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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Delta
import OAI.Probability.CriticalPercolation.GraphGeometry.ForceFalseSubset
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*}
theorem contact_pivot_iff (G : BondGraph V E) (ω : Set E) (B : Set V)
(a u v : V) (f e : E) (hf : G.ends f = s(u,v)) (hef : e ≠ 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)
(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) ↔
ω e ∧ ¬(G.openGraph (Function.update ω e False)).Reachable a u := by
let F := fun σ : Set E => ∃ b ∈ B, (G.openGraph σ).Reachable a b
have hav : ¬(G.openGraph ω).Reachable a v := by
intro h
obtain ⟨b,hb,hvb⟩ := hv
exact hn ⟨b,hb,h.trans hvb⟩
have hopen : F (Function.update ω f True) :=
(G.target_after_open_iff ω B a u v f hf hn hv hav).2 hau
have hmon : G.openGraph (Function.update ω e False) ≤ G.openGraph ω :=
SimpleGraph.fromEdgeSet_mono (Set.image_mono (force_false_subset ω e))
have hn' : ¬∃ b ∈ B, (G.openGraph (Function.update ω e False)).Reachable a b := by
rintro ⟨b,hb,h⟩
exact hn ⟨b,hb,h.mono hmon⟩
have hav' : ¬(G.openGraph (Function.update ω e False)).Reachable a v :=
fun h => hav (h.mono hmon)
have hv' : ∃ b ∈ B, (G.openGraph (Function.update ω e False)).Reachable v b := by
obtain ⟨b,hb,hvb⟩ := hv
obtain ⟨r,hr,har⟩ := he
refine ⟨b,hb,G.reachable_force_false_of_not_reachable ω e r hr ?_ hvb⟩
intro hvr
exact hav (har.trans hvr.symm)
have hcomm : Function.update (Function.update ω f True) e False =
Function.update (Function.update ω e False) f True := by
funext g
by_cases hge : g = e
· subst g; erw [Function.update_self,Function.update_of_ne hef,Function.update_self]
· by_cases hgf : g = f
· subst g; erw [Function.update_of_ne (Ne.symm hef),Function.update_self,Function.update_self]
· erw [Function.update_of_ne hge,Function.update_of_ne hgf,
Function.update_of_ne hgf,Function.update_of_ne hge]
have hdelete : F (Function.update (Function.update ω f True) e False) ↔
(G.openGraph (Function.update ω e False)).Reachable a u := by
rw [hcomm]
exact G.target_after_open_iff _ B a u v f hf hn' hv' hav'
change Function.update ω f True e ∧ F (Function.update ω f True) ∧
¬F (Function.update (Function.update ω f True) e False) ↔ _
erw [Function.update_of_ne hef,hdelete]
exact and_congr_right (fun _ => and_iff_right hopen)
end CriticalPercolation.BondGraph
end
end
end OAI
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