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

Given a list l,

• if l has no duplicates, then List.Nodup.getEquiv is the equivalence between Fin (length l) and {x // x ∈ l} sending i to ⟨get l i, _⟩ with the inverse sending ⟨x, hx⟩ to ⟨indexOf x l, _⟩;

• if l has no duplicates and contains every element of a type α, then List.Nodup.getEquivOfForallMemList defines an equivalence between Fin (length l) and α; if α does not have decidable equality, then there is a bijection List.Nodup.getBijectionOfForallMemList;

• if l is sorted w.r.t. (<), then List.Sorted.getIso is the same bijection reinterpreted as an OrderIso.

@[simp]

theorem List.Nodup.getBijectionOfForallMemList_coe {α : Type u_1} (l : List α) (nd : List.Nodup l) (h : ∀ (x : α), x ∈ l) (i : Fin (List.length l)) : ↑(List.Nodup.getBijectionOfForallMemList l nd h) i = List.get l i

def List.Nodup.getBijectionOfForallMemList {α : Type u_1} (l : List α) (nd : List.Nodup l) (h : ∀ (x : α), x ∈ l) : { f // 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

• List.Nodup.getBijectionOfForallMemList l nd h = { val := fun i => List.get l i, property := (_ : (Function.Injective fun i => List.get l i) ∧ Function.Surjective fun i => List.get l i) }

@[simp]

theorem List.Nodup.getEquiv_symm_apply_val {α : Type u_1} [DecidableEq α] (l : List α) (H : List.Nodup l) (x : { x // 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 : List.Nodup l) (i : Fin (List.length l)) : ↑(↑(List.Nodup.getEquiv l H) i) = List.get l i

def List.Nodup.getEquiv {α : Type u_1} [DecidableEq α] (l : List α) (H : List.Nodup l) : Fin (List.length l) ≃ { x // 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

• One or more equations did not get rendered due to their size.

@[simp]

theorem List.Nodup.getEquivOfForallMemList_symm_apply_val {α : Type u_1} [DecidableEq α] (l : List α) (nd : List.Nodup l) (h : ∀ (x : α), x ∈ l) (a : α) : ↑(↑(List.Nodup.getEquivOfForallMemList l nd h).symm a) = List.indexOf a l

@[simp]

theorem List.Nodup.getEquivOfForallMemList_apply {α : Type u_1} [DecidableEq α] (l : List α) (nd : List.Nodup l) (h : ∀ (x : α), x ∈ l) (i : Fin (List.length l)) : ↑(List.Nodup.getEquivOfForallMemList l nd h) i = List.get l i

def List.Nodup.getEquivOfForallMemList {α : Type u_1} [DecidableEq α] (l : List α) (nd : List.Nodup l) (h : ∀ (x : α), x ∈ l) : Fin (List.length l) ≃ 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

• One or more equations did not get rendered due to their size.

theorem List.Sorted.get_mono {α : Type u_1} [Preorder α] {l : List α} (h : List.Sorted (fun x x_1 => x ≤ x_1) l) : Monotone (List.get l)

theorem List.Sorted.get_strictMono {α : Type u_1} [Preorder α] {l : List α} (h : List.Sorted (fun x x_1 => x < x_1) l) : StrictMono (List.get l)

def List.Sorted.getIso {α : Type u_1} [Preorder α] [DecidableEq α] (l : List α) (H : List.Sorted (fun x x_1 => x < x_1) l) : Fin (List.length l) ≃o { x // 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

• List.Sorted.getIso l H = { toEquiv := List.Nodup.getEquiv l (_ : List.Nodup l), map_rel_iff' := (_ : ∀ {a b : Fin (List.length l)}, List.get l a ≤ List.get l b ↔ a ≤ b) }

@[simp]

theorem List.Sorted.coe_getIso_apply {α : Type u_1} [Preorder α] {l : List α} [DecidableEq α] (H : List.Sorted (fun x x_1 => x < x_1) l) {i : Fin (List.length l)} : ↑(↑(List.Sorted.getIso l H) i) = List.get l i

@[simp]

theorem List.Sorted.coe_getIso_symm_apply {α : Type u_1} [Preorder α] {l : List α} [DecidableEq α] (H : List.Sorted (fun x x_1 => x < x_1) l) {x : { x // x ∈ l }} : ↑(↑(OrderIso.symm (List.Sorted.getIso l H)) x) = List.indexOf (↑x) l

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

If there is f, an order-preserving embedding of ℕ into ℕ 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} : List.Sublist l l' ↔ ∃ f, ∀ (ix : ℕ), List.get? l ix = List.get? l' (↑f ix)

A l : List α is Sublist l l' for l' : List α iff there is f, an order-preserving embedding of ℕ into ℕ 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} : List.Sublist l l' ↔ ∃ f, ∀ (ix : Fin (List.length l)), List.get l ix = List.get l' (↑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 m x, x = List.get l n ∧ x = List.get l 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.

@[deprecated List.duplicate_iff_exists_distinct_get]

theorem List.duplicate_iff_exists_distinct_nthLe {α : Type u_1} {l : List α} {x : α} : List.Duplicate x l ↔ ∃ n hn m hm x, x = List.nthLe l n hn ∧ x = List.nthLe l m hm

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.