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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Clusters.ClusterObject
import OAI.Probability.CriticalPercolation.Modular.ObjectAvoidMass
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.BondGraph
open ClusterDeletion FiniteProduct FiniteMarkov
open scoped Classical
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
theorem clusterObject_avoidance_cross_mul (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hp : ∀ e b, 0 ≤ w e b)
(hc : ∀ e, 0 < w e False) {A B : Set V} (hAB : Disjoint A B)
(f g : ClusterObject V E → ℝ) (hf : Monotone f) (hg : Monotone g) :
(expect w (fun ω => if Disjoint (G.clusterObject ω A).1 B then
f (G.clusterObject ω A) * g (G.clusterObject ω B) else 0))*G.objectAvoidTotal w A B ≤
(expect w (fun ω => if Disjoint (G.clusterObject ω A).1 B then f (G.clusterObject ω A) else 0)) *
(expect w (fun ω => if Disjoint (G.clusterObject ω A).1 B then g (G.clusterObject ω B) else 0)) := by
have hZ : 0 < G.objectAvoidTotal w A B :=
(Finset.prod_pos fun e _ => hc e).trans_le (G.objectAvoidTotal_minorization w hp hAB)
have hh := G.clusterObject_avoidance_cross w hw hp hc hAB f g hf hg
have hi (F H Z : ℝ) : (F/Z)*(H/Z) = (F*H/Z)/Z := by ring
rw [hi, div_le_div_iff_of_pos_right hZ] at hh
exact (le_div_iff₀ hZ).mp hh
omit [Fintype V] [Fintype E] in
lemma openGraph_adj_clusterObject {ω : Set E} {S : Set V} {v t : V}
(hv : v ∈ (G.clusterObject ω S).1) (ha : (G.openGraph ω).Adj v t) :
(G.openGraph (G.clusterObject ω S).2).Adj v t := by
obtain ⟨⟨e, he, heq⟩, hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp ha
exact (SimpleGraph.fromEdgeSet_adj _).mpr
⟨⟨e, ⟨he, v, hv, by simp [heq]⟩, heq⟩, hne⟩
omit [Fintype V] [Fintype E] in
lemma reachable_clusterObject {ω : Set E} {S : Set V} {v t : V} (hv : v ∈ S) :
(G.openGraph (G.clusterObject ω S).2).Reachable v t ↔ (G.openGraph ω).Reachable v t := by
constructor
· intro ht
exact ht.mono (SimpleGraph.fromEdgeSet_mono (Set.image_mono Set.inter_subset_left))
· intro ht
have h := ((G.openGraph ω).reachable_iff_reflTransGen v t).mp ht
clear ht
induction h with
| refl => exact .refl _
| @tail u t h hut ih =>
exact ih.trans (G.openGraph_adj_clusterObject
⟨v, hv, ((G.openGraph ω).reachable_iff_reflTransGen v u).mpr h⟩ hut).reachable
end CriticalPercolation.BondGraph
end
end
end OAI
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