openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /FiniteJointGluingStrict.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Density | |
| import OAI.Probability.CriticalPercolation.Clusters.Earlier | |
| import OAI.Probability.CriticalPercolation.Clusters.HitValue | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HittingSubAvoidance | |
| namespace OAI | |
| section | |
| section | |
| open scoped BigOperators | |
| open Set MeasureTheory ProbabilityTheory | |
| namespace CriticalPercolation.BondGraph | |
| open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion | |
| open CriticalPercolation.ClusterColumns | |
| open scoped Classical | |
| variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E) | |
| lemma finite_joint_gluing_strict (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ b, w e b = 1) (hpos : ∀ e b, 0 ≤ w e b) | |
| (hclosed : ∀ e, 0 < w e False) (o : V) (A T : Set V) (a : ℝ) | |
| (ha : ∀ v ∈ A, a ≤ expect w | |
| (fun ω => hitValue T (component (G.openGraph ω) v))) : | |
| expect w (fun ω => hitValue A (component (G.openGraph ω) o)) * a ≤ | |
| expect w (fun ω => jointHitValue A T (component (G.openGraph ω) o)) := by | |
| classical | |
| by_cases ho : o ∈ A | |
| · have hA (ω : Set E) : (component (G.openGraph ω) o ∩ A).Nonempty := | |
| ⟨o, component_self _ _, ho⟩ | |
| have hh : (fun ω : Set E => hitValue A (component (G.openGraph ω) o)) = | |
| (fun _ => (1 : ℝ)) := by | |
| funext ω | |
| exact ite_eq_left (hA ω) | |
| have hj : (fun ω : Set E => jointHitValue A T (component (G.openGraph ω) o)) = | |
| (fun ω => hitValue T (component (G.openGraph ω) o)) := by | |
| funext ω | |
| exact jointHitValue_eq_hitValue (hA ω) | |
| erw [hh, hj, expect_const w hw, one_mul] | |
| exact ha o ho | |
| · let n := Fintype.card A | |
| let e : Fin n ≃ A := (Fintype.equivFin A).symm | |
| let f : Fin n → V := fun j => e j | |
| let x : Fin (n+1) → V := Fin.snoc f o | |
| have hx : Function.Injective x := by | |
| apply Fin.snoc_injective_iff.mpr | |
| refine ⟨?_, ?_⟩ | |
| · exact Subtype.val_injective.comp e.injective | |
| · rintro ⟨j, hj⟩ | |
| exact ho (hj ▸ (e j).property) | |
| have hlast : x (Fin.last n) = o := Fin.snoc_last _ _ | |
| have hearlier : earlier x (Fin.last n) = A := by | |
| ext v | |
| constructor | |
| · rintro ⟨j, hj, h⟩ | |
| rcases Fin.eq_castSucc_or_eq_last j with ⟨i, rfl⟩ | rfl | |
| · have he : x i.castSucc = f i := Fin.snoc_castSucc _ _ _ | |
| exact h ▸ he ▸ (e i).property | |
| · exact (lt_irrefl _ hj).elim | |
| · intro hv | |
| let i := e.symm ⟨v, hv⟩ | |
| refine ⟨i.castSucc, Fin.castSucc_lt_last i, ?_⟩ | |
| dsimp only [x] | |
| rw [Fin.snoc_castSucc] | |
| exact congrArg Subtype.val (e.apply_symm_apply ⟨v, hv⟩) | |
| have h := G.finite_joint_gluing_list w hw hpos hclosed x hx T a (by | |
| intro j | |
| simpa only [x, Fin.snoc_castSucc] using ha (f j) (e j).property) | |
| rwa [hearlier, hlast] at h | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |