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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.PredictionFailure
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation
variable {Ω : Type*} [MeasurableSpace Ω]
lemma relay_failure_bound (μ : Measure Ω) [IsFiniteMeasure μ]
(A : ℕ → Set Ω) (κ : ℝ) (N : ℕ)
(h : ∀ i<N, μ.real (A i\A (i+1)) ≤ κ) :
μ.real (A 0\A N) ≤ N*κ := by
induction N with
| zero => simp
| succ N ih =>
have hsub : A 0\A (N+1) ⊆ (A 0\A N)∪(A N\A (N+1)) := by
intro ω hω
by_cases hn : ω∈A N
· exact Or.inr ⟨hn,hω.2⟩
· exact Or.inl ⟨hω.1,hn⟩
calc
_ ≤ μ.real ((A 0\A N)∪(A N\A (N+1))) := measureReal_mono hsub
_ ≤ μ.real (A 0\A N)+μ.real (A N\A (N+1)) := measureReal_union_le _ _
_ ≤ N*κ+κ := add_le_add (ih (fun i hi => h i (by omega))) (h N (by omega))
_ = _ := by push_cast; ring
lemma relay_miss_bound (μ : Measure Ω) [IsFiniteMeasure μ]
(A : ℕ → Set Ω) (κ η : ℝ) (N : ℕ)
(hfirst : μ.real (A 0)ᶜ ≤ η)
(h : ∀ i<N, μ.real (A i\A (i+1)) ≤ κ) :
μ.real (A N)ᶜ ≤ η+N*κ := by
have hsub : (A N)ᶜ⊆(A 0)ᶜ∪(A 0\A N) := by
intro ω hω
by_cases h0 : ω∈A 0
· exact Or.inr ⟨h0,hω⟩
· exact Or.inl h0
calc
_ ≤ μ.real ((A 0)ᶜ∪(A 0\A N)) := measureReal_mono hsub
_ ≤ μ.real (A 0)ᶜ+μ.real (A 0\A N) := measureReal_union_le _ _
_ ≤ _ := add_le_add hfirst (relay_failure_bound μ A κ N h)
namespace BernoulliProduct
variable {E : Type*}
def twoPredictions (q : E → unitInterval) (S : Set E) (C D : Set (Set E))
(δ : ℝ) : Set (Set E) :=
{σ | 1-δ ≤ (blockProbability q S C σ).toReal ∧
1-δ ≤ (blockProbability q S D σ).toReal}
lemma measurableSet_twoPredictions (q : E → unitInterval) (S : Set E)
(C D : Set (Set E)) (hC : MeasurableSet C) (hD : MeasurableSet D) (δ : ℝ) :
MeasurableSet[blockSigma S] (twoPredictions q S C D δ) :=
(measurableSet_le measurable_const
((measurable_blockProbability q S C hC).ennreal_toReal)).inter
(measurableSet_le measurable_const
((measurable_blockProbability q S D hD).ennreal_toReal))
theorem twoPredictions_positive (q : E → unitInterval) (S : Set E)
(C D : Set (Set E)) (hC : MeasurableSet C) (hD : MeasurableSet D)
(δ a : ℝ) (hδ : 0<δ) (ha : 2*a<δ)
(hc : (law q).real Cᶜ ≤ a) (hd : (law q).real Dᶜ ≤ a) :
0<(law q).real (twoPredictions q S C D δ) := by
have h1 := bad_prediction_bound q S Set.univ C MeasurableSet.univ hC δ hδ
have h2 := bad_prediction_bound q S Set.univ D MeasurableSet.univ hD δ hδ
have hcEq : Set.univ \ C = Cᶜ := by ext; simp
have hdEq : Set.univ \ D = Dᶜ := by ext; simp
simp only [Set.univ_inter,hcEq,hdEq] at h1 h2
have he : (twoPredictions q S C D δ)ᶜ=
{σ | (blockProbability q S C σ).toReal<1-δ}∪
{σ | (blockProbability q S D σ).toReal<1-δ} := by
ext σ
simp only [twoPredictions,Set.mem_compl_iff,Set.mem_ofPred_eq,Set.mem_union,not_and_or,not_le]
have hm := (blockSigma_le S) _ (measurableSet_twoPredictions q S C D hC hD δ)
have hcomp : (law q).real (twoPredictions q S C D δ)ᶜ<1 := by
rw [he]
calc
_ ≤ (law q).real {σ | (blockProbability q S C σ).toReal<1-δ}+
(law q).real {σ | (blockProbability q S D σ).toReal<1-δ} := measureReal_union_le _ _
_ ≤ (law q).real Cᶜ/δ+(law q).real Dᶜ/δ := add_le_add h1 h2
_ ≤ a/δ+a/δ := add_le_add (div_le_div_of_nonneg_right hc hδ.le)
(div_le_div_of_nonneg_right hd hδ.le)
_ = 2*a/δ := by ring
_ < 1 := (div_lt_one hδ).mpr ha
rw [probReal_compl_eq_one_sub hm] at hcomp
linarith
end BernoulliProduct
end CriticalPercolation
end
end
end
end OAI
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