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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.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
end SparseBlocks
noncomputable section
open scoped BigOperators
namespace SparsePaths
attribute [local instance] Classical.propDecidable
end SparsePaths
noncomputable section
open scoped BigOperators
namespace SparseWitness
attribute [local instance] Classical.propDecidable
end SparseWitness
noncomputable section
open Set MeasureTheory
open scoped BigOperators
namespace SparseWitness
attribute [local instance] Classical.propDecidable
end SparseWitness
namespace SparseBlocks
attribute [local instance] Classical.propDecidable
end SparseBlocks
namespace SparseBlocks
attribute [local instance] Classical.propDecidable
end SparseBlocks
namespace SparseBlocks
open Filter
open scoped Topology
end SparseBlocks
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
local instance zeroPinnedSparseDecidableEq (sort : Sort*) : DecidableEq sort :=
Classical.typeDecidableEq sort
lemma law_real_all (V : Finset Vertex) (S : Set (V → ℝ)) :
(law V).real S = (volume (body V ∩ S)).toReal / (volume (body V)).toReal := by
simp only [Measure.real,law,Measure.smul_apply, Measure.restrict_apply' (closed_body V).measurableSet,smul_eq_mul,
ENNReal.toReal_mul,ENNReal.toReal_inv]
rw [Set.inter_comm,div_eq_mul_inv,mul_comm]
lemma law_real_of_subset (V : Finset Vertex) (S : Set (V → ℝ)) (hS : S ⊆ body V) :
(law V).real S = (volume S).toReal / (volume (body V)).toReal := by
rw [law_real_all,Set.inter_eq_right.mpr hS]
theorem zero_pinned_sparse (ρ : ℝ) (hρ : 0 < ρ) (m : ℕ) :
∃ p : Parameters, ∀ (V : Finset Vertex) (A : Finset Vertex),
(volume {h : V → ℝ | h ∈ body V ∧ ∀ z ∈ A, thickBad p V h m z}).toReal /
(volume (body V)).toReal ≤ ρ^A.card := by
let P := 2*m+4
have hP : 0 < P := by dsimp [P];omega
let D : ℝ := (offsets m).card
have hD : 0 < D := by dsimp [D];rw [offsets_card];positivity
let β := min ρ 1
have hβ : 0 < β := lt_min hρ (by norm_num)
have hβ1 : β ≤ 1 := min_le_right _ _
let θ := β^(P^2)/D
have hθ : 0 < θ := div_pos (pow_pos hβ _) hD
obtain ⟨p,hp⟩ := separated_parameters θ hθ
refine ⟨p,?_⟩
intro V A
let f : A → Fin P × Fin P := fun z => color P hP z.val
obtain ⟨c,hc⟩ := SparseWitness.large_fiber f
let B : Finset A := Finset.univ.filter (fun i => f i = c)
have hsize : A.card ≤ P^2*B.card := by
simp only [Fintype.card_coe,Fintype.card_prod,Fintype.card_fin] at hc
convert hc using 1; simp [B,pow_two]
apply congrArg Finset.card
ext i
simp
let z : B → Vertex := fun i => i.val.val
have hcolor (i j : B) : color P hP (z i) = color P hP (z j) := by
exact (Finset.mem_filter.mp i.property).2.trans (Finset.mem_filter.mp j.property).2.symm
have hne (i j : B) (hij : i ≠ j) : z i ≠ z j := by
intro hh
exact hij (Subtype.ext (Subtype.ext hh))
let S : Set (V → ℝ) := {h | h ∈ body V ∧ ∀ zz ∈ A, thickBad p V h m zz}
let T : (B → offsets m) → Set (V → ℝ) := fun t =>
{h | h ∈ body V ∧ ∀ i, bad p V h (z i+(t i).val)}
have hsub : S ⊆ ⋃ t : B → offsets m, T t := by
intro h hh
have hw (i : B) : ∃ t : offsets m, bad p V h (z i+t.val) := by
obtain ⟨w,hw1,hw2,hbad⟩ := hh.2 (z i) i.val.property
let t : offsets m := ⟨w-z i,(offsets_mem m _).mpr ⟨hw1,hw2⟩⟩
refine ⟨t,?_⟩
simpa only [t,add_sub_cancel] using hbad
choose t ht using hw
exact Set.mem_iUnion.mpr ⟨t,hh.1,ht⟩
have hbound (t : B → offsets m) : (law V).real (T t) ≤ θ^B.card := by
have hdisj (i j : B) (hij : i ≠ j) :
Disjoint (sites p.L (z i+(t i).val)) (sites p.L (z j+(t j).val)) := by
apply sites_disjoint (L_pos p)
exact perturbed_color_distance m (z i) (z j) (hne i j hij) (hcolor i j) (t i) (t j)
simpa only [Fintype.card_coe] using hp V B (fun i => z i+(t i).val) hdisj
have hprob : (law V).real S ≤ ρ^A.card := by
calc
_ ≤ (law V).real (⋃ t : B → offsets m, T t) := measureReal_mono hsub (measure_ne_top _ _)
_ ≤ ∑ t : B → offsets m, (law V).real (T t) := measureReal_iUnion_fintype_le _
_ ≤ ∑ _t : B → offsets m, θ^B.card := Finset.sum_le_sum (fun t _ => hbound t)
_ = (D*θ)^B.card := by
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_coe,
nsmul_eq_mul,Nat.cast_pow,mul_pow,D]
_ = β^(P^2*B.card) := by
rw [show D*θ = β^(P^2) by dsimp [θ];field_simp, pow_mul]
_ ≤ β^A.card := pow_le_pow_of_le_one hβ.le hβ1 hsize
_ ≤ ρ^A.card := pow_le_pow_left₀ hβ.le (min_le_left _ _) _
change (volume S).toReal / (volume (body V)).toReal ≤ _
rw [← law_real_of_subset V S (fun _ hh => hh.1)]
exact hprob
end SparseBlocks
end
end
end
end
end
end
end
end OAI
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