openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /ClusterObjectAvoidanceCross.lean
| 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 _ | |
| | 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 | |