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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
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
end ExtremalEdges
namespace ExtremalEdges
attribute [local instance] Classical.propDecidable
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
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))
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
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