Documentation

Mathlib.Init.Data.List.Lemmas

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.

#align statements without corresponding declarations (i.e. because the declaration is in Batteries or Lean) can be left here. These will be deleted soon so will not significantly delay deleting otherwise empty Init files.

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 :: la = b a l

Alias of the forward direction of List.mem_cons.

theorem List.not_exists_mem_nil {α : Type u} (p : αProp) :
¬∃ (x : α), x [] p x
@[deprecated List.not_exists_mem_nil]
theorem List.not_bex_nil {α : Type u} (p : αProp) :
¬∃ (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 : αProp} {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 l₂l₁.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 : ασσ × β) :
List ασσ × List β

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

Equations
Instances For
    @[simp]
    theorem List.length_mapAccumr {α : Type u} {β : Type v} {σ : Type w₂} (f : ασσ × 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 : αβσσ × φ) :
    List αList βσσ × List φ

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

    Equations
    Instances For
      @[simp]
      theorem List.length_mapAccumr₂ {α : Type u} {β : Type v} {φ : Type w₁} {σ : Type w₂} (f : αβσσ × 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.