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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
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