mclosure_isSwap — Mathlib

English

Permutations of Option α are equivalent to pairing an element of Option α with a permutation of α and using an explicit inverse construction.

Русский

Перестановки над Option α эквивалентны паре (элемент Option α, перестановка α) с явной конструкцией обратной связи.

LaTeX

$$$ \\mathrm{Perm}(\\mathrm{Option}(\\alpha)) \\cong (\\mathrm{Option}(\\alpha) \\times \\mathrm{Perm}(\\alpha)) $$$

Lean4

theorem mclosure_isSwap [Finite α] : Submonoid.closure {σ : Perm α | IsSwap σ} = ⊤ :=
  by
  cases nonempty_fintype α
  refine top_unique fun x _ ↦ ?_
  obtain ⟨h1, h2⟩ := Subtype.mem (truncSwapFactors x).out
  rw [← h1]
  exact Submonoid.list_prod_mem _ fun y hy ↦ Submonoid.subset_closure (h2 y hy)

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