| import Mathlib |
| import OAI.Probability.LatticeSurface.Coercivity.SparseAmbient |
| import OAI.Probability.LatticeSurface.Coercivity.SparseColor |
| import OAI.Probability.LatticeSurface.Coercivity.SparseGradients |
| import OAI.Probability.LatticeSurface.Coercivity.SparseModel |
| 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 |
| 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 |
|
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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 |
|
|
| 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 |
|
|
| 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 |
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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 |
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| 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 |
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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 |
|
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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 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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| local instance (sort : Sort*) : DecidableEq sort := Classical.typeDecidableEq sort |
|
|
| def sites (L : ℕ) (z : Vertex) : Finset Vertex := |
| Finset.Icc ((L:ℤ)*(z.1-1)) ((L:ℤ)*(z.1+2)) ×ˢ |
| Finset.Icc ((L:ℤ)*(z.2-1)) ((L:ℤ)*(z.2+2)) |
|
|
| @[simp] lemma mem_sites (L : ℕ) (z x : Vertex) : x ∈ sites L z ↔ |
| (L:ℤ)*(z.1-1) ≤ x.1 ∧ x.1 ≤ (L:ℤ)*(z.1+2) ∧ |
| (L:ℤ)*(z.2-1) ≤ x.2 ∧ x.2 ≤ (L:ℤ)*(z.2+2) := by |
| simp only [sites,Finset.mem_product,Finset.mem_Icc] |
| tauto |
|
|
| lemma mem_enlarged (p : Parameters) (z x : Vertex) : |
| x ∈ enlarged p z ↔ x ∈ sites p.L z := (mem_sites p.L z x).symm |
|
|
| def origin (L : ℕ) (z : Vertex) : Vertex := ((L:ℤ)*(z.1-1),(L:ℤ)*(z.2-1)) |
|
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| def translate (L : ℕ) (z : Vertex) (x : TriangularLipschitz.BoxVertex (3*L)) : Vertex := |
| x.val + origin L z |
|
|
| lemma translate_injective (L : ℕ) (z : Vertex) : Function.Injective (translate L z) := by |
| intro x y hxy |
| exact Subtype.ext (add_right_cancel hxy) |
|
|
| lemma translate_mem (L : ℕ) (z : Vertex) (x : TriangularLipschitz.BoxVertex (3*L)) : |
| translate L z x ∈ sites L z := by |
| have hx := TriangularLipschitz.mem_box.mp x.property |
| simp only [translate,origin,mem_sites,Prod.add_def] |
| push_cast at hx |
| simp only [mul_sub,mul_add,mul_one,mul_two] |
| constructor <;> omega |
|
|
| lemma translate_surj (L : ℕ) (z : Vertex) (x : Vertex) (hx : x ∈ sites L z) : |
| ∃ y, translate L z y = x := by |
| have hx' := (mem_sites L z x).mp hx |
| let y : TriangularLipschitz.BoxVertex (3*L) := ⟨x-origin L z,by |
| simp only [TriangularLipschitz.mem_box,origin,Prod.sub_def] |
| push_cast |
| simp only [mul_sub,mul_add,mul_one,mul_two] at hx' ⊢ |
| constructor <;> constructor <;> omega⟩ |
| refine ⟨y,?_⟩ |
| simp [translate,y] |
|
|
| lemma translate_adj (L : ℕ) (z : Vertex) (x y : TriangularLipschitz.BoxVertex (3*L)) : |
| Adj (translate L z x) (translate L z y) ↔ TriangularLipschitz.Adj x.val y.val := by |
| have hh := adj_sub (translate L z x) (translate L z y) (origin L z) |
| simpa only [Adj,TriangularLipschitz.Adj] using (show Adj (translate L z x) (translate L z y) ↔ Adj x.val y.val from by simpa only [translate,add_sub_cancel_right] using hh.symm) |
|
|
| lemma translate_dist (L : ℕ) (z : Vertex) (x y : TriangularLipschitz.BoxVertex (3*L)) : |
| latticeDist (translate L z x) (translate L z y) = |
| TriangularLipschitz.latticeDist x.val y.val := by |
| simp only [latticeDist,TriangularLipschitz.latticeDist,translate,Prod.add_def, |
| Int.cast_add,add_sub_add_right_eq_sub] |
|
|
| lemma card_sites (L : ℕ) (z : Vertex) : (sites L z).card = (3*L+1)^2 := by |
| simp only [sites,Finset.card_product,Int.card_Icc] |
| have hh (a : ℤ) : ((L:ℤ)*(a+2)+1-(L:ℤ)*(a-1)).toNat = 3*L+1 := by |
| have he : (L:ℤ)*(a+2)+1-(L:ℤ)*(a-1) = (3*L+1 : ℕ) := by push_cast;ring |
| rw [he,Int.toNat_natCast] |
| rw [hh,hh,pow_two] |
|
|
| lemma sites_disjoint {L : ℕ} (hL : 0 < L) {z w : Vertex} |
| (hzw : 3 < |z.1-w.1| ∨ 3 < |z.2-w.2|) : Disjoint (sites L z) (sites L w) := by |
| apply Finset.disjoint_left.mpr |
| intro x hx hy |
| have hx' := (mem_sites L z x).mp hx |
| have hy' := (mem_sites L w x).mp hy |
| have hL' : (0:ℤ) < L := by exact_mod_cast hL |
| rcases hzw with h | h |
| all_goals simp only [abs_eq_max_neg] at h |
| · have hh : z.1-w.1 > 3 ∨ w.1-z.1 > 3 := by omega |
| rcases hh with hh | hh |
| · have := mul_lt_mul_of_pos_left (show w.1+2 < z.1-1 by omega) hL' |
| omega |
| · have := mul_lt_mul_of_pos_left (show z.1+2 < w.1-1 by omega) hL' |
| omega |
| · have hh : z.2-w.2 > 3 ∨ w.2-z.2 > 3 := by omega |
| rcases hh with hh | hh |
| · have := mul_lt_mul_of_pos_left (show w.2+2 < z.2-1 by omega) hL' |
| omega |
| · have := mul_lt_mul_of_pos_left (show z.2+2 < w.2-1 by omega) hL' |
| omega |
| |
| lemma L_pos (p : Parameters) : 0 < p.L := by |
| have hδ : 0 < p.δ₀ := p.δpos.trans_le p.δle |
| have hk : (0:ℝ) < (p.k₀:ℝ)+2 := by positivity |
| have hh := (div_pos (mul_pos (by norm_num) hk) hδ).trans_le p.Llarge |
| exact_mod_cast hh |
| |
| lemma block_subset_sites (p : Parameters) (z : Vertex) : block p z ⊆ (sites p.L z : Set Vertex) := by |
| intro x hx |
| have hL : (0:ℤ) ≤ p.L := Int.natCast_nonneg _ |
| change _ ∧ _ ∧ _ ∧ _ at hx |
| apply (mem_sites p.L z x).mpr |
| have h1 := mul_le_mul_of_nonneg_left (show z.1-1 ≤ z.1 by omega) hL |
| have h2 := mul_le_mul_of_nonneg_left (show z.1+1 ≤ z.1+2 by omega) hL |
| have h3 := mul_le_mul_of_nonneg_left (show z.2-1 ≤ z.2 by omega) hL |
| have h4 := mul_le_mul_of_nonneg_left (show z.2+1 ≤ z.2+2 by omega) hL |
| omega |
| |
| lemma face_neighbor_sites (p : Parameters) (z : Vertex) (i : Fin 4) : |
| face p z i ∈ sites p.L z ∧ neighbor p z i ∈ sites p.L z ∧ |
| Adj (neighbor p z i) (face p z i) := by |
| have hL : (0:ℤ) < p.L := by exact_mod_cast L_pos p |
| have hhalf : (0:ℤ) ≤ (p.L:ℤ)/2 ∧ (p.L:ℤ)/2 ≤ p.L := by omega |
| fin_cases i <;> |
| norm_num [face,neighbor,mem_sites,Adj,Prod.mk.injEq,Fin.ext_iff,mul_add,mul_sub,mul_one,mul_two] <;> |
| omega |
| |
| end SparseBlocks |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
