openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /ForceFalseSubset.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.Link | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| open scoped Classical | |
| variable {V E : Type*} | |
| lemma force_false_subset (ω : Set E) (e : E) : | |
| ∀ f, Function.update ω e False f → ω f := by | |
| intro f hf | |
| by_cases he : f = e | |
| · subst f; erw [Function.update_self] at hf; exact hf.elim | |
| · erw [Function.update_of_ne he] at hf; exact hf | |
| lemma reachable_force_false_of_not_reachable (G : BondGraph V E) (ω : Set E) | |
| (e : E) (r : V) (hr : r ∈ G.ends e) {x y : V} | |
| (hxr : ¬(G.openGraph ω).Reachable x r) (hxy : (G.openGraph ω).Reachable x y) : | |
| (G.openGraph (Function.update ω e False)).Reachable x y := by | |
| have hh := ((G.openGraph ω).reachable_iff_reflTransGen x y).mp hxy | |
| clear hxy | |
| induction hh with | |
| | refl => exact .refl _ | |
| | @tail v y hpath hvy ih => | |
| have hxv := ((G.openGraph ω).reachable_iff_reflTransGen x v).mpr hpath | |
| obtain ⟨⟨f,hf,hfe⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hvy | |
| have hfe' : f ≠ e := by | |
| intro he | |
| subst f | |
| have : r = v ∨ r = y := by simpa only [hfe,Sym2.mem_iff] using hr | |
| rcases this with rfl | rfl | |
| · exact hxr hxv | |
| · exact hxr (hxv.trans hvy.reachable) | |
| apply ih.trans | |
| apply SimpleGraph.Adj.reachable | |
| apply (SimpleGraph.fromEdgeSet_adj _).mpr | |
| refine ⟨⟨f,?_,hfe⟩,hne⟩ | |
| change Function.update ω e False f | |
| erw [Function.update_of_ne hfe'] | |
| exact hf | |
| lemma target_after_open_iff (G : BondGraph V E) (ω : Set E) (B : Set V) | |
| (a u v : V) (f : E) (hf : G.ends f = s(u,v)) | |
| (hn : ¬∃ b ∈ B, (G.openGraph ω).Reachable a b) | |
| (hv : ∃ b ∈ B, (G.openGraph ω).Reachable v b) | |
| (hva : ¬(G.openGraph ω).Reachable a v) : | |
| (∃ b ∈ B, (G.openGraph (Function.update ω f True)).Reachable a b) ↔ | |
| (G.openGraph ω).Reachable a u := by | |
| have updated : (Function.update ω f True : Set E) = insert f ω := by | |
| ext bond | |
| by_cases equal : bond = f | |
| · subst bond | |
| erw [Function.update_self] | |
| exact iff_of_true trivial (Set.mem_insert f ω) | |
| · erw [Function.update_of_ne equal] | |
| change bond ∈ ω ↔ bond ∈ insert f ω | |
| simp [equal] | |
| have forced : G.openGraph (Function.update ω f True) = | |
| G.openGraph ω ⊔ SimpleGraph.edge u v := by | |
| erw [updated] | |
| rw [openGraph, Set.image_insert_eq, hf, ← Set.singleton_union, | |
| SimpleGraph.fromEdgeSet_union] | |
| exact sup_comm _ _ | |
| rw [forced] | |
| constructor | |
| · rintro ⟨b,hb,hab⟩ | |
| rcases (Wiring.reachable_sup_edge (G.openGraph ω) u v a b).mp hab with h | ⟨h,_⟩ | ⟨h,_⟩ | |
| · exact (hn ⟨b,hb,h⟩).elim | |
| · exact h | |
| · exact (hva h).elim | |
| · intro hau | |
| obtain ⟨b,hb,hvb⟩ := hv | |
| exact ⟨b,hb,(Wiring.reachable_sup_edge (G.openGraph ω) u v a b).mpr | |
| (Or.inr (Or.inl ⟨hau,hvb⟩))⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |