toArray

@[simp]

theorem List.getElem_mk {α : Type u_1} {xs : List α} {i : Nat} (h : i < xs.length) : { toList := xs }[i] = xs[i]

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)

dropLast

theorem List.dropLast_eq_eraseIdx {α : Type u_1} {xs : List α} {i : Nat} (last_idx : i + 1 = xs.length) : xs.dropLast = xs.eraseIdx i

set

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

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 = take n l ++ a :: drop (n + 1) l

theorem List.modify_eq_set_getElem? {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : l.modify n f = ((fun (a : α) => l.set n (f a)) <$> l[n]?).getD l

@[deprecated List.modify_eq_set_getElem? (since := "2025-02-15")]

theorem List.modify_eq_set_get? {α : Type u_1} (f : α → α) (n : Nat) (l : List α) : l.modify n f = ((fun (a : α) => l.set n (f a)) <$> l[n]?).getD l

Alias of List.modify_eq_set_getElem?.

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

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

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

@[deprecated List.getElem?_set_of_lt (since := "2025-02-15")]

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

Alias of List.getElem?_set_of_lt.

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

@[deprecated List.getElem?_set_of_lt' (since := "2025-02-15")]

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

Alias of List.getElem?_set_of_lt'.

tail

theorem List.length_tail_add_one {α : Type u_1} (l : List α) (h : 0 < l.length) : l.tail.length + 1 = l.length

eraseP

@[simp]

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

erase

theorem List.erase_eq_self_iff_forall_bne {α : Type u_1} [BEq α] (a : α) (xs : List α) : xs.erase a = xs ↔ ∀ (x : α), x ∈ xs → ¬(x == a) = true

findIdx?

theorem List.findIdx_eq_findIdx? {α : Type u_1} (p : α → Bool) (l : List α) : findIdx p l = match findIdx? p l with | some i => i | none => l.length

replaceF

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

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

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

theorem List.exists_of_replaceF {α : Type u_1} {p : α → Option α} {l : List α} {a a' : α} : a ∈ l → 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 ∧ 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) : 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 ∧ replaceF p l = l₁ ++ a' :: l₂

@[simp]

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

disjoint

theorem List.disjoint_symm {α✝ : Type u_1} {l₁ l₂ : List α✝} (d : 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✝} (ss : l₁ ⊆ l) (d : l.Disjoint l₂) : l₁.Disjoint l₂

theorem List.disjoint_of_subset_right {α✝ : Type u_1} {l₂ l l₁ : List α✝} (ss : l₂ ⊆ l) (d : 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✝} (d : (l₁ ++ l₂).Disjoint l) : l₁.Disjoint l

theorem List.disjoint_of_disjoint_append_left_right {α✝ : Type u_1} {l₁ l₂ l : List α✝} (d : (l₁ ++ l₂).Disjoint l) : l₂.Disjoint l

theorem List.disjoint_of_disjoint_append_right_left {α✝ : Type u_1} {l l₁ l₂ : List α✝} (d : l.Disjoint (l₁ ++ l₂)) : l.Disjoint l₁

theorem List.disjoint_of_disjoint_append_right_right {α✝ : Type u_1} {l l₁ l₂ : List α✝} (d : l.Disjoint (l₁ ++ l₂)) : l.Disjoint l₂

union

theorem List.union_def {α : Type u_1} [BEq α] (l₁ l₂ : List α) : l₁ ∪ l₂ = 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 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 l₂ : List α} : x ∈ l₁ ∪ l₂ ↔ x ∈ l₁ ∨ x ∈ l₂

inter

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

@[simp]

theorem List.mem_inter_iff {α : Type u_1} [BEq α] [LawfulBEq α] {x : α} {l₁ 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.

monadic operations

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)

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 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 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 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 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 l₂ : List α) : l₁.diff l₂ = foldl List.erase l₁ l₂

@[simp]

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

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

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

drop

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

Chain

@[simp]

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

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

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

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

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

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

range', range

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

theorem List.chain_lt_range' (s n : Nat) {step : Nat} (h : 0 < step) : Chain (fun (x1 x2 : Nat) => x1 < x2) s (range' (s + step) n step)

indexOf and indexesOf

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

@[simp]

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

theorem List.findIdxs_cons_aux {α : Type u_1} {xs : List α} {s : Nat} (p : α → Bool) : foldrIdx (fun (i : Nat) (a : α) (is : List Nat) => if p a = true then (i + 1) :: is else is) [] xs s = map (fun (x : Nat) => x + 1) (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} : findIdxs p (x :: xs) = bif p x then 0 :: map (fun (x : Nat) => x + 1) (findIdxs p xs) else map (fun (x : Nat) => x + 1) (findIdxs p xs)

@[simp]

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

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

@[simp]

theorem List.eraseIdx_idxOf_eq_erase {α : Type u_1} [BEq α] (a : α) (l : List α) : l.eraseIdx (idxOf a l) = l.erase a

@[deprecated List.eraseIdx_idxOf_eq_erase (since := "2025-01-30")]

theorem List.eraseIdx_indexOf_eq_erase {α : Type u_1} [BEq α] (a : α) (l : List α) : l.eraseIdx (idxOf a l) = l.erase a

Alias of List.eraseIdx_idxOf_eq_erase.

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

@[deprecated List.idxOf_mem_indexesOf (since := "2025-01-30")]

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

Alias of List.idxOf_mem_indexesOf.

theorem List.idxOf_eq_idxOf? {α : Type u_1} [BEq α] (a : α) (l : List α) : idxOf a l = match idxOf? a l with | some i => i | none => l.length

@[deprecated List.idxOf_eq_idxOf? (since := "2025-01-30")]

theorem List.indexOf_eq_indexOf? {α : Type u_1} [BEq α] (a : α) (l : List α) : idxOf a l = match idxOf? a l with | some i => i | none => l.length

Alias of List.idxOf_eq_idxOf?.

insertP

theorem List.insertP_loop {α : Type u_1} {p : α → Bool} (a : α) (l r : List α) : insertP.loop p a l r = r.reverseAux (insertP p a l)

@[simp]

theorem List.insertP_nil {α : Type u_1} (p : α → Bool) (a : α) : insertP p a [] = [a]

@[simp]

theorem List.insertP_cons_pos {α : Type u_1} (p : α → Bool) (a b : α) (l : List α) (h : p b = true) : insertP p a (b :: l) = a :: b :: l

@[simp]

theorem List.insertP_cons_neg {α : Type u_1} (p : α → Bool) (a b : α) (l : List α) (h : ¬p b = true) : insertP p a (b :: l) = b :: insertP p a l

@[simp]

theorem List.length_insertP {α : Type u_1} (p : α → Bool) (a : α) (l : List α) : (insertP p a l).length = l.length + 1

@[simp]

theorem List.mem_insertP {α : Type u_1} (p : α → Bool) (a : α) (l : List α) : a ∈ insertP p a l

dropPrefix?, dropSuffix?, dropInfix?

@[simp]

theorem List.dropPrefix?_nil {α : Type u_1} [BEq α] {p : List α} : p.dropPrefix? [] = some p

@[irreducible]

theorem List.dropPrefix?_eq_some_iff {α : Type u_1} [BEq α] {l p s : List α} : l.dropPrefix? p = some s ↔ ∃ (p' : List α), l = p' ++ s ∧ (p' == p) = true

theorem List.dropPrefix?_append_of_beq {α : Type u_1} [BEq α] {l₁ l₂ : List α} (p : List α) (h : (l₁ == l₂) = true) : (l₁ ++ p).dropPrefix? l₂ = some p

theorem List.dropSuffix?_eq_some_iff {α : Type u_1} [BEq α] {l p s : List α} : l.dropSuffix? s = some p ↔ ∃ (s' : List α), l = p ++ s' ∧ (s' == s) = true

@[simp]

theorem List.dropSuffix?_nil {α : Type u_1} [BEq α] {s : List α} : s.dropSuffix? [] = some s

@[irreducible]

theorem List.dropInfix?_go_eq_some_iff {α : Type u_1} [BEq α] {i l acc p s : List α} : dropInfix?.go i l acc = some (p, s) ↔ ∃ (p' : List α), p = acc.reverse ++ p' ∧ (∃ (i' : List α), l = p' ++ i' ++ s ∧ (i' == i) = true) ∧ ∀ (p'' i'' s'' : List α), l = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

theorem List.dropInfix?_eq_some_iff {α : Type u_1} [BEq α] {l i p s : List α} : l.dropInfix? i = some (p, s) ↔ (∃ (i' : List α), l = p ++ i' ++ s ∧ (i' == i) = true) ∧ ∀ (p' i' s' : List α), l = p' ++ i' ++ s' → (i' == i) = true → p'.length ≥ p.length

@[simp]

theorem List.dropInfix?_nil {α : Type u_1} [BEq α] {s : List α} : s.dropInfix? [] = some ([], s)

IsPrefixOf?, IsSuffixOf?

@[simp]

theorem List.isSome_isPrefixOf?_eq_isPrefixOf {α : Type u_1} [BEq α] (xs ys : List α) : (xs.isPrefixOf? ys).isSome = xs.isPrefixOf ys

@[simp]

theorem List.isPrefixOf?_eq_some_iff_append_eq {α : Type u_1} [BEq α] [LawfulBEq α] {xs ys zs : List α} : xs.isPrefixOf? ys = some zs ↔ xs ++ zs = ys

theorem List.append_eq_of_isPrefixOf?_eq_some {α : Type u_1} [BEq α] [LawfulBEq α] {xs ys zs : List α} (h : xs.isPrefixOf? ys = some zs) : xs ++ zs = ys

@[simp]

theorem List.isSome_isSuffixOf?_eq_isSuffixOf {α : Type u_1} [BEq α] (xs ys : List α) : (xs.isSuffixOf? ys).isSome = xs.isSuffixOf ys

@[simp]

theorem List.isSuffixOf?_eq_some_iff_append_eq {α : Type u_1} [BEq α] [LawfulBEq α] {xs ys zs : List α} : xs.isSuffixOf? ys = some zs ↔ zs ++ xs = ys

theorem List.append_eq_of_isSuffixOf?_eq_some {α : Type u_1} [BEq α] [LawfulBEq α] {xs ys zs : List α} (h : xs.isSuffixOf? ys = some zs) : zs ++ xs = ys