File size: 4,158 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 99 100 101 102 103 104 105 106 107 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.SequentialExposure
namespace OAI
section
section
namespace CriticalPercolation.LazyDFS
open scoped Classical
open FiniteProduct SearchTree
variable {V E : Type*} [Fintype E] [DecidableEq V] [DecidableEq E]
lemma compile_queryDelta_incident (G : BondGraph V E) (order : List (Dart V E))
(B : Set V) (fuel : ℕ) (st : State V E) (stack : List (Dart V E))
(F : Set E → Prop) (w : E → Prop → ℝ) (ω : Set E) (a : V) (e : E)
(hst : ∀ v ∈ st.reached, (G.openGraph ω).Reachable a v)
(hq : (compile G order B fuel st stack).queryDelta F w ω e ≠ 0) :
∃ v ∈ G.ends e, (G.openGraph ω).Reachable a v := by
induction fuel generalizing st stack w with
| zero => exact (hq rfl).elim
| succ n ih =>
rw [compile.eq_def] at hq
dsimp only at hq
split_ifs at hq with hh
· exact (hq rfl).elim
· cases stack with
| nil => exact (hq rfl).elim
| cons d rest =>
dsimp only at hq
split_ifs at hq with hd
· exact ih st rest w hst hq
· have htail : d.1 ∈ st.reached := by
by_contra ht
exact hd (Or.inr (Or.inr (Or.inl ht)))
have he : G.ends d.2.1 = s(d.1,d.2.2) := by
by_contra ht
exact hd (Or.inr (Or.inr (Or.inr ht)))
by_cases hef : e=d.2.1
· subst e
exact ⟨d.1,by simp [he],hst d.1 htail⟩
· simp only [queryDelta,ite_eq_right hef] at hq
by_cases hopen : ω d.2.1
· rw [ite_eq_left hopen] at hq
apply ih _ _ _ (fun v hv => ?_) hq
rcases Finset.mem_insert.mp hv with rfl|hv
· exact (hst d.1 htail).trans (prop_openEdge_reachable G ω hopen he)
· exact hst v hv
· rw [ite_eq_right hopen] at hq
exact ih (st.closed d.2.1) rest (pin w d.2.1 False) (fun v hv => hst v hv) hq
lemma search_queryDelta_incident [Fintype V] (G : BondGraph V E)
(order : List (Dart V E)) (B : Set V) (a : V) (F : Set E → Prop)
(w : E → Prop → ℝ) (ω : Set E) (e : E)
(hq : (search G order a B).queryDelta F w ω e ≠ 0) :
∃ v ∈ G.ends e, (G.openGraph ω).Reachable a v := by
apply compile_queryDelta_incident G order B _ _ _ F w ω a e _ hq
intro v hv
have he : v=a := Finset.mem_singleton.mp hv
subst v
exact .refl _
end CriticalPercolation.LazyDFS
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph
variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
variable (G : BondGraph V E)
theorem kernel_contact_geometry (order : List (LazyDFS.Dart V E))
(S : Set V) (a : V) (p : ℝ) (w : E → Prop → ℝ) (ω : E → Prop) (e : E)
(hk : G.contactKernel order S a p w (G.clusterObject ω S) ω (some e) ≠ 0) :
a ∉ (G.clusterObject ω S).1 ∧ ¬ω e ∧ ∃ u v,
G.ends e=s(u,v) ∧ (G.openGraph ω).Reachable a u ∧ v ∈ (G.clusterObject ω S).1 := by
have ha : a ∉ (G.clusterObject ω S).1 := by
intro ha
simp only [contactKernel,ite_eq_left ha,Option.some_ne_none,ite_false] at hk
exact hk rfl
have he : e ∈ G.cutEdges (G.clusterObject ω S).1 := by
by_contra he
simp only [contactKernel,ite_eq_right ha,contactWeight,ite_eq_right he,zero_div] at hk
exact hk rfl
have hc : ¬ω e := fun hoe => G.clusterObject_cut_closed ω S e he
⟨hoe,G.cutEdges_subset_incidentEdges _ he⟩
obtain ⟨u,v,heq,hu,hv⟩ := he
have hq : (LazyDFS.search G order a S).queryDelta
(fun σ => ∃ b ∈ S, (G.openGraph σ).Reachable a b) w ω e ≠ 0 := by
intro hq
simp only [contactKernel,ite_eq_right ha,contactWeight,hq,mul_zero,ite_self,zero_div] at hk
exact hk rfl
obtain ⟨z,hz,hr⟩ := LazyDFS.search_queryDelta_incident G order S a _ w ω e hq
have hzu : z=u := by
rw [heq,Sym2.mem_iff] at hz
rcases hz with rfl|rfl
· rfl
· obtain ⟨b,hb,hbv⟩ := hv
exact (ha ⟨b,hb,hbv.trans hr.symm⟩).elim
exact ⟨ha,hc,u,v,heq,hzu ▸ hr,hv⟩
end CriticalPercolation.BinaryLaw
end
end
end OAI
|