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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma finite_seed_tree_of_rank {S : Set V} (hS : S.Finite) (o : V)
(r : V → ℕ)
(hparent : ∀ v∈S, v≠o → ∃ w∈S, G.fullGraph.Adj w v ∧ r w<r v) :
∃ F : Set E, F.Finite ∧ F⊆G.inducedEdges S ∧ F.ncard ≤ S.ncard ∧
∀ v∈S, (G.openGraph F).Reachable o v := by
let U := S\{o}
have hU : U.Finite := hS.sdiff
let := hU.fintype
have hh : ∀ v : U, ∃ (w : V) (e : E), w∈S ∧ G.ends e=s(w,v.val) ∧
w≠v.val ∧ r w<r v.val := by
intro v
obtain ⟨w,hw,⟨⟨e,_,he⟩,hne⟩,hr⟩ := hparent v.val v.property.1 v.property.2
exact ⟨w,e,hw,he,hne,hr⟩
choose w e hw hend hne hr using hh
let F : Set E := Set.range e
have hF : F.Finite := Set.finite_range e
refine ⟨F,hF,?_,?_,?_⟩
· rintro b ⟨v,rfl⟩ z hz
rw [hend v] at hz
rcases Sym2.mem_iff.mp hz with rfl|rfl
· exact hw v
· exact v.property.1
· calc
F.ncard ≤ (Set.univ : Set U).ncard := by
simpa only [F,Set.image_univ] using (Set.ncard_image_le (f := e) (Set.finite_univ : (Set.univ : Set U).Finite))
_ = U.ncard := by simp
_ ≤ S.ncard := Set.ncard_le_ncard Set.sdiff_subset hS
· intro v hv
generalize hn : r v=n
induction n using Nat.strong_induction_on generalizing v with
| h n ih =>
by_cases hvo : v=o
· subst v; exact SimpleGraph.Reachable.refl _
· let a : U := ⟨v,hv,hvo⟩
have hprev := ih (r (w a)) (by simpa only [← hn] using hr a) (w a) (hw a) rfl
apply hprev.trans
exact (show (G.openGraph F).Adj (w a) v from
⟨⟨e a,⟨a,rfl⟩,hend a⟩,hne a⟩).reachable
def ballRank (o v : V) : ℕ :=
if h : ∃ n, v∈G.vertexBall o n then Nat.find h else 0
lemma ballRank_mem {o v : V} (h : ∃ n, v∈G.vertexBall o n) :
v∈G.vertexBall o (G.ballRank o v) := by
simp only [ballRank,dite_eq_left h]
exact Nat.find_spec h
lemma ballRank_le {o v : V} {n : ℕ} (h : v∈G.vertexBall o n) : G.ballRank o v ≤ n := by
have he : ∃ k, v∈G.vertexBall o k := ⟨n,h⟩
simp only [ballRank,dite_eq_left he]
exact Nat.find_min' he h
lemma ballRank_eq_zero_iff {o v : V} (h : ∃ n, v∈G.vertexBall o n) :
G.ballRank o v=0 ↔ v=o := by
constructor
· intro hz
have hh := G.ballRank_mem h
rw [hz] at hh
exact hh
· intro hvo
subst v
exact Nat.eq_zero_of_le_zero (G.ballRank_le (show o∈G.vertexBall o 0 from rfl))
lemma ballRank_predecessor {o v : V} (h : ∃ n, v∈G.vertexBall o n) (hne : v≠o) :
∃ w, G.fullGraph.Adj w v ∧ G.ballRank o w+1=G.ballRank o v ∧
w∈G.vertexBall o (G.ballRank o w) := by
have hn : 0 < G.ballRank o v := Nat.pos_of_ne_zero (fun hh => hne ((G.ballRank_eq_zero_iff h).mp hh))
obtain ⟨n,heq⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hn)
have hv := G.ballRank_mem h
rw [heq] at hv
rcases hv with hv|hv
· have hb := G.ballRank_le hv
omega
· obtain ⟨w,hw,hvw⟩ := Set.mem_iUnion₂.mp hv
have h1 := G.ballRank_le hw
have h2 := G.ballRank_le (G.vertexBall_step (G.ballRank_mem ⟨n,hw⟩) hvw)
refine ⟨w,hvw,by omega,G.ballRank_mem ⟨n,hw⟩⟩
lemma finite_seed_tree (hlocal : G.LocallyFinite) (o : V) (m : ℕ) :
∃ F : Set E, F.Finite ∧ F⊆G.inducedEdges (G.vertexBall o m) ∧
F.ncard ≤ (G.vertexBall o m).ncard ∧
∀ v∈G.vertexBall o m, (G.openGraph F).Reachable o v := by
apply G.finite_seed_tree_of_rank (G.vertexBall_finite hlocal o m) o (G.ballRank o)
intro v hv hne
obtain ⟨w,hw,hr,hmem⟩ := G.ballRank_predecessor ⟨m,hv⟩ hne
exact ⟨w,G.vertexBall_mono o (by have := G.ballRank_le hv; omega) hmem,hw,by omega⟩
end CriticalPercolation.BondGraph
end
end
end
end OAI
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