mathlib docs: data.list.basic

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

l.reverse_rec_on H0 H1 = _.mpr (list.rec H0 (λ (hd : α) (tl : list α) (ih : C tl.reverse), _.mpr (H1 tl.reverse hd ih)) l.reverse)

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

list.bidirectional_rec H0 H1 Hn (a :: b :: l) = let l' : list α := (b :: l).init, b' : α := (b :: l).last _ in _.mpr (Hn a l' b' (list.bidirectional_rec H0 H1 Hn l'))
list.bidirectional_rec H0 H1 Hn [a] = H1 a
list.bidirectional_rec H0 H1 Hn list.nil = H0

source

def list.bidirectional_rec_on {α : Type u} {C : list α → Sort u_1} (l : list α) :

C list.nil → (Π (a : α), C [a]) → (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

l.bidirectional_rec_on H0 H1 Hn = list.bidirectional_rec H0 H1 Hn l

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@[simp]

theorem list.nil_sublist {α : Type u} (l : list α) :

list.nil <+ l

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@[simp]

theorem list.sublist.refl {α : Type u} (l : list α) :

l <+ l

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theorem list.sublist.trans {α : Type u} {l₁ l₂ l₃ : list α} :

l₁ <+ l₂ → l₂ <+ l₃ → l₁ <+ l₃

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@[simp]

theorem list.sublist_cons {α : Type u} (a : α) (l : list α) :

l <+ a :: l

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theorem list.sublist_of_cons_sublist {α : Type u} {a : α} {l₁ l₂ : list α} :

a :: l₁ <+ l₂ → l₁ <+ l₂

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theorem list.cons_sublist_cons {α : Type u} {l₁ l₂ : list α} (a : α) :

l₁ <+ l₂ → a :: l₁ <+ a :: l₂

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@[simp]

theorem list.sublist_append_left {α : Type u} (l₁ l₂ : list α) :

l₁ <+ l₁ ++ l₂

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@[simp]

theorem list.sublist_append_right {α : Type u} (l₁ l₂ : list α) :

l₂ <+ l₁ ++ l₂

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theorem list.sublist_cons_of_sublist {α : Type u} (a : α) {l₁ l₂ : list α} :

l₁ <+ l₂ → l₁ <+ a :: l₂

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theorem list.sublist_append_of_sublist_left {α : Type u} {l l₁ l₂ : list α} :

l <+ l₁ → l <+ l₁ ++ l₂

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theorem list.sublist_append_of_sublist_right {α : Type u} {l l₁ l₂ : list α} :

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

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theorem list.cons_sublist_cons_iff {α : Type u} {l₁ l₂ : list α} {a : α} :

a :: l₁ <+ a :: l₂ ↔ l₁ <+ l₂

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@[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

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theorem list.sublist_or_mem_of_sublist {α : Type u} {l l₁ l₂ : list α} {a : α} :

l <+ l₁ ++ a :: l₂ → l <+ l₁ ++ l₂ ∨ a ∈ l

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theorem list.sublist.reverse {α : Type u} {l₁ l₂ : list α} :

l₁ <+ l₂ → l₁.reverse <+ l₂.reverse

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@[simp]

theorem list.reverse_sublist_iff {α : Type u} {l₁ l₂ : list α} :

l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂

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@[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} :

l₁ <+ l₂ → r₁ <+ r₂ → l₁ ++ r₁ <+ l₂ ++ r₂

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theorem list.sublist.subset {α : Type u} {l₁ l₂ : list α} :

l₁ <+ l₂ → l₁ ⊆ l₂

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theorem list.singleton_sublist {α : Type u} {a : α} {l : list α} :

[a] <+ l ↔ a ∈ l

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theorem list.eq_nil_of_sublist_nil {α : Type u} {l : list α} :

l <+ list.nil → l = list.nil

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theorem list.repeat_sublist_repeat {α : Type u} (a : α) {m n : ℕ} :

list.repeat a m <+ list.repeat a n ↔ m ≤ n

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theorem list.eq_of_sublist_of_length_eq {α : Type u} {l₁ l₂ : list α} :

l₁ <+ l₂ → l₁.length = l₂.length → l₁ = l₂

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theorem list.eq_of_sublist_of_length_le {α : Type u} {l₁ l₂ : list α} :

l₁ <+ l₂ → l₂.length ≤ l₁.length → l₁ = l₂

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theorem list.sublist.antisymm {α : Type u} {l₁ l₂ : list α} :

l₁ <+ l₂ → l₂ <+ l₁ → l₁ = l₂

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@[instance]

def list.decidable_sublist {α : Type u} [decidable_eq α] (l₁ l₂ : list α) :

decidable (l₁ <+ l₂)

Equations

(a :: l₁).decidable_sublist (b :: l₂) = dite (a = b) (λ (h : a = b), decidable_of_decidable_of_iff (l₁.decidable_sublist l₂) _) (λ (h : ¬a = b), decidable_of_decidable_of_iff ((a :: l₁).decidable_sublist l₂) _)
(a :: l₁).decidable_sublist list.nil = is_false _
list.nil.decidable_sublist (hd :: tl) = is_true _
list.nil.decidable_sublist list.nil = is_true list.decidable_sublist._main._proof_1

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@[simp]

theorem list.index_of_nil {α : Type u} [decidable_eq α] (a : α) :

list.index_of a list.nil = 0

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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

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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

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@[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

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@[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

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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

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@[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

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theorem list.index_of_le_length {α : Type u} [decidable_eq α] {a : α} {l : list α} :

list.index_of a l ≤ l.length

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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

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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)

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theorem list.nth_le_nth {α : Type u} {l : list α} {n : ℕ} (h : n < l.length) :

l.nth n = some (l.nth_le n h)

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theorem list.nth_len_le {α : Type u} {l : list α} {n : ℕ} :

l.length ≤ n → l.nth n = none

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theorem list.nth_eq_some {α : Type u} {l : list α} {n : ℕ} {a : α} :

l.nth n = some a ↔ ∃ (h : n < l.length), l.nth_le n h = a

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@[simp]

theorem list.nth_eq_none_iff {α : Type u} {l : list α} {n : ℕ} :

l.nth n = none ↔ l.length ≤ n

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theorem list.nth_of_mem {α : Type u} {a : α} {l : list α} :

a ∈ l → (∃ (n : ℕ), l.nth n = some a)

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theorem list.nth_le_mem {α : Type u} (l : list α) (n : ℕ) (h : n < l.length) :

l.nth_le n h ∈ l

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theorem list.nth_mem {α : Type u} {l : list α} {n : ℕ} {a : α} :

l.nth n = some a → a ∈ l

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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

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theorem list.mem_iff_nth {α : Type u} {a : α} {l : list α} :

a ∈ l ↔ ∃ (n : ℕ), l.nth n = some a

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theorem list.nth_injective {α : Type u} {xs : list α} {i j : ℕ} :

i < xs.length → xs.nodup → xs.nth i = xs.nth j → i = j

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@[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)

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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)

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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 _)

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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).

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@[simp]

theorem list.nth_le_singleton {α : Type u} (a : α) {n : ℕ} (hn : n < 1) :

[a].nth_le n hn = a

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theorem list.nth_le_zero {α : Type u} [inhabited α] {L : list α} (h : 0 < L.length) :

L.nth_le 0 h = L.head

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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

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theorem list.nth_le_append_right_aux {α : Type u} {l₁ l₂ : list α} {n : ℕ} :

l₁.length ≤ n → n < (l₁ ++ l₂).length → n - l₁.length < l₂.length

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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) _

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@[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

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theorem list.nth_append {α : Type u} {l₁ l₂ : list α} {n : ℕ} :

n < l₁.length → (l₁ ++ l₂).nth n = l₁.nth n

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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) _

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@[simp]

theorem list.nth_concat_length {α : Type u} (l : list α) (a : α) :

(l ++ [a]).nth l.length = some a

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@[ext]

theorem list.ext {α : Type u} {l₁ l₂ : list α} :

(∀ (n : ℕ), l₁.nth n = l₂.nth n) → l₁ = l₂

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theorem list.ext_le {α : Type u} {l₁ l₂ : list α} :

l₁.length = l₂.length → (∀ (n : ℕ) (h₁ : n < l₁.length) (h₂ : n < l₂.length), l₁.nth_le n h₁ = l₂.nth_le n h₂) → l₁ = l₂

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@[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

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@[simp]

theorem list.index_of_nth {α : Type u} [decidable_eq α] {a : α} {l : list α} :

a ∈ l → l.nth (list.index_of a l) = some a

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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

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theorem list.index_of_inj {α : Type u} [decidable_eq α] {l : list α} {x y : α} :

x ∈ l → y ∈ l → (list.index_of x l = list.index_of y l ↔ x = y)

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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

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@[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

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theorem list.eq_cons_of_length_one {α : Type u} {l : list α} (h : l.length = 1) :

l = [l.nth_le 0 _]

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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

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theorem list.modify_nth_tail_modify_nth_tail_le {α : Type u} {f g : list α → list α} (m n : ℕ) (l : list α) :

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

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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

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theorem list.modify_nth_tail_id {α : Type u} (n : ℕ) (l : list α) :

list.modify_nth_tail id n l = l

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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

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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

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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

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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

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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

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@[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

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@[simp]

theorem list.update_nth_length {α : Type u} (l : list α) (n : ℕ) (a : α) :

(l.update_nth n a).length = l.length

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@[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

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@[simp]

theorem list.nth_modify_nth_ne {α : Type u} (f : α → α) {m n : ℕ} (l : list α) :

m ≠ n → (list.modify_nth f m l).nth n = l.nth n

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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

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theorem list.nth_update_nth_of_lt {α : Type u} (a : α) {n : ℕ} {l : list α} :

n < l.length → (l.update_nth n a).nth n = some a

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theorem list.nth_update_nth_ne {α : Type u} (a : α) {m n : ℕ} (l : list α) :

m ≠ n → (l.update_nth m a).nth n = l.nth n

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@[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

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@[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 _

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theorem list.mem_or_eq_of_mem_update_nth {α : Type u} {l : list α} {n : ℕ} {a b : α} :

a ∈ l.update_nth n b → a ∈ l ∨ a = b

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@[simp]

theorem list.insert_nth_nil {α : Type u} (a : α) :

list.insert_nth 0 a list.nil = [a]

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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

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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

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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

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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)

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theorem list.insert_nth_comm {α : Type u} (a b : α) (i j : ℕ) (l : list α) :

i ≤ j → 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} :

n ≤ l.length → (a ∈ list.insert_nth n b l ↔ a = b ∨ a ∈ l)

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@[simp]

theorem list.map_nil {α : Type u} {β : Type v} (f : α → β) :

list.map f list.nil = list.nil

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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

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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

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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

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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)

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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} :

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

@[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.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.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

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} :

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 (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 (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_of_le_length {α : Type u} {l₁ l₂ : list α} {n : ℕ} :

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

@[simp]

theorem list.drop_nil {α : Type u} (n : ℕ) :

list.drop n list.nil = list.nil

source

theorem list.mem_of_mem_drop {α : Type u_1} {n : ℕ} {l : list α} {x : α} :

x ∈ list.drop n l → x ∈ l

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} :

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_of_le_length {α : Type u} {l₁ l₂ : list α} {n : ℕ} :

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.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

@[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} :

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) :

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 (default α) 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} :

l₁.length = n → list.take' n (l₁ ++ l₂) = l₁

source

theorem list.foldl_ext {α : Type u} {β : Type v} (f g : α → β → α) (a : α) {l : list β} :

(∀ (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} :

(∀ (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

@[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) :

(∀ (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) :

(∀ (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) :

(∀ (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) :

(∀ (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.injective_foldl_comp {α : Type u_1} {l : list (α → α)} {f : α → α} :

(∀ (f : α → α), f ∈ l → function.injective f) → function.injective f → function.injective (list.foldl function.comp f l)

source

theorem list.length_scanl {α : Type u} {β : Type u_1} {f : α → β → α} (a : α) (l : list β) :

(list.scanl f a l).length = l.length + 1

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)

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} :

mfoldl f b list.nil = 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 = 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} :

mfoldl f b (a :: l) = f b a >>= λ (b' : β), 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) (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) :

mfoldl f b l = list.foldl (λ (mb : m β) (a : α), mb >>= λ (b : β), f b a) (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} :

mfoldl f b (l₁ ++ l₂) = mfoldl f b l₁ >>= λ (x : β), 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

source

@[simp]

theorem list.sum_nil {α : Type u} [add_monoid α] :

list.nil.sum = 0

source

@[simp]

theorem list.prod_nil {α : Type u} [monoid α] :

list.nil.prod = 1

source

theorem list.sum_singleton {α : Type u} [add_monoid α] {a : α} :

[a].sum = a

source

theorem list.prod_singleton {α : Type u} [monoid α] {a : α} :

[a].prod = a

source

@[simp]

theorem list.sum_cons {α : Type u} [add_monoid α] {l : list α} {a : α} :

(a :: l).sum = a + l.sum

source

@[simp]

theorem list.prod_cons {α : Type u} [monoid α] {l : list α} {a : α} :

(a :: l).prod = a * l.prod

source

@[simp]

theorem list.prod_append {α : Type u} [monoid α] {l₁ l₂ : list α} :

(l₁ ++ l₂).prod = (l₁.prod) * l₂.prod

source

@[simp]

theorem list.sum_append {α : Type u} [add_monoid α] {l₁ l₂ : list α} :

(l₁ ++ l₂).sum = l₁.sum + l₂.sum

source

@[simp]

theorem list.sum_join {α : Type u} [add_monoid α] {l : list (list α)} :

l.join.sum = (list.map list.sum l).sum

source

@[simp]

theorem list.prod_join {α : Type u} [monoid α] {l : list (list α)} :

l.join.prod = (list.map list.prod l).prod

source

theorem list.prod_ne_zero {R : Type u_1} [domain R] {L : list R} :

(∀ (x : R), x ∈ L → x ≠ 0) → L.prod ≠ 0

source

theorem list.sum_eq_foldr {α : Type u} [add_monoid α] {l : list α} :

l.sum = list.foldr has_add.add 0 l

source

theorem list.prod_eq_foldr {α : Type u} [monoid α] {l : list α} :

l.prod = list.foldr has_mul.mul 1 l

source

theorem list.prod_hom_rel {α : Type u_1} {β : Type u_2} {γ : Type u_3} [monoid β] [monoid γ] (l : list α) {r : β → γ → Prop} {f : α → β} {g : α → γ} :

r 1 1 → (∀ ⦃a : α⦄ ⦃b : β⦄ ⦃c : γ⦄, r b c → r ((f a) * b) ((g a) * c)) → r (list.map f l).prod (list.map g l).prod

source

theorem list.sum_hom_rel {α : Type u_1} {β : Type u_2} {γ : Type u_3} [add_monoid β] [add_monoid γ] (l : list α) {r : β → γ → Prop} {f : α → β} {g : α → γ} :

r 0 0 → (∀ ⦃a : α⦄ ⦃b : β⦄ ⦃c : γ⦄, r b c → r (f a + b) (g a + c)) → r (list.map f l).sum (list.map g l).sum

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theorem list.sum_hom {α : Type u} {β : Type v} [add_monoid α] [add_monoid β] (l : list α) (f : α →+ β) :

(list.map ⇑f l).sum = ⇑f l.sum

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theorem list.prod_hom {α : Type u} {β : Type v} [monoid α] [monoid β] (l : list α) (f : α →* β) :

(list.map ⇑f l).prod = ⇑f l.prod

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@[simp]

theorem list.prod_take_mul_prod_drop {α : Type u} [monoid α] (L : list α) (i : ℕ) :

((list.take i L).prod) * (list.drop i L).prod = L.prod

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@[simp]

theorem list.prod_take_succ {α : Type u} [monoid α] (L : list α) (i : ℕ) (p : i < L.length) :

(list.take (i + 1) L).prod = ((list.take i L).prod) * L.nth_le i p

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theorem list.length_pos_of_prod_ne_one {α : Type u} [monoid α] (L : list α) :

L.prod ≠ 1 → 0 < L.length

A list with product not one must have positive length.

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theorem list.prod_update_nth {α : Type u} [monoid α] (L : list α) (n : ℕ) (a : α) :

(L.update_nth n a).prod = (((list.take n L).prod) * ite (n < L.length) a 1) * (list.drop (n + 1) L).prod

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@[simp]

theorem list.sum_take_add_sum_drop {α : Type u} [add_monoid α] (L : list α) (i : ℕ) :

(list.take i L).sum + (list.drop i L).sum = L.sum

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@[simp]

theorem list.sum_take_succ {α : Type u} [add_monoid α] (L : list α) (i : ℕ) (p : i < L.length) :

(list.take (i + 1) L).sum = (list.take i L).sum + L.nth_le i p

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theorem list.eq_of_sum_take_eq {α : Type u} [add_left_cancel_monoid α] {L L' : list α} :

L.length = L'.length → (∀ (i : ℕ), i ≤ L.length → (list.take i L).sum = (list.take i L').sum) → L = L'

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theorem list.monotone_sum_take {α : Type u} [canonically_ordered_add_monoid α] (L : list α) :

monotone (λ (i : ℕ), (list.take i L).sum)

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theorem list.sum_nonneg {α : Type u} [ordered_add_comm_monoid α] {l : list α} :

(∀ (x : α), x ∈ l → 0 ≤ x) → 0 ≤ l.sum

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theorem list.one_le_prod_of_one_le {α : Type u} [ordered_comm_monoid α] {l : list α} :

(∀ (x : α), x ∈ l → 1 ≤ x) → 1 ≤ l.prod

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theorem list.single_le_sum {α : Type u} [ordered_add_comm_monoid α] {l : list α} (hl₁ : ∀ (x : α), x ∈ l → 0 ≤ x) (x : α) :

x ∈ l → x ≤ l.sum

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theorem list.single_le_prod {α : Type u} [ordered_comm_monoid α] {l : list α} (hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x) (x : α) :

x ∈ l → x ≤ l.prod

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theorem list.all_zero_of_le_zero_le_of_sum_eq_zero {α : Type u} [ordered_add_comm_monoid α] {l : list α} (hl₁ : ∀ (x : α), x ∈ l → 0 ≤ x) (hl₂ : l.sum = 0) (x : α) :

x ∈ l → x = 0

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theorem list.all_one_of_le_one_le_of_prod_eq_one {α : Type u} [ordered_comm_monoid α] {l : list α} (hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x) (hl₂ : l.prod = 1) (x : α) :

x ∈ l → x = 1

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theorem list.sum_eq_zero_iff {α : Type u} [canonically_ordered_add_monoid α] (l : list α) :

l.sum = 0 ↔ ∀ (x : α), x ∈ l → x = 0

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theorem list.length_pos_of_sum_ne_zero {α : Type u} [add_monoid α] (L : list α) :

L.sum ≠ 0 → 0 < L.length

A list with sum not zero must have positive length.

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theorem list.length_le_sum_of_one_le (L : list ℕ) :

(∀ (i : ℕ), i ∈ L → 1 ≤ i) → L.length ≤ L.sum

If all elements in a list are bounded below by 1, then the length of the list is bounded by the sum of the elements.

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theorem list.length_pos_of_sum_pos {α : Type u} [ordered_cancel_add_comm_monoid α] (L : list α) :

0 < L.sum → 0 < L.length

A list with positive sum must have positive length.

source

@[simp]

theorem list.sum_erase {α : Type u} [decidable_eq α] [add_comm_monoid α] {a : α} {l : list α} :

a ∈ l → a + (l.erase a).sum = l.sum

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@[simp]

theorem list.prod_erase {α : Type u} [decidable_eq α] [comm_monoid α] {a : α} {l : list α} :

a ∈ l → a * (l.erase a).prod = l.prod

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theorem list.dvd_prod {α : Type u} [comm_monoid α] {a : α} {l : list α} :

a ∈ l → a ∣ l.prod

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@[simp]

theorem list.sum_const_nat (m n : ℕ) :

(list.repeat m n).sum = m * n

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theorem list.dvd_sum {α : Type u} [comm_semiring α] {a : α} {l : list α} :

(∀ (x : α), x ∈ l → a ∣ x) → a ∣ l.sum

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@[simp]

theorem list.length_join {α : Type u} (L : list (list α)) :

L.join.length = (list.map list.length L).sum

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@[simp]

theorem list.length_bind {α : Type u} {β : Type v} (l : list α) (f : α → list β) :

(l.bind f).length = (list.map (list.length ∘ f) l).sum

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theorem list.exists_lt_of_sum_lt {α : Type u} {β : Type v} [decidable_linear_ordered_cancel_add_comm_monoid β] {l : list α} (f g : α → β) :

(list.map f l).sum < (list.map g l).sum → (∃ (x : α) (H : x ∈ l), f x < g x)

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theorem list.exists_le_of_sum_le {α : Type u} {β : Type v} [decidable_linear_ordered_cancel_add_comm_monoid β] {l : list α} (hl : l ≠ list.nil) (f g : α → β) :

(list.map f l).sum ≤ (list.map g l).sum → (∃ (x : α) (H : x ∈ l), f x ≤ g x)

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theorem list.head_mul_tail_prod' {α : Type u} [monoid α] (L : list α) :

((L.nth 0).get_or_else 1) * L.tail.prod = L.prod

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theorem list.head_add_tail_sum' {α : Type u} [add_monoid α] (L : list α) :

(L.nth 0).get_or_else 0 + L.tail.sum = L.sum

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theorem list.head_add_tail_sum (L : list ℕ) :

L.head + L.tail.sum = L.sum

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theorem list.head_le_sum (L : list ℕ) :

L.head ≤ L.sum

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theorem list.tail_sum (L : list ℕ) :

L.tail.sum = L.sum - L.head

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@[simp]

theorem list.alternating_sum_nil {G : Type u_1} [add_comm_group G] :

list.nil.alternating_sum = 0

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@[simp]

theorem list.alternating_prod_nil {G : Type u_1} [comm_group G] :

list.nil.alternating_prod = 1

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@[simp]

theorem list.alternating_prod_singleton {G : Type u_1} [comm_group G] (g : G) :

[g].alternating_prod = g

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@[simp]

theorem list.alternating_sum_singleton {G : Type u_1} [add_comm_group G] (g : G) :

[g].alternating_sum = g

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@[simp]

theorem list.alternating_sum_cons_cons' {G : Type u_1} [add_comm_group G] (g h : G) (l : list G) :

(g :: h :: l).alternating_sum = g + -h + l.alternating_sum

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@[simp]

theorem list.alternating_prod_cons_cons {G : Type u_1} [comm_group G] (g h : G) (l : list G) :

(g :: h :: l).alternating_prod = (g * h⁻¹) * l.alternating_prod

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theorem list.alternating_sum_cons_cons {G : Type u_1} [add_comm_group G] (g h : G) (l : list G) :

(g :: h :: l).alternating_sum = g - h + l.alternating_sum

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theorem list.join_eq_nil {α : Type u} {L : list (list α)} :

L.join = list.nil ↔ ∀ (l : list α), l ∈ L → l = list.nil

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@[simp]

theorem list.join_append {α : Type u} (L₁ L₂ : list (list α)) :

(L₁ ++ L₂).join = L₁.join ++ L₂.join

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theorem list.join_join {α : Type u} (l : list (list (list α))) :

l.join.join = (list.map list.join l).join

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theorem list.take_sum_join {α : Type u} (L : list (list α)) (i : ℕ) :

list.take (list.take i (list.map list.length L)).sum L.join = (list.take i L).join

In a join, taking the first elements up to an index which is the sum of the lengths of the first i sublists, is the same as taking the join of the first i sublists.

source

theorem list.drop_sum_join {α : Type u} (L : list (list α)) (i : ℕ) :

list.drop (list.take i (list.map list.length L)).sum L.join = (list.drop i L).join

In a join, dropping all the elements up to an index which is the sum of the lengths of the first i sublists, is the same as taking the join after dropping the first i sublists.

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theorem list.drop_take_succ_eq_cons_nth_le {α : Type u} (L : list α) {i : ℕ} (hi : i < L.length) :

list.drop i (list.take (i + 1) L) = [L.nth_le i hi]

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

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theorem list.drop_take_succ_join_eq_nth_le {α : Type u} (L : list (list α)) {i : ℕ} (hi : i < L.length) :

list.drop (list.take i (list.map list.length L)).sum (list.take (list.take (i + 1) (list.map list.length L)).sum L.join) = L.nth_le i hi

In a join of sublists, taking the slice between the indices A and B - 1 gives back the original sublist of index i if A is the sum of the lenghts of sublists of index < i, and B is the sum of the lengths of sublists of index ≤ i.

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theorem list.sum_take_map_length_lt1 {α : Type u} (L : list (list α)) {i j : ℕ} (hi : i < L.length) :

j < (L.nth_le i hi).length → (list.take i (list.map list.length L)).sum + j < (list.take (i + 1) (list.map list.length L)).sum

Auxiliary lemma to control elements in a join.

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theorem list.sum_take_map_length_lt2 {α : Type u} (L : list (list α)) {i j : ℕ} (hi : i < L.length) :

j < (L.nth_le i hi).length → (list.take i (list.map list.length L)).sum + j < L.join.length

Auxiliary lemma to control elements in a join.

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theorem list.nth_le_join {α : Type u} (L : list (list α)) {i j : ℕ} (hi : i < L.length) (hj : j < (L.nth_le i hi).length) :

L.join.nth_le ((list.take i (list.map list.length L)).sum + j) _ = (L.nth_le i hi).nth_le j hj

The n-th element in a join of sublists is the j-th element of the ith sublist, where n can be obtained in terms of i and j by adding the lengths of all the sublists of index < i, and adding j.

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theorem list.eq_iff_join_eq {α : Type u} (L L' : list (list α)) :

L = L' ↔ L.join = L'.join ∧ list.map list.length L = list.map list.length L'

Two lists of sublists are equal iff their joins coincide, as well as the lengths of the sublists.

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inductive list.lex {α : Type u} :

(α → α → Prop) → list α → list α → Prop

nil : ∀ {α : Type u} (r : α → α → Prop) {a : α} {l : list α}, list.lex r list.nil (a :: l)
cons : ∀ {α : Type u} (r : α → α → Prop) {a : α} {l₁ l₂ : list α}, list.lex r l₁ l₂ → list.lex r (a :: l₁) (a :: l₂)
rel : ∀ {α : Type u} (r : α → α → Prop) {a₁ : α} {l₁ : list α} {a₂ : α} {l₂ : list α}, r a₁ a₂ → list.lex r (a₁ :: l₁) (a₂ :: l₂)

Given a strict order < on α, the lexicographic strict order on list α, for which [a0, ..., an] < [b0, ..., b_k] if a0 < b0 or a0 = b0 and [a1, ..., an] < [b1, ..., bk]. The definition is given for any relation r, not only strict orders.