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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def vertexCollar (B : Set V) (n : ℕ) : Set V := ⋃ b∈B, G.vertexBall b n
def distanceTo (B : Set V) (v : V) : ℕ :=
if h : ∃ n, v∈G.vertexCollar B n then Nat.find h else 0
lemma vertexCollar_exists (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty) (v : V) :
∃ n, v∈G.vertexCollar B n := by
obtain ⟨b,hb⟩ := hB
obtain ⟨n,hn⟩ := G.reachable_mem_vertexBall (hc b v)
exact ⟨n,Set.mem_iUnion₂.mpr ⟨b,hb,hn⟩⟩
lemma distanceTo_mem {B : Set V} {v : V} (h : ∃ n, v∈G.vertexCollar B n) :
v∈G.vertexCollar B (G.distanceTo B v) := by
simp only [distanceTo,dite_eq_left h]
exact Nat.find_spec h
lemma distanceTo_le {B : Set V} {v : V} {n : ℕ} (h : v∈G.vertexCollar B n) :
G.distanceTo B v ≤ n := by
have he : ∃ k, v∈G.vertexCollar B k := ⟨n,h⟩
simp only [distanceTo,dite_eq_left he]
exact Nat.find_min' he h
lemma distanceTo_le_iff (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
(v : V) (n : ℕ) : G.distanceTo B v ≤ n ↔ v∈G.vertexCollar B n := by
constructor
· intro hn
obtain ⟨b,hb,hv⟩ := Set.mem_iUnion₂.mp (G.distanceTo_mem (G.vertexCollar_exists hc hB v))
have hmono : Monotone (G.vertexBall b) :=
monotone_nat_of_le_succ fun _ => Set.subset_union_left
exact Set.mem_iUnion₂.mpr ⟨b,hb,hmono hn hv⟩
· exact G.distanceTo_le
lemma distanceTo_eq_zero_iff (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
(v : V) : G.distanceTo B v=0 ↔ v∈B := by
rw [← Nat.le_zero,G.distanceTo_le_iff hc hB]
simp [vertexCollar,vertexBall]
lemma distanceTo_adj (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
{v w : V} (hvw : G.fullGraph.Adj v w) : G.distanceTo B w ≤ G.distanceTo B v+1 := by
obtain ⟨b,hb,hv⟩ := Set.mem_iUnion₂.mp (G.distanceTo_mem (G.vertexCollar_exists hc hB v))
apply G.distanceTo_le
exact Set.mem_iUnion₂.mpr ⟨b,hb,Or.inr (Set.mem_iUnion₂.mpr ⟨v,hv,hvw⟩)⟩
lemma distanceTo_ball (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
{v w : V} {n : ℕ} (hvw : w∈G.vertexBall v n) : G.distanceTo B w ≤ G.distanceTo B v+n := by
induction n generalizing w with
| zero =>
have he : w=v := hvw
subst w
exact Nat.le_refl _
| succ n ih =>
rcases hvw with hvw|hvw
· exact (ih hvw).trans (by omega)
· obtain ⟨z,hz,hzw⟩ := Set.mem_iUnion₂.mp hvw
have h1 := ih hz
have h2 := G.distanceTo_adj hc hB hzw
omega
lemma distanceTo_predecessor (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
(v : V) (hv : G.distanceTo B v≠0) :
∃ w, G.fullGraph.Adj v w ∧ G.distanceTo B w+1=G.distanceTo B v := by
obtain ⟨n,heq⟩ := Nat.exists_eq_succ_of_ne_zero hv
have hmem := G.distanceTo_mem (G.vertexCollar_exists hc hB v)
rw [heq] at hmem
obtain ⟨b,hb,hv⟩ := Set.mem_iUnion₂.mp hmem
rcases hv with hv|hv
· have hh : v∈G.vertexCollar B n := Set.mem_iUnion₂.mpr ⟨b,hb,hv⟩
have := G.distanceTo_le hh
omega
· obtain ⟨w,hw,hwv⟩ := Set.mem_iUnion₂.mp hv
have h1 := G.distanceTo_le (B := B) (Set.mem_iUnion₂.mpr ⟨b,hb,hw⟩)
have h2 := G.distanceTo_adj hc hB hwv
exact ⟨w,hwv.symm,by omega⟩
end CriticalPercolation.BondGraph
end
end
end
end OAI
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