Lemmas about Lists and Set.range

In this file we prove lemmas about range of some operations on lists.

theorem Set.range_list_map {α : Type u_1} {β : Type u_2} (f : α → β) : Set.range (List.map f) = {l : List β | ∀ x ∈ l, x ∈ Set.range f}

theorem Set.range_list_map_coe {α : Type u_1} (s : Set α) : Set.range (List.map Subtype.val) = {l : List α | ∀ x ∈ l, x ∈ s}

@[simp]

theorem Set.range_list_nthLe {α : Type u_1} (l : List α) : (Set.range fun (k : Fin (List.length l)) => List.nthLe l ↑k ⋯) = {x : α | x ∈ l}

theorem Set.range_list_get? {α : Type u_1} (l : List α) : Set.range (List.get? l) = insert none (some '' {x : α | x ∈ l})

@[simp]

theorem Set.range_list_getD {α : Type u_1} (l : List α) (d : α) : (Set.range fun (n : ℕ) => List.getD l n d) = insert d {x : α | x ∈ l}

@[simp]

theorem Set.range_list_getI {α : Type u_1} [Inhabited α] (l : List α) : Set.range (List.getI l) = insert default {x : α | x ∈ l}

instance List.canLift {α : Type u_1} {β : Type u_2} (c : β → α) (p : α → Prop) [CanLift α β c p] : CanLift (List α) (List β) (List.map c) fun (l : List α) => ∀ x ∈ l, p x

If each element of a list can be lifted to some type, then the whole list can be lifted to this type.

Equations

• ⋯ = ⋯