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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
namespace OAI
section
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def inducedEdges (S : Set V) : Set E := {e | ∀ v ∈ G.ends e, v∈S}
def induce (S : Set V) : BondGraph S (G.inducedEdges S) where
ends e := (G.ends e.val).attachWith e.property
lemma induce_ends_map (S : Set V) (e : G.inducedEdges S) :
Sym2.map Subtype.val ((G.induce S).ends e) = G.ends e.val :=
Sym2.attachWith_map_subtypeVal _
lemma induce_ends_eq (S : Set V) (e : G.inducedEdges S) (u v : S) :
(G.induce S).ends e = s(u,v) ↔ G.ends e.val = s(u.val,v.val) := by
rw [← G.induce_ends_map S e,← Sym2.map_mk]
exact (Sym2.map.injective Subtype.val_injective).eq_iff.symm
lemma finite_incident (hG : G.LocallyFinite) {S : Set V} (hS : S.Finite) :
(G.incidentEdges S).Finite := by
have hh := hS.biUnion (fun v _ => hG v)
convert hh using 1
ext e
simp only [incidentEdges,Set.mem_ofPred_eq,Set.mem_iUnion,exists_prop]
lemma finite_inducedEdges (hG : G.LocallyFinite) {S : Set V} (hS : S.Finite) :
(G.inducedEdges S).Finite := by
apply (G.finite_incident hG hS).subset
intro e he
obtain ⟨⟨u,v⟩,huv⟩ := Sym2.mk_surjective (G.ends e)
have hu : u ∈ G.ends e := by rw [← huv]; simp
exact ⟨u,he u hu,hu⟩
lemma induce_adj (S : Set V) (ω : Set (G.inducedEdges S)) (u v : S) :
((G.induce S).openGraph ω).Adj u v ↔
(G.openGraph (Subtype.val '' ω)).Adj u.val v.val := by
simp only [openGraph,SimpleGraph.fromEdgeSet_adj,Set.mem_image]
constructor
· rintro ⟨⟨e,he,hend⟩,hne⟩
exact ⟨⟨e.val,⟨e,he,rfl⟩,(G.induce_ends_eq S e u v).mp hend⟩,
fun h => hne (Subtype.ext h)⟩
· rintro ⟨⟨e,⟨a,ha,rfl⟩,hend⟩,hne⟩
exact ⟨⟨a,ha,(G.induce_ends_eq S a u v).mpr hend⟩,
fun h => hne (congrArg Subtype.val h)⟩
lemma induced_adj_mem (S : Set V) (ω : Set (G.inducedEdges S)) {u v : V}
(h : (G.openGraph (Subtype.val '' ω)).Adj u v) : u∈S ∧ v∈S := by
obtain ⟨⟨e,⟨a,ha,rfl⟩,he⟩,_⟩ := h
constructor
· exact a.property u (by rw [he]; simp)
· exact a.property v (by rw [he]; simp)
lemma induce_reachable (S : Set V) (ω : Set (G.inducedEdges S)) (u v : S) :
((G.induce S).openGraph ω).Reachable u v ↔
(G.openGraph (Subtype.val '' ω)).Reachable u.val v.val := by
constructor
· intro h
exact h.map ⟨Subtype.val, fun {a b} hab => (G.induce_adj S ω a b).mp hab⟩
· intro h
have H : ∀ w, (G.openGraph (Subtype.val '' ω)).Reachable u.val w →
∀ hw : w∈S, ((G.induce S).openGraph ω).Reachable u ⟨w,hw⟩ := by
intro w hh
rw [SimpleGraph.reachable_iff_reflTransGen] at hh
induction hh with
| refl => intro _; exact SimpleGraph.Reachable.refl _
| @tail w z _ hz ih =>
intro hmem
have hw := (G.induced_adj_mem S ω hz).1
exact (ih hw).trans ((G.induce_adj S ω ⟨w,hw⟩ ⟨z,hmem⟩).mpr hz).reachable
exact H v.val h v.property
lemma induced_cluster_subset (S : Set V) (ω : Set (G.inducedEdges S)) (u : S) :
G.cluster (Subtype.val '' ω) u.val ⊆ S := by
intro v hv
rw [cluster,Set.mem_ofPred_eq,SimpleGraph.reachable_iff_reflTransGen] at hv
induction hv with
| refl => exact u.property
| tail _ hadj _ => exact (G.induced_adj_mem S ω hadj).2
lemma induce_cluster_image (S : Set V) (ω : Set (G.inducedEdges S)) (u : S) :
Subtype.val '' ((G.induce S).cluster ω u) = G.cluster (Subtype.val '' ω) u.val := by
ext v
constructor
· rintro ⟨a,ha,rfl⟩
exact (G.induce_reachable S ω u a).mp ha
· intro hv
have hmem := G.induced_cluster_subset S ω u hv
exact ⟨⟨v,hmem⟩,(G.induce_reachable S ω u ⟨v,hmem⟩).mpr hv,rfl⟩
end CriticalPercolation.BondGraph
end
end
end
end OAI
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