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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.ProductMeasure.Delta
import OAI.Probability.CriticalPercolation.Exploration.ReachableIncidenceImp
import OAI.Probability.CriticalPercolation.GraphGeometry.ReachableReplaceTail
import OAI.Probability.CriticalPercolation.Gluing.ContactPivot
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.BondGraph
open scoped Classical
open FiniteProduct
variable {V E : Type*}
lemma contact_exit_pivot (G : BondGraph V E) (ω : Set E) (B : Set V)
(a u v : V) (f : E) (hf : G.ends f = s(u,v)) (hclosed : ¬ω f)
(hn : ¬∃ b ∈ B, (G.openGraph ω).Reachable a b)
(hau : (G.openGraph ω).Reachable a u)
(hv : ∃ b ∈ B, (G.openGraph ω).Reachable v b) :
openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b)
f (Function.update ω f True) := by
apply (openPivot_force_true _ _ _).mpr
constructor
· apply (G.target_after_open_iff ω B a u v f hf hn hv ?_).mpr hau
intro hav
obtain ⟨b,hb,hvb⟩ := hv
exact hn ⟨b,hb,hav.trans hvb⟩
· rw [G.openGraph_force_false_of_closed ω f hclosed]
exact hn
theorem contact_pivot_incidence_congr (G : BondGraph V E) (ω η : Set E) (B : Set V)
(a u v : V) (f e : E) (hf : G.ends f = s(u,v)) (hclosed : ¬ω f)
(hn : ¬∃ b ∈ B, (G.openGraph ω).Reachable a b)
(hau : (G.openGraph ω).Reachable a u)
(hv : ∃ b ∈ B, (G.openGraph ω).Reachable v b)
(hv' : ∃ b ∈ B, (G.openGraph η).Reachable v b)
(h : ∀ g, (∃ r ∈ G.ends g, (G.openGraph ω).Reachable a r) → (ω g ↔ η g))
(he : ∃ r ∈ G.ends e, (G.openGraph ω).Reachable a r) :
openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b) e (Function.update ω f True) ↔
openPivot (fun σ => ∃ b ∈ B, (G.openGraph σ).Reachable a b) e (Function.update η f True) := by
have hr := G.reachable_incidence_congr ω η a h
have hn' : ¬∃ b ∈ B, (G.openGraph η).Reachable a b := by
rintro ⟨b,hb,hab⟩; exact hn ⟨b,hb,(hr b).mpr hab⟩
have hau' := (hr u).mp hau
by_cases hef : e = f
· subst e
have hclosed' : ¬η f := fun he => hclosed ((h f ⟨u,by simp [hf],hau⟩).mpr he)
exact iff_of_true (G.contact_exit_pivot ω B a u v f hf hclosed hn hau hv)
(G.contact_exit_pivot η B a u v f hf hclosed' hn' hau' hv')
· have he' : ∃ r ∈ G.ends e, (G.openGraph η).Reachable a r := by
obtain ⟨r,her,har⟩ := he; exact ⟨r,her,(hr r).mp har⟩
rw [G.contact_pivot_iff ω B a u v f e hf hef hn hau hv he,
G.contact_pivot_iff η B a u v f e hf hef hn' hau' hv' he',h e he,
G.reachable_delete_incidence_congr ω η a h e u]
end CriticalPercolation.BondGraph
end
end
end OAI
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