| import Mathlib |
| import OAI.Probability.LatticeSurface.Coercivity.SparseAmbient |
| import OAI.Probability.LatticeSurface.Coercivity.SparseColor |
| import OAI.Probability.LatticeSurface.Coercivity.SparseEntropy |
| import OAI.Probability.LatticeSurface.Coercivity.SparseGeodesic |
| import OAI.Probability.LatticeSurface.Coercivity.SparseGeometry |
| import OAI.Probability.LatticeSurface.Coercivity.SparseGradients |
| import OAI.Probability.LatticeSurface.Coercivity.SparseModel |
| import OAI.Probability.LatticeSurface.Coercivity.SparseParameters |
| import OAI.Probability.LatticeSurface.Coercivity.SparsePaths |
| import OAI.Probability.LatticeSurface.Coercivity.SparseProb |
| import OAI.Probability.LatticeSurface.Convexity.BoundedBase |
| import OAI.Probability.LatticeSurface.Convexity.BrunnMinkowski |
| import OAI.Probability.LatticeSurface.Convexity.CubePrekopa |
| import OAI.Probability.LatticeSurface.Convexity.GradientWindows |
| import OAI.Probability.LatticeSurface.Convexity.QuantileTransport |
| import OAI.Probability.LatticeSurface.Convexity.WeightedPrekopa |
| import OAI.Probability.LatticeSurface.Dynamics.DisplacementInequality |
| import OAI.Probability.LatticeSurface.Geometry.BlockVolume |
| import OAI.Probability.LatticeSurface.Geometry.CapCentral |
| import OAI.Probability.LatticeSurface.Geometry.CapCompact |
| import OAI.Probability.LatticeSurface.Geometry.CapContract |
| import OAI.Probability.LatticeSurface.Geometry.CapCoordinates |
| import OAI.Probability.LatticeSurface.Geometry.CapMatching |
| import OAI.Probability.LatticeSurface.Geometry.CapOrient |
| import OAI.Probability.LatticeSurface.Geometry.CapReflection |
| import OAI.Probability.LatticeSurface.Geometry.CapSmall |
| import OAI.Probability.LatticeSurface.Geometry.CapSplit |
| import OAI.Probability.LatticeSurface.Geometry.CapUnpin |
| import OAI.Probability.LatticeSurface.Geometry.CapWeld |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
| import OAI.Probability.LatticeSurface.Moments.ControlledGradients |
| import OAI.Probability.LatticeSurface.Moments.OrderedGradients |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
|
|
| open scoped Topology |
|
|
| noncomputable section |
|
|
| namespace FacetVolume |
|
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| end FacetVolume |
|
|
| namespace ExtremalEdges |
|
|
| open scoped ENNReal |
|
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| end ExtremalEdges |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| open scoped ENNReal |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| noncomputable section |
|
|
| namespace ExtremalEdges |
|
|
| open Finset |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} [Fintype V] |
|
|
| end ExtremalEdges |
|
|
| open MeasureTheory |
|
|
| open scoped ENNReal |
|
|
| noncomputable section |
|
|
| namespace TriangularLipschitz |
|
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| end TriangularLipschitz |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparsePaths |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparsePaths |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseWitness |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseWitness |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| local instance (sort : Sort*) : DecidableEq sort := Classical.typeDecidableEq sort |
|
|
| lemma q_nonneg {u : ℝ} (hu : 0 ≤ u) : 0 ≤ q u := |
| Real.rpow_nonneg (mul_nonneg (ExtremalEdges.gradientConstant_pos 6).le hu) _ |
|
|
| lemma q_pos {u : ℝ} (hu : 0 < u) : 0 < q u := |
| Real.rpow_pos_of_pos (mul_pos (ExtremalEdges.gradientConstant_pos 6) hu) _ |
|
|
| abbrev ShortChoice (L : ℕ) (z : Vertex) := ↥(sites L z) × ↥(sites L z) |
|
|
| def shortEdges {L : ℕ} {z : Vertex} (t : ShortChoice L z) : Finset (Vertex × Vertex) := |
| {(t.1.val,t.2.val)} |
|
|
| abbrev FarChoice (L : ℕ) (z : Vertex) (k : ℕ) := |
| Σ xy : TriangularLipschitz.BoxVertex (3*L) × TriangularLipschitz.BoxVertex (3*L), |
| ↥(SparseWitness.pathSubsets k (pathEdges L z xy)) |
|
|
| def farEdges {L : ℕ} {z : Vertex} {k : ℕ} (t : FarChoice L z k) : Finset (Vertex × Vertex) := t.2.val |
|
|
| def tieEdges (p : Parameters) (z : Vertex) (i : Fin 4) : Finset (Vertex × Vertex) := |
| {(neighbor p z i,face p z i)} |
|
|
| def shortTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) := |
| {h | ∃ t : ShortChoice p.L z, ∀ e ∈ shortEdges t, h ∈ safeEvent V true p.δ e} |
|
|
| def farTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) := |
| {h | ∃ t : FarChoice p.L z p.k₀, ∀ e ∈ farEdges t, h ∈ safeEvent V true (2*p.δ₀) e} |
|
|
| def tieTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) := |
| {h | ∃ i : Fin 4, ∀ e ∈ tieEdges p z i, h ∈ safeEvent V false p.s₀ e} |
|
|
| def witnessTest (V : Finset Vertex) (p : Parameters) : Fin 3 → Vertex → Set (V → ℝ) := |
| ![shortTest V p,farTest V p,tieTest V p] |
|
|
| lemma bad_witness (V : Finset Vertex) (p : Parameters) (h : V → ℝ) |
| (hh : h ∈ body V) (z : Vertex) (hz : bad p V h z) : |
| ∃ j : Fin 3, h ∈ witnessTest V p j z := by |
| rcases hz with ⟨hfree,hnear | hfar | htie⟩ |
| · obtain ⟨x,hx,y,hy,hxy,hcap⟩ := hnear |
| refine ⟨0, (⟨⟨x,block_subset_sites p z hx⟩,⟨y,block_subset_sites p z hy⟩⟩ : ShortChoice p.L z),?_⟩ |
| intro e he |
| have heq := Finset.mem_singleton.mp he |
| subst e |
| exact ⟨hxy,hfree ((mem_enlarged p z x).mpr (block_subset_sites p z hx)), |
| hfree ((mem_enlarged p z y).mpr (block_subset_sites p z hy)),hcap.le⟩ |
| · obtain ⟨x,hx,y,hy,hk,hbad⟩ := hfar |
| obtain ⟨xx,hxx⟩ := translate_surj p.L z x ((mem_enlarged p z x).mp hx) |
| obtain ⟨yy,hyy⟩ := translate_surj p.L z y ((mem_enlarged p z y).mp hy) |
| let F := pathEdges p.L z (xx,yy) |
| let S := F.filter (fun e => 1-2*p.δ₀ ≤ |height V h e.1-height V h e.2|) |
| have hε : 0 < 2*p.δ₀ := mul_pos (by norm_num) (p.δpos.trans_le p.δle) |
| have hsum : (1-(2*p.δ₀)/2)*(F.card : ℝ) < |
| ∑ e ∈ F, |height V h e.1-height V h e.2| := by |
| rw [show (F.card : ℝ) = latticeDist x y by simpa only [F,hxx,hyy] using pathEdges_card p.L z (xx,yy)] |
| calc |
| _ = (1-p.δ₀)*latticeDist x y := by ring |
| _ < |height V h x-height V h y| := hbad |
| _ ≤ _ := by simpa only [F,hxx,hyy] using pathEdges_sum p.L z (xx,yy) (height V h) |
| have hcount := SparsePaths.cap_count F (fun e => |height V h e.1-height V h e.2|) |
| (2*p.δ₀) hε (fun e he => hh _ _ (pathEdges_mem _ _ _ e he).2.2) hsum |
| have hS : S ∈ SparseWitness.pathSubsets p.k₀ F := by |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_powerset.mpr (Finset.filter_subset _ _),?_,?_⟩ |
| · have hh : (F.card : ℝ) ≤ 2*(S.card : ℝ) := by dsimp [S]; linarith |
| exact_mod_cast hh |
| · have hdist := pathEdges_card p.L z (xx,yy) |
| rw [hxx,hyy] at hdist |
| exact_mod_cast (hk.trans_eq hdist.symm) |
| refine ⟨1, ⟨(xx,yy),⟨S,hS⟩⟩,?_⟩ |
| intro e he |
| have he' : e ∈ S := he |
| obtain ⟨heF,hecap⟩ := Finset.mem_filter.mp he' |
| obtain ⟨he1,he2,heAdj⟩ := pathEdges_mem p.L z (xx,yy) e heF |
| exact ⟨heAdj,hfree ((mem_enlarged p z e.1).mpr he1), |
| hfree ((mem_enlarged p z e.2).mpr he2),hecap⟩ |
| · obtain ⟨i,hi⟩ := htie |
| obtain ⟨hface,hneighbor,hadj⟩ := face_neighbor_sites p z i |
| refine ⟨2,i,?_⟩ |
| intro e he |
| have heq := Finset.mem_singleton.mp he |
| subst e |
| exact ⟨hadj,hfree ((mem_enlarged p z _).mpr hneighbor), |
| hfree ((mem_enlarged p z _).mpr hface),hi.le⟩ |
|
|
| lemma witness_family {I : Type*} [Fintype I] (V : Finset Vertex) (L : ℕ) |
| (z : I → Vertex) (hz : ∀ i j, i ≠ j → Disjoint (sites L (z i)) (sites L (z j))) |
| (b : Bool) (u : ℝ) (hu : 0 < u) (hu8 : u ≤ 1/8) |
| {T : I → Type*} [∀ i, Fintype (T i)] (E : ∀ i, T i → Finset (Vertex × Vertex)) |
| (inside : ∀ i t e, e ∈ E i t → e.1 ∈ sites L (z i)) |
| (a : ℝ) (entropy : ∀ i, (∑ t, (q u)^(E i t).card) ≤ a) : |
| (law V).real {h | ∀ i, ∃ t, ∀ e ∈ E i t, h ∈ safeEvent V b u e} ≤ |
| a^Fintype.card I := by |
| have hbound := SparseWitness.joint_witness_bound (law V) (safeEvent V b u) E (q u) |
| (q_nonneg hu.le) (joint_safe_gradients V b u hu hu8) (by |
| intro i j hij s t |
| apply Finset.disjoint_left.mpr |
| intro e hei hej |
| exact Finset.disjoint_left.mp (hz i j hij) (inside i s e hei) (inside j t e hej)) |
| calc |
| _ ≤ ∏ i, ∑ t, (q u)^(E i t).card := hbound |
| _ ≤ ∏ _i : I, a := Finset.prod_le_prod₀ |
| (fun i _ => Finset.sum_nonneg (fun t _ => pow_nonneg (q_nonneg hu.le) _)) |
| (fun i _ => entropy i) |
| _ = _ := by simp |
|
|
| lemma short_entropy (L : ℕ) (z : Vertex) (u : ℝ) : |
| (∑ t : ShortChoice L z, (q u)^(shortEdges t).card) = ((3*L+1:ℕ):ℝ)^4 * q u := by |
| simp only [shortEdges,Finset.card_singleton,pow_one,Finset.sum_const,Finset.card_univ, Fintype.card_prod,Fintype.card_coe,card_sites,nsmul_eq_mul] |
| push_cast |
| ring |
|
|
| lemma far_entropy (L : ℕ) (z : Vertex) (k : ℕ) (u : ℝ) (hu : 0 ≤ u) |
| (hsmall : q u ≤ 1/16) : |
| (∑ t : FarChoice L z k, (q u)^(farEdges t).card) ≤ |
| ((3*L+1:ℕ):ℝ)^4 * (1/2:ℝ)^k := by |
| rw [Fintype.sum_sigma] |
| calc |
| _ ≤ ∑ _xy : TriangularLipschitz.BoxVertex (3*L) × TriangularLipschitz.BoxVertex (3*L), |
| (1/2:ℝ)^k := by |
| apply Finset.sum_le_sum |
| intro xy _ |
| change (∑ y : ↥(SparseWitness.pathSubsets k (pathEdges L z xy)), (q u)^y.val.card) ≤ _ |
| rw [Finset.sum_coe_sort (SparseWitness.pathSubsets k (pathEdges L z xy)) (fun S : Finset (Vertex × Vertex) => (q u)^S.card)] |
| exact SparseWitness.path_entropy k (pathEdges L z xy) (q u) (q_nonneg hu) hsmall |
| _ = _ := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_prod,box_card,nsmul_eq_mul] |
| push_cast |
| ring |
|
|
| end SparseBlocks |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
