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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 | 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
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