mem

@[simp]

theorem List.mem_toArray {α : Type u_1} {a : α} {l : List α} : a ∈ List.toArray l ↔ a ∈ l

drop

@[simp]

theorem List.drop_one {α : Type u_1} (l : List α) : List.drop 1 l = l.tail

zipWith

theorem List.zipWith_distrib_tail : ∀ {α : Type u_1} {α_1 : Type u_2} {α_2 : Type u_3} {f : α → α_1 → α_2} {l : List α} {l' : List α_1}, (List.zipWith f l l').tail = List.zipWith f l.tail l'.tail

List subset

theorem List.subset_def {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈ l₁ → a ∈ 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

instance List.instTransMemSubset_batteries {α : Type u_1} : Trans Membership.mem Subset Membership.mem

Equations

• List.instTransMemSubset_batteries = { trans := ⋯ }

instance List.instTransSubset_batteries {α : Type u_1} : Trans Subset Subset Subset

Equations

• List.instTransSubset_batteries = { trans := ⋯ }

@[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 ⊆ l₂ → l₁ ⊆ l₂

theorem List.subset_cons_of_subset {α : Type u_1} (a : α) {l₁ : List α} {l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ 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 ⊆ l₁ → l ⊆ l₁ ++ l₂

theorem List.subset_append_of_subset_right {α : Type u_1} {l : List α} {l₂ : List α} (l₁ : List α) : l ⊆ l₂ → l ⊆ 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.map_subset {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (f : α → β) (H : l₁ ⊆ l₂) : List.map f l₁ ⊆ List.map f l₂

sublists

@[simp]

theorem List.nil_sublist {α : Type u_1} (l : List α) : [].Sublist l

@[simp]

theorem List.Sublist.refl {α : Type u_1} (l : List α) : l.Sublist l

theorem List.Sublist.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁.Sublist l₂) (h₂ : l₂.Sublist l₃) : l₁.Sublist l₃

instance List.instTransSublist {α : Type u_1} : Trans List.Sublist List.Sublist List.Sublist

Equations

• List.instTransSublist = { trans := ⋯ }

@[simp]

theorem List.sublist_cons {α : Type u_1} (a : α) (l : List α) : l.Sublist (a :: l)

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

@[simp]

theorem List.sublist_append_left {α : Type u_1} (l₁ : List α) (l₂ : List α) : l₁.Sublist (l₁ ++ l₂)

@[simp]

theorem List.sublist_append_right {α : Type u_1} (l₁ : List α) (l₂ : List α) : l₂.Sublist (l₁ ++ l₂)

theorem List.sublist_append_of_sublist_left : ∀ {α : Type u_1} {l l₁ l₂ : List α}, l.Sublist l₁ → l.Sublist (l₁ ++ l₂)

theorem List.sublist_append_of_sublist_right : ∀ {α : Type u_1} {l l₂ l₁ : List α}, l.Sublist l₂ → l.Sublist (l₁ ++ l₂)

@[simp]

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

@[simp]

theorem List.append_sublist_append_left : ∀ {α : Type u_1} {l₁ l₂ : List α} (l : List α), (l ++ l₁).Sublist (l ++ l₂) ↔ l₁.Sublist l₂

theorem List.Sublist.append_left : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → ∀ (l : List α), (l ++ l₁).Sublist (l ++ l₂)

theorem List.Sublist.append_right : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → ∀ (l : List α), (l₁ ++ l).Sublist (l₂ ++ l)

theorem List.sublist_or_mem_of_sublist : ∀ {α : Type u_1} {l l₁ : List α} {a : α} {l₂ : List α}, l.Sublist (l₁ ++ a :: l₂) → l.Sublist (l₁ ++ l₂) ∨ a ∈ l

theorem List.Sublist.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₁.reverse.Sublist l₂.reverse

@[simp]

theorem List.reverse_sublist : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.reverse.Sublist l₂.reverse ↔ l₁.Sublist l₂

@[simp]

theorem List.append_sublist_append_right : ∀ {α : Type u_1} {l₁ l₂ : List α} (l : List α), (l₁ ++ l).Sublist (l₂ ++ l) ↔ l₁.Sublist l₂

theorem List.Sublist.append : ∀ {α : Type u_1} {l₁ l₂ r₁ r₂ : List α}, l₁.Sublist l₂ → r₁.Sublist r₂ → (l₁ ++ r₁).Sublist (l₂ ++ r₂)

theorem List.Sublist.subset : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₁ ⊆ l₂

instance List.instTransSublistSubset {α : Type u_1} : Trans List.Sublist Subset Subset

Equations

• List.instTransSublistSubset = { trans := ⋯ }

instance List.instTransSubsetSublist {α : Type u_1} : Trans Subset List.Sublist Subset

Equations

• List.instTransSubsetSublist = { trans := ⋯ }

instance List.instTransMemSublist {α : Type u_1} : Trans Membership.mem List.Sublist Membership.mem

Equations

• List.instTransMemSublist = { trans := ⋯ }

theorem List.Sublist.length_le : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₁.length ≤ l₂.length

@[simp]

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

theorem List.Sublist.eq_of_length : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₁.length = l₂.length → l₁ = l₂

theorem List.Sublist.eq_of_length_le : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂ → l₂.length ≤ l₁.length → l₁ = l₂

@[simp]

theorem List.singleton_sublist {α : Type u_1} {a : α} {l : List α} : [a].Sublist l ↔ a ∈ l

@[simp]

theorem List.replicate_sublist_replicate {α : Type u_1} {m : Nat} {n : Nat} (a : α) : (List.replicate m a).Sublist (List.replicate n a) ↔ m ≤ n

theorem List.isSublist_iff_sublist {α : Type u_1} [BEq α] [LawfulBEq α] {l₁ : List α} {l₂ : List α} : l₁.isSublist l₂ = true ↔ l₁.Sublist l₂

instance List.instDecidableSublistOfDecidableEq {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁.Sublist l₂)

Equations

• l₁.instDecidableSublistOfDecidableEq l₂ = decidable_of_iff (l₁.isSublist l₂ = true) ⋯

tail

theorem List.tail_eq_tailD {α : Type u_1} (l : List α) : l.tail = l.tailD []

theorem List.tail_eq_tail? {α : Type u_1} (l : List α) : l.tail = l.tail?.getD []

next?

@[simp]

theorem List.next?_nil {α : Type u_1} : [].next? = none

@[simp]

theorem List.next?_cons {α : Type u_1} (a : α) (l : List α) : (a :: l).next? = some (a, l)

get?

theorem List.get_eq_iff : ∀ {α : Type u_1} {l : List α} {n : Fin l.length} {x : α}, l.get n = x ↔ l.get? ↑n = some x

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

drop

theorem List.tail_drop {α : Type u_1} (l : List α) (n : Nat) : (List.drop n l).tail = List.drop (n + 1) l

modifyNth

@[simp]

theorem List.modifyNth_nil {α : Type u_1} (f : α → α) (n : Nat) : List.modifyNth f n [] = []

@[simp]

theorem List.modifyNth_zero_cons {α : Type u_1} (f : α → α) (a : α) (l : List α) : List.modifyNth f 0 (a :: l) = f a :: l

@[simp]

theorem List.modifyNth_succ_cons {α : Type u_1} (f : α → α) (a : α) (l : List α) (n : Nat) : List.modifyNth f (n + 1) (a :: l) = a :: List.modifyNth f n l

theorem List.modifyNthTail_id {α : Type u_1} (n : Nat) (l : List α) : List.modifyNthTail id n l = l

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

@[deprecated List.eraseIdx_eq_modifyNthTail]

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

Alias of List.eraseIdx_eq_modifyNthTail.

theorem List.get?_modifyNth {α : Type u_1} (f : α → α) (n : Nat) (l : List α) (m : Nat) : (List.modifyNth f n l).get? m = (fun (a : α) => if n = m then f a else a) <$> l.get? m

theorem List.modifyNthTail_length {α : Type u_1} (f : List α → List α) (H : ∀ (l : List α), (f l).length = l.length) (n : Nat) (l : List α) : (List.modifyNthTail f n l).length = l.length

theorem List.modifyNthTail_add {α : Type u_1} (f : List α → List α) (n : Nat) (l₁ : List α) (l₂ : List α) : List.modifyNthTail f (l₁.length + n) (l₁ ++ l₂) = l₁ ++ List.modifyNthTail f n l₂

theorem List.exists_of_modifyNthTail {α : Type u_1} (f : List α → List α) {n : Nat} {l : List α} (h : n ≤ l.length) : ∃ (l₁ : List α), ∃ (l₂ : List α), l = l₁ ++ l₂ ∧ l₁.length = n ∧ List.modifyNthTail f n l = l₁ ++ f l₂

@[simp]

theorem List.modify_get?_length {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : (List.modifyNth f n l).length = l.length

@[simp]

theorem List.get?_modifyNth_eq {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : (List.modifyNth f n l).get? n = f <$> l.get? n

@[simp]

theorem List.get?_modifyNth_ne {α : Type u_1} (f : α → α) {m : Nat} {n : Nat} (l : List α) (h : m ≠ n) : (List.modifyNth f m l).get? n = l.get? n

theorem List.exists_of_modifyNth {α : Type u_1} (f : α → α) {n : Nat} {l : List α} (h : n < l.length) : ∃ (l₁ : List α), ∃ (a : α), ∃ (l₂ : List α), l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ List.modifyNth f n l = l₁ ++ f a :: l₂

theorem List.modifyNthTail_eq_take_drop {α : Type u_1} (f : List α → List α) (H : f [] = []) (n : Nat) (l : List α) : List.modifyNthTail f n l = List.take n l ++ f (List.drop n l)

theorem List.modifyNth_eq_take_drop {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : List.modifyNth f n l = List.take n l ++ List.modifyHead f (List.drop n l)

theorem List.modifyNth_eq_take_cons_drop {α : Type u_1} (f : α → α) {n : Nat} {l : List α} (h : n < l.length) : List.modifyNth f n l = List.take n l ++ f (l.get ⟨n, h⟩) :: List.drop (n + 1) l

set

theorem List.set_eq_modifyNth {α : Type u_1} (a : α) (n : Nat) (l : List α) : l.set n a = List.modifyNth (fun (x : α) => a) n l

theorem List.set_eq_take_cons_drop {α : Type u_1} (a : α) {n : Nat} {l : List α} (h : n < l.length) : l.set n a = List.take n l ++ a :: List.drop (n + 1) l

theorem List.modifyNth_eq_set_get? {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : List.modifyNth f n l = ((fun (a : α) => l.set n (f a)) <$> l.get? n).getD l

theorem List.modifyNth_eq_set_get {α : Type u_1} (f : α → α) {n : Nat} {l : List α} (h : n < l.length) : List.modifyNth f n l = l.set n (f (l.get ⟨n, h⟩))

theorem List.exists_of_set {α : Type u_1} {n : Nat} {a' : α} {l : List α} (h : n < l.length) : ∃ (l₁ : List α), ∃ (a : α), ∃ (l₂ : List α), l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂

theorem List.exists_of_set' {α : Type u_1} {n : Nat} {a' : α} {l : List α} (h : n < l.length) : ∃ (l₁ : List α), ∃ (l₂ : List α), l = l₁ ++ l.get ⟨n, h⟩ :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂

@[simp]

theorem List.get?_set_eq {α : Type u_1} (a : α) (n : Nat) (l : List α) : (l.set n a).get? n = (fun (x : α) => a) <$> l.get? n

theorem List.get?_set_eq_of_lt {α : Type u_1} (a : α) {n : Nat} {l : List α} (h : n < l.length) : (l.set n a).get? n = some a

@[simp]

theorem List.get?_set_ne {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : m ≠ n) : (l.set m a).get? n = l.get? n

theorem List.get?_set {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) : (l.set m a).get? n = if m = n then (fun (x : α) => a) <$> l.get? n else l.get? n

theorem List.get?_set_of_lt {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : n < l.length) : (l.set m a).get? n = if m = n then some a else l.get? n

theorem List.get?_set_of_lt' {α : Type u_1} (a : α) {m : Nat} {n : Nat} (l : List α) (h : m < l.length) : (l.set m a).get? n = if m = n then some a else l.get? n

theorem List.drop_set_of_lt {α : Type u_1} (a : α) {n : Nat} {m : Nat} (l : List α) (h : n < m) : List.drop m (l.set n a) = List.drop m l

theorem List.take_set_of_lt {α : Type u_1} (a : α) {n : Nat} {m : Nat} (l : List α) (h : m < n) : List.take m (l.set n a) = List.take m l

removeNth

theorem List.length_eraseIdx {α : Type u_1} {l : List α} {i : Nat} : i < l.length → (l.eraseIdx i).length = l.length - 1

@[deprecated List.length_eraseIdx]

theorem List.length_removeNth {α : Type u_1} {l : List α} {i : Nat} : i < l.length → (l.eraseIdx i).length = l.length - 1

Alias of List.length_eraseIdx.

tail

@[simp]

theorem List.length_tail {α : Type u_1} (l : List α) : l.tail.length = l.length - 1

eraseP

@[simp]

theorem List.eraseP_nil : ∀ {α : Type u_1} {p : α → Bool}, List.eraseP p [] = []

theorem List.eraseP_cons {α : Type u_1} {p : α → Bool} (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 : α → Bool) (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 : α → Bool) (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 : α → Bool} {l : List α} (h : ∀ (a : α), a ∈ l → ¬p a = true) : List.eraseP p l = l

theorem List.exists_of_eraseP {α : Type u_1} {p : α → Bool} {l : List α} {a : α} (al : a ∈ l) (pa : p a = true) : ∃ (a : α), ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (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 : α → Bool) (l : List α) : List.eraseP p l = l ∨ ∃ (a : α), ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (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 : α → Bool}, a ∈ l → p a = true → (List.eraseP p l).length = l.length.pred

theorem List.eraseP_append_left {α : Type u_1} {p : α → Bool} {a : α} (pa : p a = true) {l₁ : List α} (l₂ : List α) : a ∈ l₁ → List.eraseP p (l₁ ++ l₂) = List.eraseP p l₁ ++ l₂

theorem List.eraseP_append_right {α : Type u_1} {p : α → Bool} {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 : α → Bool} (l : List α) : (List.eraseP p l).Sublist l

theorem List.eraseP_subset {α : Type u_1} {p : α → Bool} (l : List α) : List.eraseP p l ⊆ l

theorem List.Sublist.eraseP : ∀ {α : Type u_1} {l₁ l₂ : List α} {p : α → Bool}, l₁.Sublist l₂ → (List.eraseP p l₁).Sublist (List.eraseP p l₂)

theorem List.mem_of_mem_eraseP {α : Type u_1} {a : α} {p : α → Bool} {l : List α} : a ∈ List.eraseP p l → a ∈ l

@[simp]

theorem List.mem_eraseP_of_neg {α : Type u_1} {p : α → Bool} {a : α} {l : List α} (pa : ¬p a = true) : a ∈ List.eraseP p l ↔ a ∈ l

theorem List.eraseP_map {β : Type u_1} {α : Type u_2} {p : α → Bool} (f : β → α) (l : List β) : List.eraseP p (List.map f l) = List.map f (List.eraseP (p ∘ f) l)

@[simp]

theorem List.extractP_eq_find?_eraseP {α : Type u_1} {p : α → Bool} (l : List α) : List.extractP p l = (List.find? p l, List.eraseP p l)

theorem List.extractP_eq_find?_eraseP.go {α : Type u_1} {p : α → Bool} (l : List α) (acc : Array α) (xs : List α) : l = acc.data ++ xs → List.extractP.go p l xs acc = (List.find? p xs, acc.data ++ List.eraseP p xs)

erase

theorem List.erase_eq_eraseP' {α : Type u_1} [BEq α] (a : α) (l : List α) : l.erase a = List.eraseP (fun (x : α) => x == a) l

theorem List.erase_eq_eraseP {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (l : List α) : l.erase a = List.eraseP (fun (x : α) => a == x) l

theorem List.exists_erase_eq {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) : ∃ (l₁ : List α), ∃ (l₂ : List α), ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l.erase a = l₁ ++ l₂

@[simp]

theorem List.length_erase_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) : (l.erase a).length = l.length.pred

theorem List.erase_append_left {α : Type u_1} [BEq α] {a : α} [LawfulBEq α] {l₁ : List α} (l₂ : List α) (h : a ∈ l₁) : (l₁ ++ l₂).erase a = l₁.erase a ++ l₂

theorem List.erase_append_right {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : ¬a ∈ l₁) : (l₁ ++ l₂).erase a = l₁ ++ l₂.erase a

theorem List.erase_sublist {α : Type u_1} [BEq α] (a : α) (l : List α) : (l.erase a).Sublist l

theorem List.erase_subset {α : Type u_1} [BEq α] (a : α) (l : List α) : l.erase a ⊆ l

theorem List.Sublist.erase {α : Type u_1} [BEq α] (a : α) {l₁ : List α} {l₂ : List α} (h : l₁.Sublist l₂) : (l₁.erase a).Sublist (l₂.erase a)

@[deprecated List.Sublist.erase]

theorem List.sublist.erase {α : Type u_1} [BEq α] (a : α) {l₁ : List α} {l₂ : List α} (h : l₁.Sublist l₂) : (l₁.erase a).Sublist (l₂.erase a)

Alias of List.Sublist.erase.

theorem List.mem_of_mem_erase {α : Type u_1} [BEq α] {a : α} {b : α} {l : List α} (h : a ∈ l.erase b) : a ∈ l

@[simp]

theorem List.mem_erase_of_ne {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {b : α} {l : List α} (ab : a ≠ b) : a ∈ l.erase b ↔ a ∈ l

theorem List.erase_comm {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (b : α) (l : List α) : (l.erase a).erase b = (l.erase b).erase a

filter and partition

@[simp]

theorem List.filter_sublist {α : Type u_1} {p : α → Bool} (l : List α) : (List.filter p l).Sublist l

filterMap

theorem List.length_filter_le {α : Type u_1} (p : α → Bool) (l : List α) : (List.filter p l).length ≤ l.length

theorem List.length_filterMap_le {α : Type u_1} {β : Type u_2} (f : α → Option β) (l : List α) : (List.filterMap f l).length ≤ l.length

theorem List.Sublist.filterMap {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (f : α → Option β) (s : l₁.Sublist l₂) : (List.filterMap f l₁).Sublist (List.filterMap f l₂)

theorem List.Sublist.filter {α : Type u_1} (p : α → Bool) {l₁ : List α} {l₂ : List α} (s : l₁.Sublist l₂) : (List.filter p l₁).Sublist (List.filter p l₂)

@[simp]

theorem List.filter_eq_self : ∀ {α : Type u_1} {p : α → Bool} {l : List α}, List.filter p l = l ↔ ∀ (a : α), a ∈ l → p a = true

@[simp]

theorem List.filter_length_eq_length : ∀ {α : Type u_1} {p : α → Bool} {l : List α}, (List.filter p l).length = l.length ↔ ∀ (a : α), a ∈ l → p a = true

findIdx

@[simp]

theorem List.findIdx_nil {α : Type u_1} (p : α → Bool) : List.findIdx p [] = 0

theorem List.findIdx_cons {α : Type u_1} (p : α → Bool) (b : α) (l : List α) : List.findIdx p (b :: l) = bif p b then 0 else List.findIdx p l + 1

theorem List.findIdx_cons.findIdx_go_succ {α : Type u_1} (p : α → Bool) (l : List α) (n : Nat) : List.findIdx.go p l (n + 1) = List.findIdx.go p l n + 1

theorem List.findIdx_of_get?_eq_some {α : Type u_1} {p : α → Bool} {y : α} {xs : List α} (w : xs.get? (List.findIdx p xs) = some y) : p y = true

theorem List.findIdx_get {α : Type u_1} {p : α → Bool} {xs : List α} {w : List.findIdx p xs < xs.length} : p (xs.get ⟨List.findIdx p xs, w⟩) = true

theorem List.findIdx_lt_length_of_exists {α : Type u_1} {p : α → Bool} {xs : List α} (h : ∃ (x : α), x ∈ xs ∧ p x = true) : List.findIdx p xs < xs.length

theorem List.findIdx_get?_eq_get_of_exists {α : Type u_1} {p : α → Bool} {xs : List α} (h : ∃ (x : α), x ∈ xs ∧ p x = true) : xs.get? (List.findIdx p xs) = some (xs.get ⟨List.findIdx p xs, ⋯⟩)

findIdx?

@[simp]

theorem List.findIdx?_nil {α : Type u_1} {p : α → Bool} {i : Nat} : List.findIdx? p [] i = none

@[simp]

theorem List.findIdx?_cons : ∀ {α : Type u_1} {x : α} {xs : List α} {p : α → Bool} {i : Nat}, List.findIdx? p (x :: xs) i = if p x = true then some i else List.findIdx? p xs (i + 1)

@[simp]

theorem List.findIdx?_succ {α : Type u_1} {xs : List α} {p : α → Bool} {i : Nat} : List.findIdx? p xs (i + 1) = Option.map (fun (i : Nat) => i + 1) (List.findIdx? p xs i)

theorem List.findIdx?_eq_some_iff {α : Type u_1} {i : Nat} (xs : List α) (p : α → Bool) : List.findIdx? p xs = some i ↔ List.map p (List.take (i + 1) xs) = List.replicate i false ++ [true]

theorem List.findIdx?_of_eq_some {α : Type u_1} {i : Nat} {xs : List α} {p : α → Bool} (w : List.findIdx? p xs = some i) : match xs.get? i with | some a => p a = true | none => false = true

theorem List.findIdx?_of_eq_none {α : Type u_1} {xs : List α} {p : α → Bool} (w : List.findIdx? p xs = none) (i : Nat) : match xs.get? i with | some a => ¬p a = true | none => true = true

@[simp]

theorem List.findIdx?_append {α : Type u_1} {xs : List α} {ys : List α} {p : α → Bool} : List.findIdx? p (xs ++ ys) = HOrElse.hOrElse (List.findIdx? p xs) fun (x : Unit) => Option.map (fun (i : Nat) => i + xs.length) (List.findIdx? p ys)

@[simp]

theorem List.findIdx?_replicate {n : Nat} : ∀ {α : Type u_1} {a : α} {p : α → Bool}, List.findIdx? p (List.replicate n a) = if 0 < n ∧ p a = true then some 0 else none

pairwise

theorem List.Pairwise.sublist : ∀ {α : Type u_1} {l₁ l₂ : List α} {R : α → α → Prop}, l₁.Sublist l₂ → List.Pairwise R l₂ → List.Pairwise R l₁

theorem List.pairwise_map {α : Type u_1} : ∀ {α_1 : Type u_2} {f : α → α_1} {R : α_1 → α_1 → Prop} {l : List α}, List.Pairwise R (List.map f l) ↔ List.Pairwise (fun (a b : α) => R (f a) (f b)) l

theorem List.pairwise_append {α : Type u_1} {R : α → α → Prop} {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 ∈ l₂ → R a b

theorem List.pairwise_reverse {α : Type u_1} {R : α → α → Prop} {l : List α} : List.Pairwise R l.reverse ↔ List.Pairwise (fun (a b : α) => R b a) l

theorem List.Pairwise.imp {α : Type u_1} {R : α → α → Prop} {S : α → α → Prop} (H : ∀ {a b : α}, R a b → S a b) {l : List α} : List.Pairwise R l → List.Pairwise S l

replaceF

theorem List.replaceF_nil : ∀ {α : Type u_1} {p : α → Option α}, List.replaceF p [] = []

theorem List.replaceF_cons {α : Type u_1} {p : α → Option α} (a : α) (l : List α) : List.replaceF p (a :: l) = match p a with | none => a :: List.replaceF p l | some a' => a' :: l

theorem List.replaceF_cons_of_some {α : Type u_1} {a' : α} {a : α} {l : List α} (p : α → Option α) (h : p a = some a') : List.replaceF p (a :: l) = a' :: l

theorem List.replaceF_cons_of_none {α : Type u_1} {a : α} {l : List α} (p : α → Option α) (h : p a = none) : List.replaceF p (a :: l) = a :: List.replaceF p l

theorem List.replaceF_of_forall_none {α : Type u_1} {p : α → Option α} {l : List α} (h : ∀ (a : α), a ∈ l → p a = none) : List.replaceF p l = l

theorem List.exists_of_replaceF {α : Type u_1} {p : α → Option α} {l : List α} {a : α} {a' : α} (al : a ∈ l) (pa : p a = some a') : ∃ (a : α), ∃ (a' : α), ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (b : α), b ∈ l₁ → p b = none) ∧ p a = some a' ∧ l = l₁ ++ a :: l₂ ∧ List.replaceF p l = l₁ ++ a' :: l₂

theorem List.exists_or_eq_self_of_replaceF {α : Type u_1} (p : α → Option α) (l : List α) : List.replaceF p l = l ∨ ∃ (a : α), ∃ (a' : α), ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (b : α), b ∈ l₁ → p b = none) ∧ p a = some a' ∧ l = l₁ ++ a :: l₂ ∧ List.replaceF p l = l₁ ++ a' :: l₂

@[simp]

theorem List.length_replaceF : ∀ {α : Type u_1} {f : α → Option α} {l : List α}, (List.replaceF f l).length = l.length

disjoint

theorem List.disjoint_symm : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Disjoint l₂ → l₂.Disjoint l₁

theorem List.disjoint_comm : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Disjoint l₂ ↔ l₂.Disjoint l₁

theorem List.disjoint_left : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Disjoint l₂ ↔ ∀ ⦃a : α⦄, a ∈ l₁ → ¬a ∈ l₂

theorem List.disjoint_right : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Disjoint l₂ ↔ ∀ ⦃a : α⦄, a ∈ l₂ → ¬a ∈ l₁

theorem List.disjoint_iff_ne : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Disjoint l₂ ↔ ∀ (a : α), a ∈ l₁ → ∀ (b : α), b ∈ l₂ → a ≠ b

theorem List.disjoint_of_subset_left : ∀ {α : Type u_1} {l₁ l l₂ : List α}, l₁ ⊆ l → l.Disjoint l₂ → l₁.Disjoint l₂

theorem List.disjoint_of_subset_right : ∀ {α : Type u_1} {l₂ l l₁ : List α}, l₂ ⊆ l → l₁.Disjoint l → l₁.Disjoint l₂

theorem List.disjoint_of_disjoint_cons_left : ∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, (a :: l₁).Disjoint l₂ → l₁.Disjoint l₂

theorem List.disjoint_of_disjoint_cons_right : ∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, l₁.Disjoint (a :: l₂) → l₁.Disjoint l₂

@[simp]

theorem List.disjoint_nil_left {α : Type u_1} (l : List α) : [].Disjoint l

@[simp]

theorem List.disjoint_nil_right {α : Type u_1} (l : List α) : l.Disjoint []

@[simp]

theorem List.singleton_disjoint : ∀ {α : Type u_1} {a : α} {l : List α}, [a].Disjoint l ↔ ¬a ∈ l

@[simp]

theorem List.disjoint_singleton : ∀ {α : Type u_1} {l : List α} {a : α}, l.Disjoint [a] ↔ ¬a ∈ l

@[simp]

theorem List.disjoint_append_left : ∀ {α : Type u_1} {l₁ l₂ l : List α}, (l₁ ++ l₂).Disjoint l ↔ l₁.Disjoint l ∧ l₂.Disjoint l

@[simp]

theorem List.disjoint_append_right : ∀ {α : Type u_1} {l l₁ l₂ : List α}, l.Disjoint (l₁ ++ l₂) ↔ l.Disjoint l₁ ∧ l.Disjoint l₂

@[simp]

theorem List.disjoint_cons_left : ∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, (a :: l₁).Disjoint l₂ ↔ ¬a ∈ l₂ ∧ l₁.Disjoint l₂

@[simp]

theorem List.disjoint_cons_right : ∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α}, l₁.Disjoint (a :: l₂) ↔ ¬a ∈ l₁ ∧ l₁.Disjoint l₂

theorem List.disjoint_of_disjoint_append_left_left : ∀ {α : Type u_1} {l₁ l₂ l : List α}, (l₁ ++ l₂).Disjoint l → l₁.Disjoint l

theorem List.disjoint_of_disjoint_append_left_right : ∀ {α : Type u_1} {l₁ l₂ l : List α}, (l₁ ++ l₂).Disjoint l → l₂.Disjoint l

theorem List.disjoint_of_disjoint_append_right_left : ∀ {α : Type u_1} {l l₁ l₂ : List α}, l.Disjoint (l₁ ++ l₂) → l.Disjoint l₁

theorem List.disjoint_of_disjoint_append_right_right : ∀ {α : Type u_1} {l l₁ l₂ : List α}, l.Disjoint (l₁ ++ l₂) → l.Disjoint l₂

foldl / foldr

theorem List.foldl_hom {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} (f : α₁ → α₂) (g₁ : α₁ → β → α₁) (g₂ : α₂ → β → α₂) (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 : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ : α → β₂ → β₂) (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

theorem List.union_def {α : Type u_1} [BEq α] (l₁ : List α) (l₂ : List α) : l₁ ∪ l₂ = List.foldr List.insert l₂ l₁

@[simp]

theorem List.nil_union {α : Type u_1} [BEq α] (l : List α) : [] ∪ l = l

@[simp]

theorem List.cons_union {α : Type u_1} [BEq α] (a : α) (l₁ : List α) (l₂ : List α) : a :: l₁ ∪ l₂ = List.insert a (l₁ ∪ l₂)

@[simp]

theorem List.mem_union_iff {α : Type u_1} [BEq α] [LawfulBEq α] {x : α} {l₁ : List α} {l₂ : List α} : x ∈ l₁ ∪ l₂ ↔ x ∈ l₁ ∨ x ∈ l₂

inter

theorem List.inter_def {α : Type u_1} [BEq α] (l₁ : List α) (l₂ : List α) : l₁ ∩ l₂ = List.filter (fun (x : α) => List.elem x l₂) l₁

@[simp]

theorem List.mem_inter_iff {α : Type u_1} [BEq α] [LawfulBEq α] {x : α} {l₁ : List α} {l₂ : List α} : x ∈ l₁ ∩ l₂ ↔ x ∈ l₁ ∧ x ∈ l₂

product

@[simp]

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) ∈ xs.product ys ↔ x ∈ xs ∧ y ∈ ys

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

leftpad

@[simp]

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

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) : List.replicate (n - l.length) a <+: List.leftpad n a l

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

monadic operations

@[simp]

theorem List.forIn_eq_forIn {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [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} [Monad m] [LawfulMonad m] (f : α → β → m (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} [Monad m] [LawfulMonad m] (l₁ : List α) (l₂ : List α) (f : α → m PUnit) : (l₁ ++ l₂).forM f = do l₁.forM f l₂.forM f

diff

@[simp]

theorem List.diff_nil {α : Type u_1} [BEq α] (l : List α) : l.diff [] = l

@[simp]

theorem List.diff_cons {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) (a : α) : l₁.diff (a :: l₂) = (l₁.erase a).diff l₂

theorem List.diff_cons_right {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) (a : α) : l₁.diff (a :: l₂) = (l₁.diff l₂).erase a

theorem List.diff_erase {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) (a : α) : (l₁.diff l₂).erase a = (l₁.erase a).diff l₂

@[simp]

theorem List.nil_diff {α : Type u_1} [BEq α] [LawfulBEq α] (l : List α) : [].diff l = []

theorem List.cons_diff {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (l₁ : List α) (l₂ : List α) : (a :: l₁).diff l₂ = if a ∈ l₂ then l₁.diff (l₂.erase a) else a :: l₁.diff l₂

theorem List.cons_diff_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₂ : List α} (h : a ∈ l₂) (l₁ : List α) : (a :: l₁).diff l₂ = l₁.diff (l₂.erase a)

theorem List.cons_diff_of_not_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₂ : List α} (h : ¬a ∈ l₂) (l₁ : List α) : (a :: l₁).diff l₂ = a :: l₁.diff l₂

theorem List.diff_eq_foldl {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) : l₁.diff l₂ = List.foldl List.erase l₁ l₂

@[simp]

theorem List.diff_append {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) (l₃ : List α) : l₁.diff (l₂ ++ l₃) = (l₁.diff l₂).diff l₃

theorem List.diff_sublist {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) : (l₁.diff l₂).Sublist l₁

theorem List.diff_subset {α : Type u_1} [BEq α] [LawfulBEq α] (l₁ : List α) (l₂ : List α) : l₁.diff l₂ ⊆ l₁

theorem List.mem_diff_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₁ : List α} {l₂ : List α} : a ∈ l₁ → ¬a ∈ l₂ → a ∈ l₁.diff l₂

theorem List.Sublist.diff_right {α : Type u_1} [BEq α] [LawfulBEq α] {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁.Sublist l₂ → (l₁.diff l₃).Sublist (l₂.diff l₃)

theorem List.Sublist.erase_diff_erase_sublist {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₁ : List α} {l₂ : List α} : l₁.Sublist l₂ → ((l₂.erase a).diff (l₁.erase a)).Sublist (l₂.diff l₁)

prefix, suffix, infix

@[simp]

theorem List.prefix_append {α : Type u_1} (l₁ : List α) (l₂ : List α) : l₁ <+: l₁ ++ l₂

@[simp]

theorem List.suffix_append {α : Type u_1} (l₁ : List α) (l₂ : List α) : l₂ <:+ l₁ ++ l₂

theorem List.infix_append {α : Type u_1} (l₁ : List α) (l₂ : List α) (l₃ : List α) : l₂ <:+: l₁ ++ l₂ ++ l₃

@[simp]

theorem List.infix_append' {α : Type u_1} (l₁ : List α) (l₂ : List α) (l₃ : List α) : l₂ <:+: l₁ ++ (l₂ ++ l₃)

theorem List.IsPrefix.isInfix : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁ <:+: l₂

theorem List.IsSuffix.isInfix : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁ <:+: l₂

theorem List.nil_prefix {α : Type u_1} (l : List α) : [] <+: l

theorem List.nil_suffix {α : Type u_1} (l : List α) : [] <:+ l

theorem List.nil_infix {α : Type u_1} (l : List α) : [] <:+: l

theorem List.prefix_refl {α : Type u_1} (l : List α) : l <+: l

theorem List.suffix_refl {α : Type u_1} (l : List α) : l <:+ l

theorem List.infix_refl {α : Type u_1} (l : List α) : l <:+: l

@[simp]

theorem List.suffix_cons {α : Type u_1} (a : α) (l : List α) : l <:+ a :: l

theorem List.infix_cons : ∀ {α : Type u_1} {l₁ l₂ : List α} {a : α}, l₁ <:+: l₂ → l₁ <:+: a :: l₂

theorem List.infix_concat : ∀ {α : Type u_1} {l₁ l₂ : List α} {a : α}, l₁ <:+: l₂ → l₁ <:+: l₂.concat a

theorem List.IsPrefix.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <+: l₂ → l₂ <+: l₃ → l₁ <+: l₃

theorem List.IsSuffix.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <:+ l₂ → l₂ <:+ l₃ → l₁ <:+ l₃

theorem List.IsInfix.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <:+: l₂ → l₂ <:+: l₃ → l₁ <:+: l₃

theorem List.IsInfix.sublist : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁.Sublist l₂

theorem List.IsInfix.subset : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁ ⊆ l₂

theorem List.IsPrefix.sublist : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁.Sublist l₂

theorem List.IsPrefix.subset : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁ ⊆ l₂

theorem List.IsSuffix.sublist : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁.Sublist l₂

theorem List.IsSuffix.subset : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁ ⊆ l₂

@[simp]

theorem List.reverse_suffix : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.reverse <:+ l₂.reverse ↔ l₁ <+: l₂

@[simp]

theorem List.reverse_prefix : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.reverse <+: l₂.reverse ↔ l₁ <:+ l₂

@[simp]

theorem List.reverse_infix : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.reverse <:+: l₂.reverse ↔ l₁ <:+: l₂

theorem List.IsInfix.length_le : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁.length ≤ l₂.length

theorem List.IsPrefix.length_le : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁.length ≤ l₂.length

theorem List.IsSuffix.length_le : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁.length ≤ l₂.length

@[simp]

theorem List.infix_nil : ∀ {α : Type u_1} {l : List α}, l <:+: [] ↔ l = []

@[simp]

theorem List.prefix_nil : ∀ {α : Type u_1} {l : List α}, l <+: [] ↔ l = []

@[simp]

theorem List.suffix_nil : ∀ {α : Type u_1} {l : List α}, l <:+ [] ↔ l = []

theorem List.infix_iff_prefix_suffix {α : Type u_1} (l₁ : List α) (l₂ : List α) : l₁ <:+: l₂ ↔ ∃ (t : List α), l₁ <+: t ∧ t <:+ l₂

theorem List.IsInfix.eq_of_length : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁.length = l₂.length → l₁ = l₂

theorem List.IsPrefix.eq_of_length : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁.length = l₂.length → l₁ = l₂

theorem List.IsSuffix.eq_of_length : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁.length = l₂.length → l₁ = l₂

theorem List.prefix_of_prefix_length_le {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <+: l₃ → l₂ <+: l₃ → l₁.length ≤ l₂.length → l₁ <+: l₂

theorem List.prefix_or_prefix_of_prefix : ∀ {α : Type u_1} {l₁ l₃ l₂ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → l₁ <+: l₂ ∨ l₂ <+: l₁

theorem List.suffix_of_suffix_length_le : ∀ {α : Type u_1} {l₁ l₃ l₂ : List α}, l₁ <:+ l₃ → l₂ <:+ l₃ → l₁.length ≤ l₂.length → l₁ <:+ l₂

theorem List.suffix_or_suffix_of_suffix : ∀ {α : Type u_1} {l₁ l₃ l₂ : List α}, l₁ <:+ l₃ → l₂ <:+ l₃ → l₁ <:+ l₂ ∨ l₂ <:+ l₁

theorem List.suffix_cons_iff : ∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α}, l₁ <:+ a :: l₂ ↔ l₁ = a :: l₂ ∨ l₁ <:+ l₂

theorem List.infix_cons_iff : ∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α}, l₁ <:+: a :: l₂ ↔ l₁ <+: a :: l₂ ∨ l₁ <:+: l₂

theorem List.infix_of_mem_join {α : Type u_1} {l : List α} {L : List (List α)} : l ∈ L → l <:+: L.join

theorem List.prefix_append_right_inj : ∀ {α : Type u_1} {l₁ l₂ : List α} (l : List α), l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂

@[simp]

theorem List.prefix_cons_inj : ∀ {α : Type u_1} {l₁ l₂ : List α} (a : α), a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂

theorem List.take_prefix {α : Type u_1} (n : Nat) (l : List α) : List.take n l <+: l

theorem List.drop_suffix {α : Type u_1} (n : Nat) (l : List α) : List.drop n l <:+ l

theorem List.take_sublist {α : Type u_1} (n : Nat) (l : List α) : (List.take n l).Sublist l

theorem List.drop_sublist {α : Type u_1} (n : Nat) (l : List α) : (List.drop n l).Sublist l

theorem List.take_subset {α : Type u_1} (n : Nat) (l : List α) : List.take n l ⊆ l

theorem List.drop_subset {α : Type u_1} (n : Nat) (l : List α) : List.drop n l ⊆ l

theorem List.mem_of_mem_take {α : Type u_1} {a : α} {n : Nat} {l : List α} (h : a ∈ List.take n l) : a ∈ l

theorem List.IsPrefix.filter {α : Type u_1} (p : α → Bool) ⦃l₁ : List α⦄ ⦃l₂ : List α⦄ (h : l₁ <+: l₂) : List.filter p l₁ <+: List.filter p l₂

theorem List.IsSuffix.filter {α : Type u_1} (p : α → Bool) ⦃l₁ : List α⦄ ⦃l₂ : List α⦄ (h : l₁ <:+ l₂) : List.filter p l₁ <:+ List.filter p l₂

theorem List.IsInfix.filter {α : Type u_1} (p : α → Bool) ⦃l₁ : List α⦄ ⦃l₂ : List α⦄ (h : l₁ <:+: l₂) : List.filter p l₁ <:+: List.filter p l₂

drop

theorem List.mem_of_mem_drop {α : Type u_1} {a : α} {n : Nat} {l : List α} (h : a ∈ List.drop n l) : a ∈ l

theorem List.disjoint_take_drop {α : Type u_1} {m : Nat} {n : Nat} {l : List α} : l.Nodup → m ≤ n → (List.take m l).Disjoint (List.drop n l)

Chain

@[simp]

theorem List.chain_cons {α : Type u_1} {R : α → α → Prop} {a : α} {b : α} {l : List α} : List.Chain R a (b :: l) ↔ R a b ∧ List.Chain R b l

theorem List.rel_of_chain_cons {α : Type u_1} {R : α → α → Prop} {a : α} {b : α} {l : List α} (p : List.Chain R a (b :: l)) : R a b

theorem List.chain_of_chain_cons {α : Type u_1} {R : α → α → Prop} {a : α} {b : α} {l : List α} (p : List.Chain R a (b :: l)) : List.Chain R b l

theorem List.Chain.imp' {α : Type u_1} {R : α → α → Prop} {S : α → α → Prop} (HRS : ∀ ⦃a b : α⦄, R a b → S a b) {a : α} {b : α} (Hab : ∀ ⦃c : α⦄, R a c → S b c) {l : List α} (p : List.Chain R a l) : List.Chain S b l

theorem List.Chain.imp {α : Type u_1} {R : α → α → Prop} {S : α → α → Prop} (H : ∀ (a b : α), R a b → S a b) {a : α} {l : List α} (p : List.Chain R a l) : List.Chain S a l

theorem List.Pairwise.chain : ∀ {α : Type u_1} {R : α → α → Prop} {a : α} {l : List α}, List.Pairwise R (a :: l) → List.Chain R a l

range', range

@[simp]

theorem List.length_range' (s : Nat) (step : Nat) (n : Nat) : (List.range' s n step).length = n

@[simp]

theorem List.range'_eq_nil {s : Nat} {n : Nat} {step : Nat} : List.range' s n step = [] ↔ n = 0

theorem List.mem_range' {m : Nat} {s : Nat} {step : Nat} {n : Nat} : m ∈ List.range' s n step ↔ ∃ (i : Nat), i < n ∧ m = s + step * i

@[simp]

theorem List.mem_range'_1 {m : Nat} {s : Nat} {n : Nat} : m ∈ List.range' s n ↔ s ≤ m ∧ m < s + n

@[simp]

theorem List.map_add_range' (a : Nat) (s : Nat) (n : Nat) (step : Nat) : List.map (fun (x : Nat) => a + x) (List.range' s n step) = List.range' (a + s) n step

theorem List.map_sub_range' {step : Nat} (a : Nat) (s : Nat) (n : Nat) (h : a ≤ s) : List.map (fun (x : Nat) => x - a) (List.range' s n step) = List.range' (s - a) n step

theorem List.chain_succ_range' (s : Nat) (n : Nat) (step : Nat) : List.Chain (fun (a b : Nat) => b = a + step) s (List.range' (s + step) n step)

theorem List.chain_lt_range' (s : Nat) (n : Nat) {step : Nat} (h : 0 < step) : List.Chain (fun (x x_1 : Nat) => x < x_1) s (List.range' (s + step) n step)

theorem List.range'_append (s : Nat) (m : Nat) (n : Nat) (step : Nat) : List.range' s m step ++ List.range' (s + step * m) n step = List.range' s (n + m) step

@[simp]

theorem List.range'_append_1 (s : Nat) (m : Nat) (n : Nat) : List.range' s m ++ List.range' (s + m) n = List.range' s (n + m)

theorem List.range'_sublist_right {step : Nat} {s : Nat} {m : Nat} {n : Nat} : (List.range' s m step).Sublist (List.range' s n step) ↔ m ≤ n

theorem List.range'_subset_right {step : Nat} {s : Nat} {m : Nat} {n : Nat} (step0 : 0 < step) : List.range' s m step ⊆ List.range' s n step ↔ m ≤ n

theorem List.range'_subset_right_1 {s : Nat} {m : Nat} {n : Nat} : List.range' s m ⊆ List.range' s n ↔ m ≤ n

theorem List.get?_range' (s : Nat) (step : Nat) {m : Nat} {n : Nat} : m < n → (List.range' s n step).get? m = some (s + step * m)

@[simp]

theorem List.get_range' {n : Nat} {m : Nat} {step : Nat} (i : Nat) (H : i < (List.range' n m step).length) : (List.range' n m step).get ⟨i, H⟩ = n + step * i

theorem List.range'_concat {step : Nat} (s : Nat) (n : Nat) : List.range' s (n + 1) step = List.range' s n step ++ [s + step * n]

theorem List.range'_1_concat (s : Nat) (n : Nat) : List.range' s (n + 1) = List.range' s n ++ [s + n]

theorem List.range_loop_range' (s : Nat) (n : Nat) : List.range.loop s (List.range' s n) = List.range' 0 (n + s)

theorem List.range_eq_range' (n : Nat) : List.range n = List.range' 0 n

theorem List.range_succ_eq_map (n : Nat) : List.range (n + 1) = 0 :: List.map Nat.succ (List.range n)

theorem List.range'_eq_map_range (s : Nat) (n : Nat) : List.range' s n = List.map (fun (x : Nat) => s + x) (List.range n)

@[simp]

theorem List.length_range (n : Nat) : (List.range n).length = n

@[simp]

theorem List.range_eq_nil {n : Nat} : List.range n = [] ↔ n = 0

@[simp]

theorem List.range_sublist {m : Nat} {n : Nat} : (List.range m).Sublist (List.range n) ↔ m ≤ n

@[simp]

theorem List.range_subset {m : Nat} {n : Nat} : List.range m ⊆ List.range n ↔ m ≤ n

@[simp]

theorem List.mem_range {m : Nat} {n : Nat} : m ∈ List.range n ↔ m < n

theorem List.not_mem_range_self {n : Nat} : ¬n ∈ List.range n

theorem List.self_mem_range_succ (n : Nat) : n ∈ List.range (n + 1)

theorem List.get?_range {m : Nat} {n : Nat} (h : m < n) : (List.range n).get? m = some m

theorem List.range_succ (n : Nat) : List.range n.succ = List.range n ++ [n]

@[simp]

theorem List.range_zero : List.range 0 = []

theorem List.range_add (a : Nat) (b : Nat) : List.range (a + b) = List.range a ++ List.map (fun (x : Nat) => a + x) (List.range b)

theorem List.iota_eq_reverse_range' (n : Nat) : List.iota n = (List.range' 1 n).reverse

@[simp]

theorem List.length_iota (n : Nat) : (List.iota n).length = n

@[simp]

theorem List.mem_iota {m : Nat} {n : Nat} : m ∈ List.iota n ↔ 1 ≤ m ∧ m ≤ n

theorem List.reverse_range' (s : Nat) (n : Nat) : (List.range' s n).reverse = List.map (fun (x : Nat) => s + n - 1 - x) (List.range n)

@[simp]

theorem List.get_range {n : Nat} (i : Nat) (H : i < (List.range n).length) : (List.range n).get ⟨i, H⟩ = i

enum, enumFrom

@[simp]

theorem List.enumFrom_map_fst {α : Type u_1} (n : Nat) (l : List α) : List.map Prod.fst (List.enumFrom n l) = List.range' n l.length

@[simp]

theorem List.enum_map_fst {α : Type u_1} (l : List α) : List.map Prod.fst l.enum = List.range l.length

maximum?

theorem List.maximum?_eq_some_iff' {a : Nat} {xs : List Nat} : xs.maximum? = some a ↔ a ∈ xs ∧ ∀ (b : Nat), b ∈ xs → b ≤ a

indexOf and indexesOf

theorem List.foldrIdx_start {α : Type u_1} {xs : List α} : ∀ {α_1 : Sort u_2} {f : Nat → α → α_1 → α_1} {i : α_1} {s : Nat}, List.foldrIdx f i xs s = List.foldrIdx (fun (i : Nat) => f (i + s)) i xs

@[simp]

theorem List.foldrIdx_cons {α : Type u_1} {x : α} {xs : List α} : ∀ {α_1 : Sort u_2} {f : Nat → α → α_1 → α_1} {i : α_1} {s : Nat}, List.foldrIdx f i (x :: xs) s = f s x (List.foldrIdx f i xs (s + 1))

theorem List.findIdxs_cons_aux {α : Type u_1} {xs : List α} {s : Nat} (p : α → Bool) : List.foldrIdx (fun (i : Nat) (a : α) (is : List Nat) => if p a = true then (i + 1) :: is else is) [] xs s = List.map (fun (x : Nat) => x + 1) (List.foldrIdx (fun (i : Nat) (a : α) (is : List Nat) => if p a = true then i :: is else is) [] xs s)

theorem List.findIdxs_cons {α : Type u_1} {x : α} {xs : List α} {p : α → Bool} : List.findIdxs p (x :: xs) = bif p x then 0 :: List.map (fun (x : Nat) => x + 1) (List.findIdxs p xs) else List.map (fun (x : Nat) => x + 1) (List.findIdxs p xs)

@[simp]

theorem List.indexesOf_nil {α : Type u_1} {x : α} [BEq α] : List.indexesOf x [] = []

theorem List.indexesOf_cons {α : Type u_1} {x : α} {xs : List α} {y : α} [BEq α] : List.indexesOf y (x :: xs) = bif x == y then 0 :: List.map (fun (x : Nat) => x + 1) (List.indexesOf y xs) else List.map (fun (x : Nat) => x + 1) (List.indexesOf y xs)

@[simp]

theorem List.indexOf_nil {α : Type u_1} {x : α} [BEq α] : List.indexOf x [] = 0

theorem List.indexOf_cons {α : Type u_1} {x : α} {xs : List α} {y : α} [BEq α] : List.indexOf y (x :: xs) = bif x == y then 0 else List.indexOf y xs + 1

theorem List.indexOf_mem_indexesOf {α : Type u_1} {x : α} [BEq α] [LawfulBEq α] {xs : List α} (m : x ∈ xs) : List.indexOf x xs ∈ List.indexesOf x xs

theorem List.merge_loop_nil_left {α : Type u_1} (s : α → α → Bool) (r : List α) (t : List α) : List.merge.loop s [] r t = t.reverseAux r

theorem List.merge_loop_nil_right {α : Type u_1} (s : α → α → Bool) (l : List α) (t : List α) : List.merge.loop s l [] t = t.reverseAux l

theorem List.merge_loop {α : Type u_1} (s : α → α → Bool) (l : List α) (r : List α) (t : List α) : List.merge.loop s l r t = t.reverseAux (List.merge s l r)

@[simp]

theorem List.merge_nil {α : Type u_1} (s : α → α → Bool) (l : List α) : List.merge s l [] = l

@[simp]

theorem List.nil_merge {α : Type u_1} (s : α → α → Bool) (r : List α) : List.merge s [] r = r

theorem List.cons_merge_cons {α : Type u_1} (s : α → α → Bool) (a : α) (b : α) (l : List α) (r : List α) : List.merge s (a :: l) (b :: r) = if s a b = true then a :: List.merge s l (b :: r) else b :: List.merge s (a :: l) r

@[simp]

theorem List.cons_merge_cons_pos {α : Type u_1} {a : α} {b : α} (s : α → α → Bool) (l : List α) (r : List α) (h : s a b = true) : List.merge s (a :: l) (b :: r) = a :: List.merge s l (b :: r)

@[simp]

theorem List.cons_merge_cons_neg {α : Type u_1} {a : α} {b : α} (s : α → α → Bool) (l : List α) (r : List α) (h : ¬s a b = true) : List.merge s (a :: l) (b :: r) = b :: List.merge s (a :: l) r

@[simp, irreducible]

theorem List.length_merge {α : Type u_1} (s : α → α → Bool) (l : List α) (r : List α) : (List.merge s l r).length = l.length + r.length

@[simp, irreducible]

theorem List.mem_merge {α : Type u_1} {x : α} {l : List α} {r : List α} {s : α → α → Bool} : x ∈ List.merge s l r ↔ x ∈ l ∨ x ∈ r

theorem List.mem_merge_left {α : Type u_1} {x : α} {l : List α} {r : List α} (s : α → α → Bool) (h : x ∈ l) : x ∈ List.merge s l r

theorem List.mem_merge_right {α : Type u_1} {x : α} {r : List α} {l : List α} (s : α → α → Bool) (h : x ∈ r) : x ∈ List.merge s l r