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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.EraseBond
import OAI.Probability.CriticalPercolation.GraphGeometry.WiredEvent
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*} (G : BondGraph V E)
def connects (a t : V) (ω : Set E) : Prop := (G.openGraph ω).Reachable a t
lemma connects_mono (a t : V) : ∀ (ω ω' : Set E), ω ⊆ ω' → G.connects a t ω → G.connects a t ω' :=
fun _ _ h => fun hr => hr.mono (G.openGraph_mono h)
lemma eraseBond_force (f : E) (a : V) (ω : Set E) (b : Prop) :
(G.eraseBond f a).openGraph (Function.update ω f b) = (G.eraseBond f a).openGraph ω := by
erw [G.eraseBond_openGraph,G.eraseBond_openGraph]
congr 1
ext e
change (e ≠ f ∧ Function.update ω f b e) ↔ (e ≠ f ∧ e ∈ ω)
by_cases h : e = f
· subst e; simp
· erw [Function.update_of_ne h]
rfl
lemma connects_eq_wired_deleted (f : E) (a v z t : V) (ω : Set E)
(hf : f ∈ ω) (he : G.ends f = s(v,z))
(hv : ((G.eraseBond f a).openGraph ω).Reachable a v) :
G.connects a t ω ↔ (G.eraseBond f a).wiredEvent a z t ω := by
dsimp only [connects,wiredEvent]
rw [G.openGraph_eq_eraseBond_sup f a v z ω hf he]
exact Wiring.reachable_replace_tail _ a v z a t hv
lemma pivot_eq_deleted_joined (f : E) (a v z t : V) (ω : Set E)
(hf : f ∈ ω) (he : G.ends f = s(v,z))
(hv : ((G.eraseBond f a).openGraph ω).Reachable a v) :
openPivot (G.connects a t) f ω ↔ (G.eraseBond f a).joinedPair a z t ω := by
unfold openPivot
rw [G.connects_eq_wired_deleted f a v z t ω hf he hv]
have hclose : G.connects a t (Function.update ω f False) ↔
((G.eraseBond f a).openGraph ω).Reachable a t := by
dsimp only [connects]
erw [← G.eraseBond_openGraph_force_false]
erw [hclose]
exact (and_iff_right hf).trans ((G.eraseBond f a).earlier_pivot_iff a z t ω)
variable [Fintype E]
lemma deleted_joined_pin_eq (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(f : E) (h0 : w f False ≠ 0) (h1 : w f True ≠ 0) (a z t : V) :
prob (pin w f True) ((G.eraseBond f a).joinedPair a z t) =
prob (pin w f False) ((G.eraseBond f a).joinedPair a z t) := by
rw [prob_pin_transport w hw f False True h0 h1]
apply prob_congr
intro ω
dsimp [joinedPair,isolateThree]
erw [G.eraseBond_force]
lemma delta_eq_deleted_joined (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(f : E) (h0 : w f False ≠ 0) (h1 : w f True ≠ 0) (a v z t : V)
(he : G.ends f = s(v,z))
(hv : ∀ ω, density (pin w f False) ω ≠ 0 → ((G.eraseBond f a).openGraph ω).Reachable a v) :
delta w (G.connects a t) f = prob (pin w f True) ((G.eraseBond f a).joinedPair a z t) := by
rw [G.deleted_joined_pin_eq w hw f h0 h1 a z t,
delta_eq_pivotal w hw f h0 h1 _ (G.connects_mono a t)]
apply prob_congr_support
intro ω hω
have hf0 := density_pin_support w f False ω hω
have hf1 : Function.update ω f True f := by
erw [Function.update_self]
trivial
have hv' : ((G.eraseBond f a).openGraph (Function.update ω f True)).Reachable a v := by
erw [G.eraseBond_force]
exact hv ω hω
have hopen := G.connects_eq_wired_deleted f a v z t (Function.update ω f True) hf1 he hv'
have hgraph : G.openGraph ω = (G.eraseBond f a).openGraph ω := by
erw [G.eraseBond_openGraph_force_false]
have hu : Function.update ω f False = ω := by
erw [← hf0,Function.update_eq_self]
erw [hu]
erw [hopen]
dsimp only [wiredEvent,connects]
erw [G.eraseBond_force,hgraph]
exact (G.eraseBond f a).earlier_pivot_iff a z t ω
end CriticalPercolation.BondGraph
namespace CriticalPercolation.Wiring
variable {V : Type*}
lemma pivot_component_avoids (H : SimpleGraph V) (A B : Set V) (w a z : V)
(ha : a ∈ A) (hA : ∀ u ∈ A, H.Reachable w u)
(hopen : ∃ t ∈ B, (H ⊔ SimpleGraph.edge a z).Reachable w t)
(hclosed : ¬∃ t ∈ B, H.Reachable w t) :
(∃ t ∈ B, H.Reachable z t) ∧ ∀ u ∈ A, ¬H.Reachable z u := by
obtain ⟨t,ht,hwt⟩ := hopen
have hzt : H.Reachable z t := by
rcases (reachable_sup_edge H a z w t).mp hwt with h | ⟨_,h⟩ | ⟨_,h⟩
· exact (hclosed ⟨t,ht,h⟩).elim
· exact h
· exact (hclosed ⟨t,ht,(hA a ha).trans h⟩).elim
refine ⟨⟨t,ht,hzt⟩,?_⟩
intro u hu hzu
exact hclosed ⟨t,ht,(hA u hu).trans (hzu.symm.trans hzt)⟩
end CriticalPercolation.Wiring
end
end
end OAI
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