Basic properties of lists #
@[instance]
mem #
theorem
list.mem_map_of_injective
{α : Type u}
{β : Type v}
{f : α → β}
(H : function.injective f)
{a : α}
{l : list α} :
length #
@[simp]
@[simp]
set-theoretic notation of lists #
bounded quantifiers over lists #
list subset #
theorem
list.map_subset_iff
{α : Type u}
{β : Type v}
{l₁ l₂ : list α}
(f : α → β)
(h : function.injective f) :
append #
@[simp]
theorem
list.split_at_eq_take_drop
{α : Type u}
(n : ℕ)
(l : list α) :
list.split_at n l = (list.take n l, list.drop n l)
theorem
list.append_right_injective
{α : Type u}
(s : list α) :
function.injective (λ (t : list α), s ++ t)
theorem
list.append_left_injective
{α : Type u}
(t : list α) :
function.injective (λ (s : list α), s ++ t)
repeat #
@[simp]
theorem
list.repeat_succ
{α : Type u}
(a : α)
(n : ℕ) :
list.repeat a (n + 1) = a :: list.repeat a n
theorem
list.eq_repeat_of_mem
{α : Type u}
{a : α}
{l : list α} :
(∀ (b : α), b ∈ l → b = a) → l = list.repeat a l.length
theorem
list.repeat_add
{α : Type u}
(a : α)
(m n : ℕ) :
list.repeat a (m + n) = list.repeat a m ++ list.repeat a n
@[simp]
theorem
list.map_const
{α : Type u}
{β : Type v}
(l : list α)
(b : β) :
list.map (function.const α b) l = list.repeat b l.length
theorem
list.eq_of_mem_map_const
{α : Type u}
{β : Type v}
{b₁ b₂ : β}
{l : list α}
(h : b₁ ∈ list.map (function.const α b₂) l) :
b₁ = b₂
@[simp]
theorem
list.map_repeat
{α : Type u}
{β : Type v}
(f : α → β)
(a : α)
(n : ℕ) :
list.map f (list.repeat a n) = list.repeat (f a) n
@[simp]
theorem
list.tail_repeat
{α : Type u}
(a : α)
(n : ℕ) :
(list.repeat a n).tail = list.repeat a n.pred
@[simp]
theorem
list.repeat_left_injective
{α : Type u}
{n : ℕ}
(hn : n ≠ 0) :
function.injective (λ (a : α), list.repeat a n)
theorem
list.repeat_left_inj
{α : Type u}
{a b : α}
{n : ℕ}
(hn : n ≠ 0) :
list.repeat a n = list.repeat b n ↔ a = b
@[simp]
theorem
list.repeat_left_inj'
{α : Type u}
{a b : α}
{n : ℕ} :
list.repeat a n = list.repeat b n ↔ n = 0 ∨ a = b
@[simp]
theorem
list.repeat_right_inj
{α : Type u}
{a : α}
{n m : ℕ} :
list.repeat a n = list.repeat a m ↔ n = m
pure #
bind #
@[simp]
concat #
reverse #
theorem
list.map_reverse_core
{α : Type u}
{β : Type v}
(f : α → β)
(l₁ l₂ : list α) :
list.map f (l₁.reverse_core l₂) = (list.map f l₁).reverse_core (list.map f l₂)
@[simp]
theorem
list.reverse_repeat
{α : Type u}
(a : α)
(n : ℕ) :
(list.repeat a n).reverse = list.repeat a n
empty #
init #
last #
last' #
head(') and tail #
@[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'
Induction from the right #
def
list.reverse_rec_on
{α : Type u}
{C : list α → Sort u_1}
(l : list α)
(H0 : C list.nil)
(H1 : Π (l : list α) (a : α), C l → C (l ++ [a])) :
C l
Induction principle from the right for lists: if a property holds for the empty list, and
for l ++ [a] if it holds for l, then it holds for all lists. The principle is given for
a Sort-valued predicate, i.e., it can also be used to construct data.
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
def
list.bidirectional_rec_on
{α : Type u}
{C : list α → Sort u_1}
(l : list α)
(H0 : C list.nil)
(H1 : Π (a : α), C [a])
(Hn : Π (a : α) (l : list α) (b : α), C l → C (a :: (l ++ [b]))) :
C l
Like bidirectional_rec, but with the list parameter placed first.
Equations
- l.bidirectional_rec_on H0 H1 Hn = list.bidirectional_rec H0 H1 Hn l
sublists #
theorem
list.repeat_sublist_repeat
{α : Type u}
(a : α)
{m n : ℕ} :
list.repeat a m <+ list.repeat a n ↔ m ≤ n
@[instance]
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
index_of #
@[simp]
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
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
@[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
@[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
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
@[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
theorem
list.index_of_le_length
{α : Type u}
[decidable_eq α]
{a : α}
{l : list α} :
list.index_of a l ≤ l.length
theorem
list.index_of_lt_length
{α : Type u}
[decidable_eq α]
{a : α}
{l : list α} :
list.index_of a l < l.length ↔ a ∈ l
nth element #
@[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)
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).
@[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
@[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
@[simp]
theorem
list.index_of_nth
{α : Type u}
[decidable_eq α]
{a : α}
{l : list α}
(h : a ∈ l) :
l.nth (list.index_of a l) = some a
theorem
list.index_of_inj
{α : Type u}
[decidable_eq α]
{l : list α}
{x y : α}
(hx : x ∈ l)
(hy : y ∈ l) :
list.index_of x l = list.index_of y l ↔ x = y
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
theorem
list.modify_nth_tail_modify_nth_tail_le
{α : Type u}
{f g : list α → list α}
(m n : ℕ)
(l : list α)
(h : n ≤ m) :
list.modify_nth_tail g m (list.modify_nth_tail f n l) = list.modify_nth_tail (λ (l : list α), list.modify_nth_tail g (m - n) (f l)) n l
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
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
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
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)) <$> l.nth n).get_or_else l
@[simp]
theorem
list.modify_nth_length
{α : Type u}
(f : α → α)
(n : ℕ)
(l : list α) :
(list.modify_nth f n l).length = l.length
@[simp]
theorem
list.update_nth_length
{α : Type u}
(l : list α)
(n : ℕ)
(a : α) :
(l.update_nth n a).length = l.length
@[simp]
theorem
list.nth_modify_nth_eq
{α : Type u}
(f : α → α)
(n : ℕ)
(l : list α) :
(list.modify_nth f n l).nth n = f <$> l.nth n
@[simp]
theorem
list.nth_modify_nth_ne
{α : Type u}
(f : α → α)
{m n : ℕ}
(l : list α)
(h : m ≠ n) :
(list.modify_nth f m l).nth n = l.nth n
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
theorem
list.nth_update_nth_of_lt
{α : Type u}
(a : α)
{n : ℕ}
{l : list α}
(h : n < l.length) :
(l.update_nth n a).nth n = some a
theorem
list.nth_update_nth_ne
{α : Type u}
(a : α)
{m n : ℕ}
(l : list α)
(h : m ≠ n) :
(l.update_nth m a).nth n = l.nth n
@[simp]
@[simp]
theorem
list.update_nth_succ
{α : Type u}
(x : α)
(xs : list α)
(n : ℕ)
(a : α) :
(x :: xs).update_nth n.succ a = x :: xs.update_nth n a
theorem
list.update_nth_comm
{α : Type u}
(a b : α)
{n m : ℕ}
(l : list α)
(h : n ≠ m) :
(l.update_nth n a).update_nth m b = (l.update_nth m b).update_nth n a
@[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
@[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 _
theorem
list.mem_or_eq_of_mem_update_nth
{α : Type u}
{l : list α}
{n : ℕ}
{a b : α}
(h : a ∈ l.update_nth n b) :
@[simp]
@[simp]
theorem
list.insert_nth_succ_nil
{α : Type u}
(n : ℕ)
(a : α) :
list.insert_nth (n + 1) a list.nil = list.nil
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
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
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)
theorem
list.insert_nth_comm
{α : Type u}
(a b : α)
(i j : ℕ)
(l : list α)
(h : i ≤ j)
(hj : j ≤ l.length) :
list.insert_nth (j + 1) b (list.insert_nth i a l) = list.insert_nth i a (list.insert_nth j b l)
map #
@[simp]
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.
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
map₂ #
take, drop #
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)
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)
@[simp]
@[simp]
theorem
list.take'_length
{α : Type u}
[inhabited α]
(n : ℕ)
(l : list α) :
(list.take' n l).length = n
@[simp]
theorem
list.take'_nil
{α : Type u}
[inhabited α]
(n : ℕ) :
list.take' n list.nil = list.repeat (default α) n
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
@[simp]
theorem
list.take'_left
{α : Type u}
[inhabited α]
(l₁ l₂ : list α) :
list.take' l₁.length (l₁ ++ l₂) = l₁
theorem
list.take'_left'
{α : Type u}
[inhabited α]
{l₁ l₂ : list α}
{n : ℕ}
(h : l₁.length = n) :
list.take' n (l₁ ++ l₂) = l₁
foldl, foldr #
theorem
list.foldl_ext
{α : Type u}
{β : Type v}
(f g : α → β → α)
(a : α)
{l : list β}
(H : ∀ (a : α) (b : β), b ∈ l → f a b = g a b) :
list.foldl f a l = list.foldl g a l
theorem
list.foldr_ext
{α : Type u}
{β : Type v}
(f g : α → β → β)
(b : β)
{l : list α}
(H : ∀ (a : α), a ∈ l → ∀ (b : β), f a b = g a b) :
list.foldr f b l = list.foldr g b l
@[simp]
theorem
list.foldl_nil
{α : Type u}
{β : Type v}
(f : α → β → α)
(a : α) :
list.foldl f a list.nil = a
@[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
@[simp]
theorem
list.foldr_nil
{α : Type u}
{β : Type v}
(f : α → β → β)
(b : β) :
list.foldr f b list.nil = b
@[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)
@[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
@[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
@[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 (λ (l : list α) (b : β), list.foldr f b l) b L
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
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
@[simp]
@[simp]
@[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
@[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
theorem
list.foldl_map'
{α β : Type u}
(g : α → β)
(f : α → α → α)
(f' : β → β → β)
(a : α)
(l : list α)
(h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) :
list.foldl f' (g a) (list.map g l) = g (list.foldl f a l)
theorem
list.foldr_map'
{α β : Type u}
(g : α → β)
(f : α → α → α)
(f' : β → β → β)
(a : α)
(l : list α)
(h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) :
list.foldr f' (g a) (list.map g l) = g (list.foldr f a l)
theorem
list.foldl_hom
{α : Type u}
{β : Type v}
{γ : Type w}
(l : list γ)
(f : α → β)
(op : α → γ → α)
(op' : β → γ → β)
(a : α)
(h : ∀ (a : α) (x : γ), f (op a x) = op' (f a) x) :
list.foldl op' (f a) l = f (list.foldl op a l)
theorem
list.foldr_hom
{α : Type u}
{β : Type v}
{γ : Type w}
(l : list γ)
(f : α → β)
(op : γ → α → α)
(op' : γ → β → β)
(a : α)
(h : ∀ (x : γ) (a : α), f (op x a) = op' x (f a)) :
list.foldr op' (f a) l = f (list.foldr op a l)
theorem
list.injective_foldl_comp
{α : Type u_1}
{l : list (α → α)}
{f : α → α}
(hl : ∀ (f : α → α), f ∈ l → function.injective f)
(hf : function.injective f) :
def
list.foldr_rec_on
{α : Type u}
{β : Type v}
{C : β → Sort u_1}
(l : list α)
(op : α → β → β)
(b : β)
(hb : C b)
(hl : Π (b : β), C b → Π (a : α), a ∈ l → C (op a b)) :
C (list.foldr op b l)
Induction principle for values produced by a foldr: if a property holds
for the seed element b : β and for all incremental op : α → β → β
performed on the elements (a : α) ∈ l. The principle is given for
a Sort-valued predicate, i.e., it can also be used to construct data.
Equations
- l.foldr_rec_on op b hb hl = list.rec (λ (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op a b)), hb) (λ (hd : α) (tl : list α) (IH : (Π (b : β), C b → Π (a : α), a ∈ tl → C (op a b)) → C (list.foldr op b tl)) (hl : Π (b : β), C b → Π (a : α), a ∈ hd :: tl → C (op a b)), hl (list.foldr op b tl) (IH (λ (y : β) (hy : C y) (x : α) (hx : x ∈ tl), hl y hy x _)) hd _) l hl
def
list.foldl_rec_on
{α : Type u}
{β : Type v}
{C : β → Sort u_1}
(l : list α)
(op : β → α → β)
(b : β)
(hb : C b)
(hl : Π (b : β), C b → Π (a : α), a ∈ l → C (op b a)) :
C (list.foldl op b l)
Induction principle for values produced by a foldl: if a property holds
for the seed element b : β and for all incremental op : β → α → β
performed on the elements (a : α) ∈ l. The principle is given for
a Sort-valued predicate, i.e., it can also be used to construct data.
Equations
- l.foldl_rec_on op b hb hl = list.rec (λ (hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op b a)) (b : β) (hb : C b), hb) (λ (hd : α) (tl : list α) (IH : (Π (b : β), C b → Π (a : α), a ∈ tl → C (op b a)) → Π (b : β), C b → C (list.foldl op b tl)) (hl : Π (b : β), C b → Π (a : α), a ∈ hd :: tl → C (op b a)) (b : β) (hb : C b), IH (λ (y : β) (hy : C y) (x : α) (hx : x ∈ tl), hl y hy x _) (op b hd) (hl b hb hd _)) l hl b hb
@[simp]
theorem
list.foldr_rec_on_nil
{α : Type u}
{β : Type v}
{C : β → Sort u_1}
(op : α → β → β)
(b : β)
(hb : C b)
(hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op a b)) :
list.nil.foldr_rec_on op b hb hl = hb
@[simp]
theorem
list.foldr_rec_on_cons
{α : Type u}
{β : Type v}
{C : β → Sort u_1}
(x : α)
(l : list α)
(op : α → β → β)
(b : β)
(hb : C b)
(hl : Π (b : β), C b → Π (a : α), a ∈ x :: l → C (op a b)) :
(x :: l).foldr_rec_on op b hb hl = hl (list.foldr op b l) (l.foldr_rec_on op b hb (λ (b : β) (hb : C b) (a : α) (ha : a ∈ l), hl b hb a _)) x _
@[simp]
theorem
list.foldl_rec_on_nil
{α : Type u}
{β : Type v}
{C : β → Sort u_1}
(op : β → α → β)
(b : β)
(hb : C b)
(hl : Π (b : β), C b → Π (a : α), a ∈ list.nil → C (op b a)) :
list.nil.foldl_rec_on op b hb hl = hb
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 list.nil = [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.nth_zero_scanl
{α : Type u}
{β : Type v}
{f : β → α → β}
{b : β}
{l : list α} :
(list.scanl f b l).nth 0 = some b
@[simp]
theorem
list.nth_le_zero_scanl
{α : Type u}
{β : Type v}
{f : β → α → β}
{b : β}
{l : list α}
{h : 0 < (list.scanl f b l).length} :
(list.scanl f b l).nth_le 0 h = b
theorem
list.nth_succ_scanl
{α : Type u}
{β : Type v}
{f : β → α → β}
{b : β}
{l : list α}
{i : ℕ} :
(list.scanl f b l).nth (i + 1) = ((list.scanl f b l).nth i).bind (λ (x : β), option.map (λ (y : α), f x y) (l.nth i))
theorem
list.nth_le_succ_scanl
{α : Type u}
{β : Type v}
{f : β → α → β}
{b : β}
{l : list α}
{i : ℕ}
{h : i + 1 < (list.scanl f b l).length} :
(list.scanl f b l).nth_le (i + 1) h = f ((list.scanl f b l).nth_le i _) (l.nth_le i _)
@[simp]
theorem
list.scanr_nil
{α : Type u}
{β : Type v}
(f : α → β → β)
(b : β) :
list.scanr f b list.nil = [b]
@[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)
@[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 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 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 : is_associative α op]
{l : list α}
{a₁ 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 : is_associative α op]
{l : list α}
{a₁ a₂ : α} :
op (list.foldl op a₁ l) a₂ = op a₁ (list.foldr op a₂ l)
theorem
list.foldl_assoc_comm_cons
{α : Type u}
{op : α → α → α}
[ha : is_associative α op]
[hc : is_commutative α op]
{l : list α}
{a₁ a₂ : α} :
list.foldl op a₂ (a₁ :: l) = op a₁ (list.foldl op a₂ l)
mfoldl, mfoldr, mmap #
@[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
@[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
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
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
@[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
prod and sum #
@[simp]
@[simp]
If zero is an element of a list L, then list.prod L = 0. If the domain is a nontrivial
monoid with zero with no divisors, then this implication becomes an iff, see
list.prod_eq_zero_iff.
@[simp]
theorem
list.prod_eq_zero_iff
{M₀ : Type u_1}
[monoid_with_zero M₀]
[nontrivial M₀]
[no_zero_divisors M₀]
{L : list M₀} :
Product of elements of a list L equals zero if and only if 0 ∈ L. See also
list.prod_eq_zero for an implication that needs weaker typeclass assumptions.
theorem
list.prod_ne_zero
{M₀ : Type u_1}
[monoid_with_zero M₀]
[nontrivial M₀]
[no_zero_divisors M₀]
{L : list M₀}
(hL : 0 ∉ L) :
theorem
list.sum_eq_foldr
{α : Type u}
[add_monoid α]
{l : list α} :
l.sum = list.foldr has_add.add 0 l
theorem
list.prod_eq_foldr
{α : Type u}
[monoid α]
{l : list α} :
l.prod = list.foldr has_mul.mul 1 l
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}
(h₁ : r 0 0)
(h₂ : ∀ ⦃a : α⦄ ⦃b : β⦄ ⦃c : γ⦄, r b c → r (f a + b) (g a + c)) :
theorem
list.sum_hom
{α : Type u}
{β : Type v}
[add_monoid α]
[add_monoid β]
(l : list α)
(f : α →+ β) :
theorem
list.sum_nonneg
{α : Type u}
[ordered_add_comm_monoid α]
{l : list α}
(hl₁ : ∀ (x : α), x ∈ l → 0 ≤ x) :
theorem
list.one_le_prod_of_one_le
{α : Type u}
[ordered_comm_monoid α]
{l : list α}
(hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x) :
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)
(H : x ∈ l) :
theorem
list.single_le_prod
{α : Type u}
[ordered_comm_monoid α]
{l : list α}
(hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x)
(x : α)
(H : x ∈ l) :
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)
(H : x ∈ l) :
x = 0
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)
(H : x ∈ l) :
x = 1
A list with sum not zero must have positive length.
theorem
list.length_pos_of_sum_pos
{α : Type u}
[ordered_cancel_add_comm_monoid α]
(L : list α)
(h : 0 < L.sum) :
A list with positive sum must have positive length.
@[simp]
@[simp]
theorem
list.dvd_sum
{α : Type u}
[comm_semiring α]
{a : α}
{l : list α}
(h : ∀ (x : α), x ∈ l → a ∣ x) :