openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /SourceMissLaw.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent | |
| import OAI.Probability.CriticalPercolation.Criticality.RestrictedSeed | |
| import OAI.Probability.CriticalPercolation.Gluing.RelayMoves | |
| import OAI.Probability.CriticalPercolation.Gluing.RootCorridor | |
| import OAI.Probability.CriticalPercolation.Gluing.SourceConnectivity | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| lemma source_miss_real (A T : Set V) (q : (E⊕A) → unitInterval) | |
| (hq : ∀ a : A, q (Sum.inr a)=1) : | |
| (BernoulliProduct.law q).real ((G.withSource A).hitEvent none (some '' T))ᶜ= | |
| (BernoulliProduct.law (fun b => q (Sum.inl b))).real (G.seedHitEvent A T)ᶜ := by | |
| rw [probReal_compl_eq_one_sub ((G.withSource A).measurableSet_hitEvent _ _), | |
| probReal_compl_eq_one_sub (G.measurableSet_seedHitEvent A T), | |
| measureReal_def,measureReal_def,G.source_hit_probability A T q hq] | |
| lemma restricted_seed_probability (A T : Set V) (q : E → unitInterval) (R : Set E) : | |
| BernoulliProduct.law q (G.restrictedSeed A T R)= | |
| BernoulliProduct.law (pinnedParameters q Rᶜ ∅) (G.seedHitEvent A T) := by | |
| rw [← forceOn_map_law,Measure.map_apply (measurable_forceOn _ _) (G.measurableSet_seedHitEvent A T)] | |
| congr 1 | |
| ext ω | |
| have he : forceOn Rᶜ ∅ ω=ω∩R := by | |
| ext e | |
| by_cases he : e∈R <;> simp [he] | |
| simp only [Set.mem_preimage,restrictedSeed,Set.mem_ofPred_eq,he] | |
| lemma restricted_seed_miss_real (A T : Set V) (q : E → unitInterval) (R : Set E) : | |
| (BernoulliProduct.law q).real (G.restrictedSeed A T R)ᶜ= | |
| (BernoulliProduct.law (Sum.elim (pinnedParameters q Rᶜ ∅) (fun _ : A => 1))).real | |
| ((G.withSource A).hitEvent none (some '' T))ᶜ := by | |
| rw [G.source_miss_real _ _ _ (fun _ => rfl), | |
| probReal_compl_eq_one_sub (G.restrictedSeed_measurable A T R), | |
| probReal_compl_eq_one_sub (G.measurableSet_seedHitEvent A T), | |
| measureReal_def,measureReal_def,G.restricted_seed_probability] | |
| rfl | |
| theorem original_root_corridor_miss (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (o : V) (t z : V → ℝ) (σ e ρ J : ℝ) (L R m : ℕ) (U : Set E) | |
| (hDU : G.inducedEdges (relayDomain t z J (L+1/2) σ)⊆U) | |
| (hσ : |σ|=1) (he : 0<e) (hρ : 0 ≤ ρ) (hJ : 26200<J) (hL : 416 ≤ L) | |
| (heL : (L+3)*e<1/100) (hr : (R+m+1)*ρ<e/10) (hto : t o=0) (hzo : z o=0) | |
| (ht : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v| ≤ ρ) | |
| (hz : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v| ≤ ρ) | |
| (hcR : m<R) (pc p' : unitInterval) (hpc₀ : 0<(pc : ℝ)) (hpc : (pc : ℝ)<1) | |
| (hpp : (pc : ℝ)/2 ≤ p') (hppc : (p' : ℝ) ≤ pc) | |
| (D k M VM : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) | |
| (hM : 0<M) (hMbound : ∀ v, (G.vertexBall v (4*m+2)).ncard ≤ M) | |
| (hVM : ∀ v, (G.vertexBall v m).ncard ≤ VM) | |
| (κ η : ℝ) (hκ : 0<κ) (hκ1 : κ ≤ 1) (hη₀ : 0 ≤ η) | |
| (hη : η ≤ κ*(κ/4)/(4*(Real.log (4/κ)+1))) | |
| (hk : M*⌈Real.log (4/κ)/((pc : ℝ)/2)^(m+1+VM)⌉₊ ≤ k) | |
| (hR : 4/(κ*(1-(pc : ℝ))^(D*k))<(R : ℝ)-m) | |
| (hmove : ∀ y s, |s|=1 → 1-η ≤ (BernoulliProduct.law (fun _ : E => p')).real | |
| (G.coordinateMove t z m e y s)) : | |
| (BernoulliProduct.law (fun _ : E => p')).real | |
| (G.restrictedSeed (G.vertexBall o m) (relayMark t z (L+1/2) σ ρ) U)ᶜ ≤ η+L*κ := by | |
| rw [G.restricted_seed_miss_real] | |
| apply G.root_corridor_miss hc hl o t z σ e ρ J L R m _ (fun _ => rfl) | |
| hσ he hρ hJ hL heL hr hto hzo ht hz hcR pc p' hpc₀ hpc hpp hppc _ | |
| D k M VM hD hM hMbound hVM κ η hκ hκ1 hη₀ hη hk hR hmove | |
| intro b hb | |
| simp only [Sum.elim_inl,pinnedParameters,Set.mem_compl_iff,ite_eq_right (not_not_intro (hDU hb))] | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |