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Mathlib.Data.Fintype.Sort

Sorting a finite type #

This file provides two equivalences for linearly ordered fintypes:

def monoEquivOfFin (α : Type u_1) [Fintype α] [LinearOrder α] {k : } (h : Fintype.card α = k) :
Fin k ≃o α

Given a linearly ordered fintype α of cardinal k, the order isomorphism monoEquivOfFin α h is the increasing bijection between Fin k and α. Here, h is a proof that the cardinality of α is k. We use this instead of an isomorphism Fin (card α) ≃o α to avoid casting issues in further uses of this function.

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Instances For
    def finSumEquivOfFinset {α : Type u_1} [DecidableEq α] [Fintype α] [LinearOrder α] {m : } {n : } {s : Finset α} (hm : s.card = m) (hn : s.card = n) :
    Fin m Fin n α

    If α is a linearly ordered fintype, s : Finset α has cardinality m and its complement has cardinality n, then Fin m ⊕ Fin n ≃ α. The equivalence sends elements of Fin m to elements of s and elements of Fin n to elements of sᶜ while preserving order on each "half" of Fin m ⊕ Fin n (using Set.orderIsoOfFin).

    Equations
    Instances For
      @[simp]
      theorem finSumEquivOfFinset_inl {α : Type u_1} [DecidableEq α] [Fintype α] [LinearOrder α] {m : } {n : } {s : Finset α} (hm : s.card = m) (hn : s.card = n) (i : Fin m) :
      (finSumEquivOfFinset hm hn) (Sum.inl i) = (s.orderEmbOfFin hm) i
      @[simp]
      theorem finSumEquivOfFinset_inr {α : Type u_1} [DecidableEq α] [Fintype α] [LinearOrder α] {m : } {n : } {s : Finset α} (hm : s.card = m) (hn : s.card = n) (i : Fin n) :
      (finSumEquivOfFinset hm hn) (Sum.inr i) = (s.orderEmbOfFin hn) i