mem_closure_isSwap' — Mathlib

English

A permutation lies in the closure of the set of swaps if and only if it has finite fixed-by complement.

Русский

Перестановка лежит в замыкании множества транспозиций ⇔ её complemento фиксирования конечна.

LaTeX

$$theorem mem_closure_isSwap' {f : Equiv.Perm α} : ...$$

Lean4

/-- A permutation is a product of transpositions if and only if it has finite support. -/
theorem mem_closure_isSwap' {f : Perm α} : f ∈ closure {σ : Perm α | σ.IsSwap} ↔ (fixedBy α f)ᶜ.Finite :=
  by
  refine (mem_closure_isSwap fun _ ↦ id).trans (and_iff_left fun x ↦ ⟨⟨swap x (f x), ?_⟩, swap_apply_left x (f x)⟩)
  by_cases h : x = f x
  · rw [← h, swap_self]
    apply Subgroup.one_mem
  · exact subset_closure ⟨x, f x, h, rfl⟩

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