measurable κ (fun _ => θ ω) (fun _ => v) + measurable_const measurable_const id measurable_id) + have hid : (fun p => H.indicator Dens (ω,p)) = + (fun p => M ω*V.indicator (fun p => canonicalFiniteDensity κ + (v,Target.diskMapInv (θ ω) 0,p)) p) := by + funext p + by_cases hp : p∈V + · rw [indicator_of_mem (show (ω,p)∈H from ⟨hω,hp⟩),indicator_of_mem hp] + · rw [indicator_of_notMem (show (ω,p)∉H from fun h => hp h.2), + indicator_of_notMem hp,mul_zero] + rw [hid,lintegral_const_mul' _ _ (hMf ω),lintegral_indicator hV, + ← withDensity_apply _ hV,indicator_of_mem hω] + rfl + · have hn (p : C(ℝ≥0,ℝ)) : (ω,p)∉H := fun h => hω h.1 + simp only [indicator_of_notMem hω] + simp_rw [indicator_of_notMem (hn _)] + exact lintegral_zero + have hden := normalized_restart_disk_density hB hm hc γ G F hw hpublished τ hτ a ham ha + have hcell := C.normalized_cell_bounds_at_radius hB hc hκ F hw R he D hD hDf hDp + hgrowth hpublished τ hτ a ham ha + have hactual : ∀ᵐ ω ∂P, S.indicator Z ω≤H.indicator Dens (ω,Y ω) := by + filter_upwards [hden,hcell,hMactual] with ω hd hb hMω + by_cases hSω : ω∈S + · obtain ⟨hωE,ht⟩ := hSω + have hpol := hpolar ω hωE + have hmass := (hb (θ ω) hpol.1 z (hlive ω hωE) r hr (hR ω hωE) + hpol.2 u hu (hbase ω hωE) v hv ht.1).2 K hK hKball (hKball.trans (cell_ball_live_of_radius (hw ω) hgrowth (τ ω) z (hlive ω hωE) r hr (hR ω hωE))) + have heval : extendedCanonicalMass C.process C.rational.terminal (T ω) ω= + C.process ((T ω).untopD 0) ω := by + change (if T ω<⊤ then C.process ((T ω).untopD 0) ω else C.rational.terminal ω)=_ + exact ite_eq_left ht.1 + rw [heval] at hmass + have hfpos : 0 B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (z : ℂ) (hz : z∈upperHalfPlane) (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v) - (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) - (hlive : ∀ ω∈E, z∈survivingDomain (γ ω) (τ ω)) - (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u)) - (f : ℝ → ℝ≥0∞) (hf : Measurable f) - (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) : - (∫⁻ ω in A∩E, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f - (stoppedMarkedCenter κ B γ G τ z ω).arg ∂finiteCenteredLaw P κ B γ G z (u+v)) = - ∫⁻ ω in A∩E, f (extendedMarkedAngle κ B γ G - (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω) - ∂finiteCenteredLaw P κ B γ G z (u+v) := by - classical - have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left - let ν := P.map (sourceDriverPath B hc) - let : IsProbabilityMeasure ν := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - let a := stoppedMarkedScale κ B γ G τ z - let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg - let Y := normalizedRestartDriverPath B hc τ a - have ham : Measurable[hτ.measurableSpace] a := - source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z - have ha : ∀ ω, 0 (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) D hD - have hYlaw0 : P.map Y=ν := by - simpa only [Measure.restrict_univ,measure_univ,one_smul] using hYlaw univ MeasurableSet.univ - obtain ⟨Φ,Gc,Fc,_hΦ,hW⟩ := exists_measurable_canonical_witness _ κ hpublished - have hYw := canonical_witness_pullback hW Y hY hYlaw0 - have hpolar (ω : Ω) (hω : ω∈E) : - θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z= - (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by - have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω) - simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω), - stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2 - obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state - hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase - let q := fun ω => canonicalAngleAverage ν κ v f (θ ω) - have hq : Measurable[hτ.measurableSpace] q := - (canonicalAngleAverage_measurable ν κ v f hf).comp hθm - let K : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x => - M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2)* - f (canonicalStoppedAngle κ (v,Target.diskMapInv (θ x.1) 0,x.2)) - have hzM : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by - unfold Target.diskMapInv - fun_prop - have hpM : @Measurable (Ω × C(ℝ≥0,ℝ)) _ (hτ.measurableSpace.prod (borel _)) _ - (fun x => (v,Target.diskMapInv (θ x.1) 0,x.2)) := - measurable_const.prodMk ((hzM.comp measurable_fst).prodMk measurable_snd) - have hK : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ K := - ((hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp hpM)).mul - (hf.comp ((canonicalStoppedAngle_measurable κ).comp hpM)) - have hsec (ω : Ω) : (∫⁻ p, K (ω,p) ∂ν)=M ω*q ω := by - simp only [K,mul_assoc] - rw [lintegral_const_mul' _ _ (hMf ω)] - congr 1 - symm - exact lintegral_withDensity_eq_lintegral_mul _ - ((canonicalFiniteDensity_measurable κ).comp - (measurable_const.prodMk (measurable_const.prodMk measurable_id))) - (hf.comp ((canonicalStoppedAngle_measurable κ).comp - (measurable_const.prodMk (measurable_const.prodMk measurable_id)))) - let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal (stoppedGreen κ - (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z (u+v)/initialGreen κ z) - let g := fun ω => f (extendedMarkedAngle κ B γ G - (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω) - have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz - (u+v) (by linarith) - have hg : Measurable g := (hf.comp (extendedMarkedAngle_measurable B hm hc κ γ G F hw - _ hT z)).mono hT.measurableSpace_le le_rfl - have hZ : Measurable Z := by - convert (canonicalFiniteDensity_measurable κ).comp - (measurable_const.prodMk (measurable_const.prodMk (source_path_measurable (sourceDriverPath B hc) hm))) using 1 - funext ω - exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) (u+v) - (by linarith) z).symm - have hactual : ∀ᵐ ω ∂P, ω∈E → K (ω,Y ω)=Z ω*g ω := by - filter_upwards [hYw,hMactual] with ω hwY hMω hωE - have hp := hpolar ω hωE - change M ω*canonicalFiniteDensity κ (v,Target.diskMapInv (θ ω) 0,Y ω)* - f (canonicalStoppedAngle κ (v,Target.diskMapInv (θ ω) 0,Y ω))=Z ω*g ω - rw [hMω hωE] - have hn := (hw ω).normalized_disk_angle_test ((hc ω).const_mul _) κ (τ ω) (a ω) - (ha ω) (Y ω) hwY (normalizedRestartDriverPath_driver κ B hc τ a ω) - (θ ω) hp.1 z (hlive ω hωE) hz hp.2 u v hu hv (hbase ω hωE) f - rw [hn] - by_cases ht : radiusLevelTime (γ ω) (G ω) z (u+v)<∞ - · rw [show g ω=f ((G ω (radiusLevelTime (γ ω) (G ω) z (u+v)).toNNReal z- - ((Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z (u+v)).toNNReal ω:ℝ):ℂ)).arg) from - congrArg f (extendedMarkedAngle_at_finite_radius κ B hc γ G F hw z (u+v) ω ht)] - · simp only [Z,stoppedGreen,ite_eq_right ht,zero_div,ENNReal.ofReal_zero,zero_mul] - have hAm : MeasurableSet (A∩E) := hτ.measurableSpace_le _ (hA.inter hE) - calc - (∫⁻ ω in A∩E, q ω ∂finiteCenteredLaw P κ B γ G z (u+v)) = - ∫⁻ ω in A∩E, M ω*q ω ∂P := - (past_density_test_integral P _ hτ.measurableSpace hτ.measurableSpace_le M hM - E hE hMpast q hq A hA).symm - _ = ∫⁻ ω in A∩E, ∫⁻ p, K (ω,p) ∂ν ∂P := by simp only [hsec] - _ = ∫⁻ ω in A∩E, K (ω,Y ω) ∂P := - (restart_setLIntegral_random_test P ν hτ.measurableSpace hτ.measurableSpace_le - Y hY hYlaw K hK (A∩E) (hA.inter hE)).symm - _ = ∫⁻ ω in A∩E, Z ω*g ω ∂P := by - apply setLIntegral_congr_fun_ae hAm - filter_upwards [hactual] with ω hω hωAE - exact hω hωAE.2 - _ = ∫⁻ ω in A∩E, g ω ∂P.withDensity Z := - (setLIntegral_withDensity_eq_setLIntegral_mul _ hZ hg hAm).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Angle.CanonicalAngleKernel +import OAI.Probability.SLE.Geometry.ActualMarkedCell +import OAI.Probability.SLE.Geometry.PastDensityIntegral +import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState +import OAI.Probability.SLE.Geometry.RestartWeightedKernel + +namespace OAI + +/-! Actual finite-centered angle transitions at a finite stopped radius +state. This identifies the transition before applying any mixing bound. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +local instance centeredAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance centeredAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem finite_centered_angle_kernel + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (z : ℂ) (hz : z∈upperHalfPlane) (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v) + (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) + (hlive : ∀ ω∈E, z∈survivingDomain (γ ω) (τ ω)) + (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u)) + (f : ℝ → ℝ≥0∞) (hf : Measurable f) + (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) : + (∫⁻ ω in A∩E, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f + (stoppedMarkedCenter κ B γ G τ z ω).arg ∂finiteCenteredLaw P κ B γ G z (u+v)) = + ∫⁻ ω in A∩E, f (extendedMarkedAngle κ B γ G + (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω) + ∂finiteCenteredLaw P κ B γ G z (u+v) := by + classical + have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left + let ν := P.map (sourceDriverPath B hc) + let : IsProbabilityMeasure ν := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + let a := stoppedMarkedScale κ B γ G τ z + let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg + let Y := normalizedRestartDriverPath B hc τ a + have ham : Measurable[hτ.measurableSpace] a := + source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z + have ha : ∀ ω, 0 (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) D hD + have hYlaw0 : P.map Y=ν := by + simpa only [Measure.restrict_univ,measure_univ,one_smul] using hYlaw univ MeasurableSet.univ + obtain ⟨Φ,Gc,Fc,_hΦ,hW⟩ := exists_measurable_canonical_witness _ κ hpublished + have hYw := canonical_witness_pullback hW Y hY hYlaw0 + have hpolar (ω : Ω) (hω : ω∈E) : + θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z= + (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by + have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω) + simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω), + stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2 + obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state + hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase + let q := fun ω => canonicalAngleAverage ν κ v f (θ ω) + have hq : Measurable[hτ.measurableSpace] q := + (canonicalAngleAverage_measurable ν κ v f hf).comp hθm + let K : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x => + M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2)* + f (canonicalStoppedAngle κ (v,Target.diskMapInv (θ x.1) 0,x.2)) + have hzM : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by + unfold Target.diskMapInv + fun_prop + have hpM : @Measurable (Ω × C(ℝ≥0,ℝ)) _ (hτ.measurableSpace.prod (borel _)) _ + (fun x => (v,Target.diskMapInv (θ x.1) 0,x.2)) := + measurable_const.prodMk ((hzM.comp measurable_fst).prodMk measurable_snd) + have hK : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ K := + ((hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp hpM)).mul + (hf.comp ((canonicalStoppedAngle_measurable κ).comp hpM)) + have hsec (ω : Ω) : (∫⁻ p, K (ω,p) ∂ν)=M ω*q ω := by + simp only [K,mul_assoc] + rw [lintegral_const_mul' _ _ (hMf ω)] + congr 1 + symm + exact lintegral_withDensity_eq_lintegral_mul _ + ((canonicalFiniteDensity_measurable κ).comp + (measurable_const.prodMk (measurable_const.prodMk measurable_id))) + (hf.comp ((canonicalStoppedAngle_measurable κ).comp + (measurable_const.prodMk (measurable_const.prodMk measurable_id)))) + let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal (stoppedGreen κ + (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z (u+v)/initialGreen κ z) + let g := fun ω => f (extendedMarkedAngle κ B γ G + (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω) + have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz + (u+v) (by linarith) + have hg : Measurable g := (hf.comp (extendedMarkedAngle_measurable B hm hc κ γ G F hw + _ hT z)).mono hT.measurableSpace_le le_rfl + have hZ : Measurable Z := by + convert (canonicalFiniteDensity_measurable κ).comp + (measurable_const.prodMk (measurable_const.prodMk (source_path_measurable (sourceDriverPath B hc) hm))) using 1 + funext ω + exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) (hw ω) (u+v) + (by linarith) z).symm + have hactual : ∀ᵐ ω ∂P, ω∈E → K (ω,Y ω)=Z ω*g ω := by + filter_upwards [hYw,hMactual] with ω hwY hMω hωE + have hp := hpolar ω hωE + change M ω*canonicalFiniteDensity κ (v,Target.diskMapInv (θ ω) 0,Y ω)* + f (canonicalStoppedAngle κ (v,Target.diskMapInv (θ ω) 0,Y ω))=Z ω*g ω + rw [hMω hωE] + have hn := (hw ω).normalized_disk_angle_test ((hc ω).const_mul _) κ (τ ω) (a ω) + (ha ω) (Y ω) hwY (normalizedRestartDriverPath_driver κ B hc τ a ω) + (θ ω) hp.1 z (hlive ω hωE) hz hp.2 u v hu hv (hbase ω hωE) f + rw [hn] + by_cases ht : radiusLevelTime (γ ω) (G ω) z (u+v)<∞ + · rw [show g ω=f ((G ω (radiusLevelTime (γ ω) (G ω) z (u+v)).toNNReal z- + ((Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z (u+v)).toNNReal ω:ℝ):ℂ)).arg) from + congrArg f (extendedMarkedAngle_at_finite_radius κ B hc γ G F hw z (u+v) ω ht)] + · simp only [Z,stoppedGreen,ite_eq_right ht,zero_div,ENNReal.ofReal_zero,zero_mul] + have hAm : MeasurableSet (A∩E) := hτ.measurableSpace_le _ (hA.inter hE) + calc + (∫⁻ ω in A∩E, q ω ∂finiteCenteredLaw P κ B γ G z (u+v)) = + ∫⁻ ω in A∩E, M ω*q ω ∂P := + (past_density_test_integral P _ hτ.measurableSpace hτ.measurableSpace_le M hM + E hE hMpast q hq A hA).symm + _ = ∫⁻ ω in A∩E, ∫⁻ p, K (ω,p) ∂ν ∂P := by simp only [hsec] + _ = ∫⁻ ω in A∩E, K (ω,Y ω) ∂P := + (restart_setLIntegral_random_test P ν hτ.measurableSpace hτ.measurableSpace_le + Y hY hYlaw K hK (A∩E) (hA.inter hE)).symm + _ = ∫⁻ ω in A∩E, Z ω*g ω ∂P := by + apply setLIntegral_congr_fun_ae hAm + filter_upwards [hactual] with ω hω hωAE + exact hω hωAE.2 + _ = ∫⁻ ω in A∩E, g ω ∂P.withDensity Z := + (setLIntegral_withDensity_eq_setLIntegral_mul _ hZ hg hAm).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteCenteredConsistency.lean b/lean/OAI/Probability/SLE/Geometry/FiniteCenteredConsistency.lean index 0f88cd28da6f89b060d5c5aad4b18f3379eca806..82be2026a0496f328857de4c05844b8ff6ed0bc0 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteCenteredConsistency.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteCenteredConsistency.lean @@ -1,199 +1,199 @@ -import OAI.Probability.SLE.Geometry.FiniteDensityAtPast -import OAI.Probability.SLE.Loewner.ActualRadiusStops -import OAI.Probability.SLE.Gauge.ExtendedExperimentStop - -namespace OAI - -/-! Exact consistency of the finite centered laws on the past of an -earlier conformal-radius stop. Extended stopping times are exhausted -through finite cutoffs; the finite centered laws give the earlier stop -finite time almost surely. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.physicalStoppedGreen_at_or_before - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (κ : ℝ) (z : ℂ) (u : ℝ) (t : ℝ≥0) - (ht : (t : ℝ≥0∞) ≤ radiusLevelTime γ G z u) : - physicalStoppedGreen κ U γ G z u t = greenDensity κ U γ G t z := by - by_cases hstop : radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) - · have he : radiusLevelTime γ G z u = (t : ℝ≥0∞) := le_antisymm hstop ht - have hf : radiusLevelTime γ G z u < ∞ := by rw [he]; exact ENNReal.coe_lt_top - obtain ⟨T, hT, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hf - have hTt : T = t := by - exact_mod_cast (radiusLevelTime_eq_of_live_level hU hw z u T hT hR).symm.trans he - subst T - rw [physicalStoppedGreen, ite_eq_left hstop, - stoppedGreen_eq_of_live_level hU hw κ z u t hT hR] - · exact ite_eq_right hstop - -/-- Finite-stop consistency holds on the entire earlier stopped past, -including events on which the later radius level is never reached. -/ -theorem finiteCenteredLaw_consistent_at_finite_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) - (σ : Ω → ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) - (fun ω => (σ ω : WithTop ℝ≥0))) - (hbefore : ∀ᵐ ω ∂P, (σ ω : ℝ≥0∞) ≤ radiusLevelTime (γ ω) (G ω) z v) - (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : - finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := by - obtain ⟨Mv, Zv, _, hZvi, _, hZvn, _, hvstop, hvsrc, _⟩ := - exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F hw z hz v hv - obtain ⟨MV, ZV, _, hZVi, _, hZVn, _, hVstop, hVsrc, _⟩ := - exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F hw z hz V (hv.trans hvV) - have hvmeasure : P.withDensity (fun ω => ENNReal.ofReal (Zv ω / initialGreen κ z)) = - finiteCenteredLaw P κ B γ G z v := by - apply withDensity_congr_ae - filter_upwards [hvsrc] with ω hω - rw [hω.2] - have hVmeasure : P.withDensity (fun ω => ENNReal.ofReal (ZV ω / initialGreen κ z)) = - finiteCenteredLaw P κ B γ G z V := by - apply withDensity_congr_ae - filter_upwards [hVsrc] with ω hω - rw [hω.2] - have hev := normalizedDensity_measure_at_past P hσ.measurableSpace hσ.measurableSpace_le - Zv (fun ω => Mv (σ ω) ω) hZvi (Eventually.of_forall (fun ω => (hZvn ω).1)) - (hvstop σ hσ) (initialGreen κ z) (initialGreen_pos κ z hz) A hA - have heV := normalizedDensity_measure_at_past P hσ.measurableSpace hσ.measurableSpace_le - ZV (fun ω => MV (σ ω) ω) hZVi (Eventually.of_forall (fun ω => (hZVn ω).1)) - (hVstop σ hσ) (initialGreen κ z) (initialGreen_pos κ z hz) A hA - rw [hvmeasure] at hev - rw [hVmeasure] at heV - refine heV.trans ((lintegral_congr_ae ?_).trans hev.symm) - filter_upwards [ae_restrict_of_ae hw, ae_restrict_of_ae hbefore, - ae_restrict_of_ae hvsrc, ae_restrict_of_ae hVsrc] with ω hω hbef hvs hVs - rw [hvs.1 (σ ω), hVs.1 (σ ω), - hω.physicalStoppedGreen_at_or_before ((hc ω).const_mul _) κ z v (σ ω) hbef, - hω.physicalStoppedGreen_at_or_before ((hc ω).const_mul _) κ z V (σ ω) - (hbef.trans (radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV))] - -lemma stopped_past_cover_of_ae_finite - {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω) - (σ : Ω → WithTop ℝ≥0) (hfinite : ∀ᵐ ω ∂Q, σ ω ≠ ⊤) (A : Set Ω) : - A =ᵐ[Q] ⋃ n : ℕ, A ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} := by - filter_upwards [hfinite] with ω hω - simp only [mem_iUnion] - apply propext - constructor - · intro hωA - lift σ ω to ℝ≥0 using hω with t ht - obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) - refine ⟨n, hωA, ?_⟩ - change σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) - rw [← ht] - exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) - · rintro ⟨n, hn⟩ - exact hn.1 - -/-- Every earlier stop is finite under a later finite centered law, even -when it is infinite on a set of positive original probability. -/ -theorem finiteCenteredLaw_earlier_stop_ae_finite - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) - (σ : Ω → WithTop ℝ≥0) - (hbefore : ∀ᵐ ω ∂P, σ ω ≤ radiusLevelTime (γ ω) (G ω) z v) : - ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V, σ ω ≠ ⊤ := by - have hl := finiteCenteredLaw_live_compact hB κ hκ hκ8 γ G F hw z hz V (hv.trans hvV) - filter_upwards [hl, (withDensity_absolutelyContinuous P _).ae_le hbefore, - (withDensity_absolutelyContinuous P _).ae_le hw] with ω hω hb hW - obtain ⟨t, _, _, ht, _⟩ := hω - have hlt : σ ω < ⊤ := - (hb.trans (radiusLevelTime_mono ((hc ω).const_mul _) hW z hv hvV)).trans_lt - (by rw [ht]; exact ENNReal.coe_lt_top) - exact hlt.ne - -/-- Extended-stop consistency. The sole geometric requirement is that -this stop occur no later than the earlier conformal-radius level almost -surely under the original law. -/ -theorem finiteCenteredLaw_consistent_at_extended_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) - (σ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ) - (hbefore : ∀ᵐ ω ∂P, σ ω ≤ radiusLevelTime (γ ω) (G ω) z v) - (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : - finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := by - let Qv := finiteCenteredLaw P κ B γ G z v - let QV := finiteCenteredLaw P κ B γ G z V - have hvfinite : ∀ᵐ ω ∂Qv, σ ω ≠ ⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw - z hz v v hv le_rfl σ hbefore - have hVfinite : ∀ᵐ ω ∂QV, σ ω ≠ ⊤ := - finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw - z hz v V hv hvV σ hbefore - let An (n : ℕ) := A ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} - have hmono : Monotone An := by - intro n m hnm ω hω - refine ⟨hω.1, ?_⟩ - change σ ω ≤ ((m : ℝ≥0) : WithTop ℝ≥0) - exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnm)) - have hlocal (n : ℕ) : QV (An n) = Qv (An n) := by - let τ := cutoffStop σ (n : ℝ≥0) - have hτ := cutoffStop_stopping _ σ hσ (n : ℝ≥0) - apply finiteCenteredLaw_consistent_at_finite_stop hB hBm hc hB0 κ hκ hκ8 - γ G F hw z hz v V hv hvV τ hτ _ (An n) - (cutoffStop_success_measurable _ σ hσ _ A hA) - filter_upwards [hbefore] with ω hω - exact (le_of_eq (cutoffStop_coe σ (n : ℝ≥0) ω)).trans ((min_le_left _ _).trans hω) - calc - QV A = QV (⋃ n, An n) := measure_congr (stopped_past_cover_of_ae_finite QV σ hVfinite A) - _ = ⨆ n, QV (An n) := hmono.measure_iUnion - _ = ⨆ n, Qv (An n) := by congr 1; funext n; exact hlocal n - _ = Qv (⋃ n, An n) := hmono.measure_iUnion.symm - _ = Qv A := (measure_congr (stopped_past_cover_of_ae_finite Qv σ hvfinite A)).symm - -/-- The cover application: any stopping-time version of the actual -radius-v time has exactly the same stopped-past law under every later -finite centered measure. -/ -theorem finiteCenteredLaw_consistent_at_radius_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) - (σ : Ω → WithTop ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B - (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ) - (hactual : ∀ᵐ ω ∂P, σ ω = radiusLevelTime (γ ω) (G ω) z v) - (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : - finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := - finiteCenteredLaw_consistent_at_extended_stop hB hBm hc hB0 κ hκ hκ8 γ G F hw - z hz v V hv hvV σ hσ (hactual.mono fun _ h => h.le) A hA - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteDensityAtPast +import OAI.Probability.SLE.Loewner.ActualRadiusStops +import OAI.Probability.SLE.Gauge.ExtendedExperimentStop + +namespace OAI + +/-! Exact consistency of the finite centered laws on the past of an +earlier conformal-radius stop. Extended stopping times are exhausted +through finite cutoffs; the finite centered laws give the earlier stop +finite time almost surely. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.physicalStoppedGreen_at_or_before + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (κ : ℝ) (z : ℂ) (u : ℝ) (t : ℝ≥0) + (ht : (t : ℝ≥0∞) ≤ radiusLevelTime γ G z u) : + physicalStoppedGreen κ U γ G z u t = greenDensity κ U γ G t z := by + by_cases hstop : radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) + · have he : radiusLevelTime γ G z u = (t : ℝ≥0∞) := le_antisymm hstop ht + have hf : radiusLevelTime γ G z u < ∞ := by rw [he]; exact ENNReal.coe_lt_top + obtain ⟨T, hT, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hf + have hTt : T = t := by + exact_mod_cast (radiusLevelTime_eq_of_live_level hU hw z u T hT hR).symm.trans he + subst T + rw [physicalStoppedGreen, ite_eq_left hstop, + stoppedGreen_eq_of_live_level hU hw κ z u t hT hR] + · exact ite_eq_right hstop + +/-- Finite-stop consistency holds on the entire earlier stopped past, +including events on which the later radius level is never reached. -/ +theorem finiteCenteredLaw_consistent_at_finite_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) + (σ : Ω → ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) + (fun ω => (σ ω : WithTop ℝ≥0))) + (hbefore : ∀ᵐ ω ∂P, (σ ω : ℝ≥0∞) ≤ radiusLevelTime (γ ω) (G ω) z v) + (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : + finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := by + obtain ⟨Mv, Zv, _, hZvi, _, hZvn, _, hvstop, hvsrc, _⟩ := + exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F hw z hz v hv + obtain ⟨MV, ZV, _, hZVi, _, hZVn, _, hVstop, hVsrc, _⟩ := + exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F hw z hz V (hv.trans hvV) + have hvmeasure : P.withDensity (fun ω => ENNReal.ofReal (Zv ω / initialGreen κ z)) = + finiteCenteredLaw P κ B γ G z v := by + apply withDensity_congr_ae + filter_upwards [hvsrc] with ω hω + rw [hω.2] + have hVmeasure : P.withDensity (fun ω => ENNReal.ofReal (ZV ω / initialGreen κ z)) = + finiteCenteredLaw P κ B γ G z V := by + apply withDensity_congr_ae + filter_upwards [hVsrc] with ω hω + rw [hω.2] + have hev := normalizedDensity_measure_at_past P hσ.measurableSpace hσ.measurableSpace_le + Zv (fun ω => Mv (σ ω) ω) hZvi (Eventually.of_forall (fun ω => (hZvn ω).1)) + (hvstop σ hσ) (initialGreen κ z) (initialGreen_pos κ z hz) A hA + have heV := normalizedDensity_measure_at_past P hσ.measurableSpace hσ.measurableSpace_le + ZV (fun ω => MV (σ ω) ω) hZVi (Eventually.of_forall (fun ω => (hZVn ω).1)) + (hVstop σ hσ) (initialGreen κ z) (initialGreen_pos κ z hz) A hA + rw [hvmeasure] at hev + rw [hVmeasure] at heV + refine heV.trans ((lintegral_congr_ae ?_).trans hev.symm) + filter_upwards [ae_restrict_of_ae hw, ae_restrict_of_ae hbefore, + ae_restrict_of_ae hvsrc, ae_restrict_of_ae hVsrc] with ω hω hbef hvs hVs + rw [hvs.1 (σ ω), hVs.1 (σ ω), + hω.physicalStoppedGreen_at_or_before ((hc ω).const_mul _) κ z v (σ ω) hbef, + hω.physicalStoppedGreen_at_or_before ((hc ω).const_mul _) κ z V (σ ω) + (hbef.trans (radiusLevelTime_mono ((hc ω).const_mul _) hω z hv hvV))] + +lemma stopped_past_cover_of_ae_finite + {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω) + (σ : Ω → WithTop ℝ≥0) (hfinite : ∀ᵐ ω ∂Q, σ ω ≠ ⊤) (A : Set Ω) : + A =ᵐ[Q] ⋃ n : ℕ, A ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} := by + filter_upwards [hfinite] with ω hω + simp only [mem_iUnion] + apply propext + constructor + · intro hωA + lift σ ω to ℝ≥0 using hω with t ht + obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ) + refine ⟨n, hωA, ?_⟩ + change σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0) + rw [← ht] + exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn) + · rintro ⟨n, hn⟩ + exact hn.1 + +/-- Every earlier stop is finite under a later finite centered law, even +when it is infinite on a set of positive original probability. -/ +theorem finiteCenteredLaw_earlier_stop_ae_finite + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) + (σ : Ω → WithTop ℝ≥0) + (hbefore : ∀ᵐ ω ∂P, σ ω ≤ radiusLevelTime (γ ω) (G ω) z v) : + ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V, σ ω ≠ ⊤ := by + have hl := finiteCenteredLaw_live_compact hB κ hκ hκ8 γ G F hw z hz V (hv.trans hvV) + filter_upwards [hl, (withDensity_absolutelyContinuous P _).ae_le hbefore, + (withDensity_absolutelyContinuous P _).ae_le hw] with ω hω hb hW + obtain ⟨t, _, _, ht, _⟩ := hω + have hlt : σ ω < ⊤ := + (hb.trans (radiusLevelTime_mono ((hc ω).const_mul _) hW z hv hvV)).trans_lt + (by rw [ht]; exact ENNReal.coe_lt_top) + exact hlt.ne + +/-- Extended-stop consistency. The sole geometric requirement is that +this stop occur no later than the earlier conformal-radius level almost +surely under the original law. -/ +theorem finiteCenteredLaw_consistent_at_extended_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) + (σ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ) + (hbefore : ∀ᵐ ω ∂P, σ ω ≤ radiusLevelTime (γ ω) (G ω) z v) + (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : + finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := by + let Qv := finiteCenteredLaw P κ B γ G z v + let QV := finiteCenteredLaw P κ B γ G z V + have hvfinite : ∀ᵐ ω ∂Qv, σ ω ≠ ⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw + z hz v v hv le_rfl σ hbefore + have hVfinite : ∀ᵐ ω ∂QV, σ ω ≠ ⊤ := + finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F hw + z hz v V hv hvV σ hbefore + let An (n : ℕ) := A ∩ {ω | σ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)} + have hmono : Monotone An := by + intro n m hnm ω hω + refine ⟨hω.1, ?_⟩ + change σ ω ≤ ((m : ℝ≥0) : WithTop ℝ≥0) + exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnm)) + have hlocal (n : ℕ) : QV (An n) = Qv (An n) := by + let τ := cutoffStop σ (n : ℝ≥0) + have hτ := cutoffStop_stopping _ σ hσ (n : ℝ≥0) + apply finiteCenteredLaw_consistent_at_finite_stop hB hBm hc hB0 κ hκ hκ8 + γ G F hw z hz v V hv hvV τ hτ _ (An n) + (cutoffStop_success_measurable _ σ hσ _ A hA) + filter_upwards [hbefore] with ω hω + exact (le_of_eq (cutoffStop_coe σ (n : ℝ≥0) ω)).trans ((min_le_left _ _).trans hω) + calc + QV A = QV (⋃ n, An n) := measure_congr (stopped_past_cover_of_ae_finite QV σ hVfinite A) + _ = ⨆ n, QV (An n) := hmono.measure_iUnion + _ = ⨆ n, Qv (An n) := by congr 1; funext n; exact hlocal n + _ = Qv (⋃ n, An n) := hmono.measure_iUnion.symm + _ = Qv A := (measure_congr (stopped_past_cover_of_ae_finite Qv σ hvfinite A)).symm + +/-- The cover application: any stopping-time version of the actual +radius-v time has exactly the same stopped-past law under every later +finite centered measure. -/ +theorem finiteCenteredLaw_consistent_at_radius_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (v V : ℝ) (hv : 0 ≤ v) (hvV : v ≤ V) + (σ : Ω → WithTop ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B + (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ) + (hactual : ∀ᵐ ω ∂P, σ ω = radiusLevelTime (γ ω) (G ω) z v) + (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : + finiteCenteredLaw P κ B γ G z V A = finiteCenteredLaw P κ B γ G z v A := + finiteCenteredLaw_consistent_at_extended_stop hB hBm hc hB0 κ hκ hκ8 γ G F hw + z hz v V hv hvV σ hσ (hactual.mono fun _ h => h.le) A hA + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteCenteredPast.lean b/lean/OAI/Probability/SLE/Geometry/FiniteCenteredPast.lean index f671a5101920a0a7fe37f74b902db9895a3ddf7d..f08c396cab730b5342474222291733be6fd2df01 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteCenteredPast.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteCenteredPast.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Mass.LevelCurrentLimit -import OAI.Probability.SLE.Gauge.CoverChangeOfLaw - -namespace OAI - -/-! Strict past tests of the finite centered laws and their increasing-level limit. -The event remains strictly before the marked radius level. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - -include hB hBm hc h0 hκ hκ8 γ G F hw - -lemma finiteCenteredLaw_strict_past (z : ℂ) (hz : z ∈ upperHalfPlane) - (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : - ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z u - (E ∩ {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u}) = - ∫⁻ ω in E, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P := by - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - let A : Set Ω := {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u} - have hA : MeasurableSet A := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hj := actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw - have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const - have htt := hj.comp hp - have ht : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z u) := by - simpa only [Function.comp_def] using htt - exact measurableSet_lt (measurable_const : Measurable (fun _ : Ω => (t : ℝ≥0∞))) ht - have hEg : MeasurableSet E := (ℱ.le t) _ hE - calc - _ = ∫⁻ ω in E ∩ A, ENNReal.ofReal - (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u) ∂P := - (finiteCenteredLaw_event_identity P κ B γ G z hz u (hEg.inter hA)).symm - _ = ∫⁻ ω in E, ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂P := by - rw [inter_comm E A, ← setLIntegral_indicator hA] - apply lintegral_congr - intro ω - by_cases ha : ω ∈ A - · have hh : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [indicator_of_mem ha, levelRemainderGreen, ite_eq_left hh] - · have hh : ¬ (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [indicator_of_notMem ha, levelRemainderGreen, ite_eq_right hh, ENNReal.ofReal_zero] - _ = _ := levelRemainderGreen_setLIntegral hB hBm hc h0 hκ hκ8 γ G F - (ae_of_all _ hw) z hz u hu t E hE - -lemma finiteCenteredLaw_strict_past_tendsto - (z : ℂ) (hz : z ∈ upperHalfPlane) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : - Tendsto (fun n => ENNReal.ofReal (initialGreen κ z) * - finiteCenteredLaw P κ B γ G z (u n) - (E ∩ {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z (u n)})) atTop - (𝓝 (∫⁻ ω in E, ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P)) := by - simp_rw [finiteCenteredLaw_strict_past hB hBm hc h0 hκ hκ8 γ G F hw z hz _ (hu _) t E hE] - apply lintegral_tendsto_of_tendsto_of_monotone - · intro n - have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const - have hh := (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).comp hp - have hh' : Measurable (fun ω => levelCurrentGreen κ B γ G (u n) t z ω) := by - simpa only [Function.comp_def] using hh - exact hh'.ennreal_ofReal.aemeasurable - · exact ae_of_all _ (fun ω n m hnm => ENNReal.ofReal_le_ofReal - (levelCurrentGreen_mono ω (hc ω) (hw ω) z t (hu n) (hmono hnm))) - · exact ae_of_all _ (fun ω => tendsto_const_nhds.congr' - ((levelCurrentGreen_eventually_eq ω (hc ω) (hw ω) z t u hu hlim).mono - (fun n hn => congrArg ENNReal.ofReal hn.symm))) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelCurrentLimit +import OAI.Probability.SLE.Gauge.CoverChangeOfLaw + +namespace OAI + +/-! Strict past tests of the finite centered laws and their increasing-level limit. +The event remains strictly before the marked radius level. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + +include hB hBm hc h0 hκ hκ8 γ G F hw + +lemma finiteCenteredLaw_strict_past (z : ℂ) (hz : z ∈ upperHalfPlane) + (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : + ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z u + (E ∩ {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u}) = + ∫⁻ ω in E, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P := by + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + let A : Set Ω := {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u} + have hA : MeasurableSet A := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hj := actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw + have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const + have htt := hj.comp hp + have ht : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z u) := by + simpa only [Function.comp_def] using htt + exact measurableSet_lt (measurable_const : Measurable (fun _ : Ω => (t : ℝ≥0∞))) ht + have hEg : MeasurableSet E := (ℱ.le t) _ hE + calc + _ = ∫⁻ ω in E ∩ A, ENNReal.ofReal + (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u) ∂P := + (finiteCenteredLaw_event_identity P κ B γ G z hz u (hEg.inter hA)).symm + _ = ∫⁻ ω in E, ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂P := by + rw [inter_comm E A, ← setLIntegral_indicator hA] + apply lintegral_congr + intro ω + by_cases ha : ω ∈ A + · have hh : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [indicator_of_mem ha, levelRemainderGreen, ite_eq_left hh] + · have hh : ¬ (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [indicator_of_notMem ha, levelRemainderGreen, ite_eq_right hh, ENNReal.ofReal_zero] + _ = _ := levelRemainderGreen_setLIntegral hB hBm hc h0 hκ hκ8 γ G F + (ae_of_all _ hw) z hz u hu t E hE + +lemma finiteCenteredLaw_strict_past_tendsto + (z : ℂ) (hz : z ∈ upperHalfPlane) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : + Tendsto (fun n => ENNReal.ofReal (initialGreen κ z) * + finiteCenteredLaw P κ B γ G z (u n) + (E ∩ {ω | (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z (u n)})) atTop + (𝓝 (∫⁻ ω in E, ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂P)) := by + simp_rw [finiteCenteredLaw_strict_past hB hBm hc h0 hκ hκ8 γ G F hw z hz _ (hu _) t E hE] + apply lintegral_tendsto_of_tendsto_of_monotone + · intro n + have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const + have hh := (levelCurrentGreen_joint_measurable hBm hc κ (u n) (hu n) γ G F hw t).comp hp + have hh' : Measurable (fun ω => levelCurrentGreen κ B γ G (u n) t z ω) := by + simpa only [Function.comp_def] using hh + exact hh'.ennreal_ofReal.aemeasurable + · exact ae_of_all _ (fun ω n m hnm => ENNReal.ofReal_le_ofReal + (levelCurrentGreen_mono ω (hc ω) (hw ω) z t (hu n) (hmono hnm))) + · exact ae_of_all _ (fun ω => tendsto_const_nhds.congr' + ((levelCurrentGreen_eventually_eq ω (hc ω) (hw ω) z t u hu hlim).mono + (fun n hn => congrArg ENNReal.ofReal hn.symm))) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtPast.lean b/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtPast.lean index 31e74e2dd2e2aa9949099e9493a16b45077125b2..6b8f72a6e553b4a2b57a3dbb424c7fe03d679577 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtPast.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtPast.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Gauge.RandomPassageBounds - -namespace OAI - -/-! The finite tilted law on a stopped-past event is the integral of its -conditional density there. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal - -namespace SLEExactGauge - -theorem normalizedDensity_measure_at_past - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Z M : Ω → ℝ) (hZi : Integrable Z P) (hZn : 0 ≤ᵐ[P] Z) - (hM : M =ᵐ[P] P[Z | m]) (a : ℝ) (ha : 0 < a) - (E : Set Ω) (hE : MeasurableSet[m] E) : - P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E = - ∫⁻ ω in E, ENNReal.ofReal (M ω / a) ∂P := by - let : MeasurableSpace Ω := mΩ - have hcond := condLExp_ofReal m hZi hZn - have he : (∫⁻ ω in E, ENNReal.ofReal (M ω) ∂P) = - ∫⁻ ω in E, ENNReal.ofReal (Z ω) ∂P := by - calc - _ = ∫⁻ ω in E, P⁻[(fun ω => ENNReal.ofReal (Z ω)) | m] ω ∂P := by - apply lintegral_congr_ae - filter_upwards [ae_restrict_of_ae hM, ae_restrict_of_ae hcond] with ω hMω hcω - rw [hcω, ← hMω] - _ = _ := setLIntegral_condLExp hm P _ hE - rw [setLIntegral_eq_mul_normalizedDensity P Z ha (hm E hE), - setLIntegral_eq_mul_normalizedDensity P M ha (hm E hE)] at he - have he' : - P.withDensity (fun ω => ENNReal.ofReal (M ω / a)) E * ENNReal.ofReal a = - P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E * ENNReal.ofReal a := - (mul_comm _ _).trans (he.trans (mul_comm _ _)) - have hcancel := (ENNReal.mul_left_inj (ENNReal.ofReal_ne_zero_iff.mpr ha) - ENNReal.ofReal_ne_top).mp he' - rw [withDensity_apply _ (hm E hE)] at hcancel - exact hcancel.symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RandomPassageBounds + +namespace OAI + +/-! The finite tilted law on a stopped-past event is the integral of its +conditional density there. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal + +namespace SLEExactGauge + +theorem normalizedDensity_measure_at_past + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Z M : Ω → ℝ) (hZi : Integrable Z P) (hZn : 0 ≤ᵐ[P] Z) + (hM : M =ᵐ[P] P[Z | m]) (a : ℝ) (ha : 0 < a) + (E : Set Ω) (hE : MeasurableSet[m] E) : + P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E = + ∫⁻ ω in E, ENNReal.ofReal (M ω / a) ∂P := by + let : MeasurableSpace Ω := mΩ + have hcond := condLExp_ofReal m hZi hZn + have he : (∫⁻ ω in E, ENNReal.ofReal (M ω) ∂P) = + ∫⁻ ω in E, ENNReal.ofReal (Z ω) ∂P := by + calc + _ = ∫⁻ ω in E, P⁻[(fun ω => ENNReal.ofReal (Z ω)) | m] ω ∂P := by + apply lintegral_congr_ae + filter_upwards [ae_restrict_of_ae hM, ae_restrict_of_ae hcond] with ω hMω hcω + rw [hcω, ← hMω] + _ = _ := setLIntegral_condLExp hm P _ hE + rw [setLIntegral_eq_mul_normalizedDensity P Z ha (hm E hE), + setLIntegral_eq_mul_normalizedDensity P M ha (hm E hE)] at he + have he' : + P.withDensity (fun ω => ENNReal.ofReal (M ω / a)) E * ENNReal.ofReal a = + P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E * ENNReal.ofReal a := + (mul_comm _ _).trans (he.trans (mul_comm _ _)) + have hcancel := (ENNReal.mul_left_inj (ENNReal.ofReal_ne_zero_iff.mpr ha) + ENNReal.ofReal_ne_top).mp he' + rw [withDensity_apply _ (hm E hE)] at hcancel + exact hcancel.symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtRadiusState.lean b/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtRadiusState.lean index 3b8732f8811d6f442c59fcd139358367450e52cf..de0535527a22f026750a2aa6edb572f9a570f81a 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtRadiusState.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteDensityAtRadiusState.lean @@ -1,68 +1,68 @@ -import OAI.Probability.SLE.Geometry.FiniteDensityAtPast -import OAI.Probability.SLE.Gauge.DenseRadiusGap -import OAI.Probability.SLE.InverseMoments - -namespace OAI - -/-! The exact density on a finite stopped past at a live radius state. -The deeper finite-centered law has the original Green density there. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -theorem exists_finiteDensity_at_radius_state - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B - (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u q : ℝ) (hu : 0 ≤ u) (hq : 0 < q) - (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) - (hlive : ∀ ω ∈ E, z ∈ survivingDomain (γ ω) (τ ω)) - (hbase : ∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) : - ∃ M : Ω → ℝ≥0∞, Measurable[hτ.measurableSpace] M ∧ - (∀ ω, M ω ≠ ∞) ∧ - (∀ᵐ ω ∂P, ω ∈ E → M ω = ENNReal.ofReal - (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z / - initialGreen κ z)) ∧ - ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → - (∫⁻ ω in A ∩ E, M ω ∂P) = finiteCenteredLaw P κ B γ G z (u + q) (A ∩ E) := by - obtain ⟨V, Z, _, hZi, _, hZb, _, hVstop, hsource, _⟩ := - exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F - (Eventually.of_forall hw) z hz (u + q) (by linarith) - let C : Ω → ℝ := P[Z | hτ.measurableSpace] - let M : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal (C ω / initialGreen κ z) - have hCm : Measurable[hτ.measurableSpace] C := stronglyMeasurable_condExp.measurable - have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) = - finiteCenteredLaw P κ B γ G z (u + q) := by - apply withDensity_congr_ae - filter_upwards [hsource] with ω hω - rw [hω.2] - refine ⟨M, ENNReal.measurable_ofReal.comp (hCm.div_const _), - fun _ => ENNReal.ofReal_ne_top, ?_, ?_⟩ - · filter_upwards [hVstop τ hτ, hsource] with ω hVω hsrc hωE - have hbefore := (hw ω).before_deeper_radius_level ((hc ω).const_mul _) z u q hu hq - (τ ω) (hlive ω hωE) (hbase ω hωE) - change ENNReal.ofReal (C ω / _) = _ - rw [show C ω = V (τ ω) ω from hVω.symm, hsrc.1 (τ ω), physicalStoppedGreen, - ite_eq_right (not_le.mpr hbefore)] - · intro A hA - rw [← he] - exact (normalizedDensity_measure_at_past P hτ.measurableSpace hτ.measurableSpace_le - Z C hZi (Eventually.of_forall (fun ω => (hZb ω).1)) Filter.EventuallyEq.rfl - (initialGreen κ z) (initialGreen_pos κ z hz) (A ∩ E) (hA.inter hE)).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteDensityAtPast +import OAI.Probability.SLE.Gauge.DenseRadiusGap +import OAI.Probability.SLE.InverseMoments + +namespace OAI + +/-! The exact density on a finite stopped past at a live radius state. +The deeper finite-centered law has the original Green density there. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +theorem exists_finiteDensity_at_radius_state + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B + (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u q : ℝ) (hu : 0 ≤ u) (hq : 0 < q) + (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) + (hlive : ∀ ω ∈ E, z ∈ survivingDomain (γ ω) (τ ω)) + (hbase : ∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) : + ∃ M : Ω → ℝ≥0∞, Measurable[hτ.measurableSpace] M ∧ + (∀ ω, M ω ≠ ∞) ∧ + (∀ᵐ ω ∂P, ω ∈ E → M ω = ENNReal.ofReal + (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z / + initialGreen κ z)) ∧ + ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → + (∫⁻ ω in A ∩ E, M ω ∂P) = finiteCenteredLaw P κ B γ G z (u + q) (A ∩ E) := by + obtain ⟨V, Z, _, hZi, _, hZb, _, hVstop, hsource, _⟩ := + exists_physical_green_stopping_regular hB hBm hc hB0 κ hκ hκ8 γ G F + (Eventually.of_forall hw) z hz (u + q) (by linarith) + let C : Ω → ℝ := P[Z | hτ.measurableSpace] + let M : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal (C ω / initialGreen κ z) + have hCm : Measurable[hτ.measurableSpace] C := stronglyMeasurable_condExp.measurable + have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) = + finiteCenteredLaw P κ B γ G z (u + q) := by + apply withDensity_congr_ae + filter_upwards [hsource] with ω hω + rw [hω.2] + refine ⟨M, ENNReal.measurable_ofReal.comp (hCm.div_const _), + fun _ => ENNReal.ofReal_ne_top, ?_, ?_⟩ + · filter_upwards [hVstop τ hτ, hsource] with ω hVω hsrc hωE + have hbefore := (hw ω).before_deeper_radius_level ((hc ω).const_mul _) z u q hu hq + (τ ω) (hlive ω hωE) (hbase ω hωE) + change ENNReal.ofReal (C ω / _) = _ + rw [show C ω = V (τ ω) ω from hVω.symm, hsrc.1 (τ ω), physicalStoppedGreen, + ite_eq_right (not_le.mpr hbefore)] + · intro A hA + rw [← he] + exact (normalizedDensity_measure_at_past P hτ.measurableSpace hτ.measurableSpace_le + Z C hZi (Eventually.of_forall (fun ω => (hZb ω).1)) Filter.EventuallyEq.rfl + (initialGreen κ z) (initialGreen_pos κ z hz) (A ∩ E) (hA.inter hE)).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteEnergyArcExtension.lean b/lean/OAI/Probability/SLE/Geometry/FiniteEnergyArcExtension.lean index de97456914c6fb2c678eebc7bed911045412e7b3..8c6c4cc01958deb825a432f89e2e81852a4329a6 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteEnergyArcExtension.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteEnergyArcExtension.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Geometry.ComplexAngularEnergy - -namespace OAI - -/-! An angular section with finite derivative energy has a continuous -extension to both endpoints. The construction is its actual integral primitive. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma finite_energy_angular_extension {f : ℝ → ℂ} - (hf : ∀ θ∈Ioo 0 Real.pi, DifferentiableAt ℝ f θ) - (hd : MemLp (deriv f) 2 (volume.restrict (Ioo 0 Real.pi))) : - ∃ F : ℝ → ℂ, Continuous F ∧ EqOn F f (Ioo 0 Real.pi) := by - let a := Real.pi/2 - have ha : a∈Ioo 0 Real.pi := ⟨half_pos Real.pi_pos,by dsimp [a]; linarith [Real.pi_pos]⟩ - let g := (Ioo 0 Real.pi).indicator (deriv f) - have hi : Integrable g volume := (integrable_indicator_iff measurableSet_Ioo).mpr - (hd.integrable (by norm_num)) - refine ⟨fun θ => f a + ∫ s in a..θ, g s, - continuous_const.add (hi.continuous_primitive a),?_⟩ - intro θ hθ - have hs : uIcc a θ⊆Ioo 0 Real.pi := ordConnected_Ioo.uIcc_subset ha hθ - have hi' : IntegrableOn (deriv f) (Ioo 0 Real.pi) := hd.integrable (by norm_num) - have hid : IntervalIntegrable (deriv f) volume a θ := intervalIntegrable_iff.mpr - (hi'.mono_set (uIoc_subset_uIcc.trans hs)) - have he : (∫ s in a..θ, g s)=∫ s in a..θ, deriv f s := - intervalIntegral.integral_congr (fun s hs' => indicator_of_mem (hs hs') _) - change f a + (∫ s in a..θ, g s)=f θ - rw [he,intervalIntegral.integral_deriv_eq_sub (fun s hs' => hf s (hs hs')) hid] - exact add_sub_cancel _ _ - -lemma continuous_angular_oscillation_on_endpoints {F : ℝ → ℂ} - (hF : Continuous F) (ε : ℝ) - (hsmall : ∀ θ∈Ioo 0 Real.pi, ∀ φ∈Ioo 0 Real.pi, ‖F θ-F φ‖≤ε) : - ∀ θ∈Icc 0 Real.pi, ∀ φ∈Icc 0 Real.pi, ‖F θ-F φ‖≤ε := by - have hc : IsClosed {p : ℝ×ℝ | ‖F p.1-F p.2‖≤ε} := - isClosed_le ((hF.comp continuous_fst).sub (hF.comp continuous_snd)).norm continuous_const - have hs : (Ioo (0:ℝ) Real.pi) ×ˢ (Ioo (0:ℝ) Real.pi)⊆ - {p : ℝ×ℝ | ‖F p.1-F p.2‖≤ε} := fun p hp => hsmall p.1 hp.1 p.2 hp.2 - have hcl := hc.closure_subset_iff.mpr hs - rw [closure_prod_eq,closure_Ioo (ne_of_lt Real.pi_pos)] at hcl - intro θ hθ φ hφ - exact hcl (show (θ,φ)∈(Icc 0 Real.pi) ×ˢ (Icc 0 Real.pi) from ⟨hθ,hφ⟩) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexAngularEnergy + +namespace OAI + +/-! An angular section with finite derivative energy has a continuous +extension to both endpoints. The construction is its actual integral primitive. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma finite_energy_angular_extension {f : ℝ → ℂ} + (hf : ∀ θ∈Ioo 0 Real.pi, DifferentiableAt ℝ f θ) + (hd : MemLp (deriv f) 2 (volume.restrict (Ioo 0 Real.pi))) : + ∃ F : ℝ → ℂ, Continuous F ∧ EqOn F f (Ioo 0 Real.pi) := by + let a := Real.pi/2 + have ha : a∈Ioo 0 Real.pi := ⟨half_pos Real.pi_pos,by dsimp [a]; linarith [Real.pi_pos]⟩ + let g := (Ioo 0 Real.pi).indicator (deriv f) + have hi : Integrable g volume := (integrable_indicator_iff measurableSet_Ioo).mpr + (hd.integrable (by norm_num)) + refine ⟨fun θ => f a + ∫ s in a..θ, g s, + continuous_const.add (hi.continuous_primitive a),?_⟩ + intro θ hθ + have hs : uIcc a θ⊆Ioo 0 Real.pi := ordConnected_Ioo.uIcc_subset ha hθ + have hi' : IntegrableOn (deriv f) (Ioo 0 Real.pi) := hd.integrable (by norm_num) + have hid : IntervalIntegrable (deriv f) volume a θ := intervalIntegrable_iff.mpr + (hi'.mono_set (uIoc_subset_uIcc.trans hs)) + have he : (∫ s in a..θ, g s)=∫ s in a..θ, deriv f s := + intervalIntegral.integral_congr (fun s hs' => indicator_of_mem (hs hs') _) + change f a + (∫ s in a..θ, g s)=f θ + rw [he,intervalIntegral.integral_deriv_eq_sub (fun s hs' => hf s (hs hs')) hid] + exact add_sub_cancel _ _ + +lemma continuous_angular_oscillation_on_endpoints {F : ℝ → ℂ} + (hF : Continuous F) (ε : ℝ) + (hsmall : ∀ θ∈Ioo 0 Real.pi, ∀ φ∈Ioo 0 Real.pi, ‖F θ-F φ‖≤ε) : + ∀ θ∈Icc 0 Real.pi, ∀ φ∈Icc 0 Real.pi, ‖F θ-F φ‖≤ε := by + have hc : IsClosed {p : ℝ×ℝ | ‖F p.1-F p.2‖≤ε} := + isClosed_le ((hF.comp continuous_fst).sub (hF.comp continuous_snd)).norm continuous_const + have hs : (Ioo (0:ℝ) Real.pi) ×ˢ (Ioo (0:ℝ) Real.pi)⊆ + {p : ℝ×ℝ | ‖F p.1-F p.2‖≤ε} := fun p hp => hsmall p.1 hp.1 p.2 hp.2 + have hcl := hc.closure_subset_iff.mpr hs + rw [closure_prod_eq,closure_Ioo (ne_of_lt Real.pi_pos)] at hcl + intro θ hθ φ hφ + exact hcl (show (θ,φ)∈(Icc 0 Real.pi) ×ˢ (Icc 0 Real.pi) from ⟨hθ,hφ⟩) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteExitArcAt.lean b/lean/OAI/Probability/SLE/Geometry/FiniteExitArcAt.lean index 9cc519237a1124e02726d6cd9cc9e6f7fda775e1..28b86f410cefb8276b6688f567737be50e90a626 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteExitArcAt.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteExitArcAt.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Mass.WaypointHarmonicMeasure - -namespace OAI - -/-! Identify exit arcs from any strictly positive local departure time. -This permits a short initial departure inside a marked corner chart. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.RibbonCoordinates -open Target CrosscutSides - -theorem finite_exit_event_measure_le_arc_at - (μ : Measure (Driver × Driver)) (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ) - (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (hg : ContinuousOn g D) (E : Set (Driver × Driver)) {dir a : ℝ} - (hdir : dir = -1 ∨ dir = 1) (ha : 0 ≤ a) - (hexit : ∀ᵐ b ∂μ, ∀ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ → - ∃ x : ℝ, x ≠ 0 ∧ Tendsto (fun t => g (physicalUnitBrownianPath w b t)) - (𝓝[<] τ) (𝓝 (x : ℂ))) - (hgeom : ∀ b ∈ E, ∃ τ : ℝ, - UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ a < τ ∧ - ∀ t ∈ Ico a τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0) - (hstart : ∀ b ∈ E, 0 < dir * (h (g (physicalUnitBrownianPath w b a))).re) : - μ E ≤ μ (unitWindowExitArc D g w dir) := by - apply measure_mono_ae - filter_upwards [hexit] with b hb - intro he - obtain ⟨τ, hτ, haτ, hn⟩ := hgeom b he - obtain ⟨x, hx, hlim⟩ := hb τ hτ - refine ⟨τ, hτ, x, ?_, hlim⟩ - let ζ := physicalUnitBrownianPath w b - have hj : ContinuousOn (fun z => (h (g z)).re) D := - Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) - have hin : MapsTo ζ (Ico a τ) D := hτ.2.2.mono_left (Ico_subset_Ico_left ha) - have hs := hstart b he - rcases hdir with rfl | rfl - · have hp := exit_prefix_negative (fun z => (h (g z)).re) hj ζ - (physicalUnitBrownianPath_continuous w b).continuousOn hin hn haτ (by nlinarith) - have hxneg := real_exit_negative_of_straightened_prefix h hreal - (fun t => g (ζ t)) haτ hx hp hlim - nlinarith - · have hp := exit_prefix_positive (fun z => (h (g z)).re) hj ζ - (physicalUnitBrownianPath_continuous w b).continuousOn hin hn haτ (by simpa using hs) - have hxpos := real_exit_positive_of_straightened_prefix h hreal - (fun t => g (ζ t)) haτ hx hp hlim - simpa using hxpos - -end SLEExactGauge.RibbonCoordinates - -end - -end OAI +import OAI.Probability.SLE.Mass.WaypointHarmonicMeasure + +namespace OAI + +/-! Identify exit arcs from any strictly positive local departure time. +This permits a short initial departure inside a marked corner chart. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.RibbonCoordinates +open Target CrosscutSides + +theorem finite_exit_event_measure_le_arc_at + (μ : Measure (Driver × Driver)) (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ) + (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (hg : ContinuousOn g D) (E : Set (Driver × Driver)) {dir a : ℝ} + (hdir : dir = -1 ∨ dir = 1) (ha : 0 ≤ a) + (hexit : ∀ᵐ b ∂μ, ∀ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ → + ∃ x : ℝ, x ≠ 0 ∧ Tendsto (fun t => g (physicalUnitBrownianPath w b t)) + (𝓝[<] τ) (𝓝 (x : ℂ))) + (hgeom : ∀ b ∈ E, ∃ τ : ℝ, + UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ a < τ ∧ + ∀ t ∈ Ico a τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0) + (hstart : ∀ b ∈ E, 0 < dir * (h (g (physicalUnitBrownianPath w b a))).re) : + μ E ≤ μ (unitWindowExitArc D g w dir) := by + apply measure_mono_ae + filter_upwards [hexit] with b hb + intro he + obtain ⟨τ, hτ, haτ, hn⟩ := hgeom b he + obtain ⟨x, hx, hlim⟩ := hb τ hτ + refine ⟨τ, hτ, x, ?_, hlim⟩ + let ζ := physicalUnitBrownianPath w b + have hj : ContinuousOn (fun z => (h (g z)).re) D := + Complex.continuous_re.comp_continuousOn (h.continuous.comp_continuousOn hg) + have hin : MapsTo ζ (Ico a τ) D := hτ.2.2.mono_left (Ico_subset_Ico_left ha) + have hs := hstart b he + rcases hdir with rfl | rfl + · have hp := exit_prefix_negative (fun z => (h (g z)).re) hj ζ + (physicalUnitBrownianPath_continuous w b).continuousOn hin hn haτ (by nlinarith) + have hxneg := real_exit_negative_of_straightened_prefix h hreal + (fun t => g (ζ t)) haτ hx hp hlim + nlinarith + · have hp := exit_prefix_positive (fun z => (h (g z)).re) hj ζ + (physicalUnitBrownianPath_continuous w b).continuousOn hin hn haτ (by simpa using hs) + have hxpos := real_exit_positive_of_straightened_prefix h hreal + (fun t => g (ζ t)) haτ hx hp hlim + simpa using hxpos + +end SLEExactGauge.RibbonCoordinates + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteLawTailBounds.lean b/lean/OAI/Probability/SLE/Geometry/FiniteLawTailBounds.lean index f8645dcef3042e5be1b5c83bd8ef29ce0e892de6..6303f0036d041f330a3ad871e2a63014437fc43c 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteLawTailBounds.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteLawTailBounds.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Gauge.MomentBatches - -namespace OAI - -/-! The batch estimates need only upper moment bounds on finite laws. -Increasing finite-law tails can be passed to their supremum before summing. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge.MomentBatches - -lemma upper_finite_batch_bound {Ω : Type*} [MeasurableSpace Ω] - (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X) - (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n) - {m T : ℝ} (hm : 0 < m) (hi : Integrable (fun ω => X ω^n) P) - (hMoment : (∫ ω, X ω^n ∂P) ≤ m) (S : Finset ℤ) : - (∑ k ∈ S, (copies n m T k : ℝ)*P.real {ω | 16*dyadic k ≤ X ω}) ≤ - (4:ℝ)^n/16^n/(1-(1/2:ℝ)^n) := by - let C := (4:ℝ)^n/(m*16^n*(1-(1/2:ℝ)^n)) - have hden : 0 < 1-(1/2:ℝ)^n := denominator_pos hn - have hC : 0 ≤ C := by dsimp only [C]; positivity - have hset (k : ℤ) : MeasurableSet {ω | 16*dyadic k ≤ X ω} := - measurableSet_le measurable_const hX - have hInt (k : ℤ) : Integrable - (fun ω => if 16*dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P := - (integrable_const _).indicator (hset k) - calc - _ = ∫ ω, ∑ k ∈ S, if 16*dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P := - (integral_hit_count P X hX n m T 16 S).symm - _ ≤ ∫ ω, C*X ω^n ∂P := integral_mono - (integrable_finsetSum S (fun k _ => hInt k)) (hi.const_mul C) - (fun ω => hit_count_upper hn hm (hX0 ω) S) - _ = C*(∫ ω, X ω^n ∂P) := integral_const_mul _ _ - _ ≤ C*m := mul_le_mul_of_nonneg_left hMoment hC - _ = _ := by dsimp only [C]; field_simp - -lemma cleanup_finite_tail_bound {Ω : Type*} [MeasurableSpace Ω] - (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X) - (hX0 : ∀ ω, 0 ≤ X ω) (hi : Integrable X P) (S : Finset ℕ) : - (∑ n ∈ S, P.real {ω | 16*((n:ℝ)+1) ≤ X ω}) ≤ (∫ ω, X ω ∂P)/16 := by - classical - have hset (n : ℕ) : MeasurableSet {ω | 16*((n:ℝ)+1) ≤ X ω} := - measurableSet_le measurable_const hX - have hInt (n : ℕ) : Integrable - (fun ω => if 16*((n:ℝ)+1) ≤ X ω then (1:ℝ) else 0) P := - (integrable_const _).indicator (hset n) - calc - _ = ∫ ω, ∑ n ∈ S, if 16*((n:ℝ)+1) ≤ X ω then (1:ℝ) else 0 ∂P := by - rw [integral_finsetSum S (fun n _ => hInt n)] - apply Finset.sum_congr rfl - intro n _ - simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one] using - (integral_indicator_const (μ := P) (1:ℝ) (hset n)).symm - _ ≤ ∫ ω, X ω/16 ∂P := integral_mono - (integrable_finsetSum S (fun n _ => hInt n)) (hi.div_const 16) - (fun ω => cleanup_count_le (hX0 ω) S) - _ = _ := integral_div _ _ - -lemma finite_monotone_tail_sum_bound {ι : Type*} (S : Finset ι) - (c : ι → ℝ≥0∞) (p : ℕ → ι → ℝ≥0∞) - (hp : ∀ i, Monotone (fun n => p n i)) (A : ℝ≥0∞) - (hbound : ∀ n, (∑ i ∈ S, c i*p n i) ≤ A) : - (∑ i ∈ S, c i*(⨆ n, p n i)) ≤ A := by - simp_rw [ENNReal.mul_iSup] - rw [ENNReal.finsetSum_iSup_of_monotone (s := S) (f := fun i n => c i*p n i) - (fun i => fun n m hnm => mul_le_mul' le_rfl (hp i hnm))] - exact iSup_le hbound - -end SLEExactGauge.MomentBatches - -end - -end OAI +import OAI.Probability.SLE.Gauge.MomentBatches + +namespace OAI + +/-! The batch estimates need only upper moment bounds on finite laws. +Increasing finite-law tails can be passed to their supremum before summing. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge.MomentBatches + +lemma upper_finite_batch_bound {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X) + (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n) + {m T : ℝ} (hm : 0 < m) (hi : Integrable (fun ω => X ω^n) P) + (hMoment : (∫ ω, X ω^n ∂P) ≤ m) (S : Finset ℤ) : + (∑ k ∈ S, (copies n m T k : ℝ)*P.real {ω | 16*dyadic k ≤ X ω}) ≤ + (4:ℝ)^n/16^n/(1-(1/2:ℝ)^n) := by + let C := (4:ℝ)^n/(m*16^n*(1-(1/2:ℝ)^n)) + have hden : 0 < 1-(1/2:ℝ)^n := denominator_pos hn + have hC : 0 ≤ C := by dsimp only [C]; positivity + have hset (k : ℤ) : MeasurableSet {ω | 16*dyadic k ≤ X ω} := + measurableSet_le measurable_const hX + have hInt (k : ℤ) : Integrable + (fun ω => if 16*dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P := + (integrable_const _).indicator (hset k) + calc + _ = ∫ ω, ∑ k ∈ S, if 16*dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P := + (integral_hit_count P X hX n m T 16 S).symm + _ ≤ ∫ ω, C*X ω^n ∂P := integral_mono + (integrable_finsetSum S (fun k _ => hInt k)) (hi.const_mul C) + (fun ω => hit_count_upper hn hm (hX0 ω) S) + _ = C*(∫ ω, X ω^n ∂P) := integral_const_mul _ _ + _ ≤ C*m := mul_le_mul_of_nonneg_left hMoment hC + _ = _ := by dsimp only [C]; field_simp + +lemma cleanup_finite_tail_bound {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X) + (hX0 : ∀ ω, 0 ≤ X ω) (hi : Integrable X P) (S : Finset ℕ) : + (∑ n ∈ S, P.real {ω | 16*((n:ℝ)+1) ≤ X ω}) ≤ (∫ ω, X ω ∂P)/16 := by + classical + have hset (n : ℕ) : MeasurableSet {ω | 16*((n:ℝ)+1) ≤ X ω} := + measurableSet_le measurable_const hX + have hInt (n : ℕ) : Integrable + (fun ω => if 16*((n:ℝ)+1) ≤ X ω then (1:ℝ) else 0) P := + (integrable_const _).indicator (hset n) + calc + _ = ∫ ω, ∑ n ∈ S, if 16*((n:ℝ)+1) ≤ X ω then (1:ℝ) else 0 ∂P := by + rw [integral_finsetSum S (fun n _ => hInt n)] + apply Finset.sum_congr rfl + intro n _ + simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one] using + (integral_indicator_const (μ := P) (1:ℝ) (hset n)).symm + _ ≤ ∫ ω, X ω/16 ∂P := integral_mono + (integrable_finsetSum S (fun n _ => hInt n)) (hi.div_const 16) + (fun ω => cleanup_count_le (hX0 ω) S) + _ = _ := integral_div _ _ + +lemma finite_monotone_tail_sum_bound {ι : Type*} (S : Finset ι) + (c : ι → ℝ≥0∞) (p : ℕ → ι → ℝ≥0∞) + (hp : ∀ i, Monotone (fun n => p n i)) (A : ℝ≥0∞) + (hbound : ∀ n, (∑ i ∈ S, c i*p n i) ≤ A) : + (∑ i ∈ S, c i*(⨆ n, p n i)) ≤ A := by + simp_rw [ENNReal.mul_iSup] + rw [ENNReal.finsetSum_iSup_of_monotone (s := S) (f := fun i n => c i*p n i) + (fun i => fun n m hnm => mul_le_mul' le_rfl (hp i hnm))] + exact iSup_le hbound + +end SLEExactGauge.MomentBatches + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteLawTailENN.lean b/lean/OAI/Probability/SLE/Geometry/FiniteLawTailENN.lean index cc10fe672d3a723253ec60cbc411f17c34b78732..db2d5435167736d7dd2dc4a515bcfdddbac2460e 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteLawTailENN.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteLawTailENN.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Geometry.FiniteLawTailBounds - -namespace OAI - -/-! Summable dyadic and cleanup tails from increasing finite laws. -No limiting probability measure or future-dependent density is required. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge.MomentBatches - -lemma finite_probability_sum_eq {Ω ι : Type*} [MeasurableSpace Ω] - (Q : Measure Ω) [IsFiniteMeasure Q] (S : Finset ι) - (c : ι → ℕ) (E : ι → Set Ω) : - (∑ i ∈ S, (c i : ℝ≥0∞)*Q (E i)) = - ENNReal.ofReal (∑ i ∈ S, (c i : ℝ)*Q.real (E i)) := by - symm - rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (Nat.cast_nonneg _) measureReal_nonneg)] - apply Finset.sum_congr rfl - intro i _ - rw [ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast,ofReal_measureReal] - -lemma upper_finite_batch_bound_enn {Ω : Type*} [MeasurableSpace Ω] - (Q : Measure Ω) [IsFiniteMeasure Q] (X : Ω → ℝ) (hX : Measurable X) - (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n) - {m T : ℝ} (hm : 0 < m) (hi : Integrable (fun ω => X ω^n) Q) - (hMoment : (∫ ω, X ω^n ∂Q) ≤ m) (S : Finset ℤ) : - (∑ k ∈ S, (copies n m T k : ℝ≥0∞)*Q {ω | 16*dyadic k ≤ X ω}) ≤ - ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by - rw [finite_probability_sum_eq] - exact ENNReal.ofReal_le_ofReal (upper_finite_batch_bound Q X hX hX0 hn hm hi hMoment S) - -lemma limiting_upper_batch_bound {Ω : Type*} [MeasurableSpace Ω] - (Q : ℕ → Measure Ω) (hfin : ∀ j, IsFiniteMeasure (Q j)) - (X : ℕ → Ω → ℝ) (hX : ∀ j, Measurable (X j)) (hX0 : ∀ j ω, 0 ≤ X j ω) - (hmono : ∀ b, Monotone (fun j => Q j {ω | b ≤ X j ω})) - {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m) - (hi : ∀ j, Integrable (fun ω => X j ω^n) (Q j)) - (hMoment : ∀ j, (∫ ω, X j ω^n ∂Q j) ≤ m) : - (∑' k : ℤ, (copies n m T k : ℝ≥0∞)* - (⨆ j, Q j {ω | 16*dyadic k ≤ X j ω})) ≤ - ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by - rw [ENNReal.tsum_eq_iSup_sum] - apply iSup_le - intro S - apply finite_monotone_tail_sum_bound S (fun k => (copies n m T k : ℝ≥0∞)) - (fun j k => Q j {ω | 16*dyadic k ≤ X j ω}) (fun k => hmono _) _ - intro j - let := hfin j - exact upper_finite_batch_bound_enn (Q j) (X j) (hX j) (hX0 j) hn hm (hi j) (hMoment j) S - -lemma limiting_cleanup_tail_bound {Ω : Type*} [MeasurableSpace Ω] - (Q : ℕ → Measure Ω) (hfin : ∀ j, IsFiniteMeasure (Q j)) - (X : ℕ → Ω → ℝ) (hX : ∀ j, Measurable (X j)) (hX0 : ∀ j ω, 0 ≤ X j ω) - (hmono : ∀ b, Monotone (fun j => Q j {ω | b ≤ X j ω})) - (hi : ∀ j, Integrable (X j) (Q j)) (m : ℝ) - (hMoment : ∀ j, (∫ ω, X j ω ∂Q j) ≤ m) : - (∑' n : ℕ, ⨆ j, Q j {ω | 16*((n:ℝ)+1) ≤ X j ω}) ≤ ENNReal.ofReal (m/16) := by - rw [ENNReal.tsum_eq_iSup_sum] - apply iSup_le - intro S - have hs := finite_monotone_tail_sum_bound S (fun _ => (1:ℝ≥0∞)) - (fun j n => Q j {ω | 16*((n:ℝ)+1) ≤ X j ω}) (fun n => hmono _) (ENNReal.ofReal (m/16)) - simp only [one_mul] at hs - apply hs - intro j - let := hfin j - have he := finite_probability_sum_eq (Q j) S (fun _ => 1) - (fun n => {ω | 16*((n:ℝ)+1) ≤ X j ω}) - simp only [Nat.cast_one,one_mul] at he - rw [he] - apply ENNReal.ofReal_le_ofReal - exact (cleanup_finite_tail_bound (Q j) (X j) (hX j) (hX0 j) (hi j) S).trans - (div_le_div_of_nonneg_right (hMoment j) (by norm_num)) - -end SLEExactGauge.MomentBatches - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteLawTailBounds + +namespace OAI + +/-! Summable dyadic and cleanup tails from increasing finite laws. +No limiting probability measure or future-dependent density is required. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge.MomentBatches + +lemma finite_probability_sum_eq {Ω ι : Type*} [MeasurableSpace Ω] + (Q : Measure Ω) [IsFiniteMeasure Q] (S : Finset ι) + (c : ι → ℕ) (E : ι → Set Ω) : + (∑ i ∈ S, (c i : ℝ≥0∞)*Q (E i)) = + ENNReal.ofReal (∑ i ∈ S, (c i : ℝ)*Q.real (E i)) := by + symm + rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (Nat.cast_nonneg _) measureReal_nonneg)] + apply Finset.sum_congr rfl + intro i _ + rw [ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast,ofReal_measureReal] + +lemma upper_finite_batch_bound_enn {Ω : Type*} [MeasurableSpace Ω] + (Q : Measure Ω) [IsFiniteMeasure Q] (X : Ω → ℝ) (hX : Measurable X) + (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n) + {m T : ℝ} (hm : 0 < m) (hi : Integrable (fun ω => X ω^n) Q) + (hMoment : (∫ ω, X ω^n ∂Q) ≤ m) (S : Finset ℤ) : + (∑ k ∈ S, (copies n m T k : ℝ≥0∞)*Q {ω | 16*dyadic k ≤ X ω}) ≤ + ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by + rw [finite_probability_sum_eq] + exact ENNReal.ofReal_le_ofReal (upper_finite_batch_bound Q X hX hX0 hn hm hi hMoment S) + +lemma limiting_upper_batch_bound {Ω : Type*} [MeasurableSpace Ω] + (Q : ℕ → Measure Ω) (hfin : ∀ j, IsFiniteMeasure (Q j)) + (X : ℕ → Ω → ℝ) (hX : ∀ j, Measurable (X j)) (hX0 : ∀ j ω, 0 ≤ X j ω) + (hmono : ∀ b, Monotone (fun j => Q j {ω | b ≤ X j ω})) + {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m) + (hi : ∀ j, Integrable (fun ω => X j ω^n) (Q j)) + (hMoment : ∀ j, (∫ ω, X j ω^n ∂Q j) ≤ m) : + (∑' k : ℤ, (copies n m T k : ℝ≥0∞)* + (⨆ j, Q j {ω | 16*dyadic k ≤ X j ω})) ≤ + ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by + rw [ENNReal.tsum_eq_iSup_sum] + apply iSup_le + intro S + apply finite_monotone_tail_sum_bound S (fun k => (copies n m T k : ℝ≥0∞)) + (fun j k => Q j {ω | 16*dyadic k ≤ X j ω}) (fun k => hmono _) _ + intro j + let := hfin j + exact upper_finite_batch_bound_enn (Q j) (X j) (hX j) (hX0 j) hn hm (hi j) (hMoment j) S + +lemma limiting_cleanup_tail_bound {Ω : Type*} [MeasurableSpace Ω] + (Q : ℕ → Measure Ω) (hfin : ∀ j, IsFiniteMeasure (Q j)) + (X : ℕ → Ω → ℝ) (hX : ∀ j, Measurable (X j)) (hX0 : ∀ j ω, 0 ≤ X j ω) + (hmono : ∀ b, Monotone (fun j => Q j {ω | b ≤ X j ω})) + (hi : ∀ j, Integrable (X j) (Q j)) (m : ℝ) + (hMoment : ∀ j, (∫ ω, X j ω ∂Q j) ≤ m) : + (∑' n : ℕ, ⨆ j, Q j {ω | 16*((n:ℝ)+1) ≤ X j ω}) ≤ ENNReal.ofReal (m/16) := by + rw [ENNReal.tsum_eq_iSup_sum] + apply iSup_le + intro S + have hs := finite_monotone_tail_sum_bound S (fun _ => (1:ℝ≥0∞)) + (fun j n => Q j {ω | 16*((n:ℝ)+1) ≤ X j ω}) (fun n => hmono _) (ENNReal.ofReal (m/16)) + simp only [one_mul] at hs + apply hs + intro j + let := hfin j + have he := finite_probability_sum_eq (Q j) S (fun _ => 1) + (fun n => {ω | 16*((n:ℝ)+1) ≤ X j ω}) + simp only [Nat.cast_one,one_mul] at he + rw [he] + apply ENNReal.ofReal_le_ofReal + exact (cleanup_finite_tail_bound (Q j) (X j) (hX j) (hX0 j) (hi j) S).trans + (div_le_div_of_nonneg_right (hMoment j) (by norm_num)) + +end SLEExactGauge.MomentBatches + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FiniteParameterProjection.lean b/lean/OAI/Probability/SLE/Geometry/FiniteParameterProjection.lean index ca1fac7e2ddcdcbd0191af6ed2568ba8b48571a7..507ed15b81c87c278a4b61bc84a034eac377fa3c 100644 --- a/lean/OAI/Probability/SLE/Geometry/FiniteParameterProjection.lean +++ b/lean/OAI/Probability/SLE/Geometry/FiniteParameterProjection.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Geometry.ContinuousRangeCompletion - -namespace OAI - -/-! Transfer of Borel parameter families under a finite path law, with one -exceptional set for all parameters. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma nullMeasurableSet_forall_parameter - {X T : Type*} [MetricSpace X] [SecondCountableTopology X] - [MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] - [MeasurableSpace T] [StandardBorelSpace T] - (μ : Measure X) [IsFiniteMeasure μ] (p : T → X → Prop) - (hp : MeasurableSet {q : T × X | p q.1 q.2}) : - NullMeasurableSet {x | ∀ t, p t x} μ := by - have hbad : AnalyticSet (Prod.snd '' {q : T × X | ¬p q.1 q.2}) := - hp.compl.analyticSet_image measurable_snd - have heq : {x | ∀ t, p t x}=(Prod.snd '' {q : T × X | ¬p q.1 q.2})ᶜ := by - ext x - simp only [mem_compl_iff,mem_image,mem_ofPred_eq,Prod.exists, - exists_eq_right,not_exists,not_not] - rw [heq] - exact (nullMeasurableSet_of_analytic_finite μ hbad).compl - -lemma ae_map_iff_nullMeasurableSet - {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - {P : Measure Ω} {f : Ω → X} (hf : AEMeasurable f P) {p : X → Prop} - (hp : NullMeasurableSet {x | p x} (P.map f)) : - (∀ᵐ x ∂P.map f, p x) ↔ ∀ᵐ ω ∂P, p (f ω) := by - have hn : NullMeasurableSet {x | ¬p x} (P.map f) := hp.compl - rw [ae_iff,Measure.map_apply₀ hf hn] - rfl - -lemma ae_forall_parameter_of_hasLaw - {Ω X T : Type*} [MeasurableSpace Ω] - [MetricSpace X] [SecondCountableTopology X] - [MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] - [MeasurableSpace T] [StandardBorelSpace T] - {P : Measure Ω} {μ : Measure X} [IsFiniteMeasure μ] - {f : Ω → X} (hf : HasLaw f μ P) (p : T → X → Prop) - (hp : MeasurableSet {q : T × X | p q.1 q.2}) - (h : ∀ᵐ ω ∂P, ∀ t, p t (f ω)) : - ∀ᵐ x ∂μ, ∀ t, p t x := by - have hn := nullMeasurableSet_forall_parameter μ p hp - rw [← hf.map_eq] at hn ⊢ - exact (ae_map_iff_nullMeasurableSet hf.aemeasurable hn).mpr h - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ContinuousRangeCompletion + +namespace OAI + +/-! Transfer of Borel parameter families under a finite path law, with one +exceptional set for all parameters. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma nullMeasurableSet_forall_parameter + {X T : Type*} [MetricSpace X] [SecondCountableTopology X] + [MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] + [MeasurableSpace T] [StandardBorelSpace T] + (μ : Measure X) [IsFiniteMeasure μ] (p : T → X → Prop) + (hp : MeasurableSet {q : T × X | p q.1 q.2}) : + NullMeasurableSet {x | ∀ t, p t x} μ := by + have hbad : AnalyticSet (Prod.snd '' {q : T × X | ¬p q.1 q.2}) := + hp.compl.analyticSet_image measurable_snd + have heq : {x | ∀ t, p t x}=(Prod.snd '' {q : T × X | ¬p q.1 q.2})ᶜ := by + ext x + simp only [mem_compl_iff,mem_image,mem_ofPred_eq,Prod.exists, + exists_eq_right,not_exists,not_not] + rw [heq] + exact (nullMeasurableSet_of_analytic_finite μ hbad).compl + +lemma ae_map_iff_nullMeasurableSet + {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + {P : Measure Ω} {f : Ω → X} (hf : AEMeasurable f P) {p : X → Prop} + (hp : NullMeasurableSet {x | p x} (P.map f)) : + (∀ᵐ x ∂P.map f, p x) ↔ ∀ᵐ ω ∂P, p (f ω) := by + have hn : NullMeasurableSet {x | ¬p x} (P.map f) := hp.compl + rw [ae_iff,Measure.map_apply₀ hf hn] + rfl + +lemma ae_forall_parameter_of_hasLaw + {Ω X T : Type*} [MeasurableSpace Ω] + [MetricSpace X] [SecondCountableTopology X] + [MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] + [MeasurableSpace T] [StandardBorelSpace T] + {P : Measure Ω} {μ : Measure X} [IsFiniteMeasure μ] + {f : Ω → X} (hf : HasLaw f μ P) (p : T → X → Prop) + (hp : MeasurableSet {q : T × X | p q.1 q.2}) + (h : ∀ᵐ ω ∂P, ∀ t, p t (f ω)) : + ∀ᵐ x ∂μ, ∀ t, p t x := by + have hn := nullMeasurableSet_forall_parameter μ p hp + rw [← hf.map_eq] at hn ⊢ + exact (ae_map_iff_nullMeasurableSet hf.aemeasurable hn).mpr h + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FinitePastCellTail.lean b/lean/OAI/Probability/SLE/Geometry/FinitePastCellTail.lean index c445056106bc89115cd85a88dfe129a1e95ac8ed..6450c34448b102299256df4d1198829e5805dafb 100644 --- a/lean/OAI/Probability/SLE/Geometry/FinitePastCellTail.lean +++ b/lean/OAI/Probability/SLE/Geometry/FinitePastCellTail.lean @@ -1,41 +1,41 @@ -import OAI.Probability.SLE.Geometry.FiniteCellTailKernel -import OAI.Probability.SLE.Geometry.FinitePastEvents - -namespace OAI - -/-! A positive mass test strictly before a finite radius frontier forces -the corresponding canonical cell mass test at the frontier. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma CanonicalMass.finite_strict_past_le_cell_test - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (Q : Measure Ω) (hQP : Q.AbsolutelyContinuous P) - (T : Ω → WithTop ℝ≥0) (hT : ∀ᵐ ω ∂Q, T ω≠⊤) - (J : Finset ℝ≥0) (K : Set ℂ) (H : ℝ) : - Q (finiteStrictPastEvent J (fun t => {ω | H<(C.process t ω (Subtype.val ⁻¹' K)).toReal}) T) ≤ - Q (C.finiteCellMassTest T K (ENNReal.ofReal H)) := by - apply measure_mono_ae - filter_upwards [hQP.ae_le C.paths,hT] with ω hp hf hω - obtain ⟨t,_ht,hval,hbefore⟩ := mem_iUnion₂.mp hω - have hf' := hf - lift T ω to ℝ≥0 using hf with s hs - have hts : t ≤ s := by - apply WithTop.coe_le_coe.mp - simpa only [hs] using hbefore.le - have hmass := hp.monotone hts (Subtype.val ⁻¹' K) - refine ⟨lt_top_iff_ne_top.mpr (hs ▸ hf'),?_⟩ - have htime : (T ω).untopD 0=s := by rw [← hs,WithTop.untopD_coe] - rw [htime] - exact (ENNReal.ofReal_le_ofReal hval.le).trans (ENNReal.ofReal_toReal_le.trans hmass) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteCellTailKernel +import OAI.Probability.SLE.Geometry.FinitePastEvents + +namespace OAI + +/-! A positive mass test strictly before a finite radius frontier forces +the corresponding canonical cell mass test at the frontier. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma CanonicalMass.finite_strict_past_le_cell_test + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (Q : Measure Ω) (hQP : Q.AbsolutelyContinuous P) + (T : Ω → WithTop ℝ≥0) (hT : ∀ᵐ ω ∂Q, T ω≠⊤) + (J : Finset ℝ≥0) (K : Set ℂ) (H : ℝ) : + Q (finiteStrictPastEvent J (fun t => {ω | H<(C.process t ω (Subtype.val ⁻¹' K)).toReal}) T) ≤ + Q (C.finiteCellMassTest T K (ENNReal.ofReal H)) := by + apply measure_mono_ae + filter_upwards [hQP.ae_le C.paths,hT] with ω hp hf hω + obtain ⟨t,_ht,hval,hbefore⟩ := mem_iUnion₂.mp hω + have hf' := hf + lift T ω to ℝ≥0 using hf with s hs + have hts : t ≤ s := by + apply WithTop.coe_le_coe.mp + simpa only [hs] using hbefore.le + have hmass := hp.monotone hts (Subtype.val ⁻¹' K) + refine ⟨lt_top_iff_ne_top.mpr (hs ▸ hf'),?_⟩ + have htime : (T ω).untopD 0=s := by rw [← hs,WithTop.untopD_coe] + rw [htime] + exact (ENNReal.ofReal_le_ofReal hval.le).trans (ENNReal.ofReal_toReal_le.trans hmass) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Geometry/FinitePastEvents.lean b/lean/OAI/Probability/SLE/Geometry/FinitePastEvents.lean index b28d077e300dc1431b5499aa43931975bbc7185b..d4e8a8affb7946d98f4ab5732f5b3fd222dbb657 100644 --- a/lean/OAI/Probability/SLE/Geometry/FinitePastEvents.lean +++ b/lean/OAI/Probability/SLE/Geometry/FinitePastEvents.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Geometry.FirstSuccessfulPast - -namespace OAI - -/-! First-success partition for finite tests strictly before an extended time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint - -variable {Ω : Type*} - -def finiteStrictPastEvent (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (T : Ω → ℝ≥0∞) : Set Ω := - ⋃ t∈I, E t ∩ {ω | (t : ℝ≥0∞) < T ω} - -lemma finiteStrictPastEvent_first (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (T : Ω → ℝ≥0∞) : - finiteStrictPastEvent I E T = - ⋃ t∈I, firstPastSuccess I E t ∩ {ω | (t : ℝ≥0∞) < T ω} := by - ext ω - constructor - · intro h - obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp h - obtain ⟨s,hs,hEs,hst⟩ := exists_firstPastSuccess I E ω ⟨t,ht,hEt⟩ - exact mem_iUnion₂.mpr ⟨s,hs,hEs,(ENNReal.coe_le_coe.mpr (hst t ht hEt)).trans_lt hT⟩ - · intro h - obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp h - exact mem_iUnion₂.mpr ⟨t,ht,((mem_firstPastSuccess_iff I E t ω).mp hEt).1,hT⟩ - -lemma finiteStrictPastEvent_mono (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - {S T : Ω → ℝ≥0∞} (h : S ≤ T) : - finiteStrictPastEvent I E S ⊆ finiteStrictPastEvent I E T := by - intro ω hω - obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp hω - exact mem_iUnion₂.mpr ⟨t,ht,hEt,hT.trans_le (h ω)⟩ - -lemma finiteStrictPastEvent_measurable [MeasurableSpace Ω] - (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) (hE : ∀ t∈I, MeasurableSet (E t)) - (T : Ω → ℝ≥0∞) (hT : Measurable T) : MeasurableSet (finiteStrictPastEvent I E T) := - Finset.measurableSet_biUnion I (fun t ht => (hE t ht).inter - (measurableSet_lt measurable_const hT)) - -lemma finiteStrictPastEvent_measure [mΩ : MeasurableSpace Ω] - (Q : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) - (hE : ∀ t∈I, MeasurableSet[ℱ t] (E t)) - (T : Ω → ℝ≥0∞) (hT : Measurable T) : - Q (finiteStrictPastEvent I E T) = - ∑ t∈I, Q (firstPastSuccess I E t ∩ {ω | (t : ℝ≥0∞) < T ω}) := by - rw [finiteStrictPastEvent_first] - apply measure_biUnion_finset - · intro s hs t ht hne - exact (firstPastSuccess_disjoint I E hs ht hne).mono inter_subset_left inter_subset_left - · intro t ht - exact ((ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht)).inter - (measurableSet_lt measurable_const hT) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.FirstSuccessfulPast + +namespace OAI + +/-! First-success partition for finite tests strictly before an extended time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge.Multipoint + +variable {Ω : Type*} + +def finiteStrictPastEvent (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (T : Ω → ℝ≥0∞) : Set Ω := + ⋃ t∈I, E t ∩ {ω | (t : ℝ≥0∞) < T ω} + +lemma finiteStrictPastEvent_first (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (T : Ω → ℝ≥0∞) : + finiteStrictPastEvent I E T = + ⋃ t∈I, firstPastSuccess I E t ∩ {ω | (t : ℝ≥0∞) < T ω} := by + ext ω + constructor + · intro h + obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp h + obtain ⟨s,hs,hEs,hst⟩ := exists_firstPastSuccess I E ω ⟨t,ht,hEt⟩ + exact mem_iUnion₂.mpr ⟨s,hs,hEs,(ENNReal.coe_le_coe.mpr (hst t ht hEt)).trans_lt hT⟩ + · intro h + obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp h + exact mem_iUnion₂.mpr ⟨t,ht,((mem_firstPastSuccess_iff I E t ω).mp hEt).1,hT⟩ + +lemma finiteStrictPastEvent_mono (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + {S T : Ω → ℝ≥0∞} (h : S ≤ T) : + finiteStrictPastEvent I E S ⊆ finiteStrictPastEvent I E T := by + intro ω hω + obtain ⟨t,ht,hEt,hT⟩ := mem_iUnion₂.mp hω + exact mem_iUnion₂.mpr ⟨t,ht,hEt,hT.trans_le (h ω)⟩ + +lemma finiteStrictPastEvent_measurable [MeasurableSpace Ω] + (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) (hE : ∀ t∈I, MeasurableSet (E t)) + (T : Ω → ℝ≥0∞) (hT : Measurable T) : MeasurableSet (finiteStrictPastEvent I E T) := + Finset.measurableSet_biUnion I (fun t ht => (hE t ht).inter + (measurableSet_lt measurable_const hT)) + +lemma finiteStrictPastEvent_measure [mΩ : MeasurableSpace Ω] + (Q : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ) (I : Finset ℝ≥0) (E : ℝ≥0 → Set Ω) + (hE : ∀ t∈I, MeasurableSet[ℱ t] (E t)) + (T : Ω → ℝ≥0∞) (hT : Measurable T) : + Q (finiteStrictPastEvent I E T) = + ∑ t∈I, Q (firstPastSuccess I E t ∩ {ω | (t : ℝ≥0∞) < T ω}) := by + rw [finiteStrictPastEvent_first] + apply measure_biUnion_finset + · intro s hs t ht hne + exact (firstPastSuccess_disjoint I E hs ht hne).mono inter_subset_left inter_subset_left + · intro t ht + exact ((ℱ.le t) _ (firstPastSuccess_measurable ℱ I E hE ht)).inter + (measurableSet_lt measurable_const hT) + +end SLEExactGauge.Multipoint + +end + +end OAI