Basic properties of lists

instance List.uniqueOfIsEmpty {α : Type u} [IsEmpty α] : Unique (List α)

There is only one list of an empty type

Equations

• List.uniqueOfIsEmpty = let __src := instInhabitedList; { toInhabited := __src, uniq := ⋯ }

instance List.instLawfulIdentityAppendNil_mathlib {α : Type u} : Std.LawfulIdentity Append.append []

Equations

• ⋯ = ⋯

instance List.instAssociativeAppend_mathlib {α : Type u} : Std.Associative Append.append

Equations

• ⋯ = ⋯

@[simp]

theorem List.cons_injective {α : Type u} {a : α} : Function.Injective (List.cons a)

theorem List.singleton_injective {α : Type u} : Function.Injective fun (a : α) => [a]

theorem List.singleton_inj {α : Type u} {a : α} {b : α} : [a] = [b] ↔ a = b

theorem List.set_of_mem_cons {α : Type u} (l : List α) (a : α) : {x : α | x ∈ a :: l} = insert a {x : α | x ∈ l}

mem

theorem Decidable.List.eq_or_ne_mem_of_mem {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} (h : a ∈ b :: l) : a = b ∨ a ≠ b ∧ a ∈ l

theorem List.mem_pair {α : Type u} {a : α} {b : α} {c : α} : a ∈ [b, c] ↔ a = b ∨ a = c

@[deprecated List.append_of_mem]

theorem List.mem_split {α : Type u_1} {a : α} {l : List α} : a ∈ l → ∃ (s : List α), ∃ (t : List α), l = s ++ a :: t

Alias of List.append_of_mem.

@[simp]

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

@[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

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

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 fun (a : α) => g (f a)

length

theorem List.length_pos_of_ne_nil {α : Type u_1} {l : List α} : l ≠ [] → 0 < l.length

Alias of the reverse direction of List.length_pos.

theorem List.ne_nil_of_length_pos {α : Type u_1} {l : List α} : 0 < l.length → l ≠ []

Alias of the forward direction of List.length_pos.

theorem List.length_pos_iff_ne_nil {α : Type u} {l : List α} : 0 < l.length ↔ l ≠ []

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

@[simp]

theorem List.length_injective_iff {α : Type u} : Function.Injective List.length ↔ Subsingleton α

@[simp]

theorem List.length_injective {α : Type u} [Subsingleton α] : Function.Injective List.length

theorem List.length_eq_two {α : Type u} {l : List α} : l.length = 2 ↔ ∃ (a : α), ∃ (b : α), l = [a, b]

theorem List.length_eq_three {α : Type u} {l : List α} : l.length = 3 ↔ ∃ (a : α), ∃ (b : α), ∃ (c : α), l = [a, b, c]

set-theoretic notation of lists

instance List.instSingletonList {α : Type u} : Singleton α (List α)

Equations

• List.instSingletonList = { singleton := fun (x : α) => [x] }

instance List.instInsertOfDecidableEq_mathlib {α : Type u} [DecidableEq α] : Insert α (List α)

Equations

• List.instInsertOfDecidableEq_mathlib = { insert := List.insert }

instance List.instLawfulSingleton_mathlib {α : Type u} [DecidableEq α] : LawfulSingleton α (List α)

Equations

• ⋯ = ⋯

theorem List.singleton_eq {α : Type u} (x : α) : {x} = [x]

theorem List.insert_neg {α : Type u} [DecidableEq α] {x : α} {l : List α} (h : ¬x ∈ l) : insert x l = x :: l

theorem List.insert_pos {α : Type u} [DecidableEq α] {x : α} {l : List α} (h : x ∈ l) : insert x l = l

theorem List.doubleton_eq {α : Type u} [DecidableEq α] {x : α} {y : α} (h : x ≠ y) : {x, y} = [x, y]

bounded quantifiers over lists

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) : x ∈ l → p x

theorem List.exists_mem_cons_of {α : Type u} {p : α → Prop} {a : α} (l : List α) (h : p a) : ∃ (x : α), x ∈ a :: l ∧ p x

theorem List.exists_mem_cons_of_exists {α : Type u} {p : α → Prop} {a : α} {l : List α} : (∃ (x : α), x ∈ l ∧ p x) → ∃ (x : α), x ∈ a :: l ∧ p x

theorem List.or_exists_of_exists_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

theorem List.exists_mem_cons_iff {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

list subset

instance List.instIsTransSubset {α : Type u} : IsTrans (List α) Subset

Equations

• ⋯ = ⋯

theorem List.cons_subset_of_subset_of_mem {α : Type u} {a : α} {l : List α} {m : List α} (ainm : a ∈ m) (lsubm : l ⊆ m) : a :: l ⊆ m

theorem List.append_subset_of_subset_of_subset {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) : l₁ ++ l₂ ⊆ l

theorem List.eq_nil_of_subset_nil {α : Type u_1} {l : List α} : l ⊆ [] → l = []

Alias of the forward direction of List.subset_nil.

theorem List.map_subset_iff {α : Type u} {β : Type v} {l₁ : List α} {l₂ : List α} (f : α → β) (h : Function.Injective f) : List.map f l₁ ⊆ List.map f l₂ ↔ l₁ ⊆ l₂

append

theorem List.append_eq_has_append {α : Type u} {L₁ : List α} {L₂ : List α} : L₁.append L₂ = L₁ ++ L₂

@[deprecated List.append_eq_cons]

theorem List.append_eq_cons_iff : ∀ {α : Type u_1} {a b : List α} {x : α} {c : List α}, a ++ b = x :: c ↔ a = [] ∧ b = x :: c ∨ ∃ (a' : List α), a = x :: a' ∧ c = a' ++ b

Alias of List.append_eq_cons.

@[deprecated List.cons_eq_append]

theorem List.cons_eq_append_iff : ∀ {α : Type u_1} {x : α} {c a b : List α}, x :: c = a ++ b ↔ a = [] ∧ b = x :: c ∨ ∃ (a' : List α), a = x :: a' ∧ c = a' ++ b

Alias of List.cons_eq_append.

@[deprecated List.append_cancel_left]

theorem List.append_left_cancel {α : Type u_1} {as : List α} {bs : List α} {cs : List α} (h : as ++ bs = as ++ cs) : bs = cs

Alias of List.append_cancel_left.

@[deprecated List.append_cancel_right]

theorem List.append_right_cancel {α : Type u_1} {as : List α} {bs : List α} {cs : List α} (h : as ++ bs = cs ++ bs) : as = cs

Alias of List.append_cancel_right.

@[simp]

theorem List.append_left_eq_self {α : Type u} {x : List α} {y : List α} : x ++ y = y ↔ x = []

@[simp]

theorem List.self_eq_append_left {α : Type u} {x : List α} {y : List α} : y = x ++ y ↔ x = []

@[simp]

theorem List.append_right_eq_self {α : Type u} {x : List α} {y : List α} : x ++ y = x ↔ y = []

@[simp]

theorem List.self_eq_append_right {α : Type u} {x : List α} {y : List α} : x = x ++ y ↔ y = []

theorem List.append_right_injective {α : Type u} (s : List α) : Function.Injective fun (t : List α) => s ++ t

theorem List.append_left_injective {α : Type u} (t : List α) : Function.Injective fun (s : List α) => s ++ t

replicate

@[simp]

theorem List.replicate_zero {α : Type u} (a : α) : List.replicate 0 a = []

theorem List.replicate_one {α : Type u} (a : α) : List.replicate 1 a = [a]

theorem List.eq_replicate_length {α : Type u} {a : α} {l : List α} : l = List.replicate l.length a ↔ ∀ (b : α), b ∈ l → b = a

theorem List.replicate_add {α : Type u} (m : ℕ) (n : ℕ) (a : α) : List.replicate (m + n) a = List.replicate m a ++ List.replicate n a

theorem List.replicate_succ' {α : Type u} (n : ℕ) (a : α) : List.replicate (n + 1) a = List.replicate n a ++ [a]

theorem List.replicate_subset_singleton {α : Type u} (n : ℕ) (a : α) : List.replicate n a ⊆ [a]

theorem List.subset_singleton_iff {α : Type u} {a : α} {L : List α} : L ⊆ [a] ↔ ∃ (n : ℕ), L = List.replicate n a

@[simp]

theorem List.map_replicate {α : Type u} {β : Type v} (f : α → β) (n : ℕ) (a : α) : List.map f (List.replicate n a) = List.replicate n (f a)

@[simp]

theorem List.tail_replicate {α : Type u} (a : α) (n : ℕ) : (List.replicate n a).tail = List.replicate (n - 1) a

@[simp]

theorem List.join_replicate_nil {α : Type u} (n : ℕ) : (List.replicate n []).join = []

theorem List.replicate_right_injective {α : Type u} {n : ℕ} (hn : n ≠ 0) : Function.Injective (List.replicate n)

theorem List.replicate_right_inj {α : Type u} {a : α} {b : α} {n : ℕ} (hn : n ≠ 0) : List.replicate n a = List.replicate n b ↔ a = b

@[simp]

theorem List.replicate_right_inj' {α : Type u} {a : α} {b : α} {n : ℕ} : List.replicate n a = List.replicate n b ↔ n = 0 ∨ a = b

theorem List.replicate_left_injective {α : Type u} (a : α) : Function.Injective fun (x : ℕ) => List.replicate x a

@[simp]

theorem List.replicate_left_inj {α : Type u} {a : α} {n : ℕ} {m : ℕ} : List.replicate n a = List.replicate m a ↔ n = m

@[simp]

theorem List.head_replicate {α : Type u} (n : ℕ) (a : α) (h : List.replicate n a ≠ []) : (List.replicate n a).head h = a

pure

theorem List.mem_pure {α : Type u} (x : α) (y : α) : x ∈ pure y ↔ x = y

bind

@[simp]

theorem List.bind_eq_bind {α : Type u_2} {β : Type u_2} (f : α → List β) (l : List α) : l >>= f = l.bind f

concat

reverse

theorem List.reverse_cons' {α : Type u} (a : α) (l : List α) : (a :: l).reverse = l.reverse.concat a

theorem List.reverse_concat' {α : Type u} (l : List α) (a : α) : (l ++ [a]).reverse = a :: l.reverse

theorem List.reverse_singleton {α : Type u} (a : α) : [a].reverse = [a]

@[simp]

theorem List.reverse_involutive {α : Type u} : Function.Involutive List.reverse

@[simp]

theorem List.reverse_injective {α : Type u} : Function.Injective List.reverse

theorem List.reverse_surjective {α : Type u} : Function.Surjective List.reverse

theorem List.reverse_bijective {α : Type u} : Function.Bijective List.reverse

@[simp]

theorem List.reverse_inj {α : Type u} {l₁ : List α} {l₂ : List α} : l₁.reverse = l₂.reverse ↔ l₁ = l₂

theorem List.reverse_eq_iff {α : Type u} {l : List α} {l' : List α} : l.reverse = l' ↔ l = l'.reverse

theorem List.concat_eq_reverse_cons {α : Type u} (a : α) (l : List α) : l.concat a = (a :: l.reverse).reverse

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

theorem List.map_reverseAux {α : Type u} {β : Type v} (f : α → β) (l₁ : List α) (l₂ : List α) : List.map f (l₁.reverseAux l₂) = (List.map f l₁).reverseAux (List.map f l₂)

@[simp]

theorem List.reverse_replicate {α : Type u} (n : ℕ) (a : α) : (List.replicate n a).reverse = List.replicate n a

empty

theorem List.isEmpty_iff_eq_nil {α : Type u} {l : List α} : l.isEmpty = true ↔ l = []

dropLast

getLast

@[simp]

theorem List.getLast_cons {α : Type u} {a : α} {l : List α} (h : l ≠ []) : (a :: l).getLast ⋯ = l.getLast h

theorem List.getLast_append_singleton {α : Type u} {a : α} (l : List α) : (l ++ [a]).getLast ⋯ = a

theorem List.getLast_append' {α : Type u} (l₁ : List α) (l₂ : List α) (h : l₂ ≠ []) : (l₁ ++ l₂).getLast ⋯ = l₂.getLast h

theorem List.getLast_concat' {α : Type u} {a : α} (l : List α) : (l.concat a).getLast ⋯ = a

@[simp]

theorem List.getLast_singleton' {α : Type u} (a : α) : [a].getLast ⋯ = a

theorem List.getLast_cons_cons {α : Type u} (a₁ : α) (a₂ : α) (l : List α) : (a₁ :: a₂ :: l).getLast ⋯ = (a₂ :: l).getLast ⋯

theorem List.dropLast_append_getLast {α : Type u} {l : List α} (h : l ≠ []) : l.dropLast ++ [l.getLast h] = l

theorem List.getLast_congr {α : Type u} {l₁ : List α} {l₂ : List α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : l₁.getLast h₁ = l₂.getLast h₂

theorem List.getLast_replicate_succ {α : Type u} (m : ℕ) (a : α) : (List.replicate (m + 1) a).getLast ⋯ = a

theorem List.getLast_filter {α : Type u} {p : α → Bool} (l : List α) (hlp : List.filter p l ≠ []) : p (l.getLast ⋯) = true → (List.filter p l).getLast hlp = l.getLast ⋯

If the last element of l does not satisfy p, then it is also the last element of l.filter p.

getLast?

@[simp]

theorem List.getLast?_cons_cons {α : Type u} (a : α) (b : α) (l : List α) : (a :: b :: l).getLast? = (b :: l).getLast?

@[simp]

theorem List.getLast?_isNone {α : Type u} {l : List α} : l.getLast?.isNone = true ↔ l = []

@[simp]

theorem List.getLast?_isSome {α : Type u} {l : List α} : l.getLast?.isSome = true ↔ l ≠ []

theorem List.mem_getLast?_eq_getLast {α : Type u} {l : List α} {x : α} : x ∈ l.getLast? → ∃ (h : l ≠ []), x = l.getLast h

theorem List.getLast?_eq_getLast_of_ne_nil {α : Type u} {l : List α} (h : l ≠ []) : l.getLast? = some (l.getLast h)

theorem List.mem_getLast?_cons {α : Type u} {x : α} {y : α} {l : List α} : x ∈ l.getLast? → x ∈ (y :: l).getLast?

theorem List.mem_of_mem_getLast? {α : Type u} {l : List α} {a : α} (ha : a ∈ l.getLast?) : a ∈ l

theorem List.dropLast_append_getLast? {α : Type u} {l : List α} (a : α) : a ∈ l.getLast? → l.dropLast ++ [a] = l

theorem List.getLastI_eq_getLast? {α : Type u} [Inhabited α] (l : List α) : l.getLastI = l.getLast?.iget

@[simp]

theorem List.getLast?_append_cons {α : Type u} (l₁ : List α) (a : α) (l₂ : List α) : (l₁ ++ a :: l₂).getLast? = (a :: l₂).getLast?

theorem List.getLast?_append_of_ne_nil {α : Type u} (l₁ : List α) {l₂ : List α} : l₂ ≠ [] → (l₁ ++ l₂).getLast? = l₂.getLast?

theorem List.getLast?_append {α : Type u} {l₁ : List α} {l₂ : List α} {x : α} (h : x ∈ l₂.getLast?) : x ∈ (l₁ ++ l₂).getLast?

head(!?) and tail

@[simp]

theorem List.head!_nil {α : Type u} [Inhabited α] : [].head! = default

@[simp]

theorem List.head_cons_tail {α : Type u} (x : List α) (h : x ≠ []) : x.head h :: x.tail = x

theorem List.head!_eq_head? {α : Type u} [Inhabited α] (l : List α) : l.head! = l.head?.iget

theorem List.surjective_head! {α : Type u} [Inhabited α] : Function.Surjective List.head!

theorem List.surjective_head? {α : Type u} : Function.Surjective List.head?

theorem List.surjective_tail {α : Type u} : Function.Surjective List.tail

theorem List.eq_cons_of_mem_head? {α : Type u} {x : α} {l : List α} : x ∈ l.head? → l = x :: l.tail

theorem List.mem_of_mem_head? {α : Type u} {x : α} {l : List α} (h : x ∈ l.head?) : x ∈ l

@[simp]

theorem List.head!_cons {α : Type u} [Inhabited α] (a : α) (l : List α) : (a :: l).head! = a

@[simp]

theorem List.head!_append {α : Type u} [Inhabited α] (t : List α) {s : List α} (h : s ≠ []) : (s ++ t).head! = s.head!

theorem List.head?_append {α : Type u} {s : List α} {t : List α} {x : α} (h : x ∈ s.head?) : x ∈ (s ++ t).head?

theorem List.head?_append_of_ne_nil {α : Type u} (l₁ : List α) {l₂ : List α} : l₁ ≠ [] → (l₁ ++ l₂).head? = l₁.head?

theorem List.tail_append_singleton_of_ne_nil {α : Type u} {a : α} {l : List α} (h : l ≠ []) : (l ++ [a]).tail = l.tail ++ [a]

theorem List.cons_head?_tail {α : Type u} {l : List α} {a : α} : a ∈ l.head? → a :: l.tail = l

theorem List.head!_mem_head? {α : Type u} [Inhabited α] {l : List α} : l ≠ [] → l.head! ∈ l.head?

theorem List.cons_head!_tail {α : Type u} [Inhabited α] {l : List α} (h : l ≠ []) : l.head! :: l.tail = l

theorem List.head!_mem_self {α : Type u} [Inhabited α] {l : List α} (h : l ≠ []) : l.head! ∈ l

theorem List.head_mem {α : Type u} {l : List α} (h : l ≠ []) : l.head h ∈ l

@[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?

theorem List.tail_append_of_ne_nil {α : Type u} (l : List α) (l' : List α) (h : l ≠ []) : (l ++ l').tail = l.tail ++ l'

theorem List.get_eq_get? {α : Type u} (l : List α) (i : Fin l.length) : l.get i = (l.get? ↑i).get ⋯

@[deprecated List.get]

def List.nthLe {α : Type u} (l : List α) (n : ℕ) (h : n < l.length) : α

nth element of a list l given n < l.length.

Equations

• l.nthLe n h = l.get ⟨n, h⟩

@[simp]

theorem List.nthLe_tail {α : Type u} (l : List α) (i : ℕ) (h : i < l.tail.length) (h' : optParam (i + 1 < l.length) ⋯) : l.tail.nthLe i h = l.nthLe (i + 1) h'

theorem List.nthLe_cons_aux {α : Type u} {l : List α} {a : α} {n : ℕ} (hn : n ≠ 0) (h : n < (a :: l).length) : n - 1 < l.length

theorem List.nthLe_cons {α : Type u} {l : List α} {a : α} {n : ℕ} (hl : n < (a :: l).length) : (a :: l).nthLe n hl = if hn : n = 0 then a else l.nthLe (n - 1) ⋯

@[simp]

theorem List.modifyHead_modifyHead {α : Type u} (l : List α) (f : α → α) (g : α → α) : List.modifyHead g (List.modifyHead f l) = List.modifyHead (g ∘ f) l

Induction from the right

@[irreducible]

def List.reverseRecOn {α : Type u} {motive : List α → Sort u_2} (l : List α) (nil : motive []) (append_singleton : (l : List α) → (a : α) → motive l → motive (l ++ [a])) : motive 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

• One or more equations did not get rendered due to their size.

@[simp]

theorem List.reverseRecOn_nil {α : Type u} {motive : List α → Sort u_2} (nil : motive []) (append_singleton : (l : List α) → (a : α) → motive l → motive (l ++ [a])) : List.reverseRecOn [] nil append_singleton = nil

@[simp]

theorem List.reverseRecOn_concat {α : Type u} {motive : List α → Sort u_2} (x : α) (xs : List α) (nil : motive []) (append_singleton : (l : List α) → (a : α) → motive l → motive (l ++ [a])) : List.reverseRecOn (xs ++ [x]) nil append_singleton = append_singleton xs x (List.reverseRecOn xs nil append_singleton)

@[irreducible]

def List.bidirectionalRec {α : Type u} {motive : List α → Sort u_2} (nil : motive []) (singleton : (a : α) → motive [a]) (cons_append : (a : α) → (l : List α) → (b : α) → motive l → motive (a :: (l ++ [b]))) (l : List α) : motive 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

• One or more equations did not get rendered due to their size.
• List.bidirectionalRec nil singleton cons_append [] = nil
• List.bidirectionalRec nil singleton cons_append [a] = singleton a

@[simp]

theorem List.bidirectionalRec_nil {α : Type u} {motive : List α → Sort u_2} (nil : motive []) (singleton : (a : α) → motive [a]) (cons_append : (a : α) → (l : List α) → (b : α) → motive l → motive (a :: (l ++ [b]))) : List.bidirectionalRec nil singleton cons_append [] = nil

@[simp]

theorem List.bidirectionalRec_singleton {α : Type u} {motive : List α → Sort u_2} (nil : motive []) (singleton : (a : α) → motive [a]) (cons_append : (a : α) → (l : List α) → (b : α) → motive l → motive (a :: (l ++ [b]))) (a : α) : List.bidirectionalRec nil singleton cons_append [a] = singleton a

@[simp]

theorem List.bidirectionalRec_cons_append {α : Type u} {motive : List α → Sort u_2} (nil : motive []) (singleton : (a : α) → motive [a]) (cons_append : (a : α) → (l : List α) → (b : α) → motive l → motive (a :: (l ++ [b]))) (a : α) (l : List α) (b : α) : List.bidirectionalRec nil singleton cons_append (a :: (l ++ [b])) = cons_append a l b (List.bidirectionalRec nil singleton cons_append l)

@[reducible, inline]

abbrev List.bidirectionalRecOn {α : Type u} {C : List α → Sort u_2} (l : List α) (H0 : C []) (H1 : (a : α) → C [a]) (Hn : (a : α) → (l : List α) → (b : α) → C l → C (a :: (l ++ [b]))) : C l

Like bidirectionalRec, but with the list parameter placed first.

Equations

• l.bidirectionalRecOn H0 H1 Hn = List.bidirectionalRec H0 H1 Hn l

sublists

theorem List.Sublist.cons_cons {α : Type u} {l₁ : List α} {l₂ : List α} (a : α) (s : l₁.Sublist l₂) : (a :: l₁).Sublist (a :: l₂)

theorem List.cons_sublist_cons' {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} {b : α} : (a :: l₁).Sublist (b :: l₂) ↔ (a :: l₁).Sublist l₂ ∨ a = b ∧ l₁.Sublist l₂

theorem List.sublist_cons_of_sublist {α : Type u} {l₁ : List α} {l₂ : List α} (a : α) (h : l₁.Sublist l₂) : l₁.Sublist (a :: l₂)

theorem List.tail_sublist {α : Type u} (l : List α) : l.tail.Sublist l

theorem List.Sublist.tail {α : Type u} {l₁ : List α} {l₂ : List α} : l₁.Sublist l₂ → l₁.tail.Sublist l₂.tail

theorem List.Sublist.of_cons_cons {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} {b : α} (h : (a :: l₁).Sublist (b :: l₂)) : l₁.Sublist l₂

@[deprecated]

theorem List.sublist_of_cons_sublist_cons {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} (h : (a :: l₁).Sublist (a :: l₂)) : l₁.Sublist l₂

@[deprecated List.cons_sublist_cons]

theorem List.cons_sublist_cons_iff : ∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, (a :: l₁).Sublist (a :: l₂) ↔ l₁.Sublist l₂

Alias of List.cons_sublist_cons.

theorem List.eq_nil_of_sublist_nil {α : Type u} {l : List α} (s : l.Sublist []) : l = []

theorem List.sublist_nil_iff_eq_nil {α : Type u_1} {l : List α} : l.Sublist [] ↔ l = []

Alias of List.sublist_nil.

@[simp]

theorem List.sublist_singleton {α : Type u} {l : List α} {a : α} : l.Sublist [a] ↔ l = [] ∨ l = [a]

theorem List.sublist_replicate_iff {α : Type u} {l : List α} {a : α} {n : ℕ} : l.Sublist (List.replicate n a) ↔ ∃ (k : ℕ), k ≤ n ∧ l = List.replicate k a

theorem List.Sublist.antisymm {α : Type u} {l₁ : List α} {l₂ : List α} (s₁ : l₁.Sublist l₂) (s₂ : l₂.Sublist l₁) : l₁ = l₂

instance List.decidableSublist {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁.Sublist l₂)

Equations

• [].decidableSublist x = isTrue ⋯
• (head :: tail).decidableSublist [] = isFalse ⋯
• (a :: l₁).decidableSublist (b :: l₂) = if h : a = b then decidable_of_decidable_of_iff ⋯ else decidable_of_decidable_of_iff ⋯

indexOf

@[simp]

theorem List.indexOf_cons_self {α : Type u} [DecidableEq α] (a : α) (l : List α) : List.indexOf a (a :: l) = 0

theorem List.indexOf_cons_eq {α : Type u} [DecidableEq α] {a : α} {b : α} (l : List α) : b = a → List.indexOf a (b :: l) = 0

@[simp]

theorem List.indexOf_cons_ne {α : Type u} [DecidableEq α] {a : α} {b : α} (l : List α) : b ≠ a → List.indexOf a (b :: l) = (List.indexOf a l).succ

theorem List.indexOf_eq_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : List.indexOf a l = l.length ↔ ¬a ∈ l

@[simp]

theorem List.indexOf_of_not_mem {α : Type u} [DecidableEq α] {l : List α} {a : α} : ¬a ∈ l → List.indexOf a l = l.length

theorem List.indexOf_le_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : List.indexOf a l ≤ l.length

theorem List.indexOf_lt_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : List.indexOf a l < l.length ↔ a ∈ l

theorem List.indexOf_append_of_mem {α : Type u} {l₁ : List α} {l₂ : List α} [DecidableEq α] {a : α} (h : a ∈ l₁) : List.indexOf a (l₁ ++ l₂) = List.indexOf a l₁

theorem List.indexOf_append_of_not_mem {α : Type u} {l₁ : List α} {l₂ : List α} [DecidableEq α] {a : α} (h : ¬a ∈ l₁) : List.indexOf a (l₁ ++ l₂) = l₁.length + List.indexOf a l₂

nth element

@[deprecated List.get_of_mem]

theorem List.nthLe_of_mem {α : Type u} {a : α} {l : List α} (h : a ∈ l) : ∃ (n : ℕ), ∃ (h : n < l.length), l.nthLe n h = a

@[deprecated List.get?_eq_get]

theorem List.nthLe_get? {α : Type u} {l : List α} {n : ℕ} (h : n < l.length) : l.get? n = some (l.nthLe n h)

@[simp]

theorem List.get?_length {α : Type u} (l : List α) : l.get? l.length = none

@[deprecated List.get_mem]

theorem List.nthLe_mem {α : Type u} (l : List α) (n : ℕ) (h : n < l.length) : l.nthLe n h ∈ l

@[deprecated List.mem_iff_get]

theorem List.mem_iff_nthLe {α : Type u} {a : α} {l : List α} : a ∈ l ↔ ∃ (n : ℕ), ∃ (h : n < l.length), l.nthLe n h = a

@[deprecated List.get?_inj]

theorem List.get?_injective {i : ℕ} : ∀ {α : Type u_1} {xs : List α} {j : ℕ}, i < xs.length → xs.Nodup → xs.get? i = xs.get? j → i = j

Alias of List.get?_inj.

@[deprecated List.get_map]

theorem List.nthLe_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).nthLe n H1 = f (l.nthLe n H2)

theorem List.get_map_rev {α : Type u} {β : Type v} (f : α → β) {l : List α} {n : Fin l.length} : f (l.get n) = (List.map f l).get ⟨↑n, ⋯⟩

A version of get_map that can be used for rewriting.

@[deprecated List.get_map_rev]

theorem List.nthLe_map_rev {α : Type u} {β : Type v} (f : α → β) {l : List α} {n : ℕ} (H : n < l.length) : f (l.nthLe n H) = (List.map f l).nthLe n ⋯

A version of nthLe_map that can be used for rewriting.

@[simp, deprecated List.get_map]

theorem List.nthLe_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).nthLe n H = f (l.nthLe n ⋯)

@[simp, deprecated List.get_singleton]

theorem List.nthLe_singleton {α : Type u} (a : α) {n : ℕ} (hn : n < 1) : [a].nthLe n hn = a

@[deprecated List.get_append_right']

theorem List.nthLe_append_right {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (h₁ : l₁.length ≤ n) (h₂ : n < (l₁ ++ l₂).length) : (l₁ ++ l₂).nthLe n h₂ = l₂.nthLe (n - l₁.length) ⋯

theorem List.get_length_sub_one {α : Type u} {l : List α} (h : l.length - 1 < l.length) : l.get ⟨l.length - 1, h⟩ = l.getLast ⋯

@[deprecated List.get_cons_length]

theorem List.nthLe_cons_length {α : Type u} (x : α) (xs : List α) (n : ℕ) (h : n = xs.length) : (x :: xs).nthLe n ⋯ = (x :: xs).getLast ⋯

theorem List.take_one_drop_eq_of_lt_length {α : Type u} {l : List α} {n : ℕ} (h : n < l.length) : List.take 1 (List.drop n l) = [l.get ⟨n, h⟩]

theorem List.ext_get? {α : Type u_1} {l₁ : List α} {l₂ : List α} : (∀ (n : ℕ), l₁.get? n = l₂.get? n) → l₁ = l₂

Alias of List.ext.

theorem List.ext_get?' {α : Type u} {l₁ : List α} {l₂ : List α} (h' : ∀ (n : ℕ), n < max l₁.length l₂.length → l₁.get? n = l₂.get? n) : l₁ = l₂

theorem List.ext_get?_iff {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ = l₂ ↔ ∀ (n : ℕ), l₁.get? n = l₂.get? n

theorem List.ext_get_iff {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ = l₂ ↔ l₁.length = l₂.length ∧ ∀ (n : ℕ) (h₁ : n < l₁.length) (h₂ : n < l₂.length), l₁.get ⟨n, h₁⟩ = l₂.get ⟨n, h₂⟩

theorem List.ext_get?_iff' {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ = l₂ ↔ ∀ (n : ℕ), n < max l₁.length l₂.length → l₁.get? n = l₂.get? n

@[deprecated List.ext_get]

theorem List.ext_nthLe {α : Type u} {l₁ : List α} {l₂ : List α} (hl : l₁.length = l₂.length) (h : ∀ (n : ℕ) (h₁ : n < l₁.length) (h₂ : n < l₂.length), l₁.nthLe n h₁ = l₂.nthLe n h₂) : l₁ = l₂

@[simp]

theorem List.indexOf_get {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : List.indexOf a l < l.length) : l.get ⟨List.indexOf a l, h⟩ = a

@[simp]

theorem List.indexOf_get? {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : l.get? (List.indexOf a l) = some a

@[deprecated]

theorem List.get_reverse_aux₁ {α : Type u} (l : List α) (r : List α) (i : ℕ) (h1 : i + l.length < (l.reverseAux r).length) (h2 : i < r.length) : (l.reverseAux r).get ⟨i + l.length, h1⟩ = r.get ⟨i, h2⟩

theorem List.indexOf_inj {α : Type u} [DecidableEq α] {l : List α} {x : α} {y : α} (hx : x ∈ l) (hy : y ∈ l) : List.indexOf x l = List.indexOf y l ↔ x = y

theorem List.get_reverse_aux₂ {α : Type u} (l : List α) (r : List α) (i : ℕ) (h1 : l.length - 1 - i < (l.reverseAux r).length) (h2 : i < l.length) : (l.reverseAux r).get ⟨l.length - 1 - i, h1⟩ = l.get ⟨i, h2⟩

@[simp]

theorem List.get_reverse {α : Type u} (l : List α) (i : ℕ) (h1 : l.length - 1 - i < l.reverse.length) (h2 : i < l.length) : l.reverse.get ⟨l.length - 1 - i, h1⟩ = l.get ⟨i, h2⟩

@[simp, deprecated List.get_reverse]

theorem List.nthLe_reverse {α : Type u} (l : List α) (i : ℕ) (h1 : l.length - 1 - i < l.reverse.length) (h2 : i < l.length) : l.reverse.nthLe (l.length - 1 - i) h1 = l.nthLe i h2

@[deprecated List.get_reverse']

theorem List.nthLe_reverse' {α : Type u} (l : List α) (n : ℕ) (hn : n < l.reverse.length) (hn' : l.length - 1 - n < l.length) : l.reverse.nthLe n hn = l.nthLe (l.length - 1 - n) hn'

theorem List.get_reverse' {α : Type u} (l : List α) (n : Fin l.reverse.length) (hn' : l.length - 1 - ↑n < l.length) : l.reverse.get n = l.get ⟨l.length - 1 - ↑n, hn'⟩

theorem List.eq_cons_of_length_one {α : Type u} {l : List α} (h : l.length = 1) : l = [l.nthLe 0 ⋯]

theorem List.modifyNthTail_modifyNthTail {α : Type u} {f : List α → List α} {g : List α → List α} (m : ℕ) (n : ℕ) (l : List α) : List.modifyNthTail g (m + n) (List.modifyNthTail f n l) = List.modifyNthTail (fun (l : List α) => List.modifyNthTail g m (f l)) n l

theorem List.modifyNthTail_modifyNthTail_le {α : Type u} {f : List α → List α} {g : List α → List α} (m : ℕ) (n : ℕ) (l : List α) (h : n ≤ m) : List.modifyNthTail g m (List.modifyNthTail f n l) = List.modifyNthTail (fun (l : List α) => List.modifyNthTail g (m - n) (f l)) n l

theorem List.modifyNthTail_modifyNthTail_same {α : Type u} {f : List α → List α} {g : List α → List α} (n : ℕ) (l : List α) : List.modifyNthTail g n (List.modifyNthTail f n l) = List.modifyNthTail (g ∘ f) n l

@[deprecated List.eraseIdx_eq_modifyNthTail]

theorem List.removeNth_eq_nthTail {α : Type u_1} (n : ℕ) (l : List α) : l.eraseIdx n = List.modifyNthTail List.tail n l

Alias of List.eraseIdx_eq_modifyNthTail.

theorem List.modifyNth_eq_set {α : Type u} (f : α → α) (n : ℕ) (l : List α) : List.modifyNth f n l = ((fun (a : α) => l.set n (f a)) <$> l.get? n).getD l

theorem List.length_modifyNthTail {α : Type u} (f : List α → List α) (H : ∀ (l : List α), (f l).length = l.length) (n : ℕ) (l : List α) : (List.modifyNthTail f n l).length = l.length

theorem List.length_modifyNth {α : Type u} (f : α → α) (n : ℕ) (l : List α) : (List.modifyNth f n l).length = l.length

@[simp]

theorem List.get_set_of_ne {α : Type u} {l : List α} {i : ℕ} {j : ℕ} (h : i ≠ j) (a : α) (hj : j < (l.set i a).length) : (l.set i a).get ⟨j, hj⟩ = l.get ⟨j, ⋯⟩

map

theorem List.map_eq_foldr {α : Type u} {β : Type v} (f : α → β) (l : List α) : List.map f l = List.foldr (fun (a : α) (bs : List β) => f a :: bs) [] l

theorem List.map_congr {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : List α} : (∀ (x : α), x ∈ l → f x = g x) → List.map f l = List.map g l

theorem List.map_eq_map_iff {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : List α} : List.map f l = List.map g l ↔ ∀ (x : α), x ∈ l → f x = g x

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)

theorem List.map_id'' {α : Type u} {f : α → α} (h : ∀ (x : α), f x = x) (l : List α) : List.map f l = l

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 = []) : l = []

@[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

theorem List.bind_pure_eq_map {α : Type u} {β : Type v} (f : α → β) (l : List α) : l.bind (pure ∘ f) = List.map f l

@[deprecated List.bind_pure_eq_map]

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

theorem List.bind_congr {α : Type u} {β : Type v} {l : List α} {f : α → List β} {g : α → List β} (h : ∀ (x : α), x ∈ l → f x = g x) : l.bind f = l.bind g

theorem List.infix_bind_of_mem {α : Type u} {a : α} {as : List α} (h : a ∈ as) (f : α → List α) : f a <:+: as.bind f

@[simp]

theorem List.map_eq_map {α : Type u_2} {β : Type u_2} (f : α → β) (l : List α) : f <$> l = List.map f l

@[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

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.

@[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.

theorem Function.LeftInverse.list_map {α : Type u} {β : Type v} {f : α → β} {g : β → α} (h : Function.LeftInverse f g) : Function.LeftInverse (List.map f) (List.map g)

theorem Function.RightInverse.list_map {α : Type u} {β : Type v} {f : α → β} {g : β → α} (h : Function.RightInverse f g) : Function.RightInverse (List.map f) (List.map g)

theorem Function.Involutive.list_map {α : Type u} {f : α → α} (h : Function.Involutive f) : Function.Involutive (List.map f)

@[simp]

theorem List.map_leftInverse_iff {α : Type u} {β : Type v} {f : α → β} {g : β → α} : Function.LeftInverse (List.map f) (List.map g) ↔ Function.LeftInverse f g

@[simp]

theorem List.map_rightInverse_iff {α : Type u} {β : Type v} {f : α → β} {g : β → α} : Function.RightInverse (List.map f) (List.map g) ↔ Function.RightInverse f g

@[simp]

theorem List.map_involutive_iff {α : Type u} {f : α → α} : Function.Involutive (List.map f) ↔ Function.Involutive f

theorem Function.Injective.list_map {α : Type u} {β : Type v} {f : α → β} (h : Function.Injective f) : Function.Injective (List.map f)

@[simp]

theorem List.map_injective_iff {α : Type u} {β : Type v} {f : α → β} : Function.Injective (List.map f) ↔ Function.Injective f

theorem Function.Surjective.list_map {α : Type u} {β : Type v} {f : α → β} (h : Function.Surjective f) : Function.Surjective (List.map f)

@[simp]

theorem List.map_surjective_iff {α : Type u} {β : Type v} {f : α → β} : Function.Surjective (List.map f) ↔ Function.Surjective f

theorem Function.Bijective.list_map {α : Type u} {β : Type v} {f : α → β} (h : Function.Bijective f) : Function.Bijective (List.map f)

@[simp]

theorem List.map_bijective_iff {α : Type u} {β : Type v} {f : α → β} : Function.Bijective (List.map f) ↔ Function.Bijective f

theorem List.map_filter_eq_foldr {α : Type u} {β : Type v} (f : α → β) (p : α → Bool) (as : List α) : List.map f (List.filter p as) = List.foldr (fun (a : α) (bs : List β) => bif p a then f a :: bs else bs) [] as

theorem List.getLast_map {α : Type u} {β : Type v} (f : α → β) {l : List α} (hl : l ≠ []) : (List.map f l).getLast ⋯ = f (l.getLast hl)

theorem List.map_eq_replicate_iff {α : Type u} {β : Type v} {l : List α} {f : α → β} {b : β} : List.map f l = List.replicate l.length b ↔ ∀ (x : α), x ∈ l → f x = b

@[simp]

theorem List.map_const {α : Type u} {β : Type v} (l : List α) (b : β) : List.map (Function.const α b) l = List.replicate l.length b

@[simp]

theorem List.map_const' {α : Type u} {β : Type v} (l : List α) (b : β) : List.map (fun (x : α) => b) l = List.replicate l.length b

theorem List.eq_of_mem_map_const {α : Type u} {β : Type v} {b₁ : β} {b₂ : β} {l : List α} (h : b₁ ∈ List.map (Function.const α b₂) l) : b₁ = b₂

zipWith

theorem List.nil_zipWith {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : List β) : List.zipWith f [] l = []

theorem List.zipWith_nil {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : List α) : List.zipWith f l [] = []

@[simp]

theorem List.zipWith_flip {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (as : List α) (bs : List β) : List.zipWith (flip f) bs as = List.zipWith f as bs

take, drop

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

@[simp]

theorem List.drop_tail {α : Type u} (l : List α) (n : ℕ) : List.drop n l.tail = List.drop (n + 1) l

theorem List.cons_get_drop_succ {α : Type u} {l : List α} {n : Fin l.length} : l.get n :: List.drop (↑n + 1) l = List.drop (↑n) l

@[simp]

theorem List.takeI_length {α : Type u} [Inhabited α] (n : ℕ) (l : List α) : (List.takeI n l).length = n

@[simp]

theorem List.takeI_nil {α : Type u} [Inhabited α] (n : ℕ) : List.takeI n [] = List.replicate n default

theorem List.takeI_eq_take {α : Type u} [Inhabited α] {n : ℕ} {l : List α} : n ≤ l.length → List.takeI n l = List.take n l

@[simp]

theorem List.takeI_left {α : Type u} [Inhabited α] (l₁ : List α) (l₂ : List α) : List.takeI l₁.length (l₁ ++ l₂) = l₁

theorem List.takeI_left' {α : Type u} [Inhabited α] {l₁ : List α} {l₂ : List α} {n : ℕ} (h : l₁.length = n) : List.takeI n (l₁ ++ l₂) = l₁

@[simp]

theorem List.takeD_length {α : Type u} (n : ℕ) (l : List α) (a : α) : (List.takeD n l a).length = n

theorem List.takeD_eq_take {α : Type u} {n : ℕ} {l : List α} (a : α) : n ≤ l.length → List.takeD n l a = List.take n l

@[simp]

theorem List.takeD_left {α : Type u} (l₁ : List α) (l₂ : List α) (a : α) : List.takeD l₁.length (l₁ ++ l₂) a = l₁

theorem List.takeD_left' {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} {a : α} (h : l₁.length = n) : List.takeD n (l₁ ++ l₂) a = l₁

foldl, foldr

theorem List.foldl_ext {α : Type u} {β : Type v} (f : α → β → α) (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

theorem List.foldr_ext {α : Type u} {β : Type v} (f : α → β → β) (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

theorem List.foldl_concat {α : Type u} {β : Type v} (f : β → α → β) (b : β) (x : α) (xs : List α) : List.foldl f b (xs ++ [x]) = f (List.foldl f b xs) x

theorem List.foldr_concat {α : Type u} {β : Type v} (f : α → β → β) (b : β) (x : α) (xs : List α) : List.foldr f b (xs ++ [x]) = List.foldr f (f x b) xs

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

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

@[simp]

theorem List.foldl_fixed {α : Type u} {β : Type v} {a : α} (l : List β) : List.foldl (fun (a : α) (x : β) => a) a l = a

@[simp]

theorem List.foldr_fixed {α : Type u} {β : Type v} {b : β} (l : List α) : List.foldr (fun (x : α) (b : β) => b) b l = b

@[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

@[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 (fun (l : List α) (b : β) => List.foldr f b l) b L

theorem List.foldr_eta {α : Type u} (l : List α) : List.foldr List.cons [] l = l

@[simp]

theorem List.reverse_foldl {α : Type u} {l : List α} : (List.foldl (fun (t : List α) (h : α) => h :: t) [] l).reverse = l

theorem List.foldl_map' {α : Type u} {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)

theorem List.foldr_map' {α : Type u} {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)

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)

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)

theorem List.injective_foldl_comp {α : Type u} {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)

def List.foldrRecOn {α : Type u} {β : Type v} {C : β → Sort u_2} (l : List α) (op : α → β → β) (b : β) (hb : C b) (hl : (b : β) → C b → (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

• One or more equations did not get rendered due to their size.

def List.foldlRecOn {α : Type u} {β : Type v} {C : β → Sort u_2} (l : List α) (op : β → α → β) (b : β) (hb : C b) (hl : (b : β) → C b → (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

• One or more equations did not get rendered due to their size.

@[simp]

theorem List.foldrRecOn_nil {α : Type u} {β : Type v} {C : β → Sort u_2} (op : α → β → β) (b : β) (hb : C b) (hl : (b : β) → C b → (a : α) → a ∈ [] → C (op a b)) : [].foldrRecOn op b hb hl = hb

@[simp]

theorem List.foldrRecOn_cons {α : Type u} {β : Type v} {C : β → Sort u_2} (x : α) (l : List α) (op : α → β → β) (b : β) (hb : C b) (hl : (b : β) → C b → (a : α) → a ∈ x :: l → C (op a b)) : (x :: l).foldrRecOn op b hb hl = hl (List.foldr op b l) (l.foldrRecOn op b hb fun (b : β) (hb : C b) (a : α) (ha : a ∈ l) => hl b hb a ⋯) x ⋯

@[simp]

theorem List.foldlRecOn_nil {α : Type u} {β : Type v} {C : β → Sort u_2} (op : β → α → β) (b : β) (hb : C b) (hl : (b : β) → C b → (a : α) → a ∈ [] → C (op b a)) : [].foldlRecOn op b hb hl = hb

theorem List.append_cons_inj_of_not_mem {α : Type u} {x₁ : List α} {x₂ : List α} {z₁ : List α} {z₂ : List α} {a₁ : α} {a₂ : α} (notin_x : ¬a₂ ∈ x₁) (notin_z : ¬a₂ ∈ z₁) : x₁ ++ a₁ :: z₁ = x₂ ++ a₂ :: z₂ ↔ x₁ = x₂ ∧ a₁ = a₂ ∧ z₁ = z₂

Consider two lists l₁ and l₂ with designated elements a₁ and a₂ somewhere in them: l₁ = x₁ ++ [a₁] ++ z₁ and l₂ = x₂ ++ [a₂] ++ z₂. Assume the designated element a₂ is present in neither x₁ nor z₁. We conclude that the lists are equal (l₁ = l₂) if and only if their respective parts are equal (x₁ = x₂ ∧ a₁ = a₂ ∧ z₁ = z₂).

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

@[simp]

theorem List.scanl_nil {α : Type u} {β : Type v} {f : β → α → β} (b : β) : List.scanl f b [] = [b]

@[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

@[simp]

theorem List.get?_zero_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} : (List.scanl f b l).get? 0 = some b

@[simp]

theorem List.get_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).get ⟨0, h⟩ = b

@[simp, deprecated List.get_zero_scanl]

theorem List.nthLe_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).nthLe 0 h = b

theorem List.get?_succ_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} {i : ℕ} : (List.scanl f b l).get? (i + 1) = ((List.scanl f b l).get? i).bind fun (x : β) => Option.map (fun (y : α) => f x y) (l.get? i)

@[deprecated List.get_succ_scanl]

theorem List.nthLe_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).nthLe (i + 1) h = f ((List.scanl f b l).nthLe i ⋯) (l.nthLe i ⋯)

theorem List.get_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).get ⟨i + 1, h⟩ = f ((List.scanl f b l).get ⟨i, ⋯⟩) (l.get ⟨i, ⋯⟩)

@[simp]

theorem List.scanr_nil {α : Type u} {β : Type v} (f : α → β → β) (b : β) : List.scanr f b [] = [b]

@[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

theorem List.foldl1_eq_foldr1 {α : Type u} {f : α → α → α} (hassoc : Associative f) (a : α) (b : α) (l : List α) : List.foldl f a (l ++ [b]) = List.foldr f b (a :: l)

theorem List.foldl_eq_of_comm_of_assoc {α : Type u} {f : α → α → α} (hcomm : Commutative f) (hassoc : Associative f) (a : α) (b : α) (l : List α) : List.foldl f a (b :: l) = f b (List.foldl f a l)

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

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

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

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

theorem List.foldl_assoc {α : Type u} {op : α → α → α} [ha : Std.Associative op] {l : List α} {a₁ : α} {a₂ : α} : List.foldl op (op a₁ a₂) l = op a₁ (List.foldl op a₂ l)

theorem List.foldl_op_eq_op_foldr_assoc {α : Type u} {op : α → α → α} [ha : Std.Associative op] {l : List α} {a₁ : α} {a₂ : α} : op (List.foldl op a₁ l) a₂ = op a₁ (List.foldr (fun (x x_1 : α) => op x x_1) a₂ l)

theorem List.foldl_assoc_comm_cons {α : Type u} {op : α → α → α} [ha : Std.Associative op] [hc : Std.Commutative op] {l : List α} {a₁ : α} {a₂ : α} : List.foldl op a₂ (a₁ :: l) = op a₁ (List.foldl op a₂ l)

foldlM, foldrM, mapM

theorem List.foldrM_eq_foldr {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] [LawfulMonad m] (f : α → β → m β) (b : β) (l : List α) : List.foldrM f b l = List.foldr (fun (a : α) (mb : m β) => mb >>= f a) (pure b) l

theorem List.foldlM_eq_foldl {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] [LawfulMonad m] (f : β → α → m β) (b : β) (l : List α) : List.foldlM f b l = List.foldl (fun (mb : m β) (a : α) => do let b ← mb f b a) (pure b) l

intersperse

@[simp]

theorem List.intersperse_singleton {α : Type u} (a : α) (b : α) : List.intersperse a [b] = [b]

@[simp]

theorem List.intersperse_cons_cons {α : Type u} (a : α) (b : α) (c : α) (tl : List α) : List.intersperse a (b :: c :: tl) = b :: a :: List.intersperse a (c :: tl)

splitAt and splitOn

@[simp]

theorem List.splitAt_eq_take_drop {α : Type u} (n : ℕ) (l : List α) : List.splitAt n l = (List.take n l, List.drop n l)

theorem List.splitAt_eq_take_drop.go_eq_take_drop {α : Type u} (n : ℕ) (l : List α) (xs : List α) (acc : Array α) : List.splitAt.go l xs n acc = if n < xs.length then (acc.toList ++ List.take n xs, List.drop n xs) else (l, [])

@[simp]

theorem List.splitOn_nil {α : Type u} [DecidableEq α] (a : α) : List.splitOn a [] = [[]]

@[simp]

theorem List.splitOnP_nil {α : Type u} (p : α → Bool) : List.splitOnP p [] = [[]]

theorem List.splitOnP.go_ne_nil {α : Type u} (p : α → Bool) (xs : List α) (acc : List α) : List.splitOnP.go p xs acc ≠ []

theorem List.splitOnP.go_acc {α : Type u} (p : α → Bool) (xs : List α) (acc : List α) : List.splitOnP.go p xs acc = List.modifyHead (fun (x : List α) => acc.reverse ++ x) (List.splitOnP p xs)

theorem List.splitOnP_ne_nil {α : Type u} (p : α → Bool) (xs : List α) : List.splitOnP p xs ≠ []

@[simp]

theorem List.splitOnP_cons {α : Type u} (p : α → Bool) (x : α) (xs : List α) : List.splitOnP p (x :: xs) = if p x = true then [] :: List.splitOnP p xs else List.modifyHead (List.cons x) (List.splitOnP p xs)

theorem List.splitOnP_spec {α : Type u} (p : α → Bool) (as : List α) : (List.zipWith (fun (x x_1 : List α) => x ++ x_1) (List.splitOnP p as) (List.map (fun (x : α) => [x]) (List.filter p as) ++ [[]])).join = as

The original list L can be recovered by joining the lists produced by splitOnP p L, interspersed with the elements L.filter p.

theorem List.splitOnP_spec.join_zipWith {α : Type u} {xs : List (List α)} {ys : List (List α)} {a : α} (hxs : xs ≠ []) (hys : ys ≠ []) : (List.zipWith (fun (x x_1 : List α) => x ++ x_1) (List.modifyHead (List.cons a) xs) ys).join = a :: (List.zipWith (fun (x x_1 : List α) => x ++ x_1) xs ys).join

theorem List.splitOnP_eq_single {α : Type u} (p : α → Bool) (xs : List α) (h : ∀ (x : α), x ∈ xs → ¬p x = true) : List.splitOnP p xs = [xs]

If no element satisfies p in the list xs, then xs.splitOnP p = [xs]

theorem List.splitOnP_first {α : Type u} (p : α → Bool) (xs : List α) (h : ∀ (x : α), x ∈ xs → ¬p x = true) (sep : α) (hsep : p sep = true) (as : List α) : List.splitOnP p (xs ++ sep :: as) = xs :: List.splitOnP p as

When a list of the form [...xs, sep, ...as] is split on p, the first element is xs, assuming no element in xs satisfies p but sep does satisfy p

theorem List.intercalate_splitOn {α : Type u} (xs : List α) (x : α) [DecidableEq α] : [x].intercalate (List.splitOn x xs) = xs

intercalate [x] is the left inverse of splitOn x

theorem List.splitOn_intercalate {α : Type u} (ls : List (List α)) [DecidableEq α] (x : α) (hx : ∀ (l : List α), l ∈ ls → ¬x ∈ l) (hls : ls ≠ []) : List.splitOn x ([x].intercalate ls) = ls

splitOn x is the left inverse of intercalate [x], on the domain consisting of each nonempty list of lists ls whose elements do not contain x

modifyLast

theorem List.modifyLast.go_append_one {α : Type u} (f : α → α) (a : α) (tl : List α) (r : Array α) : List.modifyLast.go f (tl ++ [a]) r = r.toListAppend (List.modifyLast.go f (tl ++ [a]) #[])

theorem List.modifyLast_append_one {α : Type u} (f : α → α) (a : α) (l : List α) : List.modifyLast f (l ++ [a]) = l ++ [f a]

theorem List.modifyLast_append {α : Type u} (f : α → α) (l₁ : List α) (l₂ : List α) : l₂ ≠ [] → List.modifyLast f (l₁ ++ l₂) = l₁ ++ List.modifyLast f l₂

map for partial functions

@[simp]

theorem List.attach_nil {α : Type u} : [].attach = []

theorem List.sizeOf_lt_sizeOf_of_mem {α : Type u} [SizeOf α] {x : α} {l : List α} (hx : x ∈ l) : sizeOf x < sizeOf l

@[simp]

theorem List.pmap_eq_map {α : Type u} {β : Type v} (p : α → Prop) (f : α → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.pmap (fun (a : α) (x : p a) => f a) l H = List.map f l

theorem List.pmap_congr {α : Type u} {β : Type v} {p : α → Prop} {q : α → Prop} {f : (a : α) → p a → β} {g : (a : α) → q a → β} (l : List α) {H₁ : ∀ (a : α), a ∈ l → p a} {H₂ : ∀ (a : α), a ∈ l → q a} (h : ∀ (a : α), a ∈ l → ∀ (h₁ : p a) (h₂ : q a), f a h₁ = g a h₂) : List.pmap f l H₁ = List.pmap g l H₂

theorem List.map_pmap {α : Type u} {β : Type v} {γ : Type w} {p : α → Prop} (g : β → γ) (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.map g (List.pmap f l H) = List.pmap (fun (a : α) (h : p a) => g (f a h)) l H

theorem List.pmap_map {α : Type u} {β : Type v} {γ : Type w} {p : β → Prop} (g : (b : β) → p b → γ) (f : α → β) (l : List α) (H : ∀ (a : β), a ∈ List.map f l → p a) : List.pmap g (List.map f l) H = List.pmap (fun (a : α) (h : p (f a)) => g (f a) h) l ⋯

theorem List.pmap_eq_map_attach {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.pmap f l H = List.map (fun (x : { x : α // x ∈ l }) => f ↑x ⋯) l.attach

theorem List.attach_map_coe' {α : Type u} {β : Type v} (l : List α) (f : α → β) : List.map (fun (i : { i : α // i ∈ l }) => f ↑i) l.attach = List.map f l

theorem List.attach_map_val' {α : Type u} {β : Type v} (l : List α) (f : α → β) : List.map (fun (i : { x : α // x ∈ l }) => f ↑i) l.attach = List.map f l

@[simp]

theorem List.attach_map_val {α : Type u} (l : List α) : List.map Subtype.val l.attach = l

@[simp]

theorem List.mem_attach {α : Type u} (l : List α) (x : { x : α // x ∈ l }) : x ∈ l.attach

@[simp]

theorem List.mem_pmap {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} {b : β} : b ∈ List.pmap f l H ↔ ∃ (a : α), ∃ (h : a ∈ l), f a ⋯ = b

@[simp]

theorem List.length_pmap {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : (List.pmap f l H).length = l.length

@[simp]

theorem List.length_attach {α : Type u} (L : List α) : L.attach.length = L.length

@[simp]

theorem List.pmap_eq_nil {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : List.pmap f l H = [] ↔ l = []

@[simp]

theorem List.attach_eq_nil {α : Type u} (l : List α) : l.attach = [] ↔ l = []

theorem List.getLast_pmap {α : Type u} {β : Type v} (p : α → Prop) (f : (a : α) → p a → β) (l : List α) (hl₁ : ∀ (a : α), a ∈ l → p a) (hl₂ : l ≠ []) : (List.pmap f l hl₁).getLast ⋯ = f (l.getLast hl₂) ⋯

theorem List.get?_pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) (n : ℕ) : (List.pmap f l h).get? n = Option.pmap f (l.get? n) ⋯

theorem List.get_pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) {n : ℕ} (hn : n < (List.pmap f l h).length) : (List.pmap f l h).get ⟨n, hn⟩ = f (l.get ⟨n, ⋯⟩) ⋯

@[deprecated List.get_pmap]

theorem List.nthLe_pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) {n : ℕ} (hn : n < (List.pmap f l h).length) : (List.pmap f l h).nthLe n hn = f (l.nthLe n ⋯) ⋯

theorem List.pmap_append {ι : Type u_1} {α : Type u} {p : ι → Prop} (f : (a : ι) → p a → α) (l₁ : List ι) (l₂ : List ι) (h : ∀ (a : ι), a ∈ l₁ ++ l₂ → p a) : List.pmap f (l₁ ++ l₂) h = List.pmap f l₁ ⋯ ++ List.pmap f l₂ ⋯

theorem List.pmap_append' {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) (l₁ : List α) (l₂ : List α) (h₁ : ∀ (a : α), a ∈ l₁ → p a) (h₂ : ∀ (a : α), a ∈ l₂ → p a) : List.pmap f (l₁ ++ l₂) ⋯ = List.pmap f l₁ h₁ ++ List.pmap f l₂ h₂

find

@[deprecated List.mem_of_find?_eq_some]

theorem List.find?_mem : ∀ {α : Type u_1} {p : α → Bool} {a : α} {l : List α}, List.find? p l = some a → a ∈ l

Alias of List.mem_of_find?_eq_some.

lookmap

theorem List.lookmap.go_append {α : Type u} (f : α → Option α) (l : List α) (acc : Array α) : List.lookmap.go f l acc = acc.toListAppend (List.lookmap f l)

@[simp]

theorem List.lookmap_nil {α : Type u} (f : α → Option α) : List.lookmap f [] = []

@[simp]

theorem List.lookmap_cons_none {α : Type u} (f : α → Option α) {a : α} (l : List α) (h : f a = none) : List.lookmap f (a :: l) = a :: List.lookmap f l

@[simp]

theorem List.lookmap_cons_some {α : Type u} (f : α → Option α) {a : α} {b : α} (l : List α) (h : f a = some b) : List.lookmap f (a :: l) = b :: l

theorem List.lookmap_some {α : Type u} (l : List α) : List.lookmap some l = l

theorem List.lookmap_none {α : Type u} (l : List α) : List.lookmap (fun (x : α) => none) l = l

theorem List.lookmap_congr {α : Type u} {f : α → Option α} {g : α → Option α} {l : List α} : (∀ (a : α), a ∈ l → f a = g a) → List.lookmap f l = List.lookmap g l

theorem List.lookmap_of_forall_not {α : Type u} (f : α → Option α) {l : List α} (H : ∀ (a : α), a ∈ l → f a = none) : List.lookmap f l = l

theorem List.lookmap_map_eq {α : Type u} {β : Type v} (f : α → Option α) (g : α → β) (h : ∀ (a b : α), b ∈ f a → g a = g b) (l : List α) : List.map g (List.lookmap f l) = List.map g l

theorem List.lookmap_id' {α : Type u} (f : α → Option α) (h : ∀ (a b : α), b ∈ f a → a = b) (l : List α) : List.lookmap f l = l

theorem List.length_lookmap {α : Type u} (f : α → Option α) (l : List α) : (List.lookmap f l).length = l.length

filter

theorem List.length_eq_length_filter_add {α : Type u} {l : List α} (f : α → Bool) : l.length = (List.filter f l).length + (List.filter (fun (x : α) => !f x) l).length

filterMap

theorem List.Sublist.map {α : Type u} {β : Type v} (f : α → β) {l₁ : List α} {l₂ : List α} (s : l₁.Sublist l₂) : (List.map f l₁).Sublist (List.map f l₂)

theorem List.filterMap_eq_bind_toList {α : Type u} {β : Type v} (f : α → Option β) (l : List α) : List.filterMap f l = l.bind fun (a : α) => (f a).toList

theorem List.filterMap_congr {α : Type u} {β : Type v} {f : α → Option β} {g : α → Option β} {l : List α} (h : ∀ (x : α), x ∈ l → f x = g x) : List.filterMap f l = List.filterMap g l

theorem List.filterMap_eq_map_iff_forall_eq_some {α : Type u} {β : Type v} {f : α → Option β} {g : α → β} {l : List α} : List.filterMap f l = List.map g l ↔ ∀ (x : α), x ∈ l → f x = some (g x)

filter

theorem List.filter_singleton {α : Type u} {p : α → Bool} {a : α} : List.filter p [a] = bif p a then [a] else []

theorem List.filter_eq_foldr {α : Type u} (p : α → Bool) (l : List α) : List.filter p l = List.foldr (fun (a : α) (out : List α) => bif p a then a :: out else out) [] l

@[simp]

theorem List.filter_subset {α : Type u} {p : α → Bool} (l : List α) : List.filter p l ⊆ l

theorem List.of_mem_filter {α : Type u} {p : α → Bool} {a : α} {l : List α} (h : a ∈ List.filter p l) : p a = true

theorem List.mem_of_mem_filter {α : Type u} {p : α → Bool} {a : α} {l : List α} (h : a ∈ List.filter p l) : a ∈ l

theorem List.mem_filter_of_mem {α : Type u} {p : α → Bool} {a : α} {l : List α} (h₁ : a ∈ l) (h₂ : p a = true) : a ∈ List.filter p l

theorem List.monotone_filter_left {α : Type u} (p : α → Bool) ⦃l : List α⦄ ⦃l' : List α⦄ (h : l ⊆ l') : List.filter p l ⊆ List.filter p l'

theorem List.monotone_filter_right {α : Type u} (l : List α) ⦃p : α → Bool⦄ ⦃q : α → Bool⦄ (h : ∀ (a : α), p a = true → q a = true) : (List.filter p l).Sublist (List.filter q l)

theorem List.map_filter' {α : Type u} {β : Type v} (p : α → Bool) {f : α → β} (hf : Function.Injective f) (l : List α) [DecidablePred fun (b : β) => ∃ (a : α), p a = true ∧ f a = b] : List.map f (List.filter p l) = List.filter (fun (b : β) => decide (∃ (a : α), p a = true ∧ f a = b)) (List.map f l)

theorem List.filter_attach' {α : Type u} (l : List α) (p : { a : α // a ∈ l } → Bool) [DecidableEq α] : List.filter p l.attach = List.map (Subtype.map id ⋯) (List.filter (fun (x : α) => decide (∃ (h : x ∈ l), p ⟨x, h⟩ = true)) l).attach

theorem List.filter_attach {α : Type u} (l : List α) (p : α → Bool) : List.filter (fun (x : { x : α // x ∈ l }) => p ↑x) l.attach = List.map (Subtype.map id ⋯) (List.filter p l).attach

theorem List.filter_comm {α : Type u} (p : α → Bool) (q : α → Bool) (l : List α) : List.filter p (List.filter q l) = List.filter q (List.filter p l)

@[simp]

theorem List.filter_true {α : Type u} (l : List α) : List.filter (fun (x : α) => true) l = l

@[simp]

theorem List.filter_false {α : Type u} (l : List α) : List.filter (fun (x : α) => false) l = []

theorem List.span.loop_eq_take_drop {α : Type u} (p : α → Bool) (l₁ : List α) (l₂ : List α) : List.span.loop p l₁ l₂ = (l₂.reverse ++ List.takeWhile p l₁, List.dropWhile p l₁)

@[simp]

theorem List.span_eq_take_drop {α : Type u} (p : α → Bool) (l : List α) : List.span p l = (List.takeWhile p l, List.dropWhile p l)

theorem List.dropWhile_nthLe_zero_not {α : Type u} (p : α → Bool) (l : List α) (hl : 0 < (List.dropWhile p l).length) : ¬p ((List.dropWhile p l).nthLe 0 hl) = true

@[simp]

theorem List.dropWhile_eq_nil_iff {α : Type u} {p : α → Bool} {l : List α} : List.dropWhile p l = [] ↔ ∀ (x : α), x ∈ l → p x = true

@[simp]

theorem List.takeWhile_nil {α : Type u} {p : α → Bool} : List.takeWhile p [] = []

theorem List.takeWhile_cons {α : Type u} {p : α → Bool} {l : List α} {x : α} : List.takeWhile p (x :: l) = match p x with | true => x :: List.takeWhile p l | false => []

theorem List.takeWhile_cons_of_pos {α : Type u} {p : α → Bool} {l : List α} {x : α} (h : p x = true) : List.takeWhile p (x :: l) = x :: List.takeWhile p l

theorem List.takeWhile_cons_of_neg {α : Type u} {p : α → Bool} {l : List α} {x : α} (h : ¬p x = true) : List.takeWhile p (x :: l) = []

@[simp]

theorem List.takeWhile_eq_self_iff {α : Type u} {p : α → Bool} {l : List α} : List.takeWhile p l = l ↔ ∀ (x : α), x ∈ l → p x = true

@[simp]

theorem List.takeWhile_eq_nil_iff {α : Type u} {p : α → Bool} {l : List α} : List.takeWhile p l = [] ↔ ∀ (hl : 0 < l.length), ¬p (l.nthLe 0 hl) = true

theorem List.mem_takeWhile_imp {α : Type u} {p : α → Bool} {l : List α} {x : α} (hx : x ∈ List.takeWhile p l) : p x = true

theorem List.takeWhile_takeWhile {α : Type u} (p : α → Bool) (q : α → Bool) (l : List α) : List.takeWhile p (List.takeWhile q l) = List.takeWhile (fun (a : α) => decide (p a = true ∧ q a = true)) l

theorem List.takeWhile_idem {α : Type u} {p : α → Bool} {l : List α} : List.takeWhile p (List.takeWhile p l) = List.takeWhile p l

erasep

@[simp]

theorem List.length_eraseP_add_one {α : Type u} {p : α → Bool} {l : List α} {a : α} (al : a ∈ l) (pa : p a = true) : (List.eraseP p l).length + 1 = l.length

erase

@[simp]

theorem List.length_erase_add_one {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : (l.erase a).length + 1 = l.length

theorem List.map_erase {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : Function.Injective f) {a : α} (l : List α) : List.map f (l.erase a) = (List.map f l).erase (f a)

theorem List.map_foldl_erase {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : Function.Injective f) {l₁ : List α} {l₂ : List α} : List.map f (List.foldl List.erase l₁ l₂) = List.foldl (fun (l : List β) (a : α) => l.erase (f a)) (List.map f l₁) l₂

theorem List.erase_get {ι : Type u_1} [DecidableEq ι] {l : List ι} (i : Fin l.length) : (l.erase (l.get i)).Perm (l.eraseIdx ↑i)

theorem List.length_eraseIdx_add_one {ι : Type u_1} {l : List ι} {i : ℕ} (h : i < l.length) : (l.eraseIdx i).length + 1 = l.length

diff

@[simp]

theorem List.map_diff {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : Function.Injective f) {l₁ : List α} {l₂ : List α} : List.map f (l₁.diff l₂) = (List.map f l₁).diff (List.map f l₂)

theorem List.erase_diff_erase_sublist_of_sublist {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : l₁.Sublist l₂ → ((l₂.erase a).diff (l₁.erase a)).Sublist (l₂.diff l₁)

theorem List.choose_spec {α : Type u} (p : α → Prop) [DecidablePred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : List.choose p l hp ∈ l ∧ p (List.choose p l hp)

theorem List.choose_mem {α : Type u} (p : α → Prop) [DecidablePred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : List.choose p l hp ∈ l

theorem List.choose_property {α : Type u} (p : α → Prop) [DecidablePred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : p (List.choose p l hp)

map₂Left'

@[simp]

theorem List.map₂Left'_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) : List.map₂Left' f as [] = (List.map (fun (a : α) => f a none) as, [])

map₂Right'

@[simp]

theorem List.map₂Right'_nil_left {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (bs : List β) : List.map₂Right' f [] bs = (List.map (f none) bs, [])

@[simp]

theorem List.map₂Right'_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) : List.map₂Right' f as [] = ([], as)

theorem List.map₂Right'_nil_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (b : β) (bs : List β) : List.map₂Right' f [] (b :: bs) = (f none b :: List.map (f none) bs, [])

@[simp]

theorem List.map₂Right'_cons_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (a : α) (as : List α) (b : β) (bs : List β) : List.map₂Right' f (a :: as) (b :: bs) = let r := List.map₂Right' f as bs; (f (some a) b :: r.1, r.2)

zipLeft'

@[simp]

theorem List.zipLeft'_nil_right {α : Type u} {β : Type v} (as : List α) : as.zipLeft' [] = (List.map (fun (a : α) => (a, none)) as, [])

@[simp]

theorem List.zipLeft'_nil_left {α : Type u} {β : Type v} (bs : List β) : [].zipLeft' bs = ([], bs)

theorem List.zipLeft'_cons_nil {α : Type u} {β : Type v} (a : α) (as : List α) : (a :: as).zipLeft' [] = ((a, none) :: List.map (fun (a : α) => (a, none)) as, [])

@[simp]

theorem List.zipLeft'_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : (a :: as).zipLeft' (b :: bs) = let r := as.zipLeft' bs; ((a, some b) :: r.1, r.2)

zipRight'

@[simp]

theorem List.zipRight'_nil_left {α : Type u} {β : Type v} (bs : List β) : [].zipRight' bs = (List.map (fun (b : β) => (none, b)) bs, [])

@[simp]

theorem List.zipRight'_nil_right {α : Type u} {β : Type v} (as : List α) : as.zipRight' [] = ([], as)

theorem List.zipRight'_nil_cons {α : Type u} {β : Type v} (b : β) (bs : List β) : [].zipRight' (b :: bs) = ((none, b) :: List.map (fun (b : β) => (none, b)) bs, [])

@[simp]

theorem List.zipRight'_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : (a :: as).zipRight' (b :: bs) = let r := as.zipRight' bs; ((some a, b) :: r.1, r.2)

map₂Left

@[simp]

theorem List.map₂Left_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) : List.map₂Left f as [] = List.map (fun (a : α) => f a none) as

theorem List.map₂Left_eq_map₂Left' {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) (bs : List β) : List.map₂Left f as bs = (List.map₂Left' f as bs).1

theorem List.map₂Left_eq_zipWith {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) (bs : List β) : as.length ≤ bs.length → List.map₂Left f as bs = List.zipWith (fun (a : α) (b : β) => f a (some b)) as bs

map₂Right

@[simp]

theorem List.map₂Right_nil_left {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (bs : List β) : List.map₂Right f [] bs = List.map (f none) bs

@[simp]

theorem List.map₂Right_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) : List.map₂Right f as [] = []

theorem List.map₂Right_nil_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (b : β) (bs : List β) : List.map₂Right f [] (b :: bs) = f none b :: List.map (f none) bs

@[simp]

theorem List.map₂Right_cons_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (a : α) (as : List α) (b : β) (bs : List β) : List.map₂Right f (a :: as) (b :: bs) = f (some a) b :: List.map₂Right f as bs

theorem List.map₂Right_eq_map₂Right' {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) : List.map₂Right f as bs = (List.map₂Right' f as bs).1

theorem List.map₂Right_eq_zipWith {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) (h : bs.length ≤ as.length) : List.map₂Right f as bs = List.zipWith (fun (a : α) (b : β) => f (some a) b) as bs

zipLeft

@[simp]

theorem List.zipLeft_nil_right {α : Type u} {β : Type v} (as : List α) : as.zipLeft [] = List.map (fun (a : α) => (a, none)) as

@[simp]

theorem List.zipLeft_nil_left {α : Type u} {β : Type v} (bs : List β) : [].zipLeft bs = []

theorem List.zipLeft_cons_nil {α : Type u} {β : Type v} (a : α) (as : List α) : (a :: as).zipLeft [] = (a, none) :: List.map (fun (a : α) => (a, none)) as

@[simp]

theorem List.zipLeft_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : (a :: as).zipLeft (b :: bs) = (a, some b) :: as.zipLeft bs

theorem List.zipLeft_eq_zipLeft' {α : Type u} {β : Type v} (as : List α) (bs : List β) : as.zipLeft bs = (as.zipLeft' bs).1

zipRight

@[simp]

theorem List.zipRight_nil_left {α : Type u} {β : Type v} (bs : List β) : [].zipRight bs = List.map (fun (b : β) => (none, b)) bs

@[simp]

theorem List.zipRight_nil_right {α : Type u} {β : Type v} (as : List α) : as.zipRight [] = []

theorem List.zipRight_nil_cons {α : Type u} {β : Type v} (b : β) (bs : List β) : [].zipRight (b :: bs) = (none, b) :: List.map (fun (b : β) => (none, b)) bs

@[simp]

theorem List.zipRight_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : (a :: as).zipRight (b :: bs) = (some a, b) :: as.zipRight bs

theorem List.zipRight_eq_zipRight' {α : Type u} {β : Type v} (as : List α) (bs : List β) : as.zipRight bs = (as.zipRight' bs).1

toChunks

Forall

@[simp]

theorem List.forall_cons {α : Type u} (p : α → Prop) (x : α) (l : List α) : List.Forall p (x :: l) ↔ p x ∧ List.Forall p l

theorem List.forall_iff_forall_mem {α : Type u} {p : α → Prop} {l : List α} : List.Forall p l ↔ ∀ (x : α), x ∈ l → p x

theorem List.Forall.imp {α : Type u} {p : α → Prop} {q : α → Prop} (h : ∀ (x : α), p x → q x) {l : List α} : List.Forall p l → List.Forall q l

@[simp]

theorem List.forall_map_iff {α : Type u} {β : Type v} {l : List α} {p : β → Prop} (f : α → β) : List.Forall p (List.map f l) ↔ List.Forall (p ∘ f) l

instance List.instDecidablePredForall {α : Type u} (p : α → Prop) [DecidablePred p] : DecidablePred (List.Forall p)

Equations

• List.instDecidablePredForall p x = decidable_of_iff' (∀ (x_1 : α), x_1 ∈ x → p x_1) ⋯

Miscellaneous lemmas

theorem List.getLast_reverse {α : Type u} {l : List α} (hl : l.reverse ≠ []) (hl' : optParam (0 < l.length) ⋯) : l.reverse.getLast hl = l.get ⟨0, hl'⟩

@[simp]

theorem List.get_attach {α : Type u} (L : List α) (i : Fin L.attach.length) : ↑(L.attach.get i) = L.get ⟨↑i, ⋯⟩

@[simp]

theorem List.mem_map_swap {α : Type u} {β : Type v} (x : α) (y : β) (xs : List (α × β)) : (y, x) ∈ List.map Prod.swap xs ↔ (x, y) ∈ xs

theorem List.dropSlice_eq {α : Type u} (xs : List α) (n : ℕ) (m : ℕ) : List.dropSlice n m xs = List.take n xs ++ List.drop (n + m) xs

theorem List.sizeOf_dropSlice_lt {α : Type u} [SizeOf α] (i : ℕ) (j : ℕ) (hj : 0 < j) (xs : List α) (hi : i < xs.length) : sizeOf (List.dropSlice i j xs) < sizeOf xs

theorem List.disjoint_pmap {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {s : List α} {t : List α} (hs : ∀ (a : α), a ∈ s → p a) (ht : ∀ (a : α), a ∈ t → p a) (hf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a') (h : s.Disjoint t) : (List.pmap f s hs).Disjoint (List.pmap f t ht)

The images of disjoint lists under a partially defined map are disjoint

theorem List.disjoint_map {α : Type u} {β : Type v} {f : α → β} {s : List α} {t : List α} (hf : Function.Injective f) (h : s.Disjoint t) : (List.map f s).Disjoint (List.map f t)

The images of disjoint lists under an injective map are disjoint

theorem List.lookup_graph {α : Type u_2} {β : Type u_3} [BEq α] [LawfulBEq α] (f : α → β) {a : α} {as : List α} (h : a ∈ as) : List.lookup a (List.map (fun (x : α) => (x, f x)) as) = some (f a)