Documentation

Mathlib.Data.List.NodupEquivFin

Equivalence between Fin (length l) and elements of a list #

Given a list l,

@[simp]
theorem List.Nodup.getBijectionOfForallMemList_coe {α : Type u_1} (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x l) (i : Fin l.length) :
def List.Nodup.getBijectionOfForallMemList {α : Type u_1} (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x l) :
{ f : Fin l.lengthα // Function.Bijective f }

If l lists all the elements of α without duplicates, then List.get defines a bijection Fin l.length → α. See List.Nodup.getEquivOfForallMemList for a version giving an equivalence when there is decidable equality.

Equations
Instances For
    @[simp]
    theorem List.Nodup.getEquiv_symm_apply_val {α : Type u_1} [DecidableEq α] (l : List α) (H : l.Nodup) (x : { x : α // x l }) :
    ((List.Nodup.getEquiv l H).symm x) = List.indexOf (↑x) l
    @[simp]
    theorem List.Nodup.getEquiv_apply_coe {α : Type u_1} [DecidableEq α] (l : List α) (H : l.Nodup) (i : Fin l.length) :
    ((List.Nodup.getEquiv l H) i) = l.get i
    def List.Nodup.getEquiv {α : Type u_1} [DecidableEq α] (l : List α) (H : l.Nodup) :
    Fin l.length { x : α // x l }

    If l has no duplicates, then List.get defines an equivalence between Fin (length l) and the set of elements of l.

    Equations
    • List.Nodup.getEquiv l H = { toFun := fun (i : Fin l.length) => l.get i, , invFun := fun (x : { x : α // x l }) => List.indexOf (↑x) l, , left_inv := , right_inv := }
    Instances For
      @[simp]
      theorem List.Nodup.getEquivOfForallMemList_apply {α : Type u_1} [DecidableEq α] (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x l) (i : Fin l.length) :
      @[simp]
      theorem List.Nodup.getEquivOfForallMemList_symm_apply_val {α : Type u_1} [DecidableEq α] (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x l) (a : α) :
      def List.Nodup.getEquivOfForallMemList {α : Type u_1} [DecidableEq α] (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x l) :
      Fin l.length α

      If l lists all the elements of α without duplicates, then List.get defines an equivalence between Fin l.length and α.

      See List.Nodup.getBijectionOfForallMemList for a version without decidable equality.

      Equations
      Instances For
        theorem List.Sorted.get_mono {α : Type u_1} [Preorder α] {l : List α} (h : List.Sorted (fun (x1 x2 : α) => x1 x2) l) :
        Monotone l.get
        theorem List.Sorted.get_strictMono {α : Type u_1} [Preorder α] {l : List α} (h : List.Sorted (fun (x1 x2 : α) => x1 < x2) l) :
        def List.Sorted.getIso {α : Type u_1} [Preorder α] [DecidableEq α] (l : List α) (H : List.Sorted (fun (x1 x2 : α) => x1 < x2) l) :
        Fin l.length ≃o { x : α // x l }

        If l is a list sorted w.r.t. (<), then List.get defines an order isomorphism between Fin (length l) and the set of elements of l.

        Equations
        Instances For
          @[simp]
          theorem List.Sorted.coe_getIso_apply {α : Type u_1} [Preorder α] {l : List α} [DecidableEq α] (H : List.Sorted (fun (x1 x2 : α) => x1 < x2) l) {i : Fin l.length} :
          ((List.Sorted.getIso l H) i) = l.get i
          @[simp]
          theorem List.Sorted.coe_getIso_symm_apply {α : Type u_1} [Preorder α] {l : List α} [DecidableEq α] (H : List.Sorted (fun (x1 x2 : α) => x1 < x2) l) {x : { x : α // x l }} :
          ((List.Sorted.getIso l H).symm x) = List.indexOf (↑x) l
          theorem List.sublist_of_orderEmbedding_get?_eq {α : Type u_1} {l : List α} {l' : List α} (f : ↪o ) (hf : ∀ (ix : ), l.get? ix = l'.get? (f ix)) :
          l.Sublist l'

          If there is f, an order-preserving embedding of into such that any element of l found at index ix can be found at index f ix in l', then Sublist l l'.

          theorem List.sublist_iff_exists_orderEmbedding_get?_eq {α : Type u_1} {l : List α} {l' : List α} :
          l.Sublist l' ∃ (f : ↪o ), ∀ (ix : ), l.get? ix = l'.get? (f ix)

          A l : List α is Sublist l l' for l' : List α iff there is f, an order-preserving embedding of into such that any element of l found at index ix can be found at index f ix in l'.

          theorem List.sublist_iff_exists_fin_orderEmbedding_get_eq {α : Type u_1} {l : List α} {l' : List α} :
          l.Sublist l' ∃ (f : Fin l.length ↪o Fin l'.length), ∀ (ix : Fin l.length), l.get ix = l'.get (f ix)

          A l : List α is Sublist l l' for l' : List α iff there is f, an order-preserving embedding of Fin l.length into Fin l'.length such that any element of l found at index ix can be found at index f ix in l'.

          theorem List.duplicate_iff_exists_distinct_get {α : Type u_1} {l : List α} {x : α} :
          List.Duplicate x l ∃ (n : Fin l.length) (m : Fin l.length) (_ : n < m), x = l.get n x = l.get m

          An element x : α of l : List α is a duplicate iff it can be found at two distinct indices n m : ℕ inside the list l.