| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace CriticalPercolation.ClusterColumns | |
| open scoped Classical | |
| variable {V : Type*} {n : ℕ} | |
| noncomputable def hitOthers (x : Fin n → V) (k : Fin n) (U : Set V) : ℝ := | |
| if ∃ j, j ≠ k ∧ x j ∈ U then 1 else 0 | |
| lemma hitOthers_nonneg (x : Fin n → V) (k : Fin n) (U : Set V) : | |
| 0 ≤ hitOthers x k U := by unfold hitOthers; positivity | |
| lemma hitOthers_le_one (x : Fin n → V) (k : Fin n) (U : Set V) : | |
| hitOthers x k U ≤ 1 := by unfold hitOthers; split_ifs <;> norm_num | |
| lemma hitOthers_monotone (x : Fin n → V) (k : Fin n) : Monotone (hitOthers x k) := by | |
| intro U W hUW | |
| unfold hitOthers | |
| split_ifs with hU hW hW | |
| · exact le_rfl | |
| · obtain ⟨j, hj, hjU⟩ := hU | |
| exact (hW ⟨j, hj, hUW hjU⟩).elim | |
| · norm_num | |
| · exact le_rfl | |
| lemma hitOthers_singleton (x : Fin n → V) (hx : Function.Injective x) (k : Fin n) : | |
| hitOthers x k {x k} = 0 := by | |
| apply ite_eq_right | |
| rintro ⟨j, hj, hjk⟩ | |
| exact hj (hx hjk) | |
| end CriticalPercolation.ClusterColumns | |
| end | |
| end | |
| end OAI | |