subperm_iff_count — Mathlib

English

Let l, t1, t2 be lists with t1 and t2 disjoint. Then l ∩ (t1 ++ t2) is a permutation of (l ∩ t1) ++ (l ∩ t2).

Русский

Пусть списки l, t1, t2 удовлетворяют Disjoint(t1,t2). Тогда l ∩ (t1 ++ t2) перестановочно эквивалентно (l ∩ t1) ++ (l ∩ t2).

LaTeX

$$$\operatorname{Disjoint}(t_1,t_2) \Rightarrow l \cap (t_1 ++ t_2) \sim (l \cap t_1) ++ (l \cap t_2)$$$

Lean4

/-- See also `List.subperm_ext_iff`. -/
theorem subperm_iff_count [DecidableEq α] : l₁ <+~ l₂ ↔ ∀ a, count a l₁ ≤ count a l₂ :=
  subperm_ext_iff.trans <| forall_congr' fun a ↦ by by_cases ha : a ∈ l₁ <;> simp [ha, count_eq_zero_of_not_mem]

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