Note about Mathlib/Init/

The files in Mathlib/Init are leftovers from the port from Mathlib3. (They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)

We intend to move all the content of these files out into the main Mathlib directory structure. Contributions assisting with this are appreciated.

Lemmas for List not (yet) in Batteries

append

length

map

bind

mem

theorem List.mem_cons_eq {α : Type u} (a : α) (y : α) (l : List α) : (a ∈ y :: l) = (a = y ∨ a ∈ l)

theorem List.eq_or_mem_of_mem_cons : ∀ {α : Type u_1} {a b : α} {l : List α}, a ∈ b :: l → a = b ∨ a ∈ l

Alias of the forward direction of List.mem_cons.

theorem List.not_exists_mem_nil {α : Type u} (p : α → Prop) : ¬∃ (x : α), x ∈ [] ∧ p x

@[deprecated List.not_exists_mem_nil]

theorem List.not_bex_nil {α : Type u} (p : α → Prop) : ¬∃ (x : α), x ∈ [] ∧ p x

Alias of List.not_exists_mem_nil.

@[deprecated List.exists_mem_cons]

theorem List.bex_cons {α : Type u_1} {p : α → Prop} {a : α} {l : List α} : (∃ (x : α), ∃ (x_1 : x ∈ a :: l), p x) ↔ p a ∨ ∃ (x : α), ∃ (x_1 : x ∈ l), p x

Alias of List.exists_mem_cons.

list subset

sublists

theorem List.length_le_of_sublist : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₁.length ≤ l₂.length

Alias of List.Sublist.length_le.

filter

map_accumr

def List.mapAccumr {α : Type u} {β : Type v} {σ : Type w₂} (f : α → σ → σ × β) : List α → σ → σ × List β

Runs a function over a list returning the intermediate results and a final result.

Equations

• List.mapAccumr f [] x = (x, [])
• List.mapAccumr f (y :: yr) x = let r := List.mapAccumr f yr x; let z := f y r.fst; (z.fst, z.snd :: r.snd)

@[simp]

theorem List.length_mapAccumr {α : Type u} {β : Type v} {σ : Type w₂} (f : α → σ → σ × β) (x : List α) (s : σ) : (List.mapAccumr f x s).snd.length = x.length

Length of the list obtained by mapAccumr.

def List.mapAccumr₂ {α : Type u} {β : Type v} {φ : Type w₁} {σ : Type w₂} (f : α → β → σ → σ × φ) : List α → List β → σ → σ × List φ

Runs a function over two lists returning the intermediate results and a final result.

Equations

• List.mapAccumr₂ f [] x✝ x = (x, [])
• List.mapAccumr₂ f x✝ [] x = (x, [])
• List.mapAccumr₂ f (x_3 :: xr) (y :: yr) x = let r := List.mapAccumr₂ f xr yr x; let q := f x_3 y r.fst; (q.fst, q.snd :: r.snd)

@[simp]

theorem List.length_mapAccumr₂ {α : Type u} {β : Type v} {φ : Type w₁} {σ : Type w₂} (f : α → β → σ → σ × φ) (x : List α) (y : List β) (c : σ) : (List.mapAccumr₂ f x y c).snd.length = min x.length y.length

Length of a list obtained using mapAccumr₂.