| import Mathlib | |
| import OAI.Geometry.Cannon.Flows.FlowSetup | |
| import OAI.Geometry.Cannon.Boundary.BoundaryBasedRay | |
| import OAI.Geometry.Cannon.Boundary.VisualSynchronousPrefix | |
| import OAI.Geometry.Cannon.Flows.NearProbabilityBound | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| section | |
| namespace CannonRelease | |
| open CannonGraph CannonVisual CannonApprox CannonNode CannonPairs Set Metric Filter Topology Finset | |
| universe v | |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] | |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) | |
| local instance flowPunctureSetoid : Setoid (BasedRay D) := raySetoid D | |
| local instance flowPunctureMeasurableSpace : MeasurableSpace (Boundary D) := borel (Boundary D) | |
| local instance flowPunctureBorelSpace : BorelSpace (Boundary D) := ⟨rfl⟩ | |
| variable {ι : ℕ → Type*} [∀ n, Fintype (ι n)] {p : ∀ n, ι n → Boundary D} {r : ℕ → ℝ} | |
| lemma flow_scale_visual : | |
| let _ := visualMetricSpace D hδ | |
| ∀ (T : RecordData (boundaryHomeomorph D ⟨δ,hδ⟩) ι p r) | |
| {L CE LE : ℝ} {Q : ℕ} (F : FlowSetup T (2*visualExponent δ) L CE LE Q), ∀ g : ℕ, | |
| F.S.scale g=visualScale δ (F.S.kmin+160*(Q+1)*F.n*g) := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro T L CE LE Q F g | |
| dsimp [CannonNode.SourceConstruction.scale,CannonNode.radius,CannonVisual.visualScale] | |
| congr 1 | |
| push_cast | |
| ring | |
| lemma flow_center_z₀ : | |
| let _ := visualMetricSpace D hδ | |
| ∀ (T : RecordData (boundaryHomeomorph D ⟨δ,hδ⟩) ι p r) | |
| {L CE LE : ℝ} {Q : ℕ} (F : FlowSetup T (2*visualExponent δ) L CE LE Q), | |
| 0 ≤ F.B+1 → ∀ (g : ℕ) (i : Fin (F.S.flow.size g)), | |
| F.ε ≤ dist (F.S.center g i) T.z₀ := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro T L CE LE Q F hB g i | |
| by_contra hn | |
| have hi := mem_closedBall.mp (F.S.center_mem hB g i) | |
| apply T.z₀not (F.local_domain ?_) | |
| rw [mem_closedBall] | |
| have ht := dist_triangle T.z₀ (F.S.center g i) T.x₀ | |
| rw [dist_comm T.z₀ (F.S.center g i)] at ht | |
| linarith only [ht,hi,lt_of_not_ge hn] | |
| def flowPuncture : | |
| let _ := visualMetricSpace D hδ | |
| ∀ (T : RecordData (boundaryHomeomorph D ⟨δ,hδ⟩) ι p r) | |
| {L CE LE : ℝ} {Q : ℕ} (F : FlowSetup T (2*visualExponent δ) L CE LE Q), | |
| ∀ (g : ℕ), Fin (F.S.flow.size g) → Boundary D := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro T L CE LE Q F g i | |
| exact boundaryExpansion D hδ (F.S.center g i) (F.S.kmin+160*(Q+1)*F.n*g) T.z₀ | |
| theorem flow_puncture_multiplicity : | |
| let _ := visualMetricSpace D hδ | |
| ∀ (T : RecordData (boundaryHomeomorph D ⟨δ,hδ⟩) ι p r) | |
| {L CE LE : ℝ} {Q : ℕ} (F : FlowSetup T (2*visualExponent δ) L CE LE Q), | |
| 0 ≤ F.B+1 → 4*(F.B+1) < F.H → ∀ C : ℝ, 0 ≤ C → | |
| ∃ N : ℕ, ∀ g b, ((univ : Finset (Fin (F.S.flow.size g))).filter | |
| (fun i => dist (flowPuncture D hδ T F g i) b ≤ C*F.S.scale g)).card ≤ N := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro T L CE LE Q F hB hH C hC | |
| have hep (g : ℕ) (i : Fin (F.S.flow.size g)) : flowPuncture D hδ T F g i = | |
| boundaryTranslate D ⟨δ,hδ⟩ ((boundaryBasedRay D (F.S.center g i)).val | |
| (F.S.kmin+160*(Q+1)*F.n*g))⁻¹ T.z₀ := rfl | |
| have hdistance (x y : Boundary D) : dist x y=boundaryDist D hδ x y := rfl | |
| have hHp : 0 < F.H := by linarith only [hB,hH] | |
| obtain ⟨N,hN⟩ := puncture_ball_multiplicity D hδ F.εpos hHp hC | |
| refine ⟨N,fun g b => ?_⟩ | |
| have hh := hN (univ : Finset (Fin (F.S.flow.size g))) | |
| (fun i => boundaryBasedRay D (F.S.center g i)) (boundaryBasedRay D T.z₀) | |
| (F.S.kmin+160*(Q+1)*F.n*g) | |
| (fun i _ => by simpa only [boundaryBasedRay_rep] using flow_center_z₀ D hδ T F hB g i) | |
| (fun i _ j _ hij => by simpa only [boundaryBasedRay_rep,← flow_scale_visual D hδ T F g] | |
| using F.S.centers_separated F.decay hB hH g i j hij) b | |
| simpa only [boundaryBasedRay_rep,← flow_scale_visual D hδ T F g,hep,hdistance] using hh | |
| theorem flow_near_probability : | |
| let _ := visualMetricSpace D hδ | |
| ∀ (T : RecordData (boundaryHomeomorph D ⟨δ,hδ⟩) ι p r) | |
| {L CE LE : ℝ} {Q : ℕ} (F : FlowSetup T (2*visualExponent δ) L CE LE Q), | |
| 0 ≤ F.B+1 → 4*(F.B+1) < F.H → | |
| ∃ N : ℕ, ∀ g m (b : Boundary D), | |
| ∑ i ∈ (univ : Finset (Fin (F.S.flow.size (g+m)))).filter | |
| (fun i => dist (flowPuncture D hδ T F (g+m) i) b ≤ visualScale δ (160*(Q+1)*F.n*m)), | |
| F.S.flow.weight (g+m) i ≤ (N:ℝ)*(Real.exp (-(2*visualExponent δ)*F.n))^m := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| intro T L CE LE Q F hB hH | |
| have hep (g : ℕ) (i : Fin (F.S.flow.size g)) : flowPuncture D hδ T F g i = | |
| boundaryTranslate D ⟨δ,hδ⟩ ((boundaryBasedRay D (F.S.center g i)).val | |
| (F.S.kmin+160*(Q+1)*F.n*g))⁻¹ T.z₀ := rfl | |
| have hdistance (x y : Boundary D) : dist x y=boundaryDist D hδ x y := rfl | |
| have hHp : 0 < F.H := by linarith only [hB,hH] | |
| obtain ⟨N,hN⟩ := near_puncture_descendant_count D hδ F.εpos hHp (by positivity : 0 ≤ 2*(F.B+1)) | |
| refine ⟨N,fun g m b => ?_⟩ | |
| let S := (univ : Finset (Fin (F.S.flow.size (g+m)))).filter | |
| (fun i => dist (flowPuncture D hδ T F (g+m) i) b ≤ visualScale δ (160*(Q+1)*F.n*m)) | |
| have hcount (a : Fin (F.S.flow.size g)) : | |
| (S.filter (fun i => F.S.flow.ancestor g m i=a)).card ≤ N := by | |
| let V := S.filter (fun i => F.S.flow.ancestor g m i=a) | |
| have hd : F.S.kmin+160*(Q+1)*F.n*(g+m)-160*(Q+1)*F.n*m = | |
| F.S.kmin+160*(Q+1)*F.n*g := by simp only [Nat.mul_add]; omega | |
| apply hN V (fun i => boundaryBasedRay D (F.S.center (g+m) i)) | |
| (boundaryBasedRay D T.z₀) (boundaryBasedRay D (F.S.center g a)) (boundaryBasedRay D b) | |
| (F.S.kmin+160*(Q+1)*F.n*(g+m)) (160*(Q+1)*F.n*m) (by simp only [Nat.mul_add]; omega) | |
| · intro i _ | |
| simpa only [boundaryBasedRay_rep] using flow_center_z₀ D hδ T F hB (g+m) i | |
| · intro i _ j _ hij | |
| simpa only [boundaryBasedRay_rep,← flow_scale_visual D hδ T F (g+m)] | |
| using F.S.centers_separated F.decay hB hH (g+m) i j hij | |
| · intro i hi | |
| have hia := (mem_filter.mp hi).2 | |
| have hx := F.S.descendant_displacement F.decay hB g m i | |
| rw [hia,flow_scale_visual D hδ T F g] at hx | |
| simpa only [boundaryBasedRay_rep,hd] using hx | |
| · intro i hi | |
| have hip := (mem_filter.mp (mem_filter.mp hi).1).2 | |
| simpa only [boundaryBasedRay_rep,hep,hdistance] using hip | |
| have hh := near_probability_bound S (F.S.flow.ancestor g m) (F.S.flow.weight (g+m)) | |
| (F.S.flow.weight g) N ((Real.exp (-(2*visualExponent δ)*F.n))^m) | |
| (pow_nonneg (Real.exp_pos _).le _) (F.S.flow.nonneg g) (fun a => by | |
| convert hcount a using 1 | |
| congr 1 | |
| ext i | |
| simp only [mem_filter]) | |
| (fun i _ => F.S.weight_conditional g m i) | |
| simpa only [F.S.flow.total_mass,mul_one] using hh | |
| end CannonRelease | |
| end | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| namespace CannonCapacity | |
| open Finset Set | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| section | |
| open Finset | |
| namespace CannonCapacity | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| namespace CannonCapacity | |
| open Set Metric Finset | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| section | |
| namespace CannonCapacity | |
| open Set Metric Finset CannonAnnular | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| section | |
| namespace CannonCapacity | |
| open Set Metric Finset CannonAnnular CannonPacking | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| namespace CannonCapacity | |
| open Set Metric Finset CannonAnnular | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| section | |
| namespace CannonCapacity | |
| open Set Metric Finset CannonAnnular CannonPacking | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| section | |
| open Set Metric Finset | |
| namespace CannonCapacity | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| namespace CannonCapacity | |
| open Set Metric Finset CannonPacking CannonAnnular Filter Topology | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| end | |
| namespace CannonCapacity | |
| open Set Finset | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| namespace CannonCapacity | |
| open Set Metric Finset | |
| attribute [local instance] Classical.propDecidable | |
| end CannonCapacity | |
| namespace CannonNode | |
| open Set Metric Finset CannonCapacity CannonFlow | |
| attribute [local instance] Classical.propDecidable | |
| end CannonNode | |
| namespace CannonNode | |
| open Set Metric Finset CannonCapacity CannonFlow CannonPacking CannonAnnular Filter Topology | |
| attribute [local instance] Classical.propDecidable | |
| end CannonNode | |
| end | |
| namespace CannonNode | |
| open Set Metric Finset Filter Topology CannonPacking CannonAnnular CannonMagnification | |
| attribute [local instance] Classical.propDecidable | |
| end CannonNode | |
| end | |
| section | |
| section | |
| open Set Filter Finset | |
| namespace CannonFlow | |
| namespace FiniteFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable (T : FiniteFlow) | |
| end FiniteFlow | |
| namespace FiniteFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable (T : FiniteFlow) | |
| end FiniteFlow | |
| end CannonFlow | |
| namespace CannonFlow | |
| open Finset | |
| attribute [local instance] Classical.propDecidable | |
| end CannonFlow | |
| end | |
| namespace CannonNode | |
| open Set Metric Finset CannonFlow | |
| attribute [local instance] Classical.propDecidable | |
| end CannonNode | |
| end | |
| namespace CannonNode | |
| open Set Metric Finset Filter Topology CannonPacking CannonAnnular CannonMagnification CannonFlow | |
| attribute [local instance] Classical.propDecidable | |
| end CannonNode | |
| end | |
| section | |
| open Set Filter Finset | |
| namespace CannonFlow | |
| namespace FiniteFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable (T : FiniteFlow) | |
| end FiniteFlow | |
| namespace FiniteFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable (T : FiniteFlow) | |
| end FiniteFlow | |
| end CannonFlow | |
| namespace CannonFlow | |
| open Finset | |
| attribute [local instance] Classical.propDecidable | |
| end CannonFlow | |
| end | |
| section | |
| open Set Finset MeasureTheory Filter | |
| namespace CannonPairCost | |
| attribute [local instance] Classical.propDecidable | |
| end CannonPairCost | |
| end | |
| namespace CannonNode.SourceConstruction | |
| open Set Metric Finset Filter Topology CannonFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable {X : Type*} [MetricSpace X] {f : X → ℝ} {x₀ : X} | |
| {ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ} | |
| variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n) | |
| end CannonNode.SourceConstruction | |
| namespace CannonNode.SourceConstruction | |
| open Set Metric Finset MeasureTheory CannonFlow | |
| attribute [local instance] Classical.propDecidable | |
| variable {X : Type*} [MetricSpace X] {f : X → ℝ} {x₀ : X} | |
| {ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ} | |
| variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n) | |
| end CannonNode.SourceConstruction | |
| end | |
| namespace CannonNode | |
| open Set Metric CannonCapacity CannonMagnification | |
| namespace SourceConstruction | |
| attribute [local instance] Classical.propDecidable | |
| variable {X : Type*} [MetricSpace X] [ProperSpace X] {f : X → ℝ} {x₀ : X} | |
| {ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ} | |
| variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n) | |
| end SourceConstruction | |
| end CannonNode | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |