| 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.Coercivity.SparseSeparated |
| import OAI.Probability.LatticeSurface.Coercivity.SparseWitnessModel |
| 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 |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| 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 : Type*} |
|
|
| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| 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 |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
|
| noncomputable section |
|
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| namespace ExtremalEdges |
|
|
| open Finset |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| 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] |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| 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 |
|
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| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| 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 |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparsePaths |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparsePaths |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end SparseWitness |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseWitness |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| open Filter |
|
|
| open scoped Topology |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| local instance zeroPinnedSparseDecidableEq (sort : Sort*) : DecidableEq sort := |
| Classical.typeDecidableEq sort |
|
|
| lemma law_real_all (V : Finset Vertex) (S : Set (V → ℝ)) : |
| (law V).real S = (volume (body V ∩ S)).toReal / (volume (body V)).toReal := by |
| simp only [Measure.real,law,Measure.smul_apply, Measure.restrict_apply' (closed_body V).measurableSet,smul_eq_mul, |
| ENNReal.toReal_mul,ENNReal.toReal_inv] |
| rw [Set.inter_comm,div_eq_mul_inv,mul_comm] |
| |
| lemma law_real_of_subset (V : Finset Vertex) (S : Set (V → ℝ)) (hS : S ⊆ body V) : |
| (law V).real S = (volume S).toReal / (volume (body V)).toReal := by |
| rw [law_real_all,Set.inter_eq_right.mpr hS] |
| |
| theorem zero_pinned_sparse (ρ : ℝ) (hρ : 0 < ρ) (m : ℕ) : |
| ∃ p : Parameters, ∀ (V : Finset Vertex) (A : Finset Vertex), |
| (volume {h : V → ℝ | h ∈ body V ∧ ∀ z ∈ A, thickBad p V h m z}).toReal / |
| (volume (body V)).toReal ≤ ρ^A.card := by |
| let P := 2*m+4 |
| have hP : 0 < P := by dsimp [P];omega |
| let D : ℝ := (offsets m).card |
| have hD : 0 < D := by dsimp [D];rw [offsets_card];positivity |
| let β := min ρ 1 |
| have hβ : 0 < β := lt_min hρ (by norm_num) |
| have hβ1 : β ≤ 1 := min_le_right _ _ |
| let θ := β^(P^2)/D |
| have hθ : 0 < θ := div_pos (pow_pos hβ _) hD |
| obtain ⟨p,hp⟩ := separated_parameters θ hθ |
| refine ⟨p,?_⟩ |
| intro V A |
| let f : A → Fin P × Fin P := fun z => color P hP z.val |
| obtain ⟨c,hc⟩ := SparseWitness.large_fiber f |
| let B : Finset A := Finset.univ.filter (fun i => f i = c) |
| have hsize : A.card ≤ P^2*B.card := by |
| simp only [Fintype.card_coe,Fintype.card_prod,Fintype.card_fin] at hc |
| convert hc using 1; simp [B,pow_two] |
| apply congrArg Finset.card |
| ext i |
| simp |
| let z : B → Vertex := fun i => i.val.val |
| have hcolor (i j : B) : color P hP (z i) = color P hP (z j) := by |
| exact (Finset.mem_filter.mp i.property).2.trans (Finset.mem_filter.mp j.property).2.symm |
| have hne (i j : B) (hij : i ≠ j) : z i ≠ z j := by |
| intro hh |
| exact hij (Subtype.ext (Subtype.ext hh)) |
| let S : Set (V → ℝ) := {h | h ∈ body V ∧ ∀ zz ∈ A, thickBad p V h m zz} |
| let T : (B → offsets m) → Set (V → ℝ) := fun t => |
| {h | h ∈ body V ∧ ∀ i, bad p V h (z i+(t i).val)} |
| have hsub : S ⊆ ⋃ t : B → offsets m, T t := by |
| intro h hh |
| have hw (i : B) : ∃ t : offsets m, bad p V h (z i+t.val) := by |
| obtain ⟨w,hw1,hw2,hbad⟩ := hh.2 (z i) i.val.property |
| let t : offsets m := ⟨w-z i,(offsets_mem m _).mpr ⟨hw1,hw2⟩⟩ |
| refine ⟨t,?_⟩ |
| simpa only [t,add_sub_cancel] using hbad |
| choose t ht using hw |
| exact Set.mem_iUnion.mpr ⟨t,hh.1,ht⟩ |
| have hbound (t : B → offsets m) : (law V).real (T t) ≤ θ^B.card := by |
| have hdisj (i j : B) (hij : i ≠ j) : |
| Disjoint (sites p.L (z i+(t i).val)) (sites p.L (z j+(t j).val)) := by |
| apply sites_disjoint (L_pos p) |
| exact perturbed_color_distance m (z i) (z j) (hne i j hij) (hcolor i j) (t i) (t j) |
| simpa only [Fintype.card_coe] using hp V B (fun i => z i+(t i).val) hdisj |
| have hprob : (law V).real S ≤ ρ^A.card := by |
| calc |
| _ ≤ (law V).real (⋃ t : B → offsets m, T t) := measureReal_mono hsub (measure_ne_top _ _) |
| _ ≤ ∑ t : B → offsets m, (law V).real (T t) := measureReal_iUnion_fintype_le _ |
| _ ≤ ∑ _t : B → offsets m, θ^B.card := Finset.sum_le_sum (fun t _ => hbound t) |
| _ = (D*θ)^B.card := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_coe, |
| nsmul_eq_mul,Nat.cast_pow,mul_pow,D] |
| _ = β^(P^2*B.card) := by |
| rw [show D*θ = β^(P^2) by dsimp [θ];field_simp, pow_mul] |
| _ ≤ β^A.card := pow_le_pow_of_le_one hβ.le hβ1 hsize |
| _ ≤ ρ^A.card := pow_le_pow_left₀ hβ.le (min_le_left _ _) _ |
| change (volume S).toReal / (volume (body V)).toReal ≤ _ |
| rw [← law_real_of_subset V S (fun _ hh => hh.1)] |
| exact hprob |
| |
| end SparseBlocks |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
