openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /FreshContactTransfer.lean
| 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 | |