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.

def List.Nodup.getBijectionOfForallMemList {α : Type u_1} (l : List α) (nd : l.Nodup) (h : ∀ (x : α), x ∈ l) : { f : Fin l.length → α // 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 = ⟨fun (i : Fin l.length) => l.get i, ⋯⟩

@[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) : ↑(getBijectionOfForallMemList l nd 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.idxOf (↑x) l, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[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 }) : ↑((getEquiv l H).symm x) = idxOf (↑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) : ↑((getEquiv l H) i) = l.get i

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

• List.Nodup.getEquivOfForallMemList l nd h = { toFun := fun (i : Fin l.length) => l.get i, invFun := fun (a : α) => ⟨List.idxOf a l, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[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) : (getEquivOfForallMemList l nd h) i = l.get i

@[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) : ↑((getEquivOfForallMemList l nd h).symm a) = idxOf a l

theorem List.Sorted.get_mono {α : Type u_1} [Preorder α] {l : List α} (h : 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 : Sorted (fun (x1 x2 : α) => x1 < x2) l) : StrictMono l.get

def List.Sorted.getIso {α : Type u_1} [Preorder α] [DecidableEq α] (l : List α) (H : 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

• List.Sorted.getIso l H = { toEquiv := List.Nodup.getEquiv l ⋯, map_rel_iff' := ⋯ }

@[simp]

theorem List.Sorted.coe_getIso_apply {α : Type u_1} [Preorder α] {l : List α} [DecidableEq α] (H : Sorted (fun (x1 x2 : α) => x1 < x2) l) {i : Fin l.length} : ↑((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 : Sorted (fun (x1 x2 : α) => x1 < x2) l) {x : { x : α // x ∈ l }} : ↑((getIso l H).symm x) = idxOf (↑x) l

theorem List.sublist_of_orderEmbedding_getElem?_eq {α : Type u_1} {l l' : List α} (f : ℕ ↪o ℕ) (hf : ∀ (ix : ℕ), l[ix]? = l'[f ix]?) : l.Sublist 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_getElem?_eq {α : Type u_1} {l l' : List α} : l.Sublist l' ↔ ∃ (f : ℕ ↪o ℕ), ∀ (ix : ℕ), l[ix]? = 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 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} : 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.