derangementsRecursionEquiv — Mathlib

English

There is a canonical equivalence expressing the structure of Derangements on Option α via a sum over α of sub-derangements pieces.

Русский

Существует каноническое эквивалентное разложение дерangements на Option α через сумму по α частей дерangements.

LaTeX

$$see previous$$

Lean4

/-- The set of derangements on `Option α` is equivalent to the union over all `a : α` of
"derangements on `α` ⊕ derangements on `{a}ᶜ`". -/
def derangementsRecursionEquiv :
    derangements (Option α) ≃ Σ a : α, derangements (({ a }ᶜ : Set α) : Type _) ⊕ derangements α :=
  derangementsOptionEquivSigmaAtMostOneFixedPoint.trans (sigmaCongrRight atMostOneFixedPointEquivSum_derangements)

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