mathlib documentation

data.list.basic

Basic properties of lists #

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@[protected, instance]
def list.unique_of_is_empty {α : Type u} [is_empty α] :
unique (list α)

There is only one list of an empty type

Equations
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@[protected, instance]
def list.nil.is_left_id {α : Type u} :
is_left_id (list α) has_append.append list.nil
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@[protected, instance]
def list.nil.is_right_id {α : Type u} :
is_right_id (list α) has_append.append list.nil
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@[protected, instance]
def list.has_append.append.is_associative {α : Type u} :
is_associative (list α) has_append.append
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theorem list.cons_ne_nil {α : Type u} (a : α) (l : list α) :
a :: l ≠ list.nil
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theorem list.cons_ne_self {α : Type u} (a : α) (l : list α) :
a :: l ≠ l
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theorem list.head_eq_of_cons_eq {α : Type u} {h₁ h₂ : α} {t₁ t₂ : list α} :
h₁ :: t₁ = h₂ :: t₂ → h₁ = h₂
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theorem list.tail_eq_of_cons_eq {α : Type u} {h₁ h₂ : α} {t₁ t₂ : list α} :
h₁ :: t₁ = h₂ :: t₂ → t₁ = t₂
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@[simp]
theorem list.cons_injective {α : Type u} {a : α} :
function.injective (list.cons a)
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theorem list.cons_inj {α : Type u} (a : α) {l l' : list α} :
a :: l = a :: l' ↔ l = l'
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theorem list.exists_cons_of_ne_nil {α : Type u} {l : list α} (h : l ≠ list.nil) :
∃ (b : α) (L : list α), l = b :: L
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theorem list.set_of_mem_cons {α : Type u} (l : list α) (a : α) :
{x : α | x ∈ a :: l} = has_insert.insert a {x : α | x ∈ l}

mem #

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theorem list.mem_singleton_self {α : Type u} (a : α) :
a ∈ [a]
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theorem list.eq_of_mem_singleton {α : Type u} {a b : α} :
a ∈ [b] → a = b
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@[simp]
theorem list.mem_singleton {α : Type u} {a b : α} :
a ∈ [b] ↔ a = b
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theorem list.mem_of_mem_cons_of_mem {α : Type u} {a b : α} {l : list α} :
a ∈ b :: l → b ∈ l → a ∈ l
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theorem decidable.list.eq_or_ne_mem_of_mem {α : Type u} [decidable_eq α] {a b : α} {l : list α} (h : a ∈ b :: l) :
a = b ∨ a ≠ b ∧ a ∈ l
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theorem list.eq_or_ne_mem_of_mem {α : Type u} {a b : α} {l : list α} :
a ∈ b :: l → a = b ∨ a ≠ b ∧ a ∈ l
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theorem list.not_mem_append {α : Type u} {a : α} {s t : list α} (h₁ : a ∉ s) (h₂ : a ∉ t) :
a ∉ s ++ t
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theorem list.ne_nil_of_mem {α : Type u} {a : α} {l : list α} (h : a ∈ l) :
l ≠ list.nil
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theorem list.mem_split {α : Type u} {a : α} {l : list α} (h : a ∈ l) :
∃ (s t : list α), l = s ++ a :: t
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theorem list.mem_of_ne_of_mem {α : Type u} {a y : α} {l : list α} (h₁ : a ≠ y) (h₂ : a ∈ y :: l) :
a ∈ l
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theorem list.ne_of_not_mem_cons {α : Type u} {a b : α} {l : list α} :
a ∉ b :: l → a ≠ b
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theorem list.not_mem_of_not_mem_cons {α : Type u} {a b : α} {l : list α} :
a ∉ b :: l → a ∉ l
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theorem list.not_mem_cons_of_ne_of_not_mem {α : Type u} {a y : α} {l : list α} :
a ≠ y → a ∉ l → a ∉ y :: l
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theorem list.ne_and_not_mem_of_not_mem_cons {α : Type u} {a y : α} {l : list α} :
a ∉ y :: l → a ≠ y ∧ a ∉ l
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@[simp]
theorem list.mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {l : list α} :
b ∈ list.map f l ↔ ∃ (a : α), a ∈ l ∧ f a = b
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theorem list.exists_of_mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {l : list α} :
b ∈ list.map f l → (∃ (a : α), a ∈ l ∧ f a = b)

Alias of the forward direction of list.mem_map.

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theorem list.mem_map_of_mem {α : Type u} {β : Type v} (f : α → β) {a : α} {l : list α} (h : a ∈ l) :
f a ∈ list.map f l
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theorem list.mem_map_of_injective {α : Type u} {β : Type v} {f : α → β} (H : function.injective f) {a : α} {l : list α} :
f a ∈ list.map f l ↔ a ∈ l
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@[simp]
theorem function.involutive.exists_mem_and_apply_eq_iff {α : Type u} {f : α → α} (hf : function.involutive f) (x : α) (l : list α) :
(∃ (y : α), y ∈ l ∧ f y = x) ↔ f x ∈ l
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theorem list.mem_map_of_involutive {α : Type u} {f : α → α} (hf : function.involutive f) {a : α} {l : list α} :
a ∈ list.map f l ↔ f a ∈ l
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theorem list.forall_mem_map_iff {α : Type u} {β : Type v} {f : α → β} {l : list α} {P : β → Prop} :
(∀ (i : β), i ∈ list.map f l → P i) ↔ ∀ (j : α), j ∈ l → P (f j)
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@[simp]
theorem list.map_eq_nil {α : Type u} {β : Type v} {f : α → β} {l : list α} :
list.map f l = list.nil ↔ l = list.nil
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@[simp]
theorem list.mem_join {α : Type u} {a : α} {L : list (list α)} :
a ∈ L.join ↔ ∃ (l : list α), l ∈ L ∧ a ∈ l
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theorem list.exists_of_mem_join {α : Type u} {a : α} {L : list (list α)} :
a ∈ L.join → (∃ (l : list α), l ∈ L ∧ a ∈ l)
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theorem list.mem_join_of_mem {α : Type u} {a : α} {L : list (list α)} {l : list α} (lL : l ∈ L) (al : a ∈ l) :
a ∈ L.join
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@[simp]
theorem list.mem_bind {α : Type u} {β : Type v} {b : β} {l : list α} {f : α → list β} :
b ∈ l.bind f ↔ ∃ (a : α) (H : a ∈ l), b ∈ f a
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theorem list.exists_of_mem_bind {α : Type u} {β : Type v} {b : β} {l : list α} {f : α → list β} :
b ∈ l.bind f → (∃ (a : α) (H : a ∈ l), b ∈ f a)
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theorem list.mem_bind_of_mem {α : Type u} {β : Type v} {b : β} {l : list α} {f : α → list β} {a : α} (al : a ∈ l) (h : b ∈ f a) :
b ∈ l.bind f
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theorem list.bind_map {α : Type u} {β : Type v} {γ : Type w} {g : α → list β} {f : β → γ} (l : list α) :
list.map f (l.bind g) = l.bind (λ (a : α), list.map f (g a))
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theorem list.map_bind {α : Type u} {β : Type v} {γ : Type w} (g : β → list γ) (f : α → β) (l : list α) :
(list.map f l).bind g = l.bind (λ (a : α), g (f a))
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theorem list.range_map {α : Type u} {β : Type v} (f : α → β) :
set.range (list.map f) = {l : list β | ∀ (x : β), x ∈ l → x ∈ set.range f}
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theorem list.range_map_coe {α : Type u} (s : set α) :
set.range (list.map coe) = {l : list α | ∀ (x : α), x ∈ l → x ∈ s}
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@[protected, instance]
def list.can_lift {α : Type u} {β : Type v} (c : out_param (β → α)) (p : out_param (α → Prop)) [can_lift α β c p] :
can_lift (list α) (list β) (list.map c) (λ (l : list α), ∀ (x : 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

length #

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theorem list.length_eq_zero {α : Type u} {l : list α} :
l.length = 0 ↔ l = list.nil
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@[simp]
theorem list.length_singleton {α : Type u} (a : α) :
[a].length = 1
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theorem list.length_pos_of_mem {α : Type u} {a : α} {l : list α} :
a ∈ l → 0 < l.length
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theorem list.exists_mem_of_length_pos {α : Type u} {l : list α} :
0 < l.length → (∃ (a : α), a ∈ l)
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theorem list.length_pos_iff_exists_mem {α : Type u} {l : list α} :
0 < l.length ↔ ∃ (a : α), a ∈ l
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theorem list.ne_nil_of_length_pos {α : Type u} {l : list α} :
0 < l.length → l ≠ list.nil
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theorem list.length_pos_of_ne_nil {α : Type u} {l : list α} :
l ≠ list.nil → 0 < l.length
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theorem list.length_pos_iff_ne_nil {α : Type u} {l : list α} :
0 < l.length ↔ l ≠ list.nil
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theorem list.exists_mem_of_ne_nil {α : Type u} (l : list α) (h : l ≠ list.nil) :
∃ (x : α), x ∈ l
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theorem list.length_eq_one {α : Type u} {l : list α} :
l.length = 1 ↔ ∃ (a : α), l = [a]
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theorem list.exists_of_length_succ {α : Type u} {n : ℕ} (l : list α) :
l.length = n + 1 → (∃ (h : α) (t : list α), l = h :: t)
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@[simp]
theorem list.length_injective_iff {α : Type u} :
function.injective list.length ↔ subsingleton α
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@[simp]
theorem list.length_injective {α : Type u} [subsingleton α] :
function.injective list.length
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theorem list.length_eq_two {α : Type u} {l : list α} :
l.length = 2 ↔ ∃ (a b : α), l = [a, b]
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theorem list.length_eq_three {α : Type u} {l : list α} :
l.length = 3 ↔ ∃ (a b c : α), l = [a, b, c]

set-theoretic notation of lists #

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theorem list.empty_eq {α : Type u} :
∅ = list.nil
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theorem list.singleton_eq {α : Type u} (x : α) :
{x} = [x]
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theorem list.insert_neg {α : Type u} [decidable_eq α] {x : α} {l : list α} (h : x ∉ l) :
has_insert.insert x l = x :: l
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theorem list.insert_pos {α : Type u} [decidable_eq α] {x : α} {l : list α} (h : x ∈ l) :
has_insert.insert x l = l
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theorem list.doubleton_eq {α : Type u} [decidable_eq α] {x y : α} (h : x ≠ y) :
{x, y} = [x, y]

bounded quantifiers over lists #

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theorem list.forall_mem_nil {α : Type u} (p : α → Prop) (x : α) (H : x ∈ list.nil) :
p x
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theorem list.forall_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : list α} :
(∀ (x : α), x ∈ a :: l → p x) ↔ p a ∧ ∀ (x : α), x ∈ l → p x
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theorem list.forall_mem_of_forall_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : list α} (h : ∀ (x : α), x ∈ a :: l → p x) (x : α) (H : x ∈ l) :
p x
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theorem list.forall_mem_singleton {α : Type u} {p : α → Prop} {a : α} :
(∀ (x : α), x ∈ [a] → p x) ↔ p a
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theorem list.forall_mem_append {α : Type u} {p : α → Prop} {l₁ l₂ : list α} :
(∀ (x : α), x ∈ l₁ ++ l₂ → p x) ↔ (∀ (x : α), x ∈ l₁ → p x) ∧ ∀ (x : α), x ∈ l₂ → p x
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theorem list.not_exists_mem_nil {α : Type u} (p : α → Prop) :
¬∃ (x : α) (H : x ∈ list.nil), p x
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theorem list.exists_mem_cons_of {α : Type u} {p : α → Prop} {a : α} (l : list α) (h : p a) :
∃ (x : α) (H : x ∈ a :: l), p x
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theorem list.exists_mem_cons_of_exists {α : Type u} {p : α → Prop} {a : α} {l : list α} (h : ∃ (x : α) (H : x ∈ l), p x) :
∃ (x : α) (H : x ∈ a :: l), p x
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theorem list.or_exists_of_exists_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : list α} (h : ∃ (x : α) (H : x ∈ a :: l), p x) :
p a ∨ ∃ (x : α) (H : x ∈ l), p x
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theorem list.exists_mem_cons_iff {α : Type u} (p : α → Prop) (a : α) (l : list α) :
(∃ (x : α) (H : x ∈ a :: l), p x) ↔ p a ∨ ∃ (x : α) (H : x ∈ l), p x

list subset #

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theorem list.subset_def {α : Type u} {l₁ l₂ : list α} :
l₁ ⊆ l₂ ↔ ∀ ⦃a : α⦄, a ∈ l₁ → a ∈ l₂
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theorem list.subset_append_of_subset_left {α : Type u} (l l₁ l₂ : list α) :
l ⊆ l₁ → l ⊆ l₁ ++ l₂
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theorem list.subset_append_of_subset_right {α : Type u} (l l₁ l₂ : list α) :
l ⊆ l₂ → l ⊆ l₁ ++ l₂
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@[simp]
theorem list.cons_subset {α : Type u} {a : α} {l m : list α} :
a :: l ⊆ m ↔ a ∈ m ∧ l ⊆ m
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theorem list.cons_subset_of_subset_of_mem {α : Type u} {a : α} {l m : list α} (ainm : a ∈ m) (lsubm : l ⊆ m) :
a :: l ⊆ m
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theorem list.append_subset_of_subset_of_subset {α : Type u} {l₁ l₂ l : list α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) :
l₁ ++ l₂ ⊆ l
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@[simp]
theorem list.append_subset_iff {α : Type u} {l₁ l₂ l : list α} :
l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l
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theorem list.eq_nil_of_subset_nil {α : Type u} {l : list α} :
l ⊆ list.nil → l = list.nil
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theorem list.eq_nil_iff_forall_not_mem {α : Type u} {l : list α} :
l = list.nil ↔ ∀ (a : α), a ∉ l
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theorem list.map_subset {α : Type u} {β : Type v} {l₁ l₂ : list α} (f : α → β) (H : l₁ ⊆ l₂) :
list.map f l₁ ⊆ list.map f l₂
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theorem list.map_subset_iff {α : Type u} {β : Type v} {l₁ l₂ : list α} (f : α → β) (h : function.injective f) :
list.map f l₁ ⊆ list.map f l₂ ↔ l₁ ⊆ l₂

append #

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theorem list.append_eq_has_append {α : Type u} {L₁ L₂ : list α} :
L₁.append L₂ = L₁ ++ L₂
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@[simp]
theorem list.singleton_append {α : Type u} {x : α} {l : list α} :
[x] ++ l = x :: l
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theorem list.append_ne_nil_of_ne_nil_left {α : Type u} (s t : list α) :
s ≠ list.nil → s ++ t ≠ list.nil
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theorem list.append_ne_nil_of_ne_nil_right {α : Type u} (s t : list α) :
t ≠ list.nil → s ++ t ≠ list.nil
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@[simp]
theorem list.append_eq_nil {α : Type u} {p q : list α} :
p ++ q = list.nil ↔ p = list.nil ∧ q = list.nil
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@[simp]
theorem list.nil_eq_append_iff {α : Type u} {a b : list α} :
list.nil = a ++ b ↔ a = list.nil ∧ b = list.nil
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theorem list.append_eq_cons_iff {α : Type u} {a b c : list α} {x : α} :
a ++ b = x :: c ↔ a = list.nil ∧ b = x :: c ∨ ∃ (a' : list α), a = x :: a' ∧ c = a' ++ b
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theorem list.cons_eq_append_iff {α : Type u} {a b c : list α} {x : α} :
x :: c = a ++ b ↔ a = list.nil ∧ b = x :: c ∨ ∃ (a' : list α), a = x :: a' ∧ c = a' ++ b
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theorem list.append_eq_append_iff {α : Type u} {a b c d : list α} :
a ++ b = c ++ d ↔ (∃ (a' : list α), c = a ++ a' ∧ b = a' ++ d) ∨ ∃ (c' : list α), a = c ++ c' ∧ d = c' ++ b
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@[simp]
theorem list.take_append_drop {α : Type u} (n : ℕ) (l : list α) :
list.take n l ++ list.drop n l = l
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theorem list.append_inj {α : Type u} {s₁ s₂ t₁ t₂ : list α} :
s₁ ++ t₁ = s₂ ++ t₂ → s₁.length = s₂.length → s₁ = s₂ ∧ t₁ = t₂
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theorem list.append_inj_right {α : Type u} {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : s₁.length = s₂.length) :
t₁ = t₂
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theorem list.append_inj_left {α : Type u} {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : s₁.length = s₂.length) :
s₁ = s₂
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theorem list.append_inj' {α : Type u} {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : t₁.length = t₂.length) :
s₁ = s₂ ∧ t₁ = t₂
source
theorem list.append_inj_right' {α : Type u} {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : t₁.length = t₂.length) :
t₁ = t₂
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theorem list.append_inj_left' {α : Type u} {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : t₁.length = t₂.length) :
s₁ = s₂
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theorem list.append_left_cancel {α : Type u} {s t₁ t₂ : list α} (h : s ++ t₁ = s ++ t₂) :
t₁ = t₂
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theorem list.append_right_cancel {α : Type u} {s₁ s₂ t : list α} (h : s₁ ++ t = s₂ ++ t) :
s₁ = s₂
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theorem list.append_right_injective {α : Type u} (s : list α) :
function.injective (λ (t : list α), s ++ t)
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theorem list.append_right_inj {α : Type u} {t₁ t₂ : list α} (s : list α) :
s ++ t₁ = s ++ t₂ ↔ t₁ = t₂
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theorem list.append_left_injective {α : Type u} (t : list α) :
function.injective (λ (s : list α), s ++ t)
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theorem list.append_left_inj {α : Type u} {s₁ s₂ : list α} (t : list α) :
s₁ ++ t = s₂ ++ t ↔ s₁ = s₂
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theorem list.map_eq_append_split {α : Type u} {β : Type v} {f : α → β} {l : list α} {s₁ s₂ : list β} (h : list.map f l = s₁ ++ s₂) :
∃ (l₁ l₂ : list α), l = l₁ ++ l₂ ∧ list.map f l₁ = s₁ ∧ list.map f l₂ = s₂

repeat #

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@[simp]
theorem list.repeat_succ {α : Type u} (a : α) (n : ℕ) :
list.repeat a (n + 1) = a :: list.repeat a n
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theorem list.mem_repeat {α : Type u} {a b : α} {n : ℕ} :
b ∈ list.repeat a n ↔ n ≠ 0 ∧ b = a
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theorem list.eq_of_mem_repeat {α : Type u} {a b : α} {n : ℕ} (h : b ∈ list.repeat a n) :
b = a
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theorem list.eq_repeat_of_mem {α : Type u} {a : α} {l : list α} :
(∀ (b : α), b ∈ l → b = a) → l = list.repeat a l.length
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theorem list.eq_repeat' {α : Type u} {a : α} {l : list α} :
l = list.repeat a l.length ↔ ∀ (b : α), b ∈ l → b = a
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theorem list.eq_repeat {α : Type u} {a : α} {n : ℕ} {l : list α} :
l = list.repeat a n ↔ l.length = n ∧ ∀ (b : α), b ∈ l → b = a
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theorem list.repeat_add {α : Type u} (a : α) (m n : ℕ) :
list.repeat a (m + n) = list.repeat a m ++ list.repeat a n
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theorem list.repeat_subset_singleton {α : Type u} (a : α) (n : ℕ) :
list.repeat a n ⊆ [a]
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theorem list.subset_singleton_iff {α : Type u} {a : α} (L : list α) :
L ⊆ [a] ↔ ∃ (n : ℕ), L = list.repeat a n
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@[simp]
theorem list.map_const {α : Type u} {β : Type v} (l : list α) (b : β) :
list.map (function.const α b) l = list.repeat b l.length
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theorem list.eq_of_mem_map_const {α : Type u} {β : Type v} {b₁ b₂ : β} {l : list α} (h : b₁ ∈ list.map (function.const α b₂) l) :
b₁ = b₂
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@[simp]
theorem list.map_repeat {α : Type u} {β : Type v} (f : α → β) (a : α) (n : ℕ) :
list.map f (list.repeat a n) = list.repeat (f a) n
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@[simp]
theorem list.tail_repeat {α : Type u} (a : α) (n : ℕ) :
(list.repeat a n).tail = list.repeat a n.pred
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@[simp]
theorem list.join_repeat_nil {α : Type u} (n : ℕ) :
(list.repeat list.nil n).join = list.nil
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theorem list.repeat_left_injective {α : Type u} {n : ℕ} (hn : n ≠ 0) :
function.injective (λ (a : α), list.repeat a n)
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theorem list.repeat_left_inj {α : Type u} {a b : α} {n : ℕ} (hn : n ≠ 0) :
list.repeat a n = list.repeat b n ↔ a = b
source
@[simp]
theorem list.repeat_left_inj' {α : Type u} {a b : α} {n : ℕ} :
list.repeat a n = list.repeat b n ↔ n = 0 ∨ a = b
source
theorem list.repeat_right_injective {α : Type u} (a : α) :
function.injective (list.repeat a)
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@[simp]
theorem list.repeat_right_inj {α : Type u} {a : α} {n m : ℕ} :
list.repeat a n = list.repeat a m ↔ n = m

pure #

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@[simp]
theorem list.mem_pure {α : Type u_1} (x y : α) :
x ∈ has_pure.pure y ↔ x = y

bind #

source
@[simp]
theorem list.bind_eq_bind {α β : Type u_1} (f : α → list β) (l : list α) :
l >>= f = l.bind f
source
theorem list.bind_append {α : Type u} {β : Type v} (f : α → list β) (l₁ l₂ : list α) :
(l₁ ++ l₂).bind f = l₁.bind f ++ l₂.bind f
source
@[simp]
theorem list.bind_singleton {α : Type u} {β : Type v} (f : α → list β) (x : α) :
[x].bind f = f x
source
@[simp]
theorem list.bind_singleton' {α : Type u} (l : list α) :
l.bind (λ (x : α), [x]) = l
source
theorem list.map_eq_bind {α : Type u_1} {β : Type u_2} (f : α → β) (l : list α) :
list.map f l = l.bind (λ (x : α), [f x])
source
theorem list.bind_assoc {γ : Type w} {α : Type u_1} {β : Type u_2} (l : list α) (f : α → list β) (g : β → list γ) :
(l.bind f).bind g = l.bind (λ (x : α), (f x).bind g)

concat #

source
theorem list.concat_nil {α : Type u} (a : α) :
list.nil.concat a = [a]
source
theorem list.concat_cons {α : Type u} (a b : α) (l : list α) :
(a :: l).concat b = a :: l.concat b
source
@[simp]
theorem list.concat_eq_append {α : Type u} (a : α) (l : list α) :
l.concat a = l ++ [a]
source
theorem list.init_eq_of_concat_eq {α : Type u} {a : α} {l₁ l₂ : list α} :
l₁.concat a = l₂.concat a → l₁ = l₂
source
theorem list.last_eq_of_concat_eq {α : Type u} {a b : α} {l : list α} :
l.concat a = l.concat b → a = b
source
theorem list.concat_ne_nil {α : Type u} (a : α) (l : list α) :
l.concat a ≠ list.nil
source
theorem list.concat_append {α : Type u} (a : α) (l₁ l₂ : list α) :
l₁.concat a ++ l₂ = l₁ ++ a :: l₂
source
theorem list.length_concat {α : Type u} (a : α) (l : list α) :
(l.concat a).length = l.length.succ
source
theorem list.append_concat {α : Type u} (a : α) (l₁ l₂ : list α) :
l₁ ++ l₂.concat a = (l₁ ++ l₂).concat a

reverse #

source
@[simp]
theorem list.reverse_nil {α : Type u} :
list.nil.reverse = list.nil
source
@[simp]
theorem list.reverse_cons {α : Type u} (a : α) (l : list α) :
(a :: l).reverse = l.reverse ++ [a]
source
theorem list.reverse_core_eq {α : Type u} (l₁ l₂ : list α) :
l₁.reverse_core l₂ = l₁.reverse ++ l₂
source
theorem list.reverse_cons' {α : Type u} (a : α) (l : list α) :
(a :: l).reverse = l.reverse.concat a
source
@[simp]
theorem list.reverse_singleton {α : Type u} (a : α) :
[a].reverse = [a]
source
@[simp]
theorem list.reverse_append {α : Type u} (s t : list α) :
(s ++ t).reverse = t.reverse ++ s.reverse
source
theorem list.reverse_concat {α : Type u} (l : list α) (a : α) :
(l.concat a).reverse = a :: l.reverse
source
@[simp]
theorem list.reverse_reverse {α : Type u} (l : list α) :
l.reverse.reverse = l
source
@[simp]
theorem list.reverse_involutive {α : Type u} :
function.involutive list.reverse
source
@[simp]
theorem list.reverse_injective {α : Type u} :
function.injective list.reverse
source
theorem list.reverse_surjective {α : Type u} :
function.surjective list.reverse
source
theorem list.reverse_bijective {α : Type u} :
function.bijective list.reverse
source
@[simp]
theorem list.reverse_inj {α : Type u} {l₁ l₂ : list α} :
l₁.reverse = l₂.reverse ↔ l₁ = l₂
source
theorem list.reverse_eq_iff {α : Type u} {l l' : list α} :
l.reverse = l' ↔ l = l'.reverse
source
@[simp]
theorem list.reverse_eq_nil {α : Type u} {l : list α} :
l.reverse = list.nil ↔ l = list.nil
source
theorem list.concat_eq_reverse_cons {α : Type u} (a : α) (l : list α) :
l.concat a = (a :: l.reverse).reverse
source
@[simp]
theorem list.length_reverse {α : Type u} (l : list α) :
l.reverse.length = l.length
source
@[simp]
theorem list.map_reverse {α : Type u} {β : Type v} (f : α → β) (l : list α) :
list.map f l.reverse = (list.map f l).reverse
source
theorem list.map_reverse_core {α : Type u} {β : Type v} (f : α → β) (l₁ l₂ : list α) :
list.map f (l₁.reverse_core l₂) = (list.map f l₁).reverse_core (list.map f l₂)
source
@[simp]
theorem list.mem_reverse {α : Type u} {a : α} {l : list α} :
a ∈ l.reverse ↔ a ∈ l
source
@[simp]
theorem list.reverse_repeat {α : Type u} (a : α) (n : ℕ) :
(list.repeat a n).reverse = list.repeat a n

empty #

source
theorem list.empty_iff_eq_nil {α : Type u} {l : list α} :
↥(l.empty) ↔ l = list.nil

init #

source
@[simp]
theorem list.length_init {α : Type u} (l : list α) :
l.init.length = l.length - 1

last #

source
@[simp]
theorem list.last_cons {α : Type u} {a : α} {l : list α} (h : l ≠ list.nil) :
(a :: l).last _ = l.last h
source
@[simp]
theorem list.last_append_singleton {α : Type u} {a : α} (l : list α) :
(l ++ [a]).last _ = a
source
theorem list.last_append {α : Type u} (l₁ l₂ : list α) (h : l₂ ≠ list.nil) :
(l₁ ++ l₂).last _ = l₂.last h
source
theorem list.last_concat {α : Type u} {a : α} (l : list α) :
(l.concat a).last _ = a
source
@[simp]
theorem list.last_singleton {α : Type u} (a : α) :
[a].last _ = a
source
@[simp]
theorem list.last_cons_cons {α : Type u} (a₁ a₂ : α) (l : list α) :
(a₁ :: a₂ :: l).last _ = (a₂ :: l).last _
source
theorem list.init_append_last {α : Type u} {l : list α} (h : l ≠ list.nil) :
l.init ++ [l.last h] = l
source
theorem list.last_congr {α : Type u} {l₁ l₂ : list α} (h₁ : l₁ ≠ list.nil) (h₂ : l₂ ≠ list.nil) (h₃ : l₁ = l₂) :
l₁.last h₁ = l₂.last h₂
source
theorem list.last_mem {α : Type u} {l : list α} (h : l ≠ list.nil) :
l.last h ∈ l
source
theorem list.last_repeat_succ (a m : ℕ) :
(list.repeat a m.succ).last _ = a

last' #

source
@[simp]
theorem list.last'_is_none {α : Type u} {l : list α} :
↥(l.last'.is_none) ↔ l = list.nil
source
@[simp]
theorem list.last'_is_some {α : Type u} {l : list α} :
↥(l.last'.is_some) ↔ l ≠ list.nil
source
theorem list.mem_last'_eq_last {α : Type u} {l : list α} {x : α} :
x ∈ l.last' → (∃ (h : l ≠ list.nil), x = l.last h)
source
theorem list.last'_eq_last_of_ne_nil {α : Type u} {l : list α} (h : l ≠ list.nil) :
l.last' = option.some (l.last h)
source
theorem list.mem_last'_cons {α : Type u} {x y : α} {l : list α} (h : x ∈ l.last') :
x ∈ (y :: l).last'
source
theorem list.mem_of_mem_last' {α : Type u} {l : list α} {a : α} (ha : a ∈ l.last') :
a ∈ l
source
theorem list.init_append_last' {α : Type u} {l : list α} (a : α) (H : a ∈ l.last') :
l.init ++ [a] = l
source
theorem list.ilast_eq_last' {α : Type u} [inhabited α] (l : list α) :
l.ilast = l.last'.iget
source
@[simp]
theorem list.last'_append_cons {α : Type u} (l₁ : list α) (a : α) (l₂ : list α) :
(l₁ ++ a :: l₂).last' = (a :: l₂).last'
source
@[simp]
theorem list.last'_cons_cons {α : Type u} (x y : α) (l : list α) :
(x :: y :: l).last' = (y :: l).last'
source
theorem list.last'_append_of_ne_nil {α : Type u} (l₁ : list α) {l₂ : list α} (hl₂ : l₂ ≠ list.nil) :
(l₁ ++ l₂).last' = l₂.last'
source
theorem list.last'_append {α : Type u} {l₁ l₂ : list α} {x : α} (h : x ∈ l₂.last') :
x ∈ (l₁ ++ l₂).last'

head(') and tail #

source
theorem list.head_eq_head' {α : Type u} [inhabited α] (l : list α) :
l.head = l.head'.iget
source
theorem list.surjective_head {α : Type u} [inhabited α] :
function.surjective list.head
source
theorem list.surjective_head' {α : Type u} :
function.surjective list.head'
source
theorem list.surjective_tail {α : Type u} :
function.surjective list.tail
source
theorem list.eq_cons_of_mem_head' {α : Type u} {x : α} {l : list α} :
x ∈ l.head' → l = x :: l.tail
source
theorem list.mem_of_mem_head' {α : Type u} {x : α} {l : list α} (h : x ∈ l.head') :
x ∈ l
source
@[simp]
theorem list.head_cons {α : Type u} [inhabited α] (a : α) (l : list α) :
(a :: l).head = a
source
@[simp]
theorem list.tail_nil {α : Type u} :
list.nil.tail = list.nil
source
@[simp]
theorem list.tail_cons {α : Type u} (a : α) (l : list α) :
(a :: l).tail = l
source
@[simp]
theorem list.head_append {α : Type u} [inhabited α] (t : list α) {s : list α} (h : s ≠ list.nil) :
(s ++ t).head = s.head
source
theorem list.head'_append {α : Type u} {s t : list α} {x : α} (h : x ∈ s.head') :
x ∈ (s ++ t).head'
source
theorem list.head'_append_of_ne_nil {α : Type u} (l₁ : list α) {l₂ : list α} (hl₁ : l₁ ≠ list.nil) :
(l₁ ++ l₂).head' = l₁.head'
source
theorem list.tail_append_singleton_of_ne_nil {α : Type u} {a : α} {l : list α} (h : l ≠ list.nil) :
(l ++ [a]).tail = l.tail ++ [a]
source
theorem list.cons_head'_tail {α : Type u} {l : list α} {a : α} (h : a ∈ l.head') :
a :: l.tail = l
source
theorem list.head_mem_head' {α : Type u} [inhabited α] {l : list α} (h : l ≠ list.nil) :
l.head ∈ l.head'
source
theorem list.cons_head_tail {α : Type u} [inhabited α] {l : list α} (h : l ≠ list.nil) :
l.head :: l.tail = l
source
theorem list.head_mem_self {α : Type u} [inhabited α] {l : list α} (h : l ≠ list.nil) :
l.head ∈ l
source
@[simp]
theorem list.head'_map {α : Type u} {β : Type v} (f : α → β) (l : list α) :
(list.map f l).head' = option.map f l.head'
source
theorem list.tail_append_of_ne_nil {α : Type u} (l l' : list α) (h : l ≠ list.nil) :
(l ++ l').tail = l.tail ++ l'
source
@[simp]
theorem list.nth_le_tail {α : Type u} (l : list α) (i : ℕ) (h : i < l.tail.length) (h' : i + 1 < l.length := _) :
l.tail.nth_le i h = l.nth_le (i + 1) h'
source
theorem list.nth_le_cons_aux {α : Type u} {l : list α} {a : α} {n : ℕ} (hn : n ≠ 0) (h : n < (a :: l).length) :
n - 1 < l.length
source
theorem list.nth_le_cons {α : Type u} {l : list α} {a : α} {n : ℕ} (hl : n < (a :: l).length) :
(a :: l).nth_le n hl = dite (n = 0) (λ (hn : n = 0), a) (λ (hn : ¬n = 0), l.nth_le (n - 1) _)
source
@[simp]
theorem list.modify_head_modify_head {α : Type u} (l : list α) (f g : α → α) :
list.modify_head g (list.modify_head f l) = list.modify_head (g ∘ f) l

Induction from the right #

source
def list.reverse_rec_on {α : Type u} {C : list α → Sort u_1} (l : list α) (H0 : C list.nil) (H1 : Π (l : list α) (a : α), C l → C (l ++ [a])) :
C l

Induction principle from the right for lists: if a property holds for the empty list, and for l ++ [a] if it holds for l, then it holds for all lists. The principle is given for a Sort-valued predicate, i.e., it can also be used to construct data.

Equations
source
def list.bidirectional_rec {α : Type u} {C : list α → Sort u_1} (H0 : C list.nil) (H1 : Π (a : α), C [a]) (Hn : Π (a : α) (l : list α) (b : α), C l → C (a :: (l ++ [b]))) (l : list α) :
C l

Bidirectional induction principle for lists: if a property holds for the empty list, the singleton list, and a :: (l ++ [b]) from l, then it holds for all lists. This can be used to prove statements about palindromes. The principle is given for a Sort-valued predicate, i.e., it can also be used to construct data.

Equations
source
def list.bidirectional_rec_on {α : Type u} {C : list α → Sort u_1} (l : list α) (H0 : C list.nil) (H1 : Π (a : α), C [a]) (Hn : Π (a : α) (l : list α) (b : α), C l → C (a :: (l ++ [b]))) :
C l

Like bidirectional_rec, but with the list parameter placed first.

Equations

sublists #

source
@[simp]
theorem list.nil_sublist {α : Type u} (l : list α) :
list.nil <+ l
source
@[simp, refl]
theorem list.sublist.refl {α : Type u} (l : list α) :
l <+ l
source
@[trans]
theorem list.sublist.trans {α : Type u} {l₁ l₂ l₃ : list α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) :
l₁ <+ l₃
source
@[simp]
theorem list.sublist_cons {α : Type u} (a : α) (l : list α) :
l <+ a :: l
source
theorem list.sublist_of_cons_sublist {α : Type u} {a : α} {l₁ l₂ : list α} :
a :: l₁ <+ l₂ → l₁ <+ l₂
source
theorem list.sublist.cons_cons {α : Type u} {l₁ l₂ : list α} (a : α) (s : l₁ <+ l₂) :
a :: l₁ <+ a :: l₂
source
@[simp]
theorem list.sublist_append_left {α : Type u} (l₁ l₂ : list α) :
l₁ <+ l₁ ++ l₂
source
@[simp]
theorem list.sublist_append_right {α : Type u} (l₁ l₂ : list α) :
l₂ <+ l₁ ++ l₂
source
theorem list.sublist_cons_of_sublist {α : Type u} (a : α) {l₁ l₂ : list α} :
l₁ <+ l₂ → l₁ <+ a :: l₂
source
theorem list.sublist_append_of_sublist_left {α : Type u} {l l₁ l₂ : list α} (s : l <+ l₁) :
l <+ l₁ ++ l₂
source
theorem list.sublist_append_of_sublist_right {α : Type u} {l l₁ l₂ : list α} (s : l <+ l₂) :
l <+ l₁ ++ l₂
source
theorem list.sublist_of_cons_sublist_cons {α : Type u} {l₁ l₂ : list α} {a : α} :
a :: l₁ <+ a :: l₂ → l₁ <+ l₂
source
theorem list.cons_sublist_cons_iff {α : Type u} {l₁ l₂ : list α} {a : α} :
a :: l₁ <+ a :: l₂ ↔ l₁ <+ l₂
source
@[simp]
theorem list.append_sublist_append_left {α : Type u} {l₁ l₂ : list α} (l : list α) :
l ++ l₁ <+ l ++ l₂ ↔ l₁ <+ l₂
source
theorem list.sublist.append_right {α : Type u} {l₁ l₂ : list α} (h : l₁ <+ l₂) (l : list α) :
l₁ ++ l <+ l₂ ++ l
source
theorem list.sublist_or_mem_of_sublist {α : Type u} {l l₁ l₂ : list α} {a : α} (h : l <+ l₁ ++ a :: l₂) :
l <+ l₁ ++ l₂ ∨ a ∈ l
source
theorem list.sublist.reverse {α : Type u} {l₁ l₂ : list α} (h : l₁ <+ l₂) :
l₁.reverse <+ l₂.reverse
source
@[simp]
theorem list.reverse_sublist_iff {α : Type u} {l₁ l₂ : list α} :
l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂
source
@[simp]
theorem list.append_sublist_append_right {α : Type u} {l₁ l₂ : list α} (l : list α) :
l₁ ++ l <+ l₂ ++ l ↔ l₁ <+ l₂
source
theorem list.sublist.append {α : Type u} {l₁ l₂ r₁ r₂ : list α} (hl : l₁ <+ l₂) (hr : r₁ <+ r₂) :
l₁ ++ r₁ <+ l₂ ++ r₂
source
theorem list.sublist.subset {α : Type u} {l₁ l₂ : list α} :
l₁ <+ l₂ → l₁ ⊆ l₂
source
@[simp]
theorem list.singleton_sublist {α : Type u} {a : α} {l : list α} :
[a] <+ l ↔ a ∈ l
source
theorem list.eq_nil_of_sublist_nil {α : Type u} {l : list α} (s : l <+ list.nil) :
l = list.nil
source
@[simp]
theorem list.sublist_nil_iff_eq_nil {α : Type u} {l : list α} :
l <+ list.nil ↔ l = list.nil
source
@[simp]
theorem list.repeat_sublist_repeat {α : Type u} (a : α) {m n : ℕ} :
list.repeat a m <+ list.repeat a n ↔ m ≤ n
source
theorem list.eq_of_sublist_of_length_eq {α : Type u} {l₁ l₂ : list α} :
l₁ <+ l₂ → l₁.length = l₂.length → l₁ = l₂
source
theorem list.eq_of_sublist_of_length_le {α : Type u} {l₁ l₂ : list α} (s : l₁ <+ l₂) (h : l₂.length ≤ l₁.length) :
l₁ = l₂
source
theorem list.sublist.antisymm {α : Type u} {l₁ l₂ : list α} (s₁ : l₁ <+ l₂) (s₂ : l₂ <+ l₁) :
l₁ = l₂
source
@[protected, instance]
def list.decidable_sublist {α : Type u} [decidable_eq α] (l₁ l₂ : list α) :
decidable (l₁ <+ l₂)
Equations

index_of #

source
@[simp]
theorem list.index_of_nil {α : Type u} [decidable_eq α] (a : α) :
list.index_of a list.nil = 0
source
theorem list.index_of_cons {α : Type u} [decidable_eq α] (a b : α) (l : list α) :
list.index_of a (b :: l) = ite (a = b) 0 (list.index_of a l).succ
source
theorem list.index_of_cons_eq {α : Type u} [decidable_eq α] {a b : α} (l : list α) :
a = b → list.index_of a (b :: l) = 0
source
@[simp]
theorem list.index_of_cons_self {α : Type u} [decidable_eq α] (a : α) (l : list α) :
list.index_of a (a :: l) = 0
source
@[simp]
theorem list.index_of_cons_ne {α : Type u} [decidable_eq α] {a b : α} (l : list α) :
a ≠ b → list.index_of a (b :: l) = (list.index_of a l).succ
source
theorem list.index_of_eq_length {α : Type u} [decidable_eq α] {a : α} {l : list α} :
list.index_of a l = l.length ↔ a ∉ l
source
@[simp]
theorem list.index_of_of_not_mem {α : Type u} [decidable_eq α] {l : list α} {a : α} :
a ∉ l → list.index_of a l = l.length
source
theorem list.index_of_le_length {α : Type u} [decidable_eq α] {a : α} {l : list α} :
list.index_of a l ≤ l.length
source
theorem list.index_of_lt_length {α : Type u} [decidable_eq α] {a : α} {l : list α} :
list.index_of a l < l.length ↔ a ∈ l

nth element #

source
theorem list.nth_le_of_mem {α : Type u} {a : α} {l : list α} :
a ∈ l → (∃ (n : ℕ) (h : n < l.length), l.nth_le n h = a)
source
theorem list.nth_le_nth {α : Type u} {l : list α} {n : ℕ} (h : n < l.length) :
l.nth n = option.some (l.nth_le n h)
source
theorem list.nth_len_le {α : Type u} {l : list α} {n : ℕ} :
l.length ≤ n → l.nth n = option.none
source
@[simp]
theorem list.nth_length {α : Type u} (l : list α) :
l.nth l.length = option.none
source
theorem list.nth_eq_some {α : Type u} {l : list α} {n : ℕ} {a : α} :
l.nth n = option.some a ↔ ∃ (h : n < l.length), l.nth_le n h = a
source
@[simp]
theorem list.nth_eq_none_iff {α : Type u} {l : list α} {n : ℕ} :
l.nth n = option.none ↔ l.length ≤ n
source
theorem list.nth_of_mem {α : Type u} {a : α} {l : list α} (h : a ∈ l) :
∃ (n : ℕ), l.nth n = option.some a
source
theorem list.nth_le_mem {α : Type u} (l : list α) (n : ℕ) (h : n < l.length) :
l.nth_le n h ∈ l
source
theorem list.nth_mem {α : Type u} {l : list α} {n : ℕ} {a : α} (e : l.nth n = option.some a) :
a ∈ l
source
theorem list.mem_iff_nth_le {α : Type u} {a : α} {l : list α} :
a ∈ l ↔ ∃ (n : ℕ) (h : n < l.length), l.nth_le n h = a
source
theorem list.mem_iff_nth {α : Type u} {a : α} {l : list α} :
a ∈ l ↔ ∃ (n : ℕ), l.nth n = option.some a
source
theorem list.nth_zero {α : Type u} (l : list α) :
l.nth 0 = l.head'
source
theorem list.nth_injective {α : Type u} {xs : list α} {i j : ℕ} (h₀ : i < xs.length) (h₁ : xs.nodup) (h₂ : xs.nth i = xs.nth j) :
i = j
source
@[simp]
theorem list.nth_map {α : Type u} {β : Type v} (f : α → β) (l : list α) (n : ℕ) :
(list.map f l).nth n = option.map f (l.nth n)
source
theorem list.nth_le_map {α : Type u} {β : Type v} (f : α → β) {l : list α} {n : ℕ} (H1 : n < (list.map f l).length) (H2 : n < l.length) :
(list.map f l).nth_le n H1 = f (l.nth_le n H2)
source
theorem list.nth_le_map_rev {α : Type u} {β : Type v} (f : α → β) {l : list α} {n : ℕ} (H : n < l.length) :
f (l.nth_le n H) = (list.map f l).nth_le n _

A version of nth_le_map that can be used for rewriting.

source
@[simp]
theorem list.nth_le_map' {α : Type u} {β : Type v} (f : α → β) {l : list α} {n : ℕ} (H : n < (list.map f l).length) :
(list.map f l).nth_le n H = f (l.nth_le n _)
source
theorem list.nth_le_of_eq {α : Type u} {L L' : list α} (h : L = L') {i : ℕ} (hi : i < L.length) :
L.nth_le i hi = L'.nth_le i _

If one has nth_le L i hi in a formula and h : L = L', one can not rw h in the formula as hi gives i < L.length and not i < L'.length. The lemma nth_le_of_eq can be used to make such a rewrite, with rw (nth_le_of_eq h).

source
@[simp]
theorem list.nth_le_singleton {α : Type u} (a : α) {n : ℕ} (hn : n < 1) :
[a].nth_le n hn = a
source
theorem list.nth_le_zero {α : Type u} [inhabited α] {L : list α} (h : 0 < L.length) :
L.nth_le 0 h = L.head
source
theorem list.nth_le_append {α : Type u} {l₁ l₂ : list α} {n : ℕ} (hn₁ : n < (l₁ ++ l₂).length) (hn₂ : n < l₁.length) :
(l₁ ++ l₂).nth_le n hn₁ = l₁.nth_le n hn₂
source
theorem list.nth_le_append_right_aux {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h₁ : l₁.length ≤ n) (h₂ : n < (l₁ ++ l₂).length) :
n - l₁.length < l₂.length
source
theorem list.nth_le_append_right {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h₁ : l₁.length ≤ n) (h₂ : n < (l₁ ++ l₂).length) :
(l₁ ++ l₂).nth_le n h₂ = l₂.nth_le (n - l₁.length) _
source
@[simp]
theorem list.nth_le_repeat {α : Type u} (a : α) {n m : ℕ} (h : m < (list.repeat a n).length) :
(list.repeat a n).nth_le m h = a
source
theorem list.nth_append {α : Type u} {l₁ l₂ : list α} {n : ℕ} (hn : n < l₁.length) :
(l₁ ++ l₂).nth n = l₁.nth n
source
theorem list.nth_append_right {α : Type u} {l₁ l₂ : list α} {n : ℕ} (hn : l₁.length ≤ n) :
(l₁ ++ l₂).nth n = l₂.nth (n - l₁.length)
source
theorem list.last_eq_nth_le {α : Type u} (l : list α) (h : l ≠ list.nil) :
l.last h = l.nth_le (l.length - 1) _
source
@[simp]
theorem list.nth_concat_length {α : Type u} (l : list α) (a : α) :
(l ++ [a]).nth l.length = option.some a
source
theorem list.nth_le_cons_length {α : Type u} (x : α) (xs : list α) (n : ℕ) (h : n = xs.length) :
(x :: xs).nth_le n _ = (x :: xs).last _
source
@[ext]
theorem list.ext {α : Type u} {l₁ l₂ : list α} :
(∀ (n : ℕ), l₁.nth n = l₂.nth n) → l₁ = l₂
source
theorem list.ext_le {α : Type u} {l₁ l₂ : list α} (hl : l₁.length = l₂.length) (h : ∀ (n : ℕ) (h₁ : n < l₁.length) (h₂ : n < l₂.length), l₁.nth_le n h₁ = l₂.nth_le n h₂) :
l₁ = l₂
source
@[simp]
theorem list.index_of_nth_le {α : Type u} [decidable_eq α] {a : α} {l : list α} (h : list.index_of a l < l.length) :
l.nth_le (list.index_of a l) h = a
source
@[simp]
theorem list.index_of_nth {α : Type u} [decidable_eq α] {a : α} {l : list α} (h : a ∈ l) :
l.nth (list.index_of a l) = option.some a
source
theorem list.nth_le_reverse_aux1 {α : Type u} (l r : list α) (i : ℕ) (h1 : i + l.length < (l.reverse_core r).length) (h2 : i < r.length) :
(l.reverse_core r).nth_le (i + l.length) h1 = r.nth_le i h2
source
theorem list.index_of_inj {α : Type u} [decidable_eq α] {l : list α} {x y : α} (hx : x ∈ l) (hy : y ∈ l) :
list.index_of x l = list.index_of y l ↔ x = y
source
theorem list.nth_le_reverse_aux2 {α : Type u} (l r : list α) (i : ℕ) (h1 : l.length - 1 - i < (l.reverse_core r).length) (h2 : i < l.length) :
(l.reverse_core r).nth_le (l.length - 1 - i) h1 = l.nth_le i h2
source
@[simp]
theorem list.nth_le_reverse {α : Type u} (l : list α) (i : ℕ) (h1 : l.length - 1 - i < l.reverse.length) (h2 : i < l.length) :
l.reverse.nth_le (l.length - 1 - i) h1 = l.nth_le i h2
source
theorem list.nth_le_reverse' {α : Type u} (l : list α) (n : ℕ) (hn : n < l.reverse.length) (hn' : l.length - 1 - n < l.length) :
l.reverse.nth_le n hn = l.nth_le (l.length - 1 - n) hn'
source
theorem list.eq_cons_of_length_one {α : Type u} {l : list α} (h : l.length = 1) :
l = [l.nth_le 0 _]
source
theorem list.nth_le_eq_iff {α : Type u} {l : list α} {n : ℕ} {x : α} {h : n < l.length} :
l.nth_le n h = x ↔ l.nth n = option.some x
source
theorem list.some_nth_le_eq {α : Type u} {l : list α} {n : ℕ} {h : n < l.length} :
option.some (l.nth_le n h) = l.nth n
source
theorem list.modify_nth_tail_modify_nth_tail {α : Type u} {f g : list α → list α} (m n : ℕ) (l : list α) :
list.modify_nth_tail g (m + n) (list.modify_nth_tail f n l) = list.modify_nth_tail (λ (l : list α), list.modify_nth_tail g m (f l)) n l
source
theorem list.modify_nth_tail_modify_nth_tail_le {α : Type u} {f g : list α → list α} (m n : ℕ) (l : list α) (h : n ≤ m) :
list.modify_nth_tail g m (list.modify_nth_tail f n l) = list.modify_nth_tail (λ (l : list α), list.modify_nth_tail g (m - n) (f l)) n l
source
theorem list.modify_nth_tail_modify_nth_tail_same {α : Type u} {f g : list α → list α} (n : ℕ) (l : list α) :
list.modify_nth_tail g n (list.modify_nth_tail f n l) = list.modify_nth_tail (g ∘ f) n l
source
theorem list.modify_nth_tail_id {α : Type u} (n : ℕ) (l : list α) :
list.modify_nth_tail id n l = l
source
theorem list.remove_nth_eq_nth_tail {α : Type u} (n : ℕ) (l : list α) :
l.remove_nth n = list.modify_nth_tail list.tail n l
source
theorem list.update_nth_eq_modify_nth {α : Type u} (a : α) (n : ℕ) (l : list α) :
l.update_nth n a = list.modify_nth (λ (_x : α), a) n l
source
theorem list.modify_nth_eq_update_nth {α : Type u} (f : α → α) (n : ℕ) (l : list α) :
list.modify_nth f n l = ((λ (a : α), l.update_nth n (f a)) <$> l.nth n).get_or_else l
source
theorem list.nth_modify_nth {α : Type u} (f : α → α) (n : ℕ) (l : list α) (m : ℕ) :
(list.modify_nth f n l).nth m = (λ (a : α), ite (n = m) (f a) a) <$> l.nth m
source
theorem list.modify_nth_tail_length {α : Type u} (f : list α → list α) (H : ∀ (l : list α), (f l).length = l.length) (n : ℕ) (l : list α) :
(list.modify_nth_tail f n l).length = l.length
source
@[simp]
theorem list.modify_nth_length {α : Type u} (f : α → α) (n : ℕ) (l : list α) :
(list.modify_nth f n l).length = l.length
source
@[simp]
theorem list.update_nth_length {α : Type u} (l : list α) (n : ℕ) (a : α) :
(l.update_nth n a).length = l.length
source
@[simp]
theorem list.nth_modify_nth_eq {α : Type u} (f : α → α) (n : ℕ) (l : list α) :
(list.modify_nth f n l).nth n = f <$> l.nth n
source
@[simp]
theorem list.nth_modify_nth_ne {α : Type u} (f : α → α) {m n : ℕ} (l : list α) (h : m ≠ n) :
(list.modify_nth f m l).nth n = l.nth n
source
theorem list.nth_update_nth_eq {α : Type u} (a : α) (n : ℕ) (l : list α) :
(l.update_nth n a).nth n = (λ (_x : α), a) <$> l.nth n
source
theorem list.nth_update_nth_of_lt {α : Type u} (a : α) {n : ℕ} {l : list α} (h : n < l.length) :
(l.update_nth n a).nth n = option.some a
source
theorem list.nth_update_nth_ne {α : Type u} (a : α) {m n : ℕ} (l : list α) (h : m ≠ n) :
(l.update_nth m a).nth n = l.nth n
source
@[simp]
theorem list.update_nth_nil {α : Type u} (n : ℕ) (a : α) :
list.nil.update_nth n a = list.nil
source
@[simp]
theorem list.update_nth_succ {α : Type u} (x : α) (xs : list α) (n : ℕ) (a : α) :
(x :: xs).update_nth n.succ a = x :: xs.update_nth n a
source
theorem list.update_nth_comm {α : Type u} (a b : α) {n m : ℕ} (l : list α) (h : n ≠ m) :
(l.update_nth n a).update_nth m b = (l.update_nth m b).update_nth n a
source
@[simp]
theorem list.nth_le_update_nth_eq {α : Type u} (l : list α) (i : ℕ) (a : α) (h : i < (l.update_nth i a).length) :
(l.update_nth i a).nth_le i h = a
source
@[simp]
theorem list.nth_le_update_nth_of_ne {α : Type u} {l : list α} {i j : ℕ} (h : i ≠ j) (a : α) (hj : j < (l.update_nth i a).length) :
(l.update_nth i a).nth_le j hj = l.nth_le j _
source
theorem list.mem_or_eq_of_mem_update_nth {α : Type u} {l : list α} {n : ℕ} {a b : α} (h : a ∈ l.update_nth n b) :
a ∈ l ∨ a = b
source
@[simp]
theorem list.insert_nth_zero {α : Type u} (s : list α) (x : α) :
list.insert_nth 0 x s = x :: s
source
@[simp]
theorem list.insert_nth_succ_nil {α : Type u} (n : ℕ) (a : α) :
list.insert_nth (n + 1) a list.nil = list.nil
source
@[simp]
theorem list.insert_nth_succ_cons {α : Type u} (s : list α) (hd x : α) (n : ℕ) :
list.insert_nth (n + 1) x (hd :: s) = hd :: list.insert_nth n x s
source
theorem list.length_insert_nth {α : Type u} {a : α} (n : ℕ) (as : list α) :
n ≤ as.length → (list.insert_nth n a as).length = as.length + 1
source
theorem list.remove_nth_insert_nth {α : Type u} {a : α} (n : ℕ) (l : list α) :
(list.insert_nth n a l).remove_nth n = l
source
theorem list.insert_nth_remove_nth_of_ge {α : Type u} {a : α} (n m : ℕ) (as : list α) :
n < as.length → n ≤ m → list.insert_nth m a (as.remove_nth n) = (list.insert_nth (m + 1) a as).remove_nth n
source
theorem list.insert_nth_remove_nth_of_le {α : Type u} {a : α} (n m : ℕ) (as : list α) :
n < as.length → m ≤ n → list.insert_nth m a (as.remove_nth n) = (list.insert_nth m a as).remove_nth (n + 1)
source
theorem list.insert_nth_comm {α : Type u} (a b : α) (i j : ℕ) (l : list α) (h : i ≤ j) (hj : j ≤ l.length) :
list.insert_nth (j + 1) b (list.insert_nth i a l) = list.insert_nth i a (list.insert_nth j b l)
source
theorem list.mem_insert_nth {α : Type u} {a b : α} {n : ℕ} {l : list α} (hi : n ≤ l.length) :
a ∈ list.insert_nth n b l ↔ a = b ∨ a ∈ l
source
theorem list.inj_on_insert_nth_index_of_not_mem {α : Type u} (l : list α) (x : α) (hx : x ∉ l) :
set.inj_on (λ (k : ℕ), list.insert_nth k x l) {n : ℕ | n ≤ l.length}
source
theorem list.insert_nth_of_length_lt {α : Type u} (l : list α) (x : α) (n : ℕ) (h : l.length < n) :
list.insert_nth n x l = l
source
@[simp]
theorem list.insert_nth_length_self {α : Type u} (l : list α) (x : α) :
list.insert_nth l.length x l = l ++ [x]
source
theorem list.length_le_length_insert_nth {α : Type u} (l : list α) (x : α) (n : ℕ) :
l.length ≤ (list.insert_nth n x l).length
source
theorem list.length_insert_nth_le_succ {α : Type u} (l : list α) (x : α) (n : ℕ) :
(list.insert_nth n x l).length ≤ l.length + 1
source
theorem list.nth_le_insert_nth_of_lt {α : Type u} (l : list α) (x : α) (n k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : k < (list.insert_nth n x l).length := _) :
(list.insert_nth n x l).nth_le k hk' = l.nth_le k hk
source
@[simp]
theorem list.nth_le_insert_nth_self {α : Type u} (l : list α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : n < (list.insert_nth n x l).length := _) :
(list.insert_nth n x l).nth_le n hn' = x
source
theorem list.nth_le_insert_nth_add_succ {α : Type u} (l : list α) (x : α) (n k : ℕ) (hk' : n + k < l.length) (hk : n + k + 1 < (list.insert_nth n x l).length := _) :
(list.insert_nth n x l).nth_le (n + k + 1) hk = l.nth_le (n + k) hk'
source
theorem list.insert_nth_injective {α : Type u} (n : ℕ) (x : α) :
function.injective (list.insert_nth n x)

map #

source
@[simp]
theorem list.map_nil {α : Type u} {β : Type v} (f : α → β) :
list.map f list.nil = list.nil
source
theorem list.map_eq_foldr {α : Type u} {β : Type v} (f : α → β) (l : list α) :
list.map f l = list.foldr (λ (a : α) (bs : list β), f a :: bs) list.nil l
source
theorem list.map_congr {α : Type u} {β : Type v} {f g : α → β} {l : list α} :
(∀ (x : α), x ∈ l → f x = g x) → list.map f l = list.map g l
source
theorem list.map_eq_map_iff {α : Type u} {β : Type v} {f g : α → β} {l : list α} :
list.map f l = list.map g l ↔ ∀ (x : α), x ∈ l → f x = g x
source
theorem list.map_concat {α : Type u} {β : Type v} (f : α → β) (a : α) (l : list α) :
list.map f (l.concat a) = (list.map f l).concat (f a)
source
@[simp]
theorem list.map_id'' {α : Type u} (l : list α) :
list.map (λ (x : α), x) l = l
source
theorem list.map_id' {α : Type u} {f : α → α} (h : ∀ (x : α), f x = x) (l : list α) :
list.map f l = l
source
theorem list.eq_nil_of_map_eq_nil {α : Type u} {β : Type v} {f : α → β} {l : list α} (h : list.map f l = list.nil) :
l = list.nil
source
@[simp]
theorem list.map_join {α : Type u} {β : Type v} (f : α → β) (L : list (list α)) :
list.map f L.join = (list.map (list.map f) L).join
source
theorem list.bind_ret_eq_map {α : Type u} {β : Type v} (f : α → β) (l : list α) :
l.bind (list.ret ∘ f) = list.map f l
source
theorem list.bind_congr {α : Type u} {β : Type v} {l : list α} {f g : α → list β} (h : ∀ (x : α), x ∈ l → f x = g x) :
l.bind f = l.bind g
source
@[simp]
theorem list.map_eq_map {α β : Type u_1} (f : α → β) (l : list α) :
f <$> l = list.map f l
source
@[simp]
theorem list.map_tail {α : Type u} {β : Type v} (f : α → β) (l : list α) :
list.map f l.tail = (list.map f l).tail
source
@[simp]
theorem list.map_injective_iff {α : Type u} {β : Type v} {f : α → β} :
function.injective (list.map f) ↔ function.injective f
source
theorem list.comp_map {α : Type u} {β : Type v} {γ : Type w} (h : β → γ) (g : α → β) (l : list α) :
list.map (h ∘ g) l = list.map h (list.map g l)

A single list.map of a composition of functions is equal to composing a list.map with another list.map, fully applied. This is the reverse direction of list.map_map.

source
@[simp]
theorem list.map_comp_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : α → β) :
list.map g ∘ list.map f = list.map (g ∘ f)

Composing a list.map with another list.map is equal to a single list.map of composed functions.

source
theorem list.map_filter_eq_foldr {α : Type u} {β : Type v} (f : α → β) (p : α → Prop) [decidable_pred p] (as : list α) :
list.map f (list.filter p as) = list.foldr (λ (a : α) (bs : list β), ite (p a) (f a :: bs) bs) list.nil as
source
theorem list.last_map {α : Type u} {β : Type v} (f : α → β) {l : list α} (hl : l ≠ list.nil) :
(list.map f l).last _ = f (l.last hl)

map₂ #

source
theorem list.nil_map₂ {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : list β) :
list.map₂ f list.nil l = list.nil
source
theorem list.map₂_nil {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : list α) :
list.map₂ f l list.nil = list.nil
source
@[simp]
theorem list.map₂_flip {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (as : list α) (bs : list β) :
list.map₂ (flip f) bs as = list.map₂ f as bs

take, drop #

source
@[simp]
theorem list.take_zero {α : Type u} (l : list α) :
list.take 0 l = list.nil
source
@[simp]
theorem list.take_nil {α : Type u} (n : ℕ) :
list.take n list.nil = list.nil
source
theorem list.take_cons {α : Type u} (n : ℕ) (a : α) (l : list α) :
list.take n.succ (a :: l) = a :: list.take n l
source
@[simp]
theorem list.take_length {α : Type u} (l : list α) :
list.take l.length l = l
source
theorem list.take_all_of_le {α : Type u} {n : ℕ} {l : list α} :
l.length ≤ n → list.take n l = l
source
@[simp]
theorem list.take_left {α : Type u} (l₁ l₂ : list α) :
list.take l₁.length (l₁ ++ l₂) = l₁
source
theorem list.take_left' {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h : l₁.length = n) :
list.take n (l₁ ++ l₂) = l₁
source
theorem list.take_take {α : Type u} (n m : ℕ) (l : list α) :
list.take n (list.take m l) = list.take (linear_order.min n m) l
source
theorem list.take_repeat {α : Type u} (a : α) (n m : ℕ) :
list.take n (list.repeat a m) = list.repeat a (linear_order.min n m)
source
theorem list.map_take {α : Type u_1} {β : Type u_2} (f : α → β) (L : list α) (i : ℕ) :
list.map f (list.take i L) = list.take i (list.map f L)
source
theorem list.take_append_eq_append_take {α : Type u} {l₁ l₂ : list α} {n : ℕ} :
list.take n (l₁ ++ l₂) = list.take n l₁ ++ list.take (n - l₁.length) l₂

Taking the first n elements in l₁ ++ l₂ is the same as appending the first n elements of l₁ to the first n - l₁.length elements of l₂.

source
theorem list.take_append_of_le_length {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h : n ≤ l₁.length) :
list.take n (l₁ ++ l₂) = list.take n l₁
source
theorem list.take_append {α : Type u} {l₁ l₂ : list α} (i : ℕ) :
list.take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ list.take i l₂

Taking the first l₁.length + i elements in l₁ ++ l₂ is the same as appending the first i elements of l₂ to l₁.

source
theorem list.nth_le_take {α : Type u} (L : list α) {i j : ℕ} (hi : i < L.length) (hj : i < j) :
L.nth_le i hi = (list.take j L).nth_le i _

The i-th element of a list coincides with the i-th element of any of its prefixes of length > i. Version designed to rewrite from the big list to the small list.

source
theorem list.nth_le_take' {α : Type u} (L : list α) {i j : ℕ} (hi : i < (list.take j L).length) :
(list.take j L).nth_le i hi = L.nth_le i _

The i-th element of a list coincides with the i-th element of any of its prefixes of length > i. Version designed to rewrite from the small list to the big list.

source
theorem list.nth_take {α : Type u} {l : list α} {n m : ℕ} (h : m < n) :
(list.take n l).nth m = l.nth m
source
@[simp]
theorem list.nth_take_of_succ {α : Type u} {l : list α} {n : ℕ} :
(list.take (n + 1) l).nth n = l.nth n
source
theorem list.take_succ {α : Type u} {l : list α} {n : ℕ} :
list.take (n + 1) l = list.take n l ++ (l.nth n).to_list
source
@[simp]
theorem list.take_eq_nil_iff {α : Type u} {l : list α} {k : ℕ} :
list.take k l = list.nil ↔ l = list.nil ∨ k = 0
source
theorem list.init_eq_take {α : Type u} (l : list α) :
l.init = list.take l.length.pred l
source
theorem list.init_take {α : Type u} {n : ℕ} {l : list α} (h : n < l.length) :
(list.take n l).init = list.take n.pred l
source
@[simp]
theorem list.init_cons_of_ne_nil {α : Type u_1} {x : α} {l : list α} (h : l ≠ list.nil) :
(x :: l).init = x :: l.init
source
@[simp]
theorem list.init_append_of_ne_nil {α : Type u_1} {l : list α} (l' : list α) (h : l ≠ list.nil) :
(l' ++ l).init = l' ++ l.init
source
@[simp]
theorem list.drop_eq_nil_of_le {α : Type u} {l : list α} {k : ℕ} (h : l.length ≤ k) :
list.drop k l = list.nil
source
theorem list.drop_eq_nil_iff_le {α : Type u} {l : list α} {k : ℕ} :
list.drop k l = list.nil ↔ l.length ≤ k
source
theorem list.tail_drop {α : Type u} (l : list α) (n : ℕ) :
(list.drop n l).tail = list.drop (n + 1) l
source
theorem list.cons_nth_le_drop_succ {α : Type u} {l : list α} {n : ℕ} (hn : n < l.length) :
l.nth_le n hn :: list.drop (n + 1) l = list.drop n l
source
theorem list.drop_nil {α : Type u} (n : ℕ) :
list.drop n list.nil = list.nil
source
@[simp]
theorem list.drop_one {α : Type u} (l : list α) :
list.drop 1 l = l.tail
source
theorem list.drop_add {α : Type u} (m n : ℕ) (l : list α) :
list.drop (m + n) l = list.drop m (list.drop n l)
source
@[simp]
theorem list.drop_left {α : Type u} (l₁ l₂ : list α) :
list.drop l₁.length (l₁ ++ l₂) = l₂
source
theorem list.drop_left' {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h : l₁.length = n) :
list.drop n (l₁ ++ l₂) = l₂
source
theorem list.drop_eq_nth_le_cons {α : Type u} {n : ℕ} {l : list α} (h : n < l.length) :
list.drop n l = l.nth_le n h :: list.drop (n + 1) l
source
@[simp]
theorem list.drop_length {α : Type u} (l : list α) :
list.drop l.length l = list.nil
source
theorem list.drop_append_eq_append_drop {α : Type u} {l₁ l₂ : list α} {n : ℕ} :
list.drop n (l₁ ++ l₂) = list.drop n l₁ ++ list.drop (n - l₁.length) l₂

Dropping the elements up to n in l₁ ++ l₂ is the same as dropping the elements up to n in l₁, dropping the elements up to n - l₁.length in l₂, and appending them.

source
theorem list.drop_append_of_le_length {α : Type u} {l₁ l₂ : list α} {n : ℕ} (h : n ≤ l₁.length) :
list.drop n (l₁ ++ l₂) = list.drop n l₁ ++ l₂
source
theorem list.drop_append {α : Type u} {l₁ l₂ : list α} (i : ℕ) :
list.drop (l₁.length + i) (l₁ ++ l₂) = list.drop i l₂

Dropping the elements up to l₁.length + i in l₁ + l₂ is the same as dropping the elements up to i in l₂.

source
theorem list.drop_sizeof_le {α : Type u} [has_sizeof α] (l : list α) (n : ℕ) :
(list.drop n l).sizeof ≤ l.sizeof
source
theorem list.nth_le_drop {α : Type u} (L : list α) {i j : ℕ} (h : i + j < L.length) :
L.nth_le (i + j) h = (list.drop i L).nth_le j _

The i + j-th element of a list coincides with the j-th element of the list obtained by dropping the first i elements. Version designed to rewrite from the big list to the small list.

source
theorem list.nth_le_drop' {α : Type u} (L : list α) {i j : ℕ} (h : j < (list.drop i L).length) :
(list.drop i L).nth_le j h = L.nth_le (i + j) _

The i + j-th element of a list coincides with the j-th element of the list obtained by dropping the first i elements. Version designed to rewrite from the small list to the big list.

source
theorem list.nth_drop {α : Type u} (L : list α) (i j : ℕ) :
(list.drop i L).nth j = L.nth (i + j)
source
@[simp]
theorem list.drop_drop {α : Type u} (n m : ℕ) (l : list α) :
list.drop n (list.drop m l) = list.drop (n + m) l
source
theorem list.drop_take {α : Type u} (m n : ℕ) (l : list α) :
list.drop m (list.take (m + n) l) = list.take n (list.drop m l)
source
theorem list.map_drop {α : Type u_1} {β : Type u_2} (f : α → β) (L : list α) (i : ℕ) :
list.map f (list.drop i L) = list.drop i (list.map f L)
source
theorem list.modify_nth_tail_eq_take_drop {α : Type u} (f : list α → list α) (H : f list.nil = list.nil) (n : ℕ) (l : list α) :
list.modify_nth_tail f n l = list.take n l ++ f (list.drop n l)
source
theorem list.modify_nth_eq_take_drop {α : Type u} (f : α → α) (n : ℕ) (l : list α) :
list.modify_nth f n l = list.take n l ++ list.modify_head f (list.drop n l)
source
theorem list.modify_nth_eq_take_cons_drop {α : Type u} (f : α → α) {n : ℕ} {l : list α} (h : n < l.length) :
list.modify_nth f n l = list.take n l ++ f (l.nth_le n h) :: list.drop (n + 1) l
source
theorem list.update_nth_eq_take_cons_drop {α : Type u} (a : α) {n : ℕ} {l : list α} (h : n < l.length) :
l.update_nth n a = list.take n l ++ a :: list.drop (n + 1) l
source
theorem list.reverse_take {α : Type u_1} {xs : list α} (n : ℕ) (h : n ≤ xs.length) :
list.take n xs.reverse = (list.drop (xs.length - n) xs).reverse
source
@[simp]
theorem list.update_nth_eq_nil {α : Type u} (l : list α) (n : ℕ) (a : α) :
l.update_nth n a = list.nil ↔ l = list.nil
source
@[simp]
theorem list.take'_length {α : Type u} [inhabited α] (n : ℕ) (l : list α) :
(list.take' n l).length = n
source
@[simp]
theorem list.take'_nil {α : Type u} [inhabited α] (n : ℕ) :
list.take' n list.nil = list.repeat inhabited.default n
source
theorem list.take'_eq_take {α : Type u} [inhabited α] {n : ℕ} {l : list α} :
n ≤ l.length → list.take' n l = list.take n l
source
@[simp]
theorem list.take'_left {α : Type u} [inhabited α] (l₁ l₂ : list α) :
list.take' l₁.length (l₁ ++ l₂) = l₁
source
theorem list.take'_left' {α : Type u} [inhabited α] {l₁ l₂ : list α} {n : ℕ} (h : l₁.length = n) :
list.take' n (l₁ ++ l₂) = l₁

foldl, foldr #

source
theorem list.foldl_ext {α : Type u} {β : Type v} (f g : α → β → α) (a : α) {l : list β} (H : ∀ (a : α) (b : β), b ∈ l → f a b = g a b) :
list.foldl f a l = list.foldl g a l
source
theorem list.foldr_ext {α : Type u} {β : Type v} (f g : α → β → β) (b : β) {l : list α} (H : ∀ (a : α), a ∈ l → ∀ (b : β), f a b = g a b) :
list.foldr f b l = list.foldr g b l
source
@[simp]
theorem list.foldl_nil {α : Type u} {β : Type v} (f : α → β → α) (a : α) :
list.foldl f a list.nil = a
source
@[simp]
theorem list.foldl_cons {α : Type u} {β : Type v} (f : α → β → α) (a : α) (b : β) (l : list β) :
list.foldl f a (b :: l) = list.foldl f (f a b) l
source
@[simp]
theorem list.foldr_nil {α : Type u} {β : Type v} (f : α → β → β) (b : β) :
list.foldr f b list.nil = b
source
@[simp]
theorem list.foldr_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : list α) :
list.foldr f b (a :: l) = f a (list.foldr f b l)
source
@[simp]
theorem list.foldl_append {α : Type u} {β : Type v} (f : α → β → α) (a : α) (l₁ l₂ : list β) :
list.foldl f a (l₁ ++ l₂) = list.foldl f (list.foldl f a l₁) l₂
source
@[simp]
theorem list.foldr_append {α : Type u} {β : Type v} (f : α → β → β) (b : β) (l₁ l₂ : list α) :
list.foldr f b (l₁ ++ l₂) = list.foldr f (list.foldr f b l₂) l₁
source
theorem list.foldl_fixed' {α : Type u} {β : Type v} {f : α → β → α} {a : α} (hf : ∀ (b : β), f a b = a) (l : list β) :
list.foldl f a l = a
source
theorem list.foldr_fixed' {α : Type u} {β : Type v} {f : α → β → β} {b : β} (hf : ∀ (a : α), f a b = b) (l : list α) :
list.foldr f b l = b
source
@[simp]
theorem list.foldl_fixed {α : Type u} {β : Type v} {a : α} (l : list β) :
list.foldl (λ (a : α) (b : β), a) a l = a
source
@[simp]
theorem list.foldr_fixed {α : Type u} {β : Type v} {b : β} (l : list α) :
list.foldr (λ (a : α) (b : β), b) b l = b
source
@[simp]
theorem list.foldl_combinator_K {α : Type u} {β : Type v} {a : α} (l : list β) :
list.foldl combinator.K a l = a
source
@[simp]
theorem list.foldl_join {α : Type u} {β : Type v} (f : α → β → α) (a : α) (L : list (list β)) :
list.foldl f a L.join = list.foldl (list.foldl f) a L
source
@[simp]
theorem list.foldr_join {α : Type u} {β : Type v} (f : α → β → β) (b : β) (L : list (list α)) :
list.foldr f b L.join = list.foldr (λ (l : list α) (b : β), list.foldr f b l) b L
source
theorem list.foldl_reverse {α : Type u} {β : Type v} (f : α → β → α) (a : α) (l : list β) :
list.foldl f a l.reverse = list.foldr (λ (x : β) (y : α), f y x) a l
source
theorem list.foldr_reverse {α : Type u} {β : Type v} (f : α → β → β) (a : β) (l : list α) :
list.foldr f a l.reverse = list.foldl (λ (x : β) (y : α), f y x) a l
source
@[simp]
theorem list.foldr_eta {α : Type u} (l : list α) :
list.foldr list.cons list.nil l = l
source
@[simp]
theorem list.reverse_foldl {α : Type u} {l : list α} :
(list.foldl (λ (t : list α) (h : α), h :: t) list.nil l).reverse = l
source
@[simp]
theorem list.foldl_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : α → γ → α) (a : α) (l : list β) :
list.foldl f a (list.map g l) = list.foldl (λ (x : α) (y : β), f x (g y)) a l
source
@[simp]
theorem list.foldr_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : γ → α → α) (a : α) (l : list β) :
list.foldr f a (list.map g l) = list.foldr (f ∘ g) a l
source
theorem list.foldl_map' {α β : Type u} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : list α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) :
list.foldl f' (g a) (list.map g l) = g (list.foldl f a l)
source
theorem list.foldr_map' {α β : Type u} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : list α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) :
list.foldr f' (g a) (list.map g l) = g (list.foldr f a l)
source
theorem list.foldl_hom {α : Type u} {β : Type v} {γ : Type w} (l : list γ) (f : α → β) (op : α → γ → α) (op' : β → γ → β) (a : α) (h : ∀ (a : α) (x : γ), f (op a x) = op' (f a) x) :
list.foldl op' (f a) l = f (list.foldl op a l)
source
theorem list.foldr_hom {α : Type u} {β : Type v} {γ : Type w} (l : list γ) (f : α → β) (op : γ → α → α) (op' : γ → β → β) (a : α) (h : ∀ (x : γ) (a : α), f (op x a) = op' x (f a)) :
list.foldr op' (f a) l = f (list.foldr op a l)
source
theorem list.foldl_hom₂ {ι : Type u_1} {α : Type u} {β : Type v} {γ : Type w} (l : list ι) (f : α → β → γ) (op₁ : α → ι → α) (op₂ : β → ι → β) (op₃ : γ → ι → γ) (a : α) (b : β) (h : ∀ (a : α) (b : β) (i : ι), f (op₁ a i) (op₂ b i) = op₃ (f a b) i) :
list.foldl op₃ (f a b) l = f (list.foldl op₁ a l) (list.foldl op₂ b l)
source
theorem list.foldr_hom₂ {ι : Type u_1} {α : Type u} {β : Type v} {γ : Type w} (l : list ι) (f : α → β → γ) (op₁ : ι → α → α) (op₂ : ι → β → β) (op₃ : ι → γ → γ) (a : α) (b : β) (h : ∀ (a : α) (b : β) (i : ι), f (op₁ i a) (op₂ i b) = op₃ i (f a b)) :
list.foldr op₃ (f a b) l = f (list.foldr op₁ a l) (list.foldr op₂ b l)
source
theorem list.injective_foldl_comp {α : Type u_1} {l : list (α → α)} {f : α → α} (hl : ∀ (f : α → α), f ∈ l → function.injective f) (hf : function.injective f) :
function.injective (list.foldl function.comp f l)
source
def list.foldr_rec_on {α : Type u} {β : Type v} {C : β → Sort u_1} (l : list α) (op : α → β → β) (b : β) (hb : C b) (hl : Π (b : β), C b → Π (a : α), a ∈ l → C (op a b)) :
C (list.foldr op b l)

Induction principle for values produced by a foldr: if a property holds for the seed element b : β and for all incremental op : α → β → β performed on the elements (a : α) ∈ l. The principle is given for a Sort-valued predicate, i.e., it can also be used to construct data.

Equations
  • l.foldr_rec_on op b hb hl = list.rec (λ (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op a b)), hb) (λ (hd : α) (tl : list α) (IH : (Π (b : β), C b → Π (a : α), a ∈ tl → C (op a b)) → C (list.foldr op b tl)) (hl : Π (b : β), C b → Π (a : α), a ∈ hd :: tl → C (op a b)), hl (list.foldr op b tl) (IH (λ (y : β) (hy : C y) (x : α) (hx : x ∈ tl), hl y hy x _)) hd _) l hl
source
def list.foldl_rec_on {α : Type u} {β : Type v} {C : β → Sort u_1} (l : list α) (op : β → α → β) (b : β) (hb : C b) (hl : Π (b : β), C b → Π (a : α), a ∈ l → C (op b a)) :
C (list.foldl op b l)

Induction principle for values produced by a foldl: if a property holds for the seed element b : β and for all incremental op : β → α → β performed on the elements (a : α) ∈ l. The principle is given for a Sort-valued predicate, i.e., it can also be used to construct data.

Equations
  • l.foldl_rec_on op b hb hl = list.rec (λ (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op b a)) (b : β) (hb : C b), hb) (λ (hd : α) (tl : list α) (IH : (Π (b : β), C b → Π (a : α), a ∈ tl → C (op b a)) → Π (b : β), C b → C (list.foldl op b tl)) (hl : Π (b : β), C b → Π (a : α), a ∈ hd :: tl → C (op b a)) (b : β) (hb : C b), IH (λ (y : β) (hy : C y) (x : α) (hx : x ∈ tl), hl y hy x _) (op b hd) (hl b hb hd _)) l hl b hb
source
@[simp]
theorem list.foldr_rec_on_nil {α : Type u} {β : Type v} {C : β → Sort u_1} (op : α → β → β) (b : β) (hb : C b) (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op a b)) :
list.nil.foldr_rec_on op b hb hl = hb
source
@[simp]
theorem list.foldr_rec_on_cons {α : Type u} {β : Type v} {C : β → Sort u_1} (x : α) (l : list α) (op : α → β → β) (b : β) (hb : C b) (hl : Π (b : β), C b → Π (a : α), a ∈ x :: l → C (op a b)) :
(x :: l).foldr_rec_on op b hb hl = hl (list.foldr op b l) (l.foldr_rec_on op b hb (λ (b : β) (hb : C b) (a : α) (ha : a ∈ l), hl b hb a _)) x _
source
@[simp]
theorem list.foldl_rec_on_nil {α : Type u} {β : Type v} {C : β → Sort u_1} (op : β → α → β) (b : β) (hb : C b) (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op b a)) :
list.nil.foldl_rec_on op b hb hl = hb
source
theorem list.length_scanl {α : Type u} {β : Type v} {f : β → α → β} (a : β) (l : list α) :
(list.scanl f a l).length = l.length + 1
source
@[simp]
theorem list.scanl_nil {α : Type u} {β : Type v} {f : β → α → β} (b : β) :
list.scanl f b list.nil = [b]
source
@[simp]
theorem list.scanl_cons {α : Type u} {β : Type v} {f : β → α → β} {b : β} {a : α} {l : list α} :
list.scanl f b (a :: l) = [b] ++ list.scanl f (f b a) l
source
@[simp]
theorem list.nth_zero_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : list α} :
(list.scanl f b l).nth 0 = option.some b
source
@[simp]
theorem list.nth_le_zero_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : list α} {h : 0 < (list.scanl f b l).length} :
(list.scanl f b l).nth_le 0 h = b
source
theorem list.nth_succ_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : list α} {i : ℕ} :
(list.scanl f b l).nth (i + 1) = ((list.scanl f b l).nth i).bind (λ (x : β), option.map (λ (y : α), f x y) (l.nth i))
source
theorem list.nth_le_succ_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : list α} {i : ℕ} {h : i + 1 < (list.scanl f b l).length} :
(list.scanl f b l).nth_le (i + 1) h = f ((list.scanl f b l).nth_le i _) (l.nth_le i _)
source
@[simp]
theorem list.scanr_nil {α : Type u} {β : Type v} (f : α → β → β) (b : β) :
list.scanr f b list.nil = [b]
source
@[simp]
theorem list.scanr_aux_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : list α) :
list.scanr_aux f b (a :: l) = (list.foldr f b (a :: l), list.scanr f b l)
source
@[simp]
theorem list.scanr_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : list α) :
list.scanr f b (a :: l) = list.foldr f b (a :: l) :: list.scanr f b l
source
theorem list.foldl1_eq_foldr1 {α : Type u} {f : α → α → α} (hassoc : associative f) (a b : α) (l : list α) :
list.foldl f a (l ++ [b]) = list.foldr f b (a :: l)
source
theorem list.foldl_eq_of_comm_of_assoc {α : Type u} {f : α → α → α} (hcomm : commutative f) (hassoc : associative f) (a b : α) (l : list α) :
list.foldl f a (b :: l) = f b (list.foldl f a l)
source
theorem list.foldl_eq_foldr {α : Type u} {f : α → α → α} (hcomm : commutative f) (hassoc : associative f) (a : α) (l : list α) :
list.foldl f a l = list.foldr f a l
source
theorem list.foldl_eq_of_comm' {α : Type u} {β : Type v} {f : α → β → α} (hf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b) (a : α) (b : β) (l : list β) :
list.foldl f a (b :: l) = f (list.foldl f a l) b
source
theorem list.foldl_eq_foldr' {α : Type u} {β : Type v} {f : α → β → α} (hf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b) (a : α) (l : list β) :
list.foldl f a l = list.foldr (flip f) a l
source
theorem list.foldr_eq_of_comm' {α : Type u} {β : Type v} {f : α → β → β} (hf : ∀ (a b : α) (c : β), f a (f b c) = f b (f a c)) (a : β) (b : α) (l : list α) :
list.foldr f a (b :: l) = list.foldr f (f b a) l
source
theorem list.foldl_assoc {α : Type u} {op : α → α → α} [ha : is_associative α op] {l : list α} {a₁ a₂ : α} :
list.foldl op (op a₁ a₂) l = op a₁ (list.foldl op a₂ l)
source
theorem list.foldl_op_eq_op_foldr_assoc {α : Type u} {op : α → α → α} [ha : is_associative α op] {l : list α} {a₁ a₂ : α} :
op (list.foldl op a₁ l) a₂ = op a₁ (list.foldr op a₂ l)
source
theorem list.foldl_assoc_comm_cons {α : Type u} {op : α → α → α} [ha : is_associative α op] [hc : is_commutative α op] {l : list α} {a₁ a₂ : α} :
list.foldl op a₂ (a₁ :: l) = op a₁ (list.foldl op a₂ l)

mfoldl, mfoldr, mmap #

source
@[simp]
theorem list.mfoldl_nil {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] (f : β → α → m β) {b : β} :
list.mfoldl f b list.nil = has_pure.pure b
source
@[simp]
theorem list.mfoldr_nil {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] (f : α → β → m β) {b : β} :
list.mfoldr f b list.nil = has_pure.pure b
source
@[simp]
theorem list.mfoldl_cons {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] {f : β → α → m β} {b : β} {a : α} {l : list α} :
list.mfoldl f b (a :: l) = f b a >>= λ (b' : β), list.mfoldl f b' l
source
@[simp]
theorem list.mfoldr_cons {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] {f : α → β → m β} {b : β} {a : α} {l : list α} :
list.mfoldr f b (a :: l) = list.mfoldr f b l >>= f a
source
theorem list.mfoldr_eq_foldr {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] (f : α → β → m β) (b : β) (l : list α) :
list.mfoldr f b l = list.foldr (λ (a : α) (mb : m β), mb >>= f a) (has_pure.pure b) l
source
theorem list.mfoldl_eq_foldl {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] [is_lawful_monad m] (f : β → α → m β) (b : β) (l : list α) :
list.mfoldl f b l = list.foldl (λ (mb : m β) (a : α), mb >>= λ (b : β), f b a) (has_pure.pure b) l
source
@[simp]
theorem list.mfoldl_append {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] [is_lawful_monad m] {f : β → α → m β} {b : β} {l₁ l₂ : list α} :
list.mfoldl f b (l₁ ++ l₂) = list.mfoldl f b l₁ >>= λ (x : β), list.mfoldl f x l₂
source
@[simp]
theorem list.mfoldr_append {α : Type u} {β : Type v} {m : Type v → Type w} [monad m] [is_lawful_monad m] {f : α → β → m β} {b : β} {l₁ l₂ : list α} :
list.mfoldr f b (l₁ ++ l₂) = list.mfoldr f b l₂ >>= λ (x : β), list.mfoldr f x l₁

intersperse #

source
@[simp]
theorem list.intersperse_nil {α : Type u} (a : α) :
list.intersperse a list.nil = list.nil
source
@[simp]
theorem list.intersperse_singleton {α : Type u} (a b : α) :
list.intersperse a [b] = [b]
source
@[simp]