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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Gluing.SelectedTrials
import OAI.Probability.CriticalPercolation.Gluing.ConditionalGluing
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {E V : Type*} [Countable E] [Countable V] (G : BondGraph V E)
theorem fresh_contact_transfer (q : E → unitInterval) (X Y : Set E)
(hXY : Disjoint X Y) (d : Set E → Finset V) (F : V → Set E)
(B : V → Set V) (L T Input : Set V) (o : V)
(hd : ∀ U, MeasurableSet[blockSigma X] {ω | d ω=U})
(hF : ∀ v, F v ⊆ Y)
(hconnect : ∀ v∈L, ∀ b∈B v, (G.openGraph (F v)).Reachable v b)
(C Small : Set (Set E)) (hC : MeasurableSet[blockSigma X] C)
(hinput : G.hitEvent o Input ⊆ Small∪C)
(hcontact : ∀ ω∈C, ∀ v∈d ω, v∈L ∧ (G.openGraph ω).Reachable o v)
(K η lam : ℝ) (hK : 0 ≤ K+1) (hη : 0 ≤ η) (hlam : 0 < lam)
(hmean : ∀ v∈L, 1-η ≤ (BernoulliProduct.law q).real (G.seedHitEvent (B v) T))
(hdis : ∀ σ∈C, (d σ : Set V).Pairwise (fun v w => Disjoint (F v) (F w)))
(hs : ∀ σ∈C, K ≤ ∑ v∈d σ, (BernoulliProduct.law q).real (allOpen (F v)))
(hs' : ∀ σ∈C, (∑ v∈d σ, (BernoulliProduct.law q).real (allOpen (F v))) ≤ K+1) :
(BernoulliProduct.law q).real (G.hitEvent o Input\G.hitEvent o T) ≤
(BernoulliProduct.law q).real Small+Real.exp (-K)+(K+1)*η/lam+lam := by
let No := C∩{ω | ∀ v∈d ω, ω∉allOpen (F v)}
let Bad := C∩{ω | ∃ v∈d ω, ω∈allOpen (F v) ∧
(blockProbability q Y (G.seedHitEvent (B v) T) ω).toReal<1-lam}
let Glue := {ω : Set E | ω∈G.hitEvent o (G.goodRelay q Y L F B T lam ω) ∧
ω∉G.hitEvent o T}
have hcover : G.hitEvent o Input\G.hitEvent o T ⊆ ((Small∪No)∪Bad)∪Glue := by
intro ω hω
obtain hSmall|hCω := hinput hω.1
· exact Or.inl (Or.inl (Or.inl hSmall))
· by_cases hno : ∀ v∈d ω, ω∉allOpen (F v)
· exact Or.inl (Or.inl (Or.inr ⟨hCω,hno⟩))
· push Not at hno
obtain ⟨v,hv,hopen⟩ := hno
by_cases hbad : (blockProbability q Y (G.seedHitEvent (B v) T) ω).toReal<1-lam
· exact Or.inl (Or.inr ⟨hCω,v,hv,hopen,hbad⟩)
· exact Or.inr ⟨⟨v,(hcontact ω hCω v hv).2,
(hcontact ω hCω v hv).1,hopen,le_of_not_gt hbad⟩,hω.2⟩
have hno := selected_no_success_bound q X Y hXY d F hd hF C hC K hdis hs
have hbad := selected_bad_trial_bound q X Y hXY d F (fun v => G.seedHitEvent (B v) T)
(fun v => G.measurableSet_seedHitEvent (B v) T) (fun v => G.seedHitEvent_upper (B v) T)
hd hF C hC K η lam hK hη hlam
(fun σ hσ v hv => hmean v (hcontact σ hσ v hv).1) hs'
have hglue := G.goodRelay_gluing q Y L F B T hF hconnect o lam hlam.le
have hno' : (BernoulliProduct.law q).real No ≤ Real.exp (-K) :=
hno.trans (mul_le_of_le_one_right (Real.exp_nonneg _) measureReal_le_one)
have hbad' : (BernoulliProduct.law q).real Bad ≤ (K+1)*η/lam :=
hbad.trans (mul_le_of_le_one_right (div_nonneg (mul_nonneg hK hη) hlam.le)
measureReal_le_one)
have hu := measureReal_union_le ((Small∪No)∪Bad) Glue (μ := BernoulliProduct.law q)
have hu' := measureReal_union_le (Small∪No) Bad (μ := BernoulliProduct.law q)
have hu'' := measureReal_union_le Small No (μ := BernoulliProduct.law q)
have hmono := measureReal_mono (μ := BernoulliProduct.law q) hcover
change (BernoulliProduct.law q).real Glue ≤ lam at hglue
linarith
end CriticalPercolation.BondGraph
end
end
end
end OAI
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