| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FreshRegionLaw |
| import OAI.Probability.CriticalPercolation.Gluing.CollarErrorBudget |
| import OAI.Probability.CriticalPercolation.Gluing.SourceCollarTransfer |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| theorem fresh_collar_transfer (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (A Input T : Set V) (hA : A.Finite) (q : (E⊕A) → unitInterval) |
| (m R : ℕ) (hcR : m<R) (Dstar : Set E) |
| (houtside : ∀ a∈A, a∉G.vertexCollar Input R) |
| (hregion : G.inducedEdges (G.vertexCollar Input R)⊆Dstar) |
| (pc p' : unitInterval) (hpc₀ : 0<(pc : ℝ)) (hpc : (pc : ℝ)<1) |
| (hpp : (pc : ℝ)/2 ≤ p') (hppc : (p' : ℝ) ≤ pc) |
| (hq : ∀ e∈Dstar, q (Sum.inl e)=p') |
| (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) |
| (hmean : ∀ y∈G.vertexCollar Input R, |
| 1-η ≤ (BernoulliProduct.law (fun _ : E => p')).real |
| ((forceOn Dstarᶜ ∅) ⁻¹' G.seedHitEvent (G.vertexBall y m) T)) : |
| (BernoulliProduct.law q).real ((G.withSource A).hitEvent none (some '' Input)\ |
| (G.withSource A).hitEvent none (some '' T)) ≤ κ := by |
| by_cases hI : Input.Nonempty |
| · let p₀ : unitInterval := ⟨(pc : ℝ)/2,by constructor <;> linarith [pc.2.1,pc.2.2]⟩ |
| let J : Finset ℤ := Finset.Icc ((m : ℤ)+1) (R : ℤ) |
| have hJ : J.Nonempty := Finset.nonempty_Icc.mpr (by omega) |
| have hJcard : (J.card : ℝ)=(R : ℝ)-m := by |
| have hjc : J.card=R-m := by |
| change (Finset.Icc ((m : ℤ)+1) (R : ℤ)).card=R-m |
| rw [Int.card_Icc] |
| omega |
| rw [hjc,Nat.cast_sub hcR.le] |
| have hK : 0 ≤ Real.log (4/κ) := Real.log_nonneg ((le_div_iff₀ hκ).mpr (by linarith)) |
| have hs : 0<((pc : ℝ)/2)^(m+1+VM) := positive_trial_power _ hpc₀ _ _ |
| have hh := G.source_distance_collar_transfer hc hl A Input T hA q hI m R J hJ |
| (by intro j hj; have hh := Finset.mem_Icc.mp hj; omega) houtside p₀ pc hpc |
| (by |
| intro e he |
| rw [hq e (hregion he)] |
| exact ⟨hpp,hppc⟩) |
| D k M ⌈Real.log (4/κ)/((pc : ℝ)/2)^(m+1+VM)⌉₊ VM hD hM hk hMbound hVM |
| (Real.log (4/κ)) η (κ/4) hK hη₀ (by linarith) |
| (trial_ceiling_budget _ _ hs) |
| (by |
| intro y hy |
| exact (hmean y hy).trans (fresh_region_lower (fun e => q (Sum.inl e)) |
| (fun _ => p') Dstar hq _ (G.measurableSet_seedHitEvent _ _) |
| (G.seedHitEvent_upper _ _))) |
| rw [hJcard] at hh |
| exact hh.trans (collar_error_budget κ η ((R : ℝ)-m) _ hκ hκ1 hη |
| (pow_pos (by linarith) _) hR) |
| · have he : Input=∅ := Set.not_nonempty_iff_eq_empty.mp hI |
| subst Input |
| have he : (G.withSource A).hitEvent none (some '' (∅ : Set V))=∅ := by |
| ext ω |
| simp [hitEvent] |
| rw [he,Set.empty_sdiff,measureReal_empty] |
| exact hκ.le |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
