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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
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