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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.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
local instance (sort : Sort*) : DecidableEq sort := Classical.typeDecidableEq sort
lemma q_nonneg {u : ℝ} (hu : 0 ≤ u) : 0 ≤ q u :=
Real.rpow_nonneg (mul_nonneg (ExtremalEdges.gradientConstant_pos 6).le hu) _
lemma q_pos {u : ℝ} (hu : 0 < u) : 0 < q u :=
Real.rpow_pos_of_pos (mul_pos (ExtremalEdges.gradientConstant_pos 6) hu) _
abbrev ShortChoice (L : ℕ) (z : Vertex) := ↥(sites L z) × ↥(sites L z)
def shortEdges {L : ℕ} {z : Vertex} (t : ShortChoice L z) : Finset (Vertex × Vertex) :=
{(t.1.val,t.2.val)}
abbrev FarChoice (L : ℕ) (z : Vertex) (k : ℕ) :=
Σ xy : TriangularLipschitz.BoxVertex (3*L) × TriangularLipschitz.BoxVertex (3*L),
↥(SparseWitness.pathSubsets k (pathEdges L z xy))
def farEdges {L : ℕ} {z : Vertex} {k : ℕ} (t : FarChoice L z k) : Finset (Vertex × Vertex) := t.2.val
def tieEdges (p : Parameters) (z : Vertex) (i : Fin 4) : Finset (Vertex × Vertex) :=
{(neighbor p z i,face p z i)}
def shortTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) :=
{h | ∃ t : ShortChoice p.L z, ∀ e ∈ shortEdges t, h ∈ safeEvent V true p.δ e}
def farTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) :=
{h | ∃ t : FarChoice p.L z p.k₀, ∀ e ∈ farEdges t, h ∈ safeEvent V true (2*p.δ₀) e}
def tieTest (V : Finset Vertex) (p : Parameters) (z : Vertex) : Set (V → ℝ) :=
{h | ∃ i : Fin 4, ∀ e ∈ tieEdges p z i, h ∈ safeEvent V false p.s₀ e}
def witnessTest (V : Finset Vertex) (p : Parameters) : Fin 3 → Vertex → Set (V → ℝ) :=
![shortTest V p,farTest V p,tieTest V p]
lemma bad_witness (V : Finset Vertex) (p : Parameters) (h : V → ℝ)
(hh : h ∈ body V) (z : Vertex) (hz : bad p V h z) :
∃ j : Fin 3, h ∈ witnessTest V p j z := by
rcases hz with ⟨hfree,hnear | hfar | htie⟩
· obtain ⟨x,hx,y,hy,hxy,hcap⟩ := hnear
refine ⟨0, (⟨⟨x,block_subset_sites p z hx⟩,⟨y,block_subset_sites p z hy⟩⟩ : ShortChoice p.L z),?_⟩
intro e he
have heq := Finset.mem_singleton.mp he
subst e
exact ⟨hxy,hfree ((mem_enlarged p z x).mpr (block_subset_sites p z hx)),
hfree ((mem_enlarged p z y).mpr (block_subset_sites p z hy)),hcap.le⟩
· obtain ⟨x,hx,y,hy,hk,hbad⟩ := hfar
obtain ⟨xx,hxx⟩ := translate_surj p.L z x ((mem_enlarged p z x).mp hx)
obtain ⟨yy,hyy⟩ := translate_surj p.L z y ((mem_enlarged p z y).mp hy)
let F := pathEdges p.L z (xx,yy)
let S := F.filter (fun e => 1-2*p.δ₀ ≤ |height V h e.1-height V h e.2|)
have hε : 0 < 2*p.δ₀ := mul_pos (by norm_num) (p.δpos.trans_le p.δle)
have hsum : (1-(2*p.δ₀)/2)*(F.card : ℝ) <
∑ e ∈ F, |height V h e.1-height V h e.2| := by
rw [show (F.card : ℝ) = latticeDist x y by simpa only [F,hxx,hyy] using pathEdges_card p.L z (xx,yy)]
calc
_ = (1-p.δ₀)*latticeDist x y := by ring
_ < |height V h x-height V h y| := hbad
_ ≤ _ := by simpa only [F,hxx,hyy] using pathEdges_sum p.L z (xx,yy) (height V h)
have hcount := SparsePaths.cap_count F (fun e => |height V h e.1-height V h e.2|)
(2*p.δ₀) hε (fun e he => hh _ _ (pathEdges_mem _ _ _ e he).2.2) hsum
have hS : S ∈ SparseWitness.pathSubsets p.k₀ F := by
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_powerset.mpr (Finset.filter_subset _ _),?_,?_⟩
· have hh : (F.card : ℝ) ≤ 2*(S.card : ℝ) := by dsimp [S]; linarith
exact_mod_cast hh
· have hdist := pathEdges_card p.L z (xx,yy)
rw [hxx,hyy] at hdist
exact_mod_cast (hk.trans_eq hdist.symm)
refine ⟨1, ⟨(xx,yy),⟨S,hS⟩⟩,?_⟩
intro e he
have he' : e ∈ S := he
obtain ⟨heF,hecap⟩ := Finset.mem_filter.mp he'
obtain ⟨he1,he2,heAdj⟩ := pathEdges_mem p.L z (xx,yy) e heF
exact ⟨heAdj,hfree ((mem_enlarged p z e.1).mpr he1),
hfree ((mem_enlarged p z e.2).mpr he2),hecap⟩
· obtain ⟨i,hi⟩ := htie
obtain ⟨hface,hneighbor,hadj⟩ := face_neighbor_sites p z i
refine ⟨2,i,?_⟩
intro e he
have heq := Finset.mem_singleton.mp he
subst e
exact ⟨hadj,hfree ((mem_enlarged p z _).mpr hneighbor),
hfree ((mem_enlarged p z _).mpr hface),hi.le⟩
lemma witness_family {I : Type*} [Fintype I] (V : Finset Vertex) (L : ℕ)
(z : I → Vertex) (hz : ∀ i j, i ≠ j → Disjoint (sites L (z i)) (sites L (z j)))
(b : Bool) (u : ℝ) (hu : 0 < u) (hu8 : u ≤ 1/8)
{T : I → Type*} [∀ i, Fintype (T i)] (E : ∀ i, T i → Finset (Vertex × Vertex))
(inside : ∀ i t e, e ∈ E i t → e.1 ∈ sites L (z i))
(a : ℝ) (entropy : ∀ i, (∑ t, (q u)^(E i t).card) ≤ a) :
(law V).real {h | ∀ i, ∃ t, ∀ e ∈ E i t, h ∈ safeEvent V b u e} ≤
a^Fintype.card I := by
have hbound := SparseWitness.joint_witness_bound (law V) (safeEvent V b u) E (q u)
(q_nonneg hu.le) (joint_safe_gradients V b u hu hu8) (by
intro i j hij s t
apply Finset.disjoint_left.mpr
intro e hei hej
exact Finset.disjoint_left.mp (hz i j hij) (inside i s e hei) (inside j t e hej))
calc
_ ≤ ∏ i, ∑ t, (q u)^(E i t).card := hbound
_ ≤ ∏ _i : I, a := Finset.prod_le_prod₀
(fun i _ => Finset.sum_nonneg (fun t _ => pow_nonneg (q_nonneg hu.le) _))
(fun i _ => entropy i)
_ = _ := by simp
lemma short_entropy (L : ℕ) (z : Vertex) (u : ℝ) :
(∑ t : ShortChoice L z, (q u)^(shortEdges t).card) = ((3*L+1:ℕ):ℝ)^4 * q u := by
simp only [shortEdges,Finset.card_singleton,pow_one,Finset.sum_const,Finset.card_univ, Fintype.card_prod,Fintype.card_coe,card_sites,nsmul_eq_mul]
push_cast
ring
lemma far_entropy (L : ℕ) (z : Vertex) (k : ℕ) (u : ℝ) (hu : 0 ≤ u)
(hsmall : q u ≤ 1/16) :
(∑ t : FarChoice L z k, (q u)^(farEdges t).card) ≤
((3*L+1:ℕ):ℝ)^4 * (1/2:ℝ)^k := by
rw [Fintype.sum_sigma]
calc
_ ≤ ∑ _xy : TriangularLipschitz.BoxVertex (3*L) × TriangularLipschitz.BoxVertex (3*L),
(1/2:ℝ)^k := by
apply Finset.sum_le_sum
intro xy _
change (∑ y : ↥(SparseWitness.pathSubsets k (pathEdges L z xy)), (q u)^y.val.card) ≤ _
rw [Finset.sum_coe_sort (SparseWitness.pathSubsets k (pathEdges L z xy)) (fun S : Finset (Vertex × Vertex) => (q u)^S.card)]
exact SparseWitness.path_entropy k (pathEdges L z xy) (q u) (q_nonneg hu) hsmall
_ = _ := by
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_prod,box_card,nsmul_eq_mul]
push_cast
ring
end SparseBlocks
end
end
end
end
end
end
end
end OAI
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