openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /BoundedPhysicalRectangles.lean
| import OAI.NumberTheory.DukePrimeDegree.Geometry.PhysicalIndependentBoxes | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.PrefixBoundedRegions | |
| namespace OAI | |
| /-! # Independent physical rectangles with compact time bounds -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem exists_shifted_root_rectangle_within | |
| {T B : Type*} [TopologicalSpace T] [TopologicalSpace B] | |
| {D : Set (T × B)} (hD : IsOpen D) | |
| {n : ℕ} {a b k : Fin n} (hab : a ≠ b) | |
| (e : D → RootSlice a b k) (he : Topology.IsOpenEmbedding e) | |
| (τ : D → ℝ) (hτ : Continuous τ) (z : D) (t : ℝ) | |
| (V : Set B) (hV : IsOpen V) (hzV : z.val.2 ∈ V) | |
| (I : Set ℝ) (hI : IsOpen I) (htI : t ∈ I) : | |
| ∃ (Q : RootTransverseRectangle D) (A : RootBox hab k) (R : ℝ) (J : Set ℝ), | |
| IsOpen J ∧ z.val ∈ Q.first ×ˢ Q.second ∧ t ∈ J ∧ | |
| Q.second ⊆ V ∧ J ⊆ I ∧ | |
| A.transverse = Set.range (e ∘ Q.inclusion) ∧ | |
| A.time ⊆ rootLeafWindow R 0 ∧ | |
| ∀ w : Q.first × Q.second, (Homeomorph.addRight (τ (Q.inclusion w))) '' J ⊆ A.time := by | |
| let D' := D ∩ (univ ×ˢ V) | |
| have hD' : IsOpen D' := hD.inter (isOpen_univ.prod hV) | |
| let i : D' → D := Set.inclusion inter_subset_left | |
| have hi : Topology.IsOpenEmbedding i := Topology.IsOpenEmbedding.inclusion inter_subset_left | |
| (hD'.preimage continuous_subtype_val) | |
| let z' : D' := ⟨z.val,⟨z.prop,mem_univ _,hzV⟩⟩ | |
| obtain ⟨Q',A,R,J,hJ,hz,ht,hcover,hbound,hw⟩ := exists_shifted_root_rectangle hD' | |
| hab (e ∘ i) (he.comp hi) (τ ∘ i) (hτ.comp hi.continuous) z' t | |
| let Q : RootTransverseRectangle D := | |
| ⟨Q'.first,Q'.second,Q'.first_open,Q'.second_open,fun _ hx => (Q'.inside hx).1⟩ | |
| refine ⟨Q,A,R,J ∩ I,hJ.inter hI,hz,⟨ht,htI⟩,?_,inter_subset_right,?_,hbound,?_⟩ | |
| · intro w hw | |
| exact (Q'.inside (show (z.val.1,w) ∈ Q'.first ×ˢ Q'.second from ⟨hz.1,hw⟩)).2.2 | |
| · exact hcover | |
| · intro w u hu | |
| obtain ⟨r,hr,rfl⟩ := hu | |
| exact hw w ⟨r,hr.1,rfl⟩ | |
| theorem exists_bounded_remainingRoot_physical_independent_box | |
| (d : ℕ) (q : InitialRootStep (d+1)) | |
| {T : Type*} [TopologicalSpace T] [MeasurableSpace T] [BorelSpace T] | |
| (C : RootPrefixCoordinates (d+1) q.processed T) (g : G (d+1)) (k : Fin (d+1)) | |
| (z : C.factorPivotDomain q g k) (t : ℝ) | |
| (μ : Measure (X (d+1))) [IsFiniteMeasure μ] | |
| (hA : ∀ u : LogCoordinates d, μ.map (fun x => rightAction x (diagonalFlow u)) = μ) : | |
| ∃ Q : RootTransverseRectangle (C.factorPivotDomain q g k), ∃ hQ : Nonempty Q.second, | |
| letI := hQ | |
| haveI : StandardBorelSpace Q.second := Q.second_open.measurableSet.standardBorel | |
| ∃ (A : RootBox q.distinct k) (R : ℝ) (J : Set ℝ) | |
| (K : Set (positiveRootGeneratedGroup (q.remaining.erase q.root))) (I : Set ℝ), | |
| IsOpen J ∧ z.val ∈ Q.first ×ˢ Q.second ∧ t ∈ J ∧ | |
| IsCompact K ∧ Q.second ⊆ K ∧ IsCompact I ∧ J ⊆ I ∧ | |
| A.time ⊆ rootLeafWindow R 0 ∧ | |
| A.transverse = Set.range (C.rectangleSlice q g k Q) ∧ | |
| (∀ w : Q.first × Q.second, | |
| (Homeomorph.addRight (C.rectangleTime q g k Q w)) '' J ⊆ A.time) ∧ | |
| let σ := shiftedRootFactorPullback q.distinct A μ | |
| (C.rectangleSlice q g k Q) (C.rectangleTime q g k Q) | |
| haveI := shiftedRootFactorPullback_finite q.distinct A μ | |
| (C.rectangleSlice q g k Q) (C.rectangleSlice_isOpenEmbedding q g k Q).measurableEmbedding | |
| (C.rectangleTime q g k Q) (C.continuous_rectangleTime q g k Q).measurable | |
| let ρ := σ.restrict (univ ×ˢ (univ ×ˢ J)) | |
| ∀ᵐ s ∂ρ.fst, ρ.condKernel s = (ρ.condKernel s).fst.prod (ρ.condKernel s).snd := by | |
| let E := q.nextGroupCoordinates | |
| let V := E ⁻¹' Metric.ball (E z.val.2) 1 | |
| let K := E.symm '' Metric.closedBall (E z.val.2) 1 | |
| have hV : IsOpen V := Metric.isOpen_ball.preimage E.continuous | |
| have hzV : z.val.2 ∈ V := by simp only [V,mem_preimage,Metric.mem_ball,dist_self]; norm_num | |
| have hK : IsCompact K := (isCompact_closedBall (E z.val.2) 1).image E.symm.continuous | |
| have hVK : V ⊆ K := by | |
| intro w hw | |
| exact ⟨E w,Metric.mem_closedBall.mpr (Metric.mem_ball.mp hw).le,E.symm_apply_apply w⟩ | |
| obtain ⟨Q,A,R,J,hJ,hz,ht,hQV,hJI,hcover,hbound,hw⟩ := | |
| exists_shifted_root_rectangle_within (C.factorPivotDomain_open q g k) q.distinct | |
| (C.factorSlice q g k) (C.factorSlice_isOpenEmbedding q g k) | |
| (C.factorTime q g k) (C.continuous_factorTime q g k) z t V hV hzV | |
| (Ioo (t-1) (t+1)) isOpen_Ioo (by constructor <;> linarith) | |
| let : Nonempty Q.second := ⟨⟨z.val.2,hz.2⟩⟩ | |
| let : StandardBorelSpace Q.second := Q.second_open.measurableSet.standardBorel | |
| refine ⟨Q,inferInstance,A,R,J,K,Icc (t-1) (t+1),hJ,hz,ht,hK,hQV.trans hVK, | |
| isCompact_Icc,hJI.trans Ioo_subset_Icc_self,hbound,hcover,hw,?_⟩ | |
| let σ := shiftedRootFactorPullback q.distinct A μ | |
| (C.rectangleSlice q g k Q) (C.rectangleTime q g k Q) | |
| let := shiftedRootFactorPullback_finite q.distinct A μ | |
| (C.rectangleSlice q g k Q) (C.rectangleSlice_isOpenEmbedding q g k Q).measurableEmbedding | |
| (C.rectangleTime q g k Q) (C.continuous_rectangleTime q g k Q).measurable | |
| apply conditional_restrict_rectangle_product σ J hJ.measurableSet | |
| exact remainingRootBox_conditional_product d q C g k Q A R hbound | |
| (by rw [hcover]; exact subset_rfl) J hJ.measurableSet hw μ hA | |
| end DukePrimeDegree | |
| end | |
| end OAI | |