Join of a list of lists

This file proves basic properties of List.flatten, which concatenates a list of lists. It is defined in Init.Data.List.Basic.

theorem List.Sublist.flatten {α : Type u_1} {l₁ l₂ : List (List α)} (h : l₁.Sublist l₂) : l₁.flatten.Sublist l₂.flatten

theorem List.Sublist.flatMap {α : Type u_1} {β : Type u_2} {l₁ l₂ : List α} (h : l₁.Sublist l₂) (f : α → List β) : (flatMap f l₁).Sublist (flatMap f l₂)

@[deprecated List.length_flatten (since := "2024-10-25")]

theorem List.length_join' {α : Type u_1} {L : List (List α)} : L.flatten.length = (map length L).sum

Alias of List.length_flatten.

@[deprecated List.countP_flatten (since := "2024-10-25")]

theorem List.countP_join' {α : Type u_1} {p : α → Bool} {l : List (List α)} : countP p l.flatten = (map (countP p) l).sum

Alias of List.countP_flatten.

@[deprecated List.count_flatten (since := "2024-10-25")]

theorem List.count_join' {α : Type u_1} [BEq α] {a : α} {l : List (List α)} : count a l.flatten = (map (count a) l).sum

Alias of List.count_flatten.

theorem List.drop_take_succ_eq_cons_getElem {α : Type u_1} (L : List α) (i : Nat) (h : i < L.length) : drop i (take (i + 1) L) = [L[i]]

Taking only the first i+1 elements in a list, and then dropping the first i ones, one is left with a list of length 1 made of the i-th element of the original list.

theorem List.append_flatten_map_append {α : Type u_1} (L : List (List α)) (x : List α) : x ++ (map (fun (x_1 : List α) => x_1 ++ x) L).flatten = (map (fun (x_1 : List α) => x ++ x_1) L).flatten ++ x

We can rebracket x ++ (l₁ ++ x) ++ (l₂ ++ x) ++ ... ++ (lₙ ++ x) to (x ++ l₁) ++ (x ++ l₂) ++ ... ++ (x ++ lₙ) ++ x where L = [l₁, l₂, ..., lₙ].