| import Mathlib |
| import OAI.Probability.LatticeSurface.Coercivity.SparseColor |
| import OAI.Probability.LatticeSurface.Coercivity.SparseParameters |
| 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 |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
|
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| open scoped Topology |
|
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| noncomputable section |
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| namespace FacetVolume |
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| end FacetVolume |
|
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| namespace ExtremalEdges |
|
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| 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 |
|
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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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| 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 |
|
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| 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 |
|
|
| 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 |
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| noncomputable section |
|
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| namespace TriangularLipschitz |
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| end TriangularLipschitz |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
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|
| def Adj (x y : Vertex) : Prop := |
| y = (x.1+1,x.2) ∨ y = (x.1-1,x.2) ∨ |
| y = (x.1,x.2+1) ∨ y = (x.1,x.2-1) ∨ |
| y = (x.1+1,x.2-1) ∨ y = (x.1-1,x.2+1) |
|
|
| def latticeDist (x y : Vertex) : ℝ := |
| max (max |(x.1 : ℝ)-y.1| |(x.2 : ℝ)-y.2|) |
| |((x.1 : ℝ)-y.1)+((x.2 : ℝ)-y.2)| |
|
|
| def height (V : Finset Vertex) (h : V → ℝ) (x : Vertex) : ℝ := |
| if hx : x ∈ V then h ⟨x,hx⟩ else 0 |
|
|
| def body (V : Finset Vertex) : Set (V → ℝ) := |
| {h | ∀ x y, Adj x y → |height V h x-height V h y| ≤ 1} |
|
|
| def block (p : Parameters) (z : Vertex) : Set Vertex := |
| {x | (p.L:ℤ)*z.1 ≤ x.1 ∧ x.1 ≤ (p.L:ℤ)*(z.1+1) ∧ |
| (p.L:ℤ)*z.2 ≤ x.2 ∧ x.2 ≤ (p.L:ℤ)*(z.2+1)} |
|
|
| def enlarged (p : Parameters) (z : Vertex) : Set Vertex := |
| {x | (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)} |
|
|
| def face (p : Parameters) (z : Vertex) (i : Fin 4) : Vertex := |
| let a := (p.L:ℤ)*z.1; let b := (p.L:ℤ)*z.2; let n : ℤ := p.L/2 |
| if i = 0 then (a,b+n) else if i = 1 then (a+p.L,b+n) |
| else if i = 2 then (a+n,b) else (a+n,b+p.L) |
|
|
| def neighbor (p : Parameters) (z : Vertex) (i : Fin 4) : Vertex := |
| let v := face p z i |
| if i = 0 then (v.1,v.2+1) else if i = 1 then (v.1,v.2-1) |
| else if i = 2 then (v.1+1,v.2) else (v.1-1,v.2) |
|
|
| def bad (p : Parameters) (V : Finset Vertex) (h : V → ℝ) (z : Vertex) : Prop := |
| enlarged p z ⊆ (V : Set Vertex) ∧ |
| ((∃ x ∈ block p z, ∃ y ∈ block p z, Adj x y ∧ |
| 1-p.δ < |height V h x-height V h y|) ∨ |
| (∃ x ∈ enlarged p z, ∃ y ∈ enlarged p z, p.k₀ ≤ latticeDist x y ∧ |
| (1-p.δ₀)*latticeDist x y < |height V h x-height V h y|) ∨ |
| (∃ i : Fin 4, |height V h (neighbor p z i)-height V h (face p z i)| < p.s₀)) |
|
|
| def thickBad (p : Parameters) (V : Finset Vertex) (h : V → ℝ) |
| (m : ℕ) (z : Vertex) : Prop := |
| ∃ w, |w.1-z.1| ≤ m ∧ |w.2-z.2| ≤ m ∧ bad p V h w |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| lemma adj_symm {x y : Vertex} (h : Adj x y) : Adj y x := |
| TriangularLipschitz.adj_symm h |
|
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| lemma adj_ne {x y : Vertex} (h : Adj x y) : x ≠ y := |
| TriangularLipschitz.adj_ne h |
|
|
| def finiteGraph (B : Finset Vertex) : SimpleGraph B := |
| TriangularLipschitz.latticeGraph.induce (B : Set Vertex) |
|
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| def fixed (B V : Finset Vertex) : Set B := {x | x.val ∉ V} |
|
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| def indexEquiv (B V : Finset Vertex) (hVB : V ⊆ B) : V ≃ ExtremalEdges.Free (fixed B V) where |
| toFun x := ⟨⟨x,hVB x.property⟩, by simp [fixed]⟩ |
| invFun x := ⟨x.val.val, by simpa [fixed] using x.property⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| def coordinates (B V : Finset Vertex) (hVB : V ⊆ B) : |
| (V → ℝ) ≃ᵐ (ExtremalEdges.Free (fixed B V) → ℝ) := |
| MeasurableEquiv.piCongrLeft (fun _ : ExtremalEdges.Free (fixed B V) => ℝ) (indexEquiv B V hVB) |
|
|
| lemma preserving_coordinates (B V : Finset Vertex) (hVB : V ⊆ B) : |
| MeasurePreserving (coordinates B V hVB) := |
| volume_measurePreserving_piCongrLeft _ (indexEquiv B V hVB) |
|
|
| lemma coordinates_height (B V : Finset Vertex) (hVB : V ⊆ B) |
| (h : V → ℝ) (x : B) : |
| ExtremalEdges.height (fixed B V) (coordinates B V hVB h) x = height V h x.val := by |
| by_cases hx : x.val ∈ V |
| · have hxp : x ∉ fixed B V := by intro hxp; exact hxp hx |
| rw [ExtremalEdges.height, dite_eq_right hxp, height, dite_eq_left hx] |
| change (Equiv.piCongrLeft (fun _ : ExtremalEdges.Free (fixed B V) => ℝ) |
| (indexEquiv B V hVB)) h ⟨x, by simpa [fixed] using hx⟩ = h ⟨x.val,hx⟩ |
| rw [Equiv.piCongrLeft_apply_eq_cast] |
| rfl |
| · simp [ExtremalEdges.height, fixed, height, hx] |
|
|
| lemma body_coordinates (B V : Finset Vertex) (hVB : V ⊆ B) |
| (hcollar : ∀ x ∈ V, ∀ y, Adj x y → y ∈ B) : |
| (coordinates B V hVB) ⁻¹' ExtremalEdges.body (finiteGraph B) (fixed B V) = body V := by |
| ext h |
| constructor |
| · intro hh x y hxy |
| by_cases hx : x ∈ V |
| · have hyB := hcollar x hx y hxy |
| have hb := hh ⟨x,hVB hx⟩ ⟨y,hyB⟩ (show (finiteGraph B).Adj _ _ from hxy) |
| simpa only [coordinates_height] using hb |
| · by_cases hy : y ∈ V |
| · have hxB := hcollar y hy x (adj_symm hxy) |
| have hb := hh ⟨x,hxB⟩ ⟨y,hVB hy⟩ (show (finiteGraph B).Adj _ _ from hxy) |
| simpa only [coordinates_height] using hb |
| · simp [height,hx,hy] |
| · intro hh x y hxy |
| simpa only [coordinates_height] using hh x.val y.val hxy |
|
|
| lemma continuous_height (V : Finset Vertex) (x : Vertex) : |
| Continuous (fun h : V → ℝ => height V h x) := by |
| unfold height |
| split_ifs with hx |
| · exact continuous_apply (⟨x,hx⟩ : V) |
| · exact continuous_const |
|
|
| lemma closed_body (V : Finset Vertex) : IsClosed (body V) := by |
| simp only [body, Set.ofPred_forall] |
| exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_iInter fun _ => |
| isClosed_le ((continuous_height V x).sub (continuous_height V y)).abs continuous_const |
|
|
| lemma volume_body_coordinates (B V : Finset Vertex) (hVB : V ⊆ B) |
| (hcollar : ∀ x ∈ V, ∀ y, Adj x y → y ∈ B) : |
| volume (ExtremalEdges.body (finiteGraph B) (fixed B V)) = volume (body V) := by |
| rw [← body_coordinates B V hVB hcollar] |
| exact ((preserving_coordinates B V hVB).measure_preimage |
| (ExtremalEdges.closed_conditionalBody (finiteGraph B) (fixed B V) (fun _ => 0)).measurableSet.nullMeasurableSet).symm |
|
|
| end SparseBlocks |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
