| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace CriticalPercolation.ClusterColumns | |
| open CriticalPercolation.ClusterDeletion | |
| open scoped Classical | |
| variable {V : Type*} | |
| noncomputable def hitValue (A U : Set V) : ℝ := if (U ∩ A).Nonempty then 1 else 0 | |
| noncomputable def jointHitValue (A T U : Set V) : ℝ := | |
| if (U ∩ A).Nonempty ∧ (U ∩ T).Nonempty then 1 else 0 | |
| lemma hitValue_nonneg (A U : Set V) : 0 ≤ hitValue A U := by unfold hitValue; positivity | |
| lemma hitValue_le_one (A U : Set V) : hitValue A U ≤ 1 := by unfold hitValue; split_ifs <;> norm_num | |
| lemma hitValue_monotone (A : Set V) : Monotone (hitValue A) := by | |
| intro U W hUW | |
| unfold hitValue | |
| split_ifs with hU hW hW | |
| · exact le_rfl | |
| · exact (hW (hU.mono (Set.inter_subset_inter_left A hUW))).elim | |
| · norm_num | |
| · exact le_rfl | |
| lemma jointHitValue_nonneg (A T U : Set V) : 0 ≤ jointHitValue A T U := by | |
| unfold jointHitValue; positivity | |
| lemma jointHitValue_monotone (A T : Set V) : Monotone (jointHitValue A T) := by | |
| intro U W hUW | |
| unfold jointHitValue | |
| split_ifs with hU hW hW | |
| · exact le_rfl | |
| · exact (hW ⟨hU.1.mono (Set.inter_subset_inter_left A hUW), | |
| hU.2.mono (Set.inter_subset_inter_left T hUW)⟩).elim | |
| · norm_num | |
| · exact le_rfl | |
| lemma jointHitValue_eq_hitValue {A T U : Set V} (hA : (U ∩ A).Nonempty) : | |
| jointHitValue A T U = hitValue T U := by simp [jointHitValue, hitValue, hA] | |
| lemma component_inter_nonempty_iff (H : SimpleGraph V) (s : V) (A : Set V) : | |
| (component H s ∩ A).Nonempty ↔ s ∈ roots H A := by | |
| constructor | |
| · rintro ⟨t, ht, hA⟩ | |
| exact ⟨t, hA, ht.symm⟩ | |
| · rintro ⟨t, hA, ht⟩ | |
| exact ⟨t, ht.symm, hA⟩ | |
| end CriticalPercolation.ClusterColumns | |
| end | |
| end | |
| end OAI | |