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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.FreshTrial
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma prediction_persists (q : E → unitInterval) (S T U σ τ : Set E)
(A : Set (Set E)) (hA : MeasurableSet[blockSigma U] A)
(hST : S∩U=T∩U) (hstates : ∀ e∈S∩U, (e∈σ ↔ e∈τ)) :
blockProbability q S A σ=blockProbability q T A τ := by
unfold blockProbability
congr 1
ext ω
have hrestrict : U∩forceOn S σ ω=U∩forceOn T τ ω := by
ext e
by_cases he : e∈U
· have heST : e∈S ↔ e∈T := by
have hh := Set.ext_iff.mp hST e
simpa only [Set.mem_inter_iff,he,and_true] using hh
by_cases heS : e∈S
· have heT := heST.mp heS
simp only [Set.mem_inter_iff,he,true_and,mem_forceOn,ite_eq_left heS,ite_eq_left heT]
exact hstates e ⟨heS,he⟩
· have heT : e∉T := fun h => heS (heST.mpr h)
simp only [Set.mem_inter_iff,he,true_and,mem_forceOn,ite_eq_right heS,ite_eq_right heT]
· simp only [Set.mem_inter_iff,he,false_and]
have hforce : forceOn U (forceOn S σ ω) (forceOn T τ ω)=forceOn T τ ω := by
ext e
by_cases he : e∈U
· have hh := Set.ext_iff.mp hrestrict e
simpa only [Set.mem_inter_iff,he,true_and,mem_forceOn,ite_true,ite_eq_left he] using hh
· simp only [mem_forceOn,ite_eq_right he]
have hh := block_event_forceOn U A hA (forceOn S σ ω) (forceOn T τ ω)
rw [hforce] at hh
exact hh.symm
lemma blockProbability_real_compl (q : E → unitInterval) (S σ : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) :
(blockProbability q S Aᶜ σ).toReal=1-(blockProbability q S A σ).toReal := by
change (law q).real ((forceOn S σ ⁻¹' A)ᶜ)=1-(law q).real (forceOn S σ ⁻¹' A)
exact probReal_compl_eq_one_sub (hA.preimage (measurable_forceOn S σ))
lemma integral_known_prediction_defect (q : E → unitInterval) (S : Set E)
(A C : Set (Set E)) (hA : MeasurableSet[blockSigma S] A) (hC : MeasurableSet C) :
(∫ σ, A.indicator (fun σ => 1-(blockProbability q S C σ).toReal) σ ∂law q)=
(law q).real (A\C) := by
have he : (fun σ => A.indicator (fun σ => 1-(blockProbability q S C σ).toReal) σ)=
(fun σ => (blockProbability q S (Cᶜ∩A) σ).toReal) := by
funext σ
rw [blockProbability_inter_block q S Cᶜ A hA]
by_cases hσ : σ∈A
· simp only [Set.indicator_of_mem hσ,blockProbability_real_compl q S σ C hC]
· simp only [Set.indicator_of_notMem hσ,ENNReal.toReal_zero]
rw [he,integral_blockProbability_real q S (Cᶜ∩A) (hC.compl.inter ((blockSigma_le S) _ hA))]
congr 1
exact Set.inter_comm _ _
lemma bad_prediction_bound (q : E → unitInterval) (S : Set E)
(A C : Set (Set E)) (hA : MeasurableSet[blockSigma S] A) (hC : MeasurableSet C)
(δ : ℝ) (hδ : 0<δ) :
(law q).real (A∩{σ : Set E | (blockProbability q S C σ).toReal<1-δ})≤
(law q).real (A\C)/δ := by
let g := fun σ => (blockProbability q S C σ).toReal
let B := A∩{σ : Set E | g σ<1-δ}
have hg : Measurable g :=
((measurable_blockProbability q S C hC).mono (blockSigma_le S) le_rfl).ennreal_toReal
have hma := (blockSigma_le S) _ hA
have hmb : MeasurableSet B := hma.inter (measurableSet_lt hg measurable_const)
have hgi : Integrable g (law q) := integrable_blockProbability_real q S C hC
have hdef : Integrable (A.indicator (fun σ => 1-g σ)) (law q) :=
((integrable_const 1).sub hgi).indicator hma
have hl : Integrable (fun σ => δ*B.indicator (1 : Set E → ℝ) σ) (law q) :=
((integrable_const 1).indicator hmb).const_mul δ
have hi := integral_mono hl hdef (fun σ => by
by_cases hσ : σ∈B
· rw [Set.indicator_of_mem hσ,Set.indicator_of_mem hσ.1]
dsimp only [Pi.one_apply]
have hh := hσ.2
dsimp only [Set.mem_ofPred_eq] at hh
linarith
· rw [Set.indicator_of_notMem hσ,mul_zero]
by_cases hσA : σ∈A
· rw [Set.indicator_of_mem hσA]
exact sub_nonneg.mpr (ENNReal.toReal_mono (by simp) (blockProbability_le_one q S C σ))
· rw [Set.indicator_of_notMem hσA])
rw [integral_const_mul,integral_indicator_one hmb] at hi
have he := integral_known_prediction_defect q S A C hA hC
change (∫ σ, A.indicator (fun σ => 1-g σ) σ ∂law q)=(law q).real (A\C) at he
rw [he] at hi
exact (le_div_iff₀ hδ).mpr (by nlinarith [hi])
theorem two_prediction_failure (q : E → unitInterval) (S : Set E)
(A C D : Set (Set E)) (hA : MeasurableSet[blockSigma S] A)
(hC : MeasurableSet C) (hD : MeasurableSet D)
(δ a : ℝ) (hδ : 0<δ) (hin : 1-δ≤(law q).real A)
(hCfail : (law q).real (A\C)≤a) (hDfail : (law q).real (A\D)≤a) :
(law q).real (Aᶜ ∪
(A∩{σ : Set E | (blockProbability q S C σ).toReal<1-δ}) ∪
(A∩{σ : Set E | (blockProbability q S D σ).toReal<1-δ}))≤δ+2*a/δ := by
have hma := (blockSigma_le S) _ hA
have hcomp : (law q).real Aᶜ≤δ := by rw [probReal_compl_eq_one_sub hma]; linarith
have hc := (bad_prediction_bound q S A C hA hC δ hδ).trans
(div_le_div_of_nonneg_right hCfail hδ.le)
have hd := (bad_prediction_bound q S A D hA hD δ hδ).trans
(div_le_div_of_nonneg_right hDfail hδ.le)
have h1 := measureReal_union_le (μ := law q) (s₁ := Aᶜ)
(s₂ := A∩{σ : Set E | (blockProbability q S C σ).toReal<1-δ})
have h2 := measureReal_union_le (μ := law q)
(s₁ := Aᶜ∪(A∩{σ : Set E | (blockProbability q S C σ).toReal<1-δ}))
(s₂ := A∩{σ : Set E | (blockProbability q S D σ).toReal<1-δ})
calc
_ ≤ _ := h2
_ ≤ (law q).real Aᶜ+(law q).real
(A∩{σ : Set E | (blockProbability q S C σ).toReal<1-δ})+
(law q).real (A∩{σ : Set E | (blockProbability q S D σ).toReal<1-δ}) :=
add_le_add h1 le_rfl
_ ≤ δ+a/δ+a/δ := add_le_add (add_le_add hcomp hc) hd
_ = δ+2*a/δ := by ring
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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