openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /DistanceCollarTransfer.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Gluing.FreshCollar | |
| import OAI.Probability.CriticalPercolation.Gluing.InwardTrial | |
| import OAI.Probability.CriticalPercolation.Gluing.OpenTrialBound | |
| 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 distance_collar_transfer (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (q : E → unitInterval) (Input T : Set V) (hInput : Input.Nonempty) (o : V) | |
| (m R : ℕ) (J : Finset ℤ) (hJ : J.Nonempty) | |
| (hJrange : ∀ j∈J, (m : ℤ)<j ∧ j ≤ R) | |
| (ho : R ≤ G.distanceTo Input o) | |
| (p₀ p : unitInterval) (hp : (p : ℝ)<1) | |
| (hq : ∀ e∈G.inducedEdges (G.vertexCollar Input R), (p₀ : ℝ) ≤ q e ∧ (q e : ℝ) ≤ p) | |
| (D k M N VM : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) | |
| (hM : 0 < M) (hk : M*N ≤ k) | |
| (hMbound : ∀ v, (G.vertexBall v (4*m+2)).ncard ≤ M) | |
| (hVM : ∀ v, (G.vertexBall v m).ncard ≤ VM) | |
| (K η lam : ℝ) (hK : 0 ≤ K) (hη : 0 ≤ η) (hlam : 0 < lam) | |
| (hN : K ≤ N*(p₀ : ℝ)^(m+1+VM)) | |
| (hmean : ∀ y∈G.vertexCollar Input R, | |
| 1-η ≤ (BernoulliProduct.law q).real (G.seedHitEvent (G.vertexBall y m) T)) : | |
| (BernoulliProduct.law q).real (G.hitEvent o Input\G.hitEvent o T) ≤ | |
| 1/((J.card : ℝ)*(1-(p : ℝ))^(D*k))+Real.exp (-K)+(K+1)*η/lam+lam := by | |
| let π : V → ℤ := fun v => G.distanceTo Input v | |
| let Q : Set V := {v | m<G.distanceTo Input v} | |
| have ht : ∀ v : Q, ∃ (y : V) (F : Set E), F.Finite ∧ | |
| F.ncard ≤ m+1+(G.vertexBall y m).ncard ∧ | |
| G.distanceTo Input y+(m+1)=G.distanceTo Input v ∧ | |
| F⊆G.inducedEdges (G.vertexBall v (2*m+1)∩{z | G.distanceTo Input z ≤ G.distanceTo Input v}) ∧ | |
| F⊆(G.inducedEdges {z | G.distanceTo Input v ≤ G.distanceTo Input z})ᶜ ∧ | |
| ∀ z∈G.vertexBall y m, (G.openGraph F).Reachable v z := by | |
| intro v | |
| exact G.inward_seed_trial hc hl hInput v m v.2 | |
| choose y F hfin hcard hy hends hfresh hreach using ht | |
| let Ft (j : ℤ) (v : V) : Set E := if h : π v=j ∧ v∈Q then F ⟨v,h.2⟩ else ∅ | |
| let Bt (j : ℤ) (v : V) : Set V := if h : π v=j ∧ v∈Q then G.vertexBall (y ⟨v,h.2⟩) m else ∅ | |
| have hlevel (j : ℤ) (hj : j∈J) (v : V) (hv : π v=j) : π v=j ∧ v∈Q := by | |
| refine ⟨hv,?_⟩ | |
| change m<G.distanceTo Input v | |
| have hh := (hJrange j hj).1 | |
| change (G.distanceTo Input v : ℤ)=j at hv | |
| omega | |
| have hsupp (j : ℤ) (v : V) : Ft j v⊆G.inducedEdges (G.vertexBall v (2*m+1)) := by | |
| dsimp only [Ft] | |
| split_ifs with h | |
| · intro e he z hz | |
| exact (hends ⟨v,h.2⟩ he z hz).1 | |
| · exact Set.empty_subset _ | |
| have hcollar (j : ℤ) (hj : j∈J) (v : V) : Ft j v⊆G.inducedEdges (G.vertexCollar Input R) := by | |
| dsimp only [Ft] | |
| split_ifs with h | |
| · intro e he z hz | |
| apply (G.distanceTo_le_iff hc hInput z R).mp | |
| have h1 : G.distanceTo Input z ≤ G.distanceTo Input v := (hends ⟨v,h.2⟩ he z hz).2 | |
| have h2 := (hJrange j hj).2 | |
| have h3 : (G.distanceTo Input v : ℤ)=j := h.1 | |
| omega | |
| · exact Set.empty_subset _ | |
| apply G.fresh_collar_transfer_from_geometry hl q π o Input T J hJ | |
| (by | |
| intro j hj | |
| change j ≤ (G.distanceTo Input o : ℤ) | |
| have := (hJrange j hj).2 | |
| omega) | |
| (by | |
| intro j hj v hv | |
| have hv0 := (G.distanceTo_eq_zero_iff hc hInput v).mpr hv | |
| change (G.distanceTo Input v : ℤ)<j | |
| have := (hJrange j hj).1 | |
| omega) | |
| (by | |
| intro j hj e v w he hv hw | |
| by_cases hvw : v=w | |
| · subst w; omega | |
| · have ha : G.fullGraph.Adj w v := ⟨⟨e,Set.mem_univ _,by simpa only [Sym2.eq_swap] using he⟩,Ne.symm hvw⟩ | |
| have hs := G.distanceTo_adj hc hInput ha | |
| change j ≤ (G.distanceTo Input v : ℤ) at hv | |
| change (G.distanceTo Input w : ℤ)<j at hw | |
| change (G.distanceTo Input v : ℤ)=j | |
| omega) | |
| p hp | |
| (by | |
| intro j hj e v w he hv hw | |
| apply (hq e ?_).2 | |
| intro z hz | |
| rw [he] at hz | |
| have hz' : z=v ∨ z=w := Sym2.mem_iff.mp hz | |
| apply (G.distanceTo_le_iff hc hInput z R).mp | |
| have hjR := (hJrange j hj).2 | |
| change (G.distanceTo Input v : ℤ)=j at hv | |
| change (G.distanceTo Input w : ℤ)<j at hw | |
| rcases hz' with hz'|hz' <;> subst z <;> omega) | |
| D k M N (fun _j _hj v _hv => hD v) hM hk | |
| (fun v w => w∈G.vertexBall v (4*m+2)) | |
| (fun v => G.mem_vertexBall_self v _) (fun {_ _} h => G.vertexBall_comm.mp h) | |
| (fun v => G.vertexBall_finite hl v _) hMbound Ft Bt | |
| (by | |
| intro j hj v | |
| dsimp only [Ft] | |
| split_ifs with h | |
| · intro e he he' | |
| apply hfresh ⟨v,h.2⟩ he | |
| intro z hz | |
| have hh := he' z hz | |
| change j ≤ (G.distanceTo Input z : ℤ) at hh | |
| have hv : (G.distanceTo Input v : ℤ)=j := h.1 | |
| change G.distanceTo Input v ≤ G.distanceTo Input z | |
| omega | |
| · exact Set.empty_subset _) | |
| (by | |
| intro j hj v hv b hb | |
| have hh := hlevel j hj v hv | |
| dsimp only [Bt] at hb | |
| rw [dite_eq_left hh] at hb | |
| dsimp only [Ft] | |
| rw [dite_eq_left hh] | |
| exact hreach ⟨v,hh.2⟩ b hb) | |
| (by | |
| intro j hj v w h | |
| exact G.inward_trials_disjoint (Ft j) m (hsupp j) h) | |
| ((p₀ : ℝ)^(m+1+VM)) K η lam hK hη hlam | |
| (by | |
| intro j hj v hv | |
| have hh := hlevel j hj v hv | |
| have hFfinite : (Ft j v).Finite := by simpa only [Ft,dite_eq_left hh] using hfin ⟨v,hh.2⟩ | |
| apply law_allOpen_lower q (Ft j v) hFfinite p₀ (m+1+VM) | |
| · rw [show Ft j v=F ⟨v,hh.2⟩ by simp only [Ft,dite_eq_left hh]] | |
| exact (hcard ⟨v,hh.2⟩).trans (by have := hVM (y ⟨v,hh.2⟩); omega) | |
| · intro e he | |
| exact (hq e (hcollar j hj v he)).1) | |
| hN | |
| (by | |
| intro j hj v hv | |
| have hh := hlevel j hj v hv | |
| dsimp only [Bt] | |
| rw [dite_eq_left hh] | |
| apply hmean | |
| apply (G.distanceTo_le_iff hc hInput _ R).mp | |
| have h1 := hy ⟨v,hh.2⟩ | |
| change G.distanceTo Input (y ⟨v,hh.2⟩)+(m+1)=G.distanceTo Input v at h1 | |
| have h2 := (hJrange j hj).2 | |
| change (G.distanceTo Input v : ℤ)=j at hv | |
| omega) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |