Definitions on lists

This file contains various definitions on lists. It does not contain proofs about these definitions, those are contained in other files in Data.List

@[implicit_reducible]

instance List.instSDiffOfDecidableEq_mathlib {α : Type u_1} [DecidableEq α] : SDiff (List α)

def List.getI {α : Type u_1} [Inhabited α] (l : List α) (n : ℕ) : α

"Inhabited" get function: returns default instead of none in the case that the index is out of bounds.

def List.headI {α : Type u_1} [Inhabited α] : List α → α

The head of a list, or the default element of the type is the list is nil.

@[simp]

theorem List.headI_nil {α : Type u_1} [Inhabited α] : [].headI = default

@[simp]

theorem List.headI_cons {α : Type u_1} [Inhabited α] {h : α} {t : List α} : (h :: t).headI = h

def List.getLastI {α : Type u_1} [Inhabited α] : List α → α

The last element of a list, with the default if list empty

def List.takeI {α : Type u_1} [Inhabited α] (n : ℕ) (l : List α) : List α

"Inhabited" take function: Take n elements from a list l. If l has less than n elements, append n - length l elements default.

def List.findM {α : Type u} {m : Type u → Type v} [Alternative m] (tac : α → m PUnit.{u + 1}) : List α → m α

findM tac l returns the first element of l on which tac succeeds, and fails otherwise.

def List.findM?' {m : Type u → Type v} [Monad m] {α : Type u} (p : α → m (ULift.{u, 0} Bool)) : List α → m (Option α)

findM? p l returns the first element a of l for which p a returns true. findM? short-circuits, so p is not necessarily run on every a in l. This is a monadic version of List.find.

def List.orM {m : Type → Type v} [Monad m] : List (m Bool) → m Bool

orM xs runs the actions in xs, returning true if any of them returns true. orM short-circuits, so if an action returns true, later actions are not run.

def List.andM {m : Type → Type v} [Monad m] : List (m Bool) → m Bool

andM xs runs the actions in xs, returning true if all of them return true. andM short-circuits, so if an action returns false, later actions are not run.

def List.foldlIdxM {m : Type v → Type w} [Monad m] {α : Type u_7} {β : Type v} (f : ℕ → β → α → m β) (b : β) (as : List α) : m β

Monadic variant of foldlIdx.

def List.foldrIdxM {m : Type v → Type w} [Monad m] {α : Type u_7} {β : Type v} (f : ℕ → α → β → m β) (b : β) (as : List α) : m β

Monadic variant of foldrIdx.

def List.mapIdxMAux' {m : Type v → Type w} [Monad m] {α : Type u_7} (f : ℕ → α → m PUnit.{v + 1}) : ℕ → List α → m PUnit.{v + 1}

Auxiliary definition for mapIdxM'.

def List.mapIdxM' {m : Type v → Type w} [Monad m] {α : Type u_7} (f : ℕ → α → m PUnit.{v + 1}) (as : List α) : m PUnit.{v + 1}

A variant of mapIdxM specialised to applicative actions which return Unit.

def List.Forall {α : Type u_1} (p : α → Prop) : List α → Prop

l.Forall p is equivalent to ∀ a ∈ l, p a, but unfolds directly to a conjunction, i.e. List.Forall p [0, 1, 2] = p 0 ∧ p 1 ∧ p 2.

def List.permutationsAux2 {α : Type u_1} {β : Type u_2} (t : α) (ts : List α) (r : List β) : List α → (List α → β) → List α × List β

An auxiliary function for defining permutations. permutationsAux2 t ts r ys f is equal to (ys ++ ts, (insert_left ys t ts).map f ++ r), where insert_left ys t ts (not explicitly defined) is the list of lists of the form insert_nth n t (ys ++ ts) for 0 ≤ n < length ys.

    permutations_aux2 10 [4, 5, 6] [] [1, 2, 3] id =
      ([1, 2, 3, 4, 5, 6],
       [[10, 1, 2, 3, 4, 5, 6],
        [1, 10, 2, 3, 4, 5, 6],
        [1, 2, 10, 3, 4, 5, 6]])

@[irreducible]

def List.permutationsAux.rec {α : Type u_1} {C : List α → List α → Sort v} (H0 : (is : List α) → C [] is) (H1 : (t : α) → (ts is : List α) → C ts (t :: is) → C is [] → C (t :: ts) is) (l₁ l₂ : List α) : C l₁ l₂

A recursor for pairs of lists. To have C l₁ l₂ for all l₁, l₂, it suffices to have it for l₂ = [] and to be able to pour the elements of l₁ into l₂.

def List.permutationsAux {α : Type u_1} : List α → List α → List (List α)

An auxiliary function for defining permutations. permutationsAux ts is is the set of all permutations of is ++ ts that do not fix ts.

def List.permutations {α : Type u_1} (l : List α) : List (List α)

List of all permutations of l.

     permutations [1, 2, 3] =
       [[1, 2, 3], [2, 1, 3], [3, 2, 1],
        [2, 3, 1], [3, 1, 2], [1, 3, 2]]

def List.permutations'Aux {α : Type u_1} (t : α) : List α → List (List α)

permutations'Aux t ts inserts t into every position in ts, including the last. This function is intended for use in specifications, so it is simpler than permutationsAux2, which plays roughly the same role in permutations.

Note that (permutationsAux2 t [] [] ts id).2 is similar to this function, but skips the last position:

    permutations'Aux 10 [1, 2, 3] =
      [[10, 1, 2, 3], [1, 10, 2, 3], [1, 2, 10, 3], [1, 2, 3, 10]]
    (permutationsAux2 10 [] [] [1, 2, 3] id).2 =
      [[10, 1, 2, 3], [1, 10, 2, 3], [1, 2, 10, 3]]

def List.permutations' {α : Type u_1} : List α → List (List α)

List of all permutations of l. This version of permutations is less efficient but has simpler definitional equations. The permutations are in a different order, but are equal up to permutation, as shown by List.permutations_perm_permutations'.

     permutations [1, 2, 3] =
       [[1, 2, 3], [2, 1, 3], [2, 3, 1],
        [1, 3, 2], [3, 1, 2], [3, 2, 1]]

def List.extractp {α : Type u_1} (p : α → Prop) [DecidablePred p] : List α → Option α × List α

extractp p l returns a pair of an element a of l satisfying the predicate p, and l, with a removed. If there is no such element a it returns (none, l).

@[implicit_reducible]

instance List.instSProd {α : Type u_1} {β : Type u_2} : SProd (List α) (List β) (List (α × β))

Notation for calculating the product of a List

def List.dedup {α : Type u_1} [DecidableEq α] : List α → List α

dedup l removes duplicates from l (taking only the last occurrence). Defined as pwFilter (≠).

dedup [1, 0, 2, 2, 1] = [0, 2, 1]

def List.destutter' {α : Type u_1} (R : α → α → Prop) [DecidableRel R] : α → List α → List α

Greedily create a sublist of a :: l such that, for every two adjacent elements a, b, R a b holds. Mostly used with ≠; for example, destutter' (≠) 1 [2, 2, 1, 1] = [1, 2, 1], destutter' (≠) 1, [2, 3, 3] = [1, 2, 3], destutter' (<) 1 [2, 5, 2, 3, 4, 9] = [1, 2, 5, 9].

def List.destutter {α : Type u_1} (R : α → α → Prop) [DecidableRel R] : List α → List α

Greedily create a sublist of l such that, for every two adjacent elements a, b ∈ l, R a b holds. Mostly used with ≠; for example, destutter (≠) [1, 2, 2, 1, 1] = [1, 2, 1], destutter (≠) [1, 2, 3, 3] = [1, 2, 3], destutter (<) [1, 2, 5, 2, 3, 4, 9] = [1, 2, 5, 9].

def List.chooseX {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : List α) : (∃ a ∈ l, p a) → { a : α // a ∈ l ∧ p a }

Given a decidable predicate p and a proof of existence of a ∈ l such that p a, choose the first element with this property. This version returns both a and proofs of a ∈ l and p a.

def List.choose {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : List α) (hp : ∃ a ∈ l, p a) : α

Given a decidable predicate p and a proof of existence of a ∈ l such that p a, choose the first element with this property. This version returns a : α, and properties are given by choose_mem and choose_property.

def List.mapDiagM' {m : Type → Type u_7} [Monad m] {α : Type u_8} (f : α → α → m Unit) : List α → m Unit

mapDiagM' f l calls f on all elements in the upper triangular part of l × l. That is, for each e ∈ l, it will run f e e and then f e e' for each e' that appears after e in l.

Example: suppose l = [1, 2, 3]. mapDiagM' f l will evaluate, in this order, f 1 1, f 1 2, f 1 3, f 2 2, f 2 3, f 3 3.

def List.map₂Left' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β → γ) : List α → List β → List γ × List β

Left-biased version of List.map₂. map₂Left' f as bs applies f to each pair of elements aᵢ ∈ as and bᵢ ∈ bs. If bs is shorter than as, f is applied to none for the remaining aᵢ. Returns the results of the f applications and the remaining bs.

map₂Left' prod.mk [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])

map₂Left' prod.mk [1] ['a', 'b'] = ([(1, some 'a')], ['b'])

def List.map₂Right' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : Option α → β → γ) (as : List α) (bs : List β) : List γ × List α

Right-biased version of List.map₂. map₂Right' f as bs applies f to each pair of elements aᵢ ∈ as and bᵢ ∈ bs. If as is shorter than bs, f is applied to none for the remaining bᵢ. Returns the results of the f applications and the remaining as.

map₂Right' prod.mk [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])

map₂Right' prod.mk [1, 2] ['a'] = ([(some 1, 'a')], [2])

def List.map₂Left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β → γ) : List α → List β → List γ

Left-biased version of List.map₂. map₂Left f as bs applies f to each pair aᵢ ∈ as and bᵢ ∈ bs. If bs is shorter than as, f is applied to none for the remaining aᵢ.

map₂Left Prod.mk [1, 2] ['a'] = [(1, some 'a'), (2, none)]

map₂Left Prod.mk [1] ['a', 'b'] = [(1, some 'a')]

map₂Left f as bs = (map₂Left' f as bs).fst

def List.map₂Right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : Option α → β → γ) (as : List α) (bs : List β) : List γ

Right-biased version of List.map₂. map₂Right f as bs applies f to each pair aᵢ ∈ as and bᵢ ∈ bs. If as is shorter than bs, f is applied to none for the remaining bᵢ.

map₂Right Prod.mk [1, 2] ['a'] = [(some 1, 'a')]

map₂Right Prod.mk [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]

map₂Right f as bs = (map₂Right' f as bs).fst

def List.mapAsyncChunked {α : Type u_7} {β : Type u_8} (f : α → β) (xs : List α) (chunk_size : ℕ := 1024) : List β

Asynchronous version of List.map.

We add some n-ary versions of List.zipWith for functions with more than two arguments. These can also be written in terms of List.zip or List.zipWith. For example, zipWith3 f xs ys zs could also be written as zipWith id (zipWith f xs ys) zs or as (zip xs <| zip ys zs).map <| fun ⟨x, y, z⟩ ↦ f x y z.

def List.zipWith3 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β → γ → δ) : List α → List β → List γ → List δ

Ternary version of List.zipWith.

def List.zipWith4 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} (f : α → β → γ → δ → ε) : List α → List β → List γ → List δ → List ε

Quaternary version of list.zipWith.

def List.zipWith5 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} {ζ : Type u_6} (f : α → β → γ → δ → ε → ζ) : List α → List β → List γ → List δ → List ε → List ζ

Quinary version of list.zipWith.

def List.replaceIf {α : Type u_1} : List α → List Bool → List α → List α

Given a starting list old, a list of Booleans and a replacement list new, read the items in old in succession and either replace them with the next element of new or not, according as to whether the corresponding Boolean is true or false.

def List.iterate {α : Type u_1} (f : α → α) (a : α) (n : ℕ) : List α

iterate f a n is [a, f a, ..., f^[n - 1] a].

@[inline]

def List.iterateTR {α : Type u_1} (f : α → α) (a : α) (n : ℕ) : List α

Tail-recursive version of List.iterate.

@[specialize #[]]

def List.iterateTR.loop {α : Type u_1} (f : α → α) (a : α) (n : ℕ) (l : List α) : List α

iterateTR.loop f a n l := iterate f a n ++ reverse l.

theorem List.iterateTR_loop_eq {α : Type u_1} (f : α → α) (a : α) (n : ℕ) (l : List α) : iterateTR.loop f a n l = l.reverse ++ iterate f a n

@[csimp]

theorem List.iterate_eq_iterateTR : @iterate = @iterateTR

def List.mapAccumr {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → γ → γ × β) : List α → γ → γ × List β

Runs a function over a list returning the intermediate results and a final result.

@[simp]

theorem List.length_mapAccumr {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → γ → γ × β) (x : List α) (s : γ) : (mapAccumr f x s).2.length = x.length

Length of the list obtained by mapAccumr.

def List.mapAccumr₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β → γ → γ × δ) : List α → List β → γ → γ × List δ

Runs a function over two lists returning the intermediate results and a final result.

@[simp]

theorem List.length_mapAccumr₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β → γ → γ × δ) (x : List α) (y : List β) (c : γ) : (mapAccumr₂ f x y c).2.length = min x.length y.length

Length of a list obtained using mapAccumr₂.

@[reducible, inline]

abbrev List.consecutivePairs {α : Type u_1} (l : List α) : List (α × α)

consecutivePairs [a, b, c, d] is [(a, b), (b, c), (c, d)].