| 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.Coercivity.ZeroPinnedSparse |
| 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.ProductVolume |
| import OAI.Probability.LatticeSurface.Convexity.QuantileTransport |
| import OAI.Probability.LatticeSurface.Convexity.RetainedBody |
| import OAI.Probability.LatticeSurface.Convexity.RetainedCoordinates |
| import OAI.Probability.LatticeSurface.Convexity.RetainedShell |
| 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 |
|
|
| end FacetVolume |
|
|
| namespace ExtremalEdges |
|
|
| open scoped ENNReal |
|
|
| 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 |
|
|
| namespace ExtremalEdges |
|
|
| 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 |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| 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 |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| noncomputable section |
|
|
| namespace ExtremalEdges |
|
|
| open Finset |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| 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 |
|
|
| 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 |
|
|
| 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 |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparsePaths |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparsePaths |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseWitness |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
|
|
| open scoped BigOperators |
|
|
| namespace SparseWitness |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseWitness |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| open Filter |
|
|
| open scoped Topology |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
| attribute [local instance_reducible, local instance] Classical.typeDecidableEq |
|
|
| end SparseBlocks |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end RetainedShell |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end RetainedShell |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| end RetainedShell |
|
|
| namespace ProductVolume |
|
|
| universe u v |
|
|
| variable {ι : Type u} [Fintype ι] {J : ι → Type v} [∀ i, Fintype (J i)] |
|
|
| end ProductVolume |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end RetainedShell |
|
|
| namespace SparseBlocks |
|
|
| end SparseBlocks |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end RetainedShell |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def ordinary (p : SparseBlocks.Parameters) (V : Finset Vertex) : Finset Vertex := |
| (V.image (fun x => (x.1/(p.L:ℤ),x.2/(p.L:ℤ)))).filter (fun z => SparseBlocks.enlarged p z ⊆ (V:Set Vertex)) |
|
|
| end RetainedShell |
|
|
| namespace TriangularLipschitz |
|
|
| lemma global_lipschitz (h : Vertex → ℝ) (he : ∀ x y,Adj x y → |h x-h y|≤1) |
| (x y : Vertex) : |h x-h y|≤latticeDist x y := by |
| suffices ∀ n x y,(intDist x y).toNat=n → |h x-h y|≤latticeDist x y from this _ x y rfl |
| intro n |
| induction n using Nat.strong_induction_on with |
| | h n ih => |
| intro x y hn |
| by_cases hxy : x=y |
| · subst y;simp [latticeDist] |
| obtain ⟨z,hxz,hz1,hz2,hd⟩ := geodesic_step hxy |
| have hlt : (intDist z y).toNat<n := by |
| have := intDist_nonneg z y |
| have := intDist_nonneg x y |
| omega |
| have hz := ih _ hlt z y rfl |
| have hr : latticeDist z y+1=latticeDist x y := by |
| simp only [← intDist_cast] |
| exact_mod_cast hd |
| calc |
| |h x-h y|≤|h x-h z|+|h z-h y| := abs_sub_le _ _ _ |
| _ ≤ 1+latticeDist z y := add_le_add (he x z hxz) hz |
| _ = latticeDist x y := by linarith |
|
|
| lemma latticeDist_add_left (a x y : Vertex) : latticeDist (a+x) (a+y)=latticeDist x y := by |
| simp only [latticeDist,Prod.fst_add,Prod.snd_add,Int.cast_add] |
| congr 2 <;> ring_nf |
|
|
| end TriangularLipschitz |
|
|
| namespace RetainedShell |
|
|
| open TriangularLipschitz |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| lemma base_enlarged (p : SparseBlocks.Parameters) (z : Vertex) : base p.L z∈SparseBlocks.enlarged p z := by |
| have hL : (0:ℤ)≤p.L := Int.natCast_nonneg _ |
| simp only [SparseBlocks.enlarged,Set.mem_ofPred_eq,base] |
| refine ⟨?_,?_,?_,?_⟩ <;> nlinarith |
|
|
| lemma mem_ordinary (p : SparseBlocks.Parameters) (V : Finset Vertex) (z : Vertex) : |
| z∈ordinary p V ↔ SparseBlocks.enlarged p z⊆(V:Set Vertex) := by |
| classical |
| constructor |
| · intro hz;exact (Finset.mem_filter.mp hz).2 |
| · intro hz |
| have hL : (p.L:ℤ)≠0 := by exact_mod_cast (SparseBlocks.L_pos p).ne' |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_image.mpr ⟨base p.L z,hz (base_enlarged p z),?_⟩,hz⟩ |
| simp [base,Int.mul_ediv_cancel_left _ hL] |
| |
| lemma block_enlarged (p : SparseBlocks.Parameters) (z : Vertex) {x : Vertex} (hx : x∈box p.L) : |
| base p.L z+x∈SparseBlocks.enlarged p z := by |
| have hL : (0:ℤ)≤p.L := Int.natCast_nonneg _ |
| have hx := mem_box.mp hx |
| simp only [SparseBlocks.enlarged,Set.mem_ofPred_eq,base,Prod.fst_add,Prod.snd_add] |
| refine ⟨?_,?_,?_,?_⟩ <;> nlinarith |
| |
| lemma block_translate (p : SparseBlocks.Parameters) (z : Vertex) {x : Vertex} (hx : x∈box p.L) : |
| base p.L z+x∈SparseBlocks.block p z := by |
| have hx := mem_box.mp hx |
| simp only [SparseBlocks.block,Set.mem_ofPred_eq,base,Prod.fst_add,Prod.snd_add] |
| refine ⟨?_,?_,?_,?_⟩ <;> nlinarith |
| |
| lemma ordinary_free (p : SparseBlocks.Parameters) (V : Finset Vertex) : |
| ∀ z∈ordinary p V,∀ x∈box p.L,base p.L z+x∈V := by |
| intro z hz x hx |
| exact (mem_ordinary p V z).mp hz (block_enlarged p z hx) |
| |
| lemma face_source (p : SparseBlocks.Parameters) (z : Vertex) (i : Fin 4) : |
| base p.L z+faceCenter p.L i=SparseBlocks.face p z i := by |
| fin_cases i <;> simp [base,faceCenter,SparseBlocks.face,Int.natCast_ediv] |
| |
| lemma neighbor_source (p : SparseBlocks.Parameters) (z : Vertex) (i : Fin 4) : |
| base p.L z+compressedNeighbor p.L i=SparseBlocks.neighbor p z i := by |
| fin_cases i <;> simp [base,compressedNeighbor,SparseBlocks.neighbor,SparseBlocks.face,Int.natCast_ediv] <;> ring |
| |
| lemma face_in_box (L : ℕ) (i : Fin 4) : faceCenter L i∈box L := by |
| have he : L/2≤L := Nat.div_le_self _ _ |
| fin_cases i <;> simp [faceCenter,mem_box] <;> omega |
| |
| end RetainedShell |
| |
| namespace RetainedShell |
| |
| open TriangularLipschitz |
| |
| theorem pin_block_faces (p : SparseBlocks.Parameters) (V : Finset Vertex) (χ : Vertex → ℝ) |
| (hlip : ∀ x y,Adj x y → |χ x-χ y|≤1) |
| (hzero : ∀ x,x∉V → χ x=0) (z : Vertex) (hz : z∉ordinary p V) : |
| ∀ i : Fin 4,|χ (base p.L z+faceCenter p.L i)|≤4*(p.L:ℝ) := by |
| classical |
| have hn : ¬ SparseBlocks.enlarged p z⊆(V:Set Vertex) := by |
| exact fun h => hz ((mem_ordinary p V z).mpr h) |
| obtain ⟨x,hxe,hxV⟩ := Set.not_subset.mp hn |
| intro i |
| have hd := global_lipschitz χ hlip (base p.L z+faceCenter p.L i) x |
| rw [hzero x hxV,sub_zero] at hd |
| apply hd.trans |
| have hfb := mem_box.mp (face_in_box p.L i) |
| have hxe : (p.L:ℤ)*(z.1-1)≤x.1 ∧ x.1≤(p.L:ℤ)*(z.1+2) ∧ |
| (p.L:ℤ)*(z.2-1)≤x.2 ∧ x.2≤(p.L:ℤ)*(z.2+2) := hxe |
| have h1 : |((base p.L z+faceCenter p.L i).1:ℝ)-x.1|≤2*(p.L:ℝ) := by |
| have hh : |(base p.L z+faceCenter p.L i).1-x.1|≤2*(p.L:ℤ) := by |
| simp only [Prod.fst_add,base] |
| rw [abs_le] |
| constructor <;> nlinarith [hfb.1.1,hfb.1.2,hxe.1,hxe.2.1] |
| exact_mod_cast hh |
| have h2 : |((base p.L z+faceCenter p.L i).2:ℝ)-x.2|≤2*(p.L:ℝ) := by |
| have hh : |(base p.L z+faceCenter p.L i).2-x.2|≤2*(p.L:ℤ) := by |
| simp only [Prod.snd_add,base] |
| rw [abs_le] |
| constructor <;> nlinarith [hfb.2.1,hfb.2.2,hxe.2.2.1,hxe.2.2.2] |
| exact_mod_cast hh |
| apply max_le |
| · exact max_le (by linarith) (by linarith) |
| · exact (abs_add_le _ _).trans (by linarith) |
| |
| theorem ordinary_good (p : SparseBlocks.Parameters) (V : Finset Vertex) (h : V → ℝ) |
| (z : Vertex) (hz : z∈ordinary p V) (hb : ¬ SparseBlocks.bad p V h z) : |
| ConvexGood p.L p.k₀ p.δ p.δ₀ p.s₀ (fun x => SparseBlocks.height V h (base p.L z+x)) := by |
| have hf := (mem_ordinary p V z).mp hz |
| refine ⟨?_,?_,?_⟩ |
| · intro x hx y hy hxy |
| apply le_of_not_gt |
| intro he |
| apply hb |
| exact ⟨hf,Or.inl ⟨base p.L z+x,block_translate p z hx, |
| base p.L z+y,block_translate p z hy,(adj_translate p.L z x y).mpr hxy,he⟩⟩ |
| · intro x hx y hy hdist |
| apply le_of_not_gt |
| intro he |
| apply hb |
| refine ⟨hf,Or.inr (Or.inl ⟨base p.L z+x,block_enlarged p z hx, |
| base p.L z+y,block_enlarged p z hy,?_,?_⟩)⟩ |
| · change (p.k₀:ℝ)≤TriangularLipschitz.latticeDist (base p.L z+x) (base p.L z+y) |
| rwa [latticeDist_add_left] |
| · change (1-p.δ₀)*TriangularLipschitz.latticeDist (base p.L z+x) (base p.L z+y)<_ |
| rwa [latticeDist_add_left] |
| · intro i |
| change p.s₀≤|SparseBlocks.height V h (base p.L z+compressedNeighbor p.L i)- |
| SparseBlocks.height V h (base p.L z+faceCenter p.L i)| |
| rw [face_source,neighbor_source] |
| apply le_of_not_gt |
| intro he |
| exact hb ⟨hf,Or.inr (Or.inr ⟨i,he⟩)⟩ |
| |
| end RetainedShell |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
