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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.SpecifiedFailures
import OAI.Probability.CriticalPercolation.ProductMeasure.ContourProbability
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.Boundary
def gridSlot (v : Grid) : ℕ := Nat.pair v.1.toNat v.2.toNat
def quadrant (v : Grid) : Prop := 0≤v.1 ∧ 0≤v.2
lemma gridSlot_injective : Set.InjOn gridSlot {v | quadrant v} := by
intro v hv w hw he
have hh := Nat.pairEquiv.injective (show Nat.pairEquiv (v.1.toNat,v.2.toNat)=
Nat.pairEquiv (w.1.toNat,w.2.toNat) from he)
change (v.1.toNat,v.2.toNat)=(w.1.toNat,w.2.toNat) at hh
have hx := congrArg Prod.fst hh
have hy := congrArg Prod.snd hh
apply Prod.ext <;> dsimp only [quadrant,Set.mem_ofPred_eq] at hv hw <;> omega
lemma gridSlot_east (v : Grid) (hv : quadrant v) : gridSlot v<gridSlot (v+(1,0)) := by
change Nat.pair v.1.toNat v.2.toNat<Nat.pair (v.1+1).toNat (v.2+0).toNat
rw [add_zero]
exact Nat.pair_lt_pair_left _ (by dsimp [quadrant] at hv; omega)
lemma gridSlot_north (v : Grid) (hv : quadrant v) : gridSlot v<gridSlot (v+(0,1)) := by
change Nat.pair v.1.toNat v.2.toNat<Nat.pair (v.1+0).toNat (v.2+1).toNat
rw [add_zero]
exact Nat.pair_lt_pair_right _ (by dsimp [quadrant] at hv; omega)
def gridFailure {Ω : Type*} (F : ℕ → Set Ω) (v : Grid) : Set Ω :=
if quadrant v then F (gridSlot v) else ∅
lemma specified_grid_failures {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(M : ℕ → MeasurableSpace Ω) (hM : Monotone M)
(F : ℕ → Set Ω) (hF : ∀ n, MeasurableSet[M (n+1)] (F n))
(B : Set Ω) (hB : MeasurableSet[M 0] B)
(ε : ℝ) (hε : 0≤ε)
(hcond : ∀ n A, MeasurableSet[M n] A → A⊆B → μ.real (A∩F n)≤ε*μ.real A)
(s : Finset Grid) : μ.real (B∩⋂ z∈s, gridFailure F z)≤μ.real B*ε^s.card := by
by_cases hs : ∀ z∈s, quadrant z
· have he : (⋂ z∈s, gridFailure F z)=⋂ n∈s.image gridSlot, F n := by
ext ω
constructor
· intro h
refine Set.mem_iInter.mpr fun n => Set.mem_iInter.mpr fun hn => ?_
obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hn
have hh := Set.mem_iInter.mp (Set.mem_iInter.mp h z) hz
simpa only [gridFailure,ite_eq_left (hs z hz)] using hh
· intro h
refine Set.mem_iInter.mpr fun z => Set.mem_iInter.mpr fun hz => ?_
have hh := Set.mem_iInter.mp (Set.mem_iInter.mp h (gridSlot z))
(Finset.mem_image.mpr ⟨z,hz,rfl⟩)
simpa only [gridFailure,ite_eq_left (hs z hz)] using hh
rw [he]
have hc : (s.image gridSlot).card=s.card := Finset.card_image_iff.mpr (by
intro z hz w hw he
exact gridSlot_injective (hs z hz) (hs w hw) he)
have h := specified_failures_with_initialization μ M hM F hF B hB ε hε hcond
(s.image gridSlot)
rwa [hc] at h
· push Not at hs
obtain ⟨z,hz,hq⟩ := hs
have he : (⋂ z∈s, gridFailure F z)=∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
intro ω h
have hh := Set.mem_iInter.mp (Set.mem_iInter.mp h z) hz
simp only [gridFailure,ite_eq_right hq,Set.mem_empty_iff_false] at hh
rw [he,Set.inter_empty,measureReal_empty]
positivity
theorem infinitely_many_good_from_tests {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(M : ℕ → MeasurableSpace Ω) (hM : Monotone M)
(F : ℕ → Set Ω) (hF : ∀ n, MeasurableSet[M (n+1)] (F n))
(B : Set Ω) (hmB : MeasurableSet[M 0] B) (hB : 0<μ.real B)
(S : Ω → Set Grid)
(hroot : ∀ ω∈B, (0,0)∈S ω)
(hquad : ∀ ω∈B, ∀ v∈S ω, quadrant v)
(hchild : ∀ ω∈B, ∀ z, z∉S ω →
(∃ y∈S ω, z=y+(1,0) ∨ z=y+(0,1)) → ω∈F (gridSlot z))
(q : ℝ) (hq₀ : 0≤q) (hq : 8*q<1)
(hcond : ∀ n A, MeasurableSet[M n] A → A⊆B →
μ.real (A∩F n)≤q^4*μ.real A) :
0<μ.real (B∩{ω | (S ω).Infinite}) := by
apply infinite_good_positive μ B S (gridFailure F) hB hroot hquad ?_ q hq₀ hq
(specified_grid_failures μ M hM F hF B hmB (q^4) (by positivity) hcond)
intro ω hω z hz hc
have hqz : quadrant z := by
obtain ⟨y,hy,he|he⟩ := hc
· rw [he]
have h := hquad ω hω y hy
change 0≤y.1+1 ∧ 0≤y.2+0
exact ⟨by linarith [h.1],by linarith [h.2]⟩
· rw [he]
have h := hquad ω hω y hy
change 0≤y.1+0 ∧ 0≤y.2+1
exact ⟨by linarith [h.1],by linarith [h.2]⟩
simpa only [gridFailure,ite_eq_left hqz] using hchild ω hω z hz hc
end CriticalPercolation.Boundary
end
end
end
end OAI
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