Documentation

Std.Data.List.Lemmas

Basic properties of Lists #

theorem List.cons_ne_nil {α : Type u_1} (a : α) (l : List α) :
a :: l []
theorem List.cons_ne_self {α : Type u_1} (a : α) (l : List α) :
a :: l l
theorem List.head_eq_of_cons_eq :
∀ {α : Type u_1} {h₁ : α} {t₁ : List α} {h₂ : α} {t₂ : List α}, h₁ :: t₁ = h₂ :: t₂h₁ = h₂
theorem List.tail_eq_of_cons_eq :
∀ {α : Type u_1} {h₁ : α} {t₁ : List α} {h₂ : α} {t₂ : List α}, h₁ :: t₁ = h₂ :: t₂t₁ = t₂
theorem List.cons_inj {α : Type u_1} (a : α) {l : List α} {l' : List α} :
a :: l = a :: l' l = l'
theorem List.exists_cons_of_ne_nil {α : Type u_1} {l : List α} :
l []b L, l = b :: L

length #

@[simp]
theorem List.length_singleton {α : Type u_1} (a : α) :
theorem List.length_pos_of_mem {α : Type u_1} {a : α} {l : List α} :
a l0 < List.length l
theorem List.exists_mem_of_length_pos {α : Type u_1} {l : List α} :
0 < List.length la, a l
theorem List.length_pos_iff_exists_mem {α : Type u_1} {l : List α} :
0 < List.length l a, a l
theorem List.length_pos {α : Type u_1} {l : List α} :
0 < List.length l l []
theorem List.exists_mem_of_ne_nil {α : Type u_1} (l : List α) (h : l []) :
x, x l
theorem List.length_eq_one {α : Type u_1} {l : List α} :
List.length l = 1 a, l = [a]

mem #

@[simp]
theorem List.not_mem_nil {α : Type u_1} (a : α) :
¬a []
theorem List.mem_nil_iff {α : Type u_1} (a : α) :
@[simp]
theorem List.mem_cons :
∀ {α : Type u_1} {a b : α} {l : List α}, a b :: l a = b a l
theorem List.mem_cons_self {α : Type u_1} (a : α) (l : List α) :
a a :: l
theorem List.mem_cons_of_mem {α : Type u_1} (y : α) {a : α} {l : List α} :
a la y :: l
theorem List.mem_singleton_self {α : Type u_1} (a : α) :
a [a]
theorem List.eq_of_mem_singleton {α : Type u_1} {a : α} {b : α} :
a [b]a = b
@[simp]
theorem List.mem_singleton {α : Type u_1} {a : α} {b : α} :
a [b] a = b
theorem List.mem_of_mem_cons_of_mem {α : Type u_1} {a : α} {b : α} {l : List α} :
a b :: lb la l
theorem List.eq_or_ne_mem_of_mem {α : Type u_1} {a : α} {b : α} {l : List α} (h' : a b :: l) :
a = b a b a l
theorem List.ne_nil_of_mem {α : Type u_1} {a : α} {l : List α} (h : a l) :
l []
theorem List.mem_constructor {α : Type u_1} {a : α} {l : List α} :
a ls t, l = s ++ a :: t

append #

theorem List.append_eq_append :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.append l₁ l₂ = l₁ ++ l₂
@[simp]
theorem List.append_eq_nil :
∀ {α : Type u_1} {p q : List α}, p ++ q = [] p = [] q = []
theorem List.append_ne_nil_of_ne_nil_left {α : Type u_1} (s : List α) (t : List α) :
s []s ++ t []
theorem List.append_ne_nil_of_ne_nil_right {α : Type u_1} (s : List α) (t : List α) :
t []s ++ t []
@[simp]
theorem List.nil_eq_append :
∀ {α : Type u_1} {a b : List α}, [] = a ++ b a = [] b = []
theorem List.append_ne_nil_of_left_ne_nil {α : Type u_1} (a : List α) (b : List α) (h0 : a []) :
a ++ b []
theorem List.append_eq_cons :
∀ {α : Type u_1} {a b : List α} {x : α} {c : List α}, a ++ b = x :: c a = [] b = x :: c a', a = x :: a' c = a' ++ b
theorem List.cons_eq_append :
∀ {α : Type u_1} {x : α} {c a b : List α}, x :: c = a ++ b a = [] b = x :: c a', a = x :: a' c = a' ++ b
theorem List.append_eq_append_iff {α : Type u_1} {a : List α} {b : List α} {c : List α} {d : List α} :
a ++ b = c ++ d (a', c = a ++ a' b = a' ++ d) c', a = c ++ c' d = c' ++ b
@[simp]
theorem List.mem_append {α : Type u_1} {a : α} {s : List α} {t : List α} :
a s ++ t a s a t
theorem List.mem_append_eq {α : Type u_1} (a : α) (s : List α) (t : List α) :
(a s ++ t) = (a s a t)
theorem List.mem_append_left {α : Type u_1} {a : α} {l₁ : List α} (l₂ : List α) (h : a l₁) :
a l₁ ++ l₂
theorem List.mem_append_right {α : Type u_1} {a : α} (l₁ : List α) {l₂ : List α} (h : a l₂) :
a l₁ ++ l₂

map #

theorem List.map_singleton {α : Type u_1} {β : Type u_2} (f : αβ) (a : α) :
List.map f [a] = [f a]
theorem List.mem_map {α : Type u_1} {β : Type u_2} {b : β} {f : αβ} {l : List α} :
b List.map f l a, a l b = f a
theorem List.mem_map_of_mem {α : Type u_1} {β : Type u_2} {a : α} {l : List α} (f : αβ) (h : a l) :
f a List.map f l
theorem List.exists_of_mem_map :
∀ {α : Type u_1} {b : α} {α_1 : Type u_2} {f : α_1α} {l : List α_1}, b List.map f la, a l b = f a
theorem List.forall_mem_map_iff {α : Type u_1} {β : Type u_2} {f : αβ} {l : List α} {P : βProp} :
((i : β) → i List.map f lP i) (j : α) → j lP (f j)
@[simp]
theorem List.map_eq_nil {α : Type u_1} {β : Type u_2} {f : αβ} {l : List α} :
List.map f l = [] l = []
@[simp]
theorem List.length_zipWith {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβγ) (l₁ : List α) (l₂ : List β) :
List.length (List.zipWith f l₁ l₂) = min (List.length l₁) (List.length l₂)

join #

theorem List.join_nil {α : Type u_1} :
List.join [] = []
theorem List.join_cons {α : Type u_1} {a : List α} {l : List (List α)} :
theorem List.mem_join {α : Type u_1} {a : α} {L : List (List α)} :
a List.join L l, l L a l
theorem List.exists_of_mem_join :
∀ {α : Type u_1} {a : α} {L : List (List α)}, a List.join Ll, l L a l
theorem List.mem_join_of_mem :
∀ {α : Type u_1} {l : List α} {L : List (List α)} {a : α}, l La la List.join L

bind #

theorem List.mem_bind {α : Type u_1} {β : Type u_2} {f : αList β} {b : β} {l : List α} :
b List.bind l f a, a l b f a
theorem List.exists_of_mem_bind {β : Type u_1} {α : Type u_2} {b : β} {l : List α} {f : αList β} :
b List.bind l fa, a l b f a
theorem List.mem_bind_of_mem {β : Type u_1} {α : Type u_2} {b : β} {l : List α} {f : αList β} {a : α} (al : a l) (h : b f a) :
theorem List.bind_map {β : Type u_1} {γ : Type u_2} {α : Type u_3} (f : βγ) (g : αList β) (l : List α) :
List.map f (List.bind l g) = List.bind l fun a => List.map f (g a)

set-theoretic notation of Lists #

@[simp]
theorem List.empty_eq {α : Type u_1} :
= []

bounded quantifiers over Lists #

theorem List.exists_mem_nil {α : Type u_1} (p : αProp) :
¬x, x [] p x
theorem List.forall_mem_nil {α : Type u_1} (p : αProp) (x : α) :
x []p x
theorem List.exists_mem_cons {α : Type u_1} {p : αProp} {a : α} {l : List α} :
(x, x a :: l p x) p a x, x l p x
theorem List.forall_mem_cons {α : Type u_1} {p : αProp} {a : α} {l : List α} :
((x : α) → x a :: lp x) p a ((x : α) → x lp x)
theorem List.forall_mem_singleton {α : Type u_1} {p : αProp} {a : α} :
((x : α) → x [a]p x) p a
theorem List.forall_mem_append {α : Type u_1} {p : αProp} {l₁ : List α} {l₂ : List α} :
((x : α) → x l₁ ++ l₂p x) ((x : α) → x l₁p x) ((x : α) → x l₂p x)

List subset #

theorem List.subset_def {α : Type u_1} {l₁ : List α} {l₂ : List α} :
l₁ l₂ ∀ {a : α}, a l₁a l₂
@[simp]
theorem List.nil_subset {α : Type u_1} (l : List α) :
[] l
@[simp]
theorem List.Subset.refl {α : Type u_1} (l : List α) :
l l
theorem List.Subset.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁ l₂) (h₂ : l₂ l₃) :
l₁ l₃
@[simp]
theorem List.subset_cons {α : Type u_1} (a : α) (l : List α) :
l a :: l
theorem List.subset_of_cons_subset {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} :
a :: l₁ l₂l₁ l₂
theorem List.subset_cons_of_subset {α : Type u_1} (a : α) {l₁ : List α} {l₂ : List α} :
l₁ l₂l₁ a :: l₂
theorem List.cons_subset_cons {α : Type u_1} {l₁ : List α} {l₂ : List α} (a : α) (s : l₁ l₂) :
a :: l₁ a :: l₂
@[simp]
theorem List.subset_append_left {α : Type u_1} (l₁ : List α) (l₂ : List α) :
l₁ l₁ ++ l₂
@[simp]
theorem List.subset_append_right {α : Type u_1} (l₁ : List α) (l₂ : List α) :
l₂ l₁ ++ l₂
theorem List.subset_append_of_subset_left {α : Type u_1} {l : List α} {l₁ : List α} (l₂ : List α) :
l l₁l l₁ ++ l₂
theorem List.subset_append_of_subset_right {α : Type u_1} {l : List α} {l₂ : List α} (l₁ : List α) :
l l₂l l₁ ++ l₂
@[simp]
theorem List.cons_subset :
∀ {α : Type u_1} {a : α} {l m : List α}, a :: l m a m l m
@[simp]
theorem List.append_subset {α : Type u_1} {l₁ : List α} {l₂ : List α} {l : List α} :
l₁ ++ l₂ l l₁ l l₂ l
theorem List.subset_nil {α : Type u_1} {l : List α} :
l [] l = []
theorem List.eq_nil_iff_forall_not_mem {α : Type u_1} {l : List α} :
l = [] ∀ (a : α), ¬a l
theorem List.map_subset {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (f : αβ) (H : l₁ l₂) :
List.map f l₁ List.map f l₂

replicate #

theorem List.replicate_succ {α : Type u_1} (a : α) (n : Nat) :
theorem List.mem_replicate {α : Type u_1} {a : α} {b : α} {n : Nat} :
b List.replicate n a n 0 b = a
theorem List.eq_of_mem_replicate {α : Type u_1} {a : α} {b : α} {n : Nat} (h : b List.replicate n a) :
b = a

getLast #

theorem List.getLast_cons' {α : Type u_1} {a : α} {l : List α} (h₁ : a :: l []) (h₂ : l []) :
List.getLast (a :: l) h₁ = List.getLast l h₂
@[simp]
theorem List.getLast_append {α : Type u_1} {a : α} (l : List α) (h : l ++ [a] []) :
List.getLast (l ++ [a]) h = a
theorem List.getLast_concat :
∀ {α : Type u_1} {l : List α} {a : α} (h : List.concat l a []), List.getLast (List.concat l a) h = a

sublists #

@[simp]
theorem List.nil_sublist {α : Type u_1} (l : List α) :
@[simp]
theorem List.Sublist.refl {α : Type u_1} (l : List α) :
theorem List.Sublist.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : List.Sublist l₁ l₂) (h₂ : List.Sublist l₂ l₃) :
List.Sublist l₁ l₃
instance List.instTransListSublist {α : Type u_1} :
Trans List.Sublist List.Sublist List.Sublist
Equations
@[simp]
theorem List.sublist_cons {α : Type u_1} (a : α) (l : List α) :
theorem List.sublist_of_cons_sublist :
∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, List.Sublist (a :: l₁) l₂List.Sublist l₁ l₂
@[simp]
theorem List.sublist_append_left {α : Type u_1} (l₁ : List α) (l₂ : List α) :
List.Sublist l₁ (l₁ ++ l₂)
@[simp]
theorem List.sublist_append_right {α : Type u_1} (l₁ : List α) (l₂ : List α) :
List.Sublist l₂ (l₁ ++ l₂)
theorem List.sublist_append_of_sublist_left :
∀ {α : Type u_1} {l l₁ l₂ : List α}, List.Sublist l l₁List.Sublist l (l₁ ++ l₂)
theorem List.sublist_append_of_sublist_right :
∀ {α : Type u_1} {l l₂ l₁ : List α}, List.Sublist l l₂List.Sublist l (l₁ ++ l₂)
theorem List.cons_sublist_cons :
∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, List.Sublist (a :: l₁) (a :: l₂) List.Sublist l₁ l₂
@[simp]
theorem List.append_sublist_append_left :
∀ {α : Type u_1} {l₁ l₂ : List α} (l : List α), List.Sublist (l ++ l₁) (l ++ l₂) List.Sublist l₁ l₂
theorem List.Sublist.append_right :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Sublist l₁ l₂∀ (l : List α), List.Sublist (l₁ ++ l) (l₂ ++ l)
theorem List.sublist_or_mem_of_sublist :
∀ {α : Type u_1} {l l₁ : List α} {a : α} {l₂ : List α}, List.Sublist l (l₁ ++ a :: l₂)List.Sublist l (l₁ ++ l₂) a l
theorem List.Sublist.reverse :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Sublist l₁ l₂List.Sublist (List.reverse l₁) (List.reverse l₂)
@[simp]
theorem List.reverse_sublist :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Sublist (List.reverse l₁) (List.reverse l₂) List.Sublist l₁ l₂
@[simp]
theorem List.append_sublist_append_right :
∀ {α : Type u_1} {l₁ l₂ : List α} (l : List α), List.Sublist (l₁ ++ l) (l₂ ++ l) List.Sublist l₁ l₂
theorem List.Sublist.append :
∀ {α : Type u_1} {l₁ l₂ r₁ r₂ : List α}, List.Sublist l₁ l₂List.Sublist r₁ r₂List.Sublist (l₁ ++ r₁) (l₂ ++ r₂)
theorem List.Sublist.subset {α : Type u_1} {l₁ : List α} {l₂ : List α} :
List.Sublist l₁ l₂l₁ l₂
theorem List.Sublist.length_le :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Sublist l₁ l₂List.length l₁ List.length l₂
theorem List.head!_of_head? {α : Type u_1} {a : α} [inst : Inhabited α] {l : List α} :
theorem List.head?_eq_head {α : Type u_1} (l : List α) (h : l []) :

tail #

@[simp]
theorem List.tailD_eq_tail? {α : Type u_1} (l : List α) (l' : List α) :
theorem List.tail_eq_tailD {α : Type u_1} (l : List α) :
theorem List.tail_eq_tail? {α : Type u_1} (l : List α) :

next? #

@[simp]
theorem List.next?_nil {α : Type u_1} :
List.next? [] = none
@[simp]
theorem List.next?_cons {α : Type u_1} (a : α) (l : List α) :
List.next? (a :: l) = some (a, l)

getLast #

@[simp]
theorem List.getLastD_nil {α : Type u_1} (a : α) :
@[simp]
theorem List.getLastD_cons {α : Type u_1} (a : α) (b : α) (l : List α) :
theorem List.getLast_eq_getLastD {α : Type u_1} (a : α) (l : List α) (h : a :: l []) :
theorem List.getLast_singleton {α : Type u_1} (a : α) (h : [a] []) :
List.getLast [a] h = a
theorem List.getLast!_cons {α : Type u_1} {a : α} {l : List α} [inst : Inhabited α] :
@[simp]
theorem List.getLast?_nil {α : Type u_1} :
theorem List.getLast?_cons {α : Type u_1} {a : α} {l : List α} :
theorem List.getLast?_eq_getLast {α : Type u_1} (l : List α) (h : l []) :

dropLast #

@[simp]
theorem List.dropLast_nil {α : Type u_1} :
@[simp]
theorem List.dropLast_single :
∀ {α : Type u_1} {a : α}, List.dropLast [a] = []
@[simp]
theorem List.dropLast_cons₂ :
∀ {α : Type u_1} {a b : α} {l : List α}, List.dropLast (a :: b :: l) = a :: List.dropLast (b :: l)
@[simp]
theorem List.dropLast_append_cons :
∀ {α : Type u_1} {l₁ : List α} {b : α} {l₂ : List α}, List.dropLast (l₁ ++ b :: l₂) = l₁ ++ List.dropLast (b :: l₂)
@[simp]
theorem List.dropLast_concat :
∀ {α : Type u_1} {l₁ : List α} {b : α}, List.dropLast (l₁ ++ [b]) = l₁

nth element #

@[simp]
theorem List.get_cons_zero {α : Type u_1} {a : α} {as : List α} :
List.get (a :: as) { val := 0, isLt := (_ : 0 < Nat.succ (List.length as)) } = a
@[simp]
theorem List.get_cons_succ {α : Type u_1} {i : Nat} {a : α} {as : List α} {h : i + 1 < List.length (a :: as)} :
List.get (a :: as) { val := i + 1, isLt := h } = List.get as { val := i, isLt := (_ : i < List.length as) }
theorem List.get_of_mem {α : Type u_1} {a : α} {l : List α} :
a ln, List.get l n = a
theorem List.get?_eq_get {α : Type u_1} {l : List α} {n : Nat} (h : n < List.length l) :
List.get? l n = some (List.get l { val := n, isLt := h })
theorem List.get?_len_le {α : Type u_1} {l : List α} {n : Nat} :
List.length l nList.get? l n = none
theorem List.get?_eq_some :
∀ {α : Type u_1} {a : α} {l : List α} {n : Nat}, List.get? l n = some a h, List.get l { val := n, isLt := h } = a
@[simp]
theorem List.get?_eq_none :
∀ {α : Type u_1} {l : List α} {n : Nat}, List.get? l n = none List.length l n
theorem List.get?_of_mem {α : Type u_1} {a : α} {l : List α} (h : a l) :
n, List.get? l n = some a
theorem List.get_mem {α : Type u_1} (l : List α) (n : Nat) (h : n < List.length l) :
List.get l { val := n, isLt := h } l
theorem List.get?_mem {α : Type u_1} {l : List α} {n : Nat} {a : α} (e : List.get? l n = some a) :
a l
theorem List.mem_iff_get {α : Type u_1} {a : α} {l : List α} :
a l n, List.get l n = a
theorem List.Fin.exists_iff {n : Nat} (p : Fin nProp) :
(i, p i) i h, p { val := i, isLt := h }
theorem List.mem_iff_get? {α : Type u_1} {a : α} {l : List α} :
a l n, List.get? l n = some a
theorem List.get?_zero {α : Type u_1} (l : List α) :
@[simp]
theorem List.getElem_eq_get {α : Type u_1} (l : List α) (i : Nat) (h : i < List.length l) :
l[i] = List.get l { val := i, isLt := h }
@[simp]
theorem List.getElem?_eq_get? {α : Type u_1} (l : List α) (i : Nat) :
l[i]? = List.get? l i
theorem List.get?_inj {i : Nat} :
∀ {α : Type u_1} {xs : List α} {j : Nat}, i < List.length xsList.Nodup xsList.get? xs i = List.get? xs ji = j
@[simp]
theorem List.get?_map {α : Type u_1} {β : Type u_2} (f : αβ) (l : List α) (n : Nat) :
@[simp]
theorem List.get_map {α : Type u_1} {β : Type u_2} (f : αβ) {l : List α} {n : Fin (List.length (List.map f l))} :
List.get (List.map f l) n = f (List.get l { val := n.val, isLt := (_ : n.val < List.length l) })
theorem List.get_of_eq {α : Type u_1} {l : List α} {l' : List α} (h : l = l') (i : Fin (List.length l)) :
List.get l i = List.get l' { val := i.val, isLt := (_ : i.val < List.length l') }

If one has get 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 theorem get_of_eq can be used to make such a rewrite, with rw (get_of_eq h).

@[simp]
theorem List.get_singleton {α : Type u_1} (a : α) (n : Fin 1) :
List.get [a] n = a
theorem List.get_zero {α : Type u_1} {l : List α} (h : 0 < List.length l) :
some (List.get l { val := 0, isLt := h }) = List.head? l
theorem List.get_append {α : Type u_1} {l₁ : List α} {l₂ : List α} (n : Nat) (h : n < List.length l₁) :
List.get (l₁ ++ l₂) { val := n, isLt := (_ : n < List.length (l₁ ++ l₂)) } = List.get l₁ { val := n, isLt := h }
theorem List.get?_append_right {α : Type u_1} {l₁ : List α} {l₂ : List α} {n : Nat} :
List.length l₁ nList.get? (l₁ ++ l₂) n = List.get? l₂ (n - List.length l₁)
theorem List.get_append_right_aux {α : Type u_1} {l₁ : List α} {l₂ : List α} {n : Nat} (h₁ : List.length l₁ n) (h₂ : n < List.length (l₁ ++ l₂)) :
theorem List.get_append_right' {α : Type u_1} {l₁ : List α} {l₂ : List α} {n : Nat} (h₁ : List.length l₁ n) (h₂ : n < List.length (l₁ ++ l₂)) :
List.get (l₁ ++ l₂) { val := n, isLt := h₂ } = List.get l₂ { val := n - List.length l₁, isLt := (_ : n - List.length l₁ < List.length l₂) }
theorem List.get_of_append_proof {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α} {n : Nat} {l : List α} (eq : l = l₁ ++ a :: l₂) (h : List.length l₁ = n) :
theorem List.get_of_append {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α} {n : Nat} {l : List α} (eq : l = l₁ ++ a :: l₂) (h : List.length l₁ = n) :
List.get l { val := n, isLt := (_ : n < List.length l) } = a
@[simp]
theorem List.get_replicate {α : Type u_1} (a : α) {n : Nat} (m : Fin (List.length (List.replicate n a))) :
theorem List.get?_append {α : Type u_1} {l₁ : List α} {l₂ : List α} {n : Nat} (hn : n < List.length l₁) :
List.get? (l₁ ++ l₂) n = List.get? l₁ n
theorem List.getLast_eq_get {α : Type u_1} (l : List α) (h : l []) :
List.getLast l h = List.get l { val := List.length l - 1, isLt := (_ : List.length l - 1 < List.length l) }
@[simp]
theorem List.get?_concat_length {α : Type u_1} (l : List α) (a : α) :
List.get? (l ++ [a]) (List.length l) = some a
@[simp]
theorem List.getLast?_concat {α : Type u_1} {a : α} (l : List α) :
theorem List.get_cons_length {α : Type u_1} (x : α) (xs : List α) (n : Nat) (h : n = List.length xs) :
List.get (x :: xs) { val := n, isLt := (_ : n < Nat.succ (List.length xs)) } = List.getLast (x :: xs) (_ : x :: xs [])
theorem List.ext {α : Type u_1} {l₁ : List α} {l₂ : List α} :
(∀ (n : Nat), List.get? l₁ n = List.get? l₂ n) → l₁ = l₂
theorem List.ext_get {α : Type u_1} {l₁ : List α} {l₂ : List α} (hl : List.length l₁ = List.length l₂) (h : ∀ (n : Nat) (h₁ : n < List.length l₁) (h₂ : n < List.length l₂), List.get l₁ { val := n, isLt := h₁ } = List.get l₂ { val := n, isLt := h₂ }) :
l₁ = l₂
theorem List.get?_reverse' {α : Type u_1} {l : List α} (i : Nat) (j : Nat) :
theorem List.get?_reverse {α : Type u_1} {l : List α} (i : Nat) (h : i < List.length l) :
theorem List.get!_of_get? {α : Type u_1} {a : α} [inst : Inhabited α] {l : List α} {n : Nat} :
List.get? l n = some aList.get! l n = a
theorem List.getD_eq_get? {α : Type u_1} (l : List α) (n : Nat) (a : α) :

take and drop #

@[simp]
theorem List.length_take {α : Type u_1} (i : Nat) (l : List α) :
theorem List.length_take_le {α : Type u_1} (n : Nat) (l : List α) :
theorem List.length_take_of_le {n : Nat} :
∀ {α : Type u_1} {l : List α}, n List.length lList.length (List.take n l) = n
theorem List.get_cons_drop {α : Type u_1} (l : List α) (i : Fin (List.length l)) :
List.get l i :: List.drop (i.val + 1) l = List.drop i.val l
theorem List.map_eq_append_split {α : Type u_1} {β : Type u_2} {f : αβ} {l : List α} {s₁ : List β} {s₂ : List β} (h : List.map f l = s₁ ++ s₂) :
l₁ l₂, l = l₁ ++ l₂ List.map f l₁ = s₁ List.map f l₂ = s₂
theorem List.mem_of_mem_drop {α : Type u_1} {a : α} {n : Nat} {l : List α} :
a List.drop n la l

modify nth #

theorem List.modifyNthTail_id {α : Type u_1} (n : Nat) (l : List α) :
theorem List.removeNth_eq_nth_tail {α : Type u_1} (n : Nat) (l : List α) :
theorem List.get?_modifyNth {α : Type u_1} (f : αα) (n : Nat) (l : List α) (m : Nat) :
List.get? (List.modifyNth f n l) m = (fun a => if n = m then f a else a) <$> List.get? l m
theorem List.modifyNthTail_length {α : Type u_1} (f : List αList α) (H : ∀ (l : List α), List.length (f l) = List.length l) (n : Nat) (l : List α) :
theorem List.modifyNthTail_add {α : Type u_1} (f : List αList α) (n : Nat) (l₁ : List α) (l₂ : List α) :
List.modifyNthTail f (List.length l₁ + n) (l₁ ++ l₂) = l₁ ++ List.modifyNthTail f n l₂
theorem List.exists_of_modifyNthTail {α : Type u_1} (f : List αList α) {n : Nat} {l : List α} (h : n List.length l) :
l₁ l₂, l = l₁ ++ l₂ List.length l₁ = n List.modifyNthTail f n l = l₁ ++ f l₂
@[simp]
theorem List.modify_get?_length {α : Type u_1} (f : αα) (n : Nat) (l : List α) :
@[simp]
theorem List.get?_modifyNth_eq {α : Type u_1} (f : αα) (n : Nat) (l : List α) :
@[simp]
theorem List.get?_modifyNth_ne {α : Type u_1} (f : αα) {m : Nat} {n : Nat} (l : List α) (h : m n) :
theorem List.exists_of_modifyNth {α : Type u_1} (f : αα) {n : Nat} {l : List α} (h : n < List.length l) :
l₁ a l₂, l = l₁ ++ a :: l₂ List.length l₁ = n List.modifyNth f n l = l₁ ++ f a :: l₂

set #

theorem List.set_eq_modifyNth {α : Type u_1} (a : α) (n : Nat) (l : List α) :
List.set l n a = List.modifyNth (fun x => a) n l
theorem List.modifyNth_eq_set_get? {α : Type u_1} (f : αα) (n : Nat) (l : List α) :
List.modifyNth f n l = Option.getD ((fun a => List.set l n (f a)) <$> List.get? l n) l
theorem List.modifyNth_eq_set_get {α : Type u_1} (f : αα) {n : Nat} {l : List α} (h : n < List.length l) :
List.modifyNth f n l = List.set l n (f (List.get l { val := n, isLt := h }))
theorem List.exists_of_set {α : Type u_1} {n : Nat} {a' : α} {l : List α} (h : n < List.length l) :
l₁ a l₂, l = l₁ ++ a :: l₂ List.length l₁ = n List.set l n a' = l₁ ++ a' :: l₂
theorem List.exists_of_set' {α : Type u_1} {n : Nat} {a' : α} {l : List α} (h : n < List.length l) :
l₁ l₂, l = l₁ ++ List.get l { val := n, isLt := h } :: l₂ List.length l₁ = n List.set l n a' = l₁ ++ a' :: l₂
theorem List.get?_set_eq {α : Type u_1} (a : α) (n : Nat) (l : List α) :
List.get? (List.set l n a) n = (fun x => a) <$> List.get? l n
theorem List.get?_set_eq_of_lt {α : Type u_1} (a : α) {n : Nat} {l : List α} (h : n < List.length l) :
List.get? (List.set l n a) n = some a
theorem List.get?_set_ne {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : m n) :
theorem List.get?_set {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) :
List.get? (List.set l m a) n = if m = n then (fun x => a) <$> List.get? l n else List.get? l n
theorem List.get?_set_of_lt {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : n < List.length l) :
List.get? (List.set l m a) n = if m = n then some a else List.get? l n
theorem List.get?_set_of_lt' {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : m < List.length l) :
List.get? (List.set l m a) n = if m = n then some a else List.get? l n
@[simp]
theorem List.set_nil {α : Type u_1} (n : Nat) (a : α) :
List.set [] n a = []
@[simp]
theorem List.set_succ {α : Type u_1} (x : α) (xs : List α) (n : Nat) (a : α) :
List.set (x :: xs) (Nat.succ n) a = x :: List.set xs n a
theorem List.set_comm {α : Type u_1} (a : α) (b : α) {n : Nat} {m : Nat} (l : List α) :
n mList.set (List.set l n a) m b = List.set (List.set l m b) n a
@[simp]
theorem List.get_set_eq {α : Type u_1} (l : List α) (i : Nat) (a : α) (h : i < List.length (List.set l i a)) :
List.get (List.set l i a) { val := i, isLt := h } = a
@[simp]
theorem List.get_set_ne {α : Type u_1} {l : List α} {i : Nat} {j : Nat} (h : i j) (a : α) (hj : j < List.length (List.set l i a)) :
List.get (List.set l i a) { val := j, isLt := hj } = List.get l { val := j, isLt := (_ : j < List.length l) }
theorem List.get_set {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : n < List.length (List.set l m a)) :
List.get (List.set l m a) { val := n, isLt := h } = if m = n then a else List.get l { val := n, isLt := (_ : n < List.length l) }
theorem List.mem_or_eq_of_mem_set {α : Type u_1} {l : List α} {n : Nat} {a : α} {b : α} :
a List.set l n ba l a = b

remove nth #

theorem List.length_removeNth {α : Type u_1} {l : List α} {i : Nat} :

tail #

@[simp]
theorem List.length_tail {α : Type u_1} (l : List α) :

all / any #

@[simp]
theorem List.all_eq_true {α : Type u_1} {p : αBool} {l : List α} :
List.all l p = true ∀ (x : α), x lp x = true
@[simp]
theorem List.any_eq_true {α : Type u_1} {p : αBool} {l : List α} :
List.any l p = true x, x l p x = true

reverse #

@[simp]
theorem List.mem_reverseAux {α : Type u_1} (x : α) (as : List α) (bs : List α) :
x List.reverseAux as bs x as x bs
@[simp]
theorem List.mem_reverse {α : Type u_1} (x : α) (as : List α) :

insert #

@[simp]
theorem List.insert_of_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : a l) :
@[simp]
theorem List.insert_of_not_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : ¬a l) :
List.insert a l = a :: l
@[simp]
theorem List.mem_insert_iff {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} {l : List α} :
a List.insert b l a = b a l
@[simp]
theorem List.mem_insert_self {α : Type u_1} [inst : DecidableEq α] (a : α) (l : List α) :
theorem List.mem_insert_of_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} {l : List α} (h : a l) :
theorem List.eq_or_mem_of_mem_insert {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} {l : List α} (h : a List.insert b l) :
a = b a l
@[simp]
theorem List.length_insert_of_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : a l) :
@[simp]
theorem List.length_insert_of_not_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : ¬a l) :

eraseP #

@[simp]
theorem List.eraseP_nil :
∀ {α : Type u_1} {p : αBool}, List.eraseP p [] = []
theorem List.eraseP_cons {α : Type u_1} {p : αBool} (a : α) (l : List α) :
List.eraseP p (a :: l) = bif p a then l else a :: List.eraseP p l
@[simp]
theorem List.eraseP_cons_of_pos {α : Type u_1} {a : α} {l : List α} (p : αBool) (h : p a = true) :
List.eraseP p (a :: l) = l
@[simp]
theorem List.eraseP_cons_of_neg {α : Type u_1} {a : α} {l : List α} (p : αBool) (h : ¬p a = true) :
List.eraseP p (a :: l) = a :: List.eraseP p l
theorem List.eraseP_of_forall_not {α : Type u_1} {p : αBool} {l : List α} (h : ∀ (a : α), a l¬p a = true) :
theorem List.exists_of_eraseP {α : Type u_1} {p : αBool} {l : List α} {a : α} (al : a l) (pa : p a = true) :
a l₁ l₂, (∀ (b : α), b l₁¬p b = true) p a = true l = l₁ ++ a :: l₂ List.eraseP p l = l₁ ++ l₂
theorem List.exists_or_eq_self_of_eraseP {α : Type u_1} (p : αBool) (l : List α) :
List.eraseP p l = l a l₁ l₂, (∀ (b : α), b l₁¬p b = true) p a = true l = l₁ ++ a :: l₂ List.eraseP p l = l₁ ++ l₂
@[simp]
theorem List.length_eraseP_of_mem :
∀ {α : Type u_1} {a : α} {l : List α} {p : αBool}, a lp a = trueList.length (List.eraseP p l) = Nat.pred (List.length l)
theorem List.eraseP_append_left {α : Type u_1} {p : αBool} {a : α} (pa : p a = true) {l₁ : List α} (l₂ : List α) :
a l₁List.eraseP p (l₁ ++ l₂) = List.eraseP p l₁ ++ l₂
theorem List.eraseP_append_right {α : Type u_1} {p : αBool} {l₁ : List α} (l₂ : List α) :
(∀ (b : α), b l₁¬p b = true) → List.eraseP p (l₁ ++ l₂) = l₁ ++ List.eraseP p l₂
theorem List.eraseP_sublist {α : Type u_1} {p : αBool} (l : List α) :
theorem List.eraseP_subset {α : Type u_1} {p : αBool} (l : List α) :
theorem List.Sublist.eraseP {α : Type u_1} {p : αBool} {l₁ : List α} {l₂ : List α} :
List.Sublist l₁ l₂List.Sublist (List.eraseP p l₁) (List.eraseP p l₂)
theorem List.mem_of_mem_eraseP {α : Type u_1} {a : α} {p : αBool} {l : List α} :
a List.eraseP p la l
@[simp]
theorem List.mem_eraseP_of_neg {α : Type u_1} {p : αBool} {a : α} {l : List α} (pa : ¬p a = true) :
theorem List.eraseP_map {β : Type u_1} {α : Type u_2} {p : αBool} (f : βα) (l : List β) :
@[simp]
theorem List.extractP_eq_find?_eraseP {α : Type u_1} {p : αBool} (l : List α) :
theorem List.extractP_eq_find?_eraseP.go {α : Type u_1} {p : αBool} (l : List α) (acc : Array α) (xs : List α) :
l = acc.data ++ xsList.extractP.go p l xs acc = (List.find? p xs, acc.data ++ List.eraseP p xs)

erase #

@[simp]
theorem List.erase_nil {α : Type u_1} [inst : DecidableEq α] (a : α) :
List.erase [] a = []
theorem List.erase_cons {α : Type u_1} [inst : DecidableEq α] (a : α) (b : α) (l : List α) :
List.erase (b :: l) a = if b = a then l else b :: List.erase l a
@[simp]
theorem List.erase_cons_head {α : Type u_1} [inst : DecidableEq α] (a : α) (l : List α) :
List.erase (a :: l) a = l
@[simp]
theorem List.erase_cons_tail {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} (l : List α) (h : b a) :
List.erase (b :: l) a = b :: List.erase l a
theorem List.erase_eq_eraseP {α : Type u_1} [inst : DecidableEq α] (a : α) (l : List α) :
List.erase l a = List.eraseP (fun b => decide (a = b)) l
theorem List.erase_of_not_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} :
¬a lList.erase l a = l
theorem List.exists_erase_eq {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : a l) :
l₁ l₂, ¬a l₁ l = l₁ ++ a :: l₂ List.erase l a = l₁ ++ l₂
@[simp]
theorem List.length_erase_of_mem {α : Type u_1} [inst : DecidableEq α] {a : α} {l : List α} (h : a l) :
theorem List.erase_append_left {α : Type u_1} [inst : DecidableEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : a l₁) :
List.erase (l₁ ++ l₂) a = List.erase l₁ a ++ l₂
theorem List.erase_append_right {α : Type u_1} [inst : DecidableEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : ¬a l₁) :
List.erase (l₁ ++ l₂) a = l₁ ++ List.erase l₂ a
theorem List.erase_sublist {α : Type u_1} [inst : DecidableEq α] (a : α) (l : List α) :
theorem List.erase_subset {α : Type u_1} [inst : DecidableEq α] (a : α) (l : List α) :
theorem List.sublist.erase {α : Type u_1} [inst : DecidableEq α] (a : α) {l₁ : List α} {l₂ : List α} (h : List.Sublist l₁ l₂) :
theorem List.mem_of_mem_erase {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} {l : List α} (h : a List.erase l b) :
a l
@[simp]
theorem List.mem_erase_of_ne {α : Type u_1} [inst : DecidableEq α] {a : α} {b : α} {l : List α} (ab : a b) :
a List.erase l b a l
theorem List.erase_comm {α : Type u_1} [inst : DecidableEq α] (a : α) (b : α) (l : List α) :

filter and partition #

theorem List.mem_filter :
∀ {α : Type u_1} {x : α} {p : αBool} {as : List α}, x List.filter p as x as p x = true
@[simp]
theorem List.partition_eq_filter_filter {α : Type u_1} (p : αBool) (l : List α) :
theorem List.partition_eq_filter_filter.aux {α : Type u_1} (p : αBool) (l : List α) {as : List α} {bs : List α} :

pairwise #

theorem List.Pairwise.sublist :
∀ {α : Type u_1} {l₁ l₂ : List α} {R : ααProp}, List.Sublist l₁ l₂List.Pairwise R l₂List.Pairwise R l₁
theorem List.pairwise_map {α : Type u_1} :
∀ {α : Type u_2} {f : αα} {R : ααProp} {l : List α}, List.Pairwise R (List.map f l) List.Pairwise (fun a b => R (f a) (f b)) l
theorem List.pairwise_append {α : Type u_1} {R : ααProp} {l₁ : List α} {l₂ : List α} :
List.Pairwise R (l₁ ++ l₂) List.Pairwise R l₁ List.Pairwise R l₂ ((a : α) → a l₁(b : α) → b l₂R a b)
theorem List.pairwise_reverse {α : Type u_1} {R : ααProp} {l : List α} :
List.Pairwise R (List.reverse l) List.Pairwise (fun a b => R b a) l
theorem List.Pairwise.imp {α : Type u_1} {R : ααProp} {S : ααProp} (H : {a b : α} → R a bS a b) {l : List α} :

replaceF #

@[simp]
theorem List.length_replaceF :
∀ {α : Type u_1} {f : αOption α} {l : List α}, List.length (List.replaceF f l) = List.length l

disjoint #

theorem List.disjoint_symm :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Disjoint l₁ l₂List.Disjoint l₂ l₁
theorem List.disjoint_comm :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Disjoint l₁ l₂ List.Disjoint l₂ l₁
theorem List.disjoint_left :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Disjoint l₁ l₂ ∀ ⦃a : α⦄, a l₁¬a l₂
theorem List.disjoint_right :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Disjoint l₁ l₂ ∀ ⦃a : α⦄, a l₂¬a l₁
theorem List.disjoint_iff_ne :
∀ {α : Type u_1} {l₁ l₂ : List α}, List.Disjoint l₁ l₂ ∀ (a : α), a l₁∀ (b : α), b l₂a b
theorem List.disjoint_of_subset_left :
∀ {α : Type u_1} {l₁ l l₂ : List α}, l₁ lList.Disjoint l l₂List.Disjoint l₁ l₂
theorem List.disjoint_of_subset_right :
∀ {α : Type u_1} {l₂ l l₁ : List α}, l₂ lList.Disjoint l₁ lList.Disjoint l₁ l₂
theorem List.disjoint_of_disjoint_cons_left :
∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, List.Disjoint (a :: l₁) l₂List.Disjoint l₁ l₂
theorem List.disjoint_of_disjoint_cons_right :
∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, List.Disjoint l₁ (a :: l₂)List.Disjoint l₁ l₂
@[simp]
theorem List.disjoint_nil_left {α : Type u_1} (l : List α) :
@[simp]
theorem List.disjoint_nil_right {α : Type u_1} (l : List α) :
@[simp]
theorem List.singleton_disjoint :
∀ {α : Type u_1} {a : α} {l : List α}, List.Disjoint [a] l ¬a l
@[simp]
theorem List.disjoint_singleton :
∀ {α : Type u_1} {l : List α} {a : α}, List.Disjoint l [a] ¬a l
@[simp]
theorem List.disjoint_append_left :
∀ {α : Type u_1} {l₁ l₂ l : List α}, List.Disjoint (l₁ ++ l₂) l List.Disjoint l₁ l List.Disjoint l₂ l
@[simp]
theorem List.disjoint_append_right :
∀ {α : Type u_1} {l l₁ l₂ : List α}, List.Disjoint l (l₁ ++ l₂) List.Disjoint l l₁ List.Disjoint l l₂
@[simp]
theorem List.disjoint_cons_left :
∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, List.Disjoint (a :: l₁) l₂ ¬a l₂ List.Disjoint l₁ l₂
@[simp]
theorem List.disjoint_cons_right :
∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α}, List.Disjoint l₁ (a :: l₂) ¬a l₁ List.Disjoint l₁ l₂
theorem List.disjoint_of_disjoint_append_left_left :
∀ {α : Type u_1} {l₁ l₂ l : List α}, List.Disjoint (l₁ ++ l₂) lList.Disjoint l₁ l
theorem List.disjoint_of_disjoint_append_left_right :
∀ {α : Type u_1} {l₁ l₂ l : List α}, List.Disjoint (l₁ ++ l₂) lList.Disjoint l₂ l
theorem List.disjoint_of_disjoint_append_right_left :
∀ {α : Type u_1} {l l₁ l₂ : List α}, List.Disjoint l (l₁ ++ l₂)List.Disjoint l l₁
theorem List.disjoint_of_disjoint_append_right_right :
∀ {α : Type u_1} {l l₁ l₂ : List α}, List.Disjoint l (l₁ ++ l₂)List.Disjoint l l₂
theorem List.disjoint_take_drop {α : Type u_1} {m : Nat} {n : Nat} {l : List α} :
List.Nodup lm nList.Disjoint (List.take m l) (List.drop n l)

foldl / foldr #

theorem List.foldl_map {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} (f : β₁β₂) (g : αβ₂α) (l : List β₁) (init : α) :
List.foldl g init (List.map f l) = List.foldl (fun x y => g x (f y)) init l
theorem List.foldr_map {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} (f : α₁α₂) (g : α₂ββ) (l : List α₁) (init : β) :
List.foldr g init (List.map f l) = List.foldr (fun x y => g (f x) y) init l
theorem List.foldl_hom {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} (f : α₁α₂) (g₁ : α₁βα₁) (g₂ : α₂βα₂) (l : List β) (init : α₁) (H : ∀ (x : α₁) (y : β), g₂ (f x) y = f (g₁ x y)) :
List.foldl g₂ (f init) l = f (List.foldl g₁ init l)
theorem List.foldr_hom {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} (f : β₁β₂) (g₁ : αβ₁β₁) (g₂ : αβ₂β₂) (l : List α) (init : β₁) (H : ∀ (x : α) (y : β₁), g₂ x (f y) = f (g₁ x y)) :
List.foldr g₂ (f init) l = f (List.foldr g₁ init l)

union #

@[simp]
theorem List.nil_union {α : Type u_1} [inst : DecidableEq α] (l : List α) :
List.union [] l = l
@[simp]
theorem List.cons_union {α : Type u_1} [inst : DecidableEq α] (a : α) (l₁ : List α) (l₂ : List α) :
List.union (a :: l₁) l₂ = List.insert a (List.union l₁ l₂)
@[simp]
theorem List.mem_union_iff {α : Type u_1} [inst : DecidableEq α] {x : α} {l₁ : List α} {l₂ : List α} :
x List.union l₁ l₂ x l₁ x l₂

inter #

@[simp]
theorem List.mem_inter_iff {α : Type u_1} [inst : DecidableEq α] {x : α} {l₁ : List α} {l₂ : List α} :
x List.inter l₁ l₂ x l₁ x l₂

product #

theorem List.pair_mem_product {α : Type u_1} {β : Type u_2} {xs : List α} {ys : List β} {x : α} {y : β} :
(x, y) List.product xs ys x xs y ys

List.prod satisfies a specification of cartesian product on lists.

leftpad #

theorem List.leftpad_length {α : Type u_1} (n : Nat) (a : α) (l : List α) :

The length of the List returned by List.leftpad n a l is equal to the larger of n and l.length

theorem List.leftpad_prefix {α : Type u_1} (n : Nat) (a : α) (l : List α) :
theorem List.leftpad_suffix {α : Type u_1} (n : Nat) (a : α) (l : List α) :

monadic operations #

@[simp]
theorem List.forIn_eq_forIn {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [inst : Monad m] :
List.forIn = forIn
theorem List.forIn_eq_bindList {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [inst : Monad m] [inst : LawfulMonad m] (f : αβm (ForInStep β)) (l : List α) (init : β) :
forIn l init f = ForInStep.run <$> ForInStep.bindList f l (ForInStep.yield init)
@[simp]
theorem List.forM_append {m : Type u_1 → Type u_2} {α : Type u_3} [inst : Monad m] [inst : LawfulMonad m] (l₁ : List α) (l₂ : List α) (f : αm PUnit) :
List.forM (l₁ ++ l₂) f = do List.forM l₁ f List.forM l₂ f