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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
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*} (G : BondGraph V E)
noncomputable def eraseBond (f : E) (o : V) : BondGraph V E :=
⟨Function.update G.ends f s(o,o)⟩
lemma eraseBond_ends_ne (f : E) (o : V) {e : E} (he : e ≠ f) :
(G.eraseBond f o).ends e = G.ends e := by simp [eraseBond,he]
lemma eraseBond_openGraph (f : E) (o : V) (ω : Set E) :
(G.eraseBond f o).openGraph ω = G.openGraph {e | e ≠ f ∧ e ∈ ω} := by
ext u v
unfold openGraph
rw [SimpleGraph.fromEdgeSet_adj,SimpleGraph.fromEdgeSet_adj]
constructor
· rintro ⟨⟨e,he,hev⟩,huv⟩
have hef : e ≠ f := by
intro hh
subst e
simp only [eraseBond,Function.update_self] at hev
have hd := (Sym2.eq_iff.mp hev)
rcases hd with ⟨h1,h2⟩ | ⟨h1,h2⟩
· exact huv (h1.symm.trans h2)
· exact huv (h2.symm.trans h1)
exact ⟨⟨e,⟨hef,he⟩,(G.eraseBond_ends_ne f o hef).symm.trans hev⟩,huv⟩
· rintro ⟨⟨e,⟨hef,he⟩,hev⟩,huv⟩
exact ⟨⟨e,he,(G.eraseBond_ends_ne f o hef).trans hev⟩,huv⟩
lemma eraseBond_subgraph (f : E) (o : V) (ω : Set E) :
(G.eraseBond f o).openGraph ω ≤ G.openGraph ω := by
rw [G.eraseBond_openGraph]
exact SimpleGraph.fromEdgeSet_mono (Set.image_mono (fun _ member => member.2))
lemma knownGraph_le_eraseBond (read : Finset E) (f : E) (o : V) (ω : Set E)
(hf : f ∉ read) :
G.openGraph {e | e ∈ read ∧ e ∈ ω} ≤ (G.eraseBond f o).openGraph ω := by
rw [G.eraseBond_openGraph]
apply SimpleGraph.fromEdgeSet_mono ∘ Set.image_mono
intro e he
exact ⟨fun hef => hf (hef ▸ he.1),he.2⟩
lemma openGraph_eq_eraseBond_sup (f : E) (o v z : V) (ω : Set E)
(hf : f ∈ ω) (he : G.ends f = s(v,z)) :
G.openGraph ω = (G.eraseBond f o).openGraph ω ⊔ SimpleGraph.edge v z := by
rw [G.eraseBond_openGraph]
have hh : ω = insert f {e | e ≠ f ∧ e ∈ ω} := by
ext e
by_cases h : e = f
· subst e; simp [hf]
· simp [h]
conv_lhs => rw [hh]
rw [openGraph,Set.image_insert_eq,he,← Set.singleton_union,SimpleGraph.fromEdgeSet_union]
exact sup_comm _ _
lemma eraseBond_openGraph_force_false (f : E) (o : V) (ω : Set E) :
(G.eraseBond f o).openGraph ω = G.openGraph (Function.update ω f False) := by
rw [G.eraseBond_openGraph]
congr 1
ext e
change (e ≠ f ∧ e ∈ ω) ↔ Function.update ω f False e
by_cases h : e = f
· subst e
erw [Function.update_self]
simp
· erw [Function.update_of_ne h]
change (e ≠ f ∧ e ∈ ω) ↔ e ∈ ω
simp [h]
end CriticalPercolation.BondGraph
end
end
end OAI
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