| 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.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 |
|
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| namespace FacetVolume |
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| end FacetVolume |
|
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| namespace ExtremalEdges |
|
|
| open scoped ENNReal |
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| end ExtremalEdges |
|
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| noncomputable section |
|
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| open MeasureTheory Set |
|
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| open scoped ENNReal |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
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| attribute [local instance] Classical.propDecidable |
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
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| end ExtremalEdges |
|
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| noncomputable section |
|
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| namespace ExtremalEdges |
|
|
| open Finset |
|
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| variable {V I : Type*} [Fintype V] [Fintype I] |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V I : Type*} [Fintype V] [Fintype I] |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
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|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} [Fintype V] |
|
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| end ExtremalEdges |
|
|
| open MeasureTheory |
|
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| open scoped ENNReal |
|
|
| noncomputable section |
|
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
|
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| namespace SparseBlocks |
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| attribute [local instance] Classical.propDecidable |
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| end SparseBlocks |
|
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| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
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| end SparseBlocks |
|
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| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
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| end SparseBlocks |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace SparseBlocks |
|
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| attribute [local instance] Classical.propDecidable |
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| end SparseBlocks |
|
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| namespace SparseBlocks |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| noncomputable section |
|
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| open scoped BigOperators |
|
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| namespace SparsePaths |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparsePaths |
|
|
| noncomputable section |
|
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| open scoped BigOperators |
|
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| namespace SparseWitness |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparseWitness |
|
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| noncomputable section |
|
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| open Set MeasureTheory |
|
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| open scoped BigOperators |
|
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| namespace SparseWitness |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparseWitness |
|
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| namespace SparseBlocks |
|
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| attribute [local instance] Classical.propDecidable |
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| local instance (sort : Sort*) : DecidableEq sort := Classical.typeDecidableEq sort |
|
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| def geodesic (L : ℕ) (x y : TriangularLipschitz.BoxVertex (3*L)) : |
| (TriangularLipschitz.boxGraph (3*L)).Walk x y := |
| Classical.choose ((TriangularLipschitz.box_connected (3*L)).exists_path_of_dist x y) |
|
|
| lemma geodesic_isPath (L : ℕ) (x y : TriangularLipschitz.BoxVertex (3*L)) : |
| (geodesic L x y).IsPath := |
| (Classical.choose_spec ((TriangularLipschitz.box_connected (3*L)).exists_path_of_dist x y)).1 |
|
|
| lemma geodesic_length (L : ℕ) (x y : TriangularLipschitz.BoxVertex (3*L)) : |
| ((geodesic L x y).length : ℝ) = TriangularLipschitz.latticeDist x.val y.val := by |
| unfold geodesic |
| rw [(Classical.choose_spec ((TriangularLipschitz.box_connected (3*L)).exists_path_of_dist x y)).2, TriangularLipschitz.box_dist] |
|
|
| def pathEdges (L : ℕ) (z : Vertex) (xy : TriangularLipschitz.BoxVertex (3*L) × |
| TriangularLipschitz.BoxVertex (3*L)) : Finset (Vertex × Vertex) := |
| SparsePaths.edges (translate L z) (geodesic L xy.1 xy.2) |
|
|
| lemma pathEdges_card (L : ℕ) (z : Vertex) (xy : TriangularLipschitz.BoxVertex (3*L) × |
| TriangularLipschitz.BoxVertex (3*L)) : |
| ((pathEdges L z xy).card : ℝ) = latticeDist (translate L z xy.1) (translate L z xy.2) := by |
| rw [pathEdges,SparsePaths.card_edges _ (translate_injective L z) _ (geodesic_isPath _ _ _), geodesic_length,translate_dist] |
|
|
| lemma pathEdges_mem (L : ℕ) (z : Vertex) (xy : TriangularLipschitz.BoxVertex (3*L) × |
| TriangularLipschitz.BoxVertex (3*L)) (e : Vertex × Vertex) (he : e ∈ pathEdges L z xy) : |
| e.1 ∈ sites L z ∧ e.2 ∈ sites L z ∧ Adj e.1 e.2 := by |
| obtain ⟨d,hd,rfl⟩ := List.mem_map.mp (List.mem_toFinset.mp he) |
| exact ⟨translate_mem _ _ _,translate_mem _ _ _,(translate_adj _ _ _ _).mpr d.adj⟩ |
|
|
| lemma pathEdges_sum (L : ℕ) (z : Vertex) (xy : TriangularLipschitz.BoxVertex (3*L) × |
| TriangularLipschitz.BoxVertex (3*L)) (h : Vertex → ℝ) : |
| |h (translate L z xy.1)-h (translate L z xy.2)| ≤ |
| ∑ e ∈ pathEdges L z xy, |h e.1-h e.2| := |
| SparsePaths.sum_edges _ (translate_injective L z) _ (geodesic_isPath _ _ _) h |
|
|
| lemma box_card (L : ℕ) : Fintype.card (TriangularLipschitz.BoxVertex L) = (L+1)^2 := by |
| change Fintype.card (↥(TriangularLipschitz.box L)) = _ |
| rw [Fintype.card_coe] |
| simp [TriangularLipschitz.box,Int.card_Icc,pow_two] |
|
|
| def safeEvent (V : Finset Vertex) (b : Bool) (u : ℝ) (e : Vertex × Vertex) : Set (V → ℝ) := |
| {h | Adj e.1 e.2 ∧ e.1 ∈ V ∧ e.2 ∈ V ∧ h ∈ edgeEvent V b u e} |
|
|
| lemma joint_safe_gradients (V : Finset Vertex) (b : Bool) (u : ℝ) |
| (hu : 0 < u) (hu8 : u ≤ 1/8) (F : Finset (Vertex × Vertex)) : |
| (law V).real {h | ∀ e ∈ F, h ∈ safeEvent V b u e} ≤ |
| ((ExtremalEdges.gradientConstant 6*u)^(1/ExtremalEdges.gradientConstant 6))^F.card := by |
| by_cases hv : ∀ e ∈ F, Adj e.1 e.2 ∧ e.1 ∈ V ∧ e.2 ∈ V |
| · calc |
| _ ≤ (law V).real (gradientEvent V b u F) := by |
| apply measureReal_mono _ (measure_ne_top _ _) |
| intro h hh e he |
| exact (hh e he).2.2.2 |
| _ ≤ _ := joint_gradients V F (fun e he => (hv e he).1) |
| (fun e he => (hv e he).2) b u hu hu8 |
| · have hempty : {h : V → ℝ | ∀ e ∈ F, h ∈ safeEvent V b u e} = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro h hh |
| exact hv (fun e he => ⟨(hh e he).1,(hh e he).2.1,(hh e he).2.2.1⟩) |
| rw [hempty,measureReal_empty] |
| exact pow_nonneg (Real.rpow_nonneg (mul_nonneg (ExtremalEdges.gradientConstant_pos 6).le hu.le) _) _ |
|
|
| end SparseBlocks |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
