File size: 4,609 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Criticality.FiniteSeedTree
import OAI.Probability.CriticalPercolation.Gluing.CollarDistance
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma inward_path (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty)
(v : V) (l : ℕ) (hl : l ≤ G.distanceTo B v) :
∃ (y : V) (P : Set E), P.Finite ∧ P.ncard ≤ l ∧
G.distanceTo B y+l=G.distanceTo B v ∧ y∈G.vertexBall v l ∧
P⊆G.inducedEdges (G.vertexBall v l∩{z | G.distanceTo B z ≤ G.distanceTo B v}) ∧
P⊆(G.inducedEdges {z | G.distanceTo B v ≤ G.distanceTo B z})ᶜ ∧
(G.openGraph P).Reachable v y := by
induction l generalizing v with
| zero =>
exact ⟨v,∅,Set.finite_empty,by simp,by omega,rfl,Set.empty_subset _,Set.empty_subset _,
SimpleGraph.Reachable.refl _⟩
| succ l ih =>
obtain ⟨w,hw,hr⟩ := G.distanceTo_predecessor hc hB v (by omega)
obtain ⟨y,P,hP,hcard,hy,hyball,hends,hfresh,hreach⟩ := ih w (by omega)
have hwball : w∈G.vertexBall v 1 := G.vertexBall_step (G.mem_vertexBall_self v 0) hw
obtain ⟨⟨e,_,he⟩,hne⟩ := hw
refine ⟨y,insert e P,hP.insert _,?_,by omega,?_,?_,?_,?_⟩
· exact (Set.ncard_insert_le _ _).trans (by omega)
· simpa only [Nat.add_comm 1] using G.vertexBall_add hwball l hyball
· intro f hf z hz
rcases hf with rfl|hf
· rw [he] at hz
have hz' : z=v ∨ z=w := Sym2.mem_iff.mp hz
rcases hz' with hz'|hz'
· subst z
exact ⟨G.mem_vertexBall_self v (l+1),show G.distanceTo B v ≤ G.distanceTo B v from le_rfl⟩
· subst z
exact ⟨G.vertexBall_mono v (by omega) hwball,by change G.distanceTo B w ≤ G.distanceTo B v; omega⟩
· have hh := hends hf z hz
have hh2 : G.distanceTo B z ≤ G.distanceTo B w := hh.2
refine ⟨?_,by change G.distanceTo B z ≤ G.distanceTo B v; omega⟩
simpa only [Nat.add_comm 1] using G.vertexBall_add hwball l hh.1
· intro f hf hf'
rcases hf with rfl|hf
· have hh := hf' w (by rw [he]; simp)
change G.distanceTo B v ≤ G.distanceTo B w at hh
omega
· apply hfresh hf
intro z hz
have hh := hf' z hz
change G.distanceTo B v ≤ G.distanceTo B z at hh
change G.distanceTo B w ≤ G.distanceTo B z
omega
· apply (show (G.openGraph (insert e P)).Adj v w from
⟨⟨e,Set.mem_insert e P,he⟩,hne⟩).reachable.trans
exact hreach.mono (G.openGraph_mono (Set.subset_insert e P))
lemma inward_seed_trial (hc : G.fullGraph.Connected) (hlocal : G.LocallyFinite)
{B : Set V} (hB : B.Nonempty) (v : V) (m : ℕ) (hm : m<G.distanceTo B v) :
∃ (y : V) (F : Set E), F.Finite ∧
F.ncard ≤ m+1+(G.vertexBall y m).ncard ∧
G.distanceTo B y+(m+1)=G.distanceTo B v ∧
F⊆G.inducedEdges (G.vertexBall v (2*m+1)∩{z | G.distanceTo B z ≤ G.distanceTo B v}) ∧
F⊆(G.inducedEdges {z | G.distanceTo B v ≤ G.distanceTo B z})ᶜ ∧
∀ z∈G.vertexBall y m, (G.openGraph F).Reachable v z := by
obtain ⟨y,P,hP,hcard,hy,hyball,hends,hfresh,hreach⟩ := G.inward_path hc hB v (m+1) (by omega)
obtain ⟨Q,hQ,hQends,hQcard,hQreach⟩ := G.finite_seed_tree hlocal y m
refine ⟨y,P∪Q,hP.union hQ,?_,hy,?_,?_,?_⟩
· exact (Set.ncard_union_le P Q).trans (by omega)
· intro e he z hz
rcases he with he|he
· have hh := hends he z hz
exact ⟨G.vertexBall_mono v (by omega) hh.1,hh.2⟩
· have hh := hQends he z hz
have hle := G.distanceTo_ball hc hB hh
refine ⟨?_,by change G.distanceTo B z ≤ G.distanceTo B v; omega⟩
have := G.vertexBall_add hyball m hh
simpa only [show m+1+m=2*m+1 by omega] using this
· intro e he he'
rcases he with he|he
· exact hfresh he he'
· obtain ⟨⟨x,z⟩,hxz⟩ := Sym2.mk_surjective (G.ends e)
have hx : x∈G.ends e := by rw [← hxz]; simp
have hh := G.distanceTo_ball hc hB (hQends he x hx)
have hh' := he' x hx
change G.distanceTo B v ≤ G.distanceTo B x at hh'
omega
· intro z hz
exact (hreach.mono (G.openGraph_mono Set.subset_union_left)).trans
((hQreach z hz).mono (G.openGraph_mono Set.subset_union_right))
end CriticalPercolation.BondGraph
end
end
end
end OAI
|