Array literal syntax

def «term#[_,]» : Lean.ParserDescr

Syntax for Array α.

Conventions for notations in identifiers:

• The recommended spelling of #[] in identifiers is empty.

• The recommended spelling of #[x] in identifiers is singleton.

Equations

• One or more equations did not get rendered due to their size.

Preliminary theorems

@[simp]

theorem Array.size_set {α : Type u} {xs : Array α} {i : Nat} {v : α} (h : i < xs.size) : (xs.set i v h).size = xs.size

@[simp]

theorem Array.size_push {α : Type u} {xs : Array α} (v : α) : (xs.push v).size = xs.size + 1

theorem Array.ext {α : Type u} {xs ys : Array α} (h₁ : xs.size = ys.size) (h₂ : ∀ (i : Nat) (hi₁ : i < xs.size) (hi₂ : i < ys.size), xs[i] = ys[i]) : xs = ys

theorem Array.ext.extAux {α : Type u} (as bs : List α) (h₁ : as.length = bs.length) (h₂ : ∀ (i : Nat) (hi₁ : i < as.length) (hi₂ : i < bs.length), as[i] = bs[i]) : as = bs

theorem Array.ext' {α : Type u} {xs ys : Array α} (h : xs.toList = ys.toList) : xs = ys

@[simp]

theorem Array.toArrayAux_eq {α : Type u} {as : List α} {acc : Array α} : (as.toArrayAux acc).toList = acc.toList ++ as

@[simp]

theorem Array.toArray_toList {α : Type u} {xs : Array α} : xs.toList.toArray = xs

@[simp]

theorem Array.getElem_toList {α : Type u} {xs : Array α} {i : Nat} (h : i < xs.size) : xs.toList[i] = xs[i]

@[simp]

theorem Array.getElem?_toList {α : Type u} {xs : Array α} {i : Nat} : xs.toList[i]? = xs[i]?

structure Array.Mem {α : Type u} (as : Array α) (a : α) : Prop

a ∈ as is a predicate which asserts that a is in the array as.

• val : a ∈ as.toList

instance Array.instMembership {α : Type u} : Membership α (Array α)

Equations

• Array.instMembership = { mem := Array.Mem }

theorem Array.mem_def {α : Type u} {a : α} {as : Array α} : a ∈ as ↔ a ∈ as.toList

@[simp]

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

@[simp]

theorem Array.getElem_mem {α : Type u} {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs

@[simp]

theorem Array.emptyWithCapacity_eq {α : Type u_1} {n : Nat} : emptyWithCapacity n = #[]

@[simp]

theorem Array.mkEmpty_eq {α : Type u_1} {n : Nat} : mkEmpty n = #[]

@[reducible, inline, deprecated Array.toArray_toList (since := "2025-02-17")]

abbrev List.toArray_toList {α : Type u_1} {xs : Array α} : xs.toList.toArray = xs

Equations

• @List.toArray_toList = @Array.toArray_toList

theorem List.toList_toArray {α : Type u} {as : List α} : as.toArray.toList = as

@[reducible, inline, deprecated List.toList_toArray (since := "2025-02-17")]

abbrev Array.toList_toArray {α : Type u_1} {as : List α} : as.toArray.toList = as

Equations

• @Array.toList_toArray = @List.toList_toArray

@[simp]

theorem List.size_toArray {α : Type u} {as : List α} : as.toArray.size = as.length

@[reducible, inline, deprecated List.size_toArray (since := "2025-02-17")]

abbrev Array.size_toArray {α : Type u_1} {as : List α} : as.toArray.size = as.length

Equations

• @Array.size_toArray = @List.size_toArray

@[simp]

theorem List.getElem_toArray {α : Type u} {xs : List α} {i : Nat} (h : i < xs.toArray.size) : xs.toArray[i] = xs[i]

@[simp]

theorem List.getElem?_toArray {α : Type u} {xs : List α} {i : Nat} : xs.toArray[i]? = xs[i]?

@[simp]

theorem List.getElem!_toArray {α : Type u} [Inhabited α] {xs : List α} {i : Nat} : xs.toArray[i]! = xs[i]!

theorem Array.size_eq_length_toList {α : Type u} {xs : Array α} : xs.size = xs.toList.length

Externs

@[extern lean_array_size]

def Array.usize {α : Type u} (a : Array α) : USize

Returns the size of the array as a platform-native unsigned integer.

This is a low-level version of Array.size that directly queries the runtime system's representation of arrays. While this is not provable, Array.usize always returns the exact size of the array since the implementation only supports arrays of size less than USize.size.

Equations

• a.usize = a.size.toUSize

@[extern lean_array_uget]

def Array.uget {α : Type u} (a : Array α) (i : USize) (h : i.toNat < a.size) : α

Low-level indexing operator which is as fast as a C array read.

This avoids overhead due to unboxing a Nat used as an index.

Equations

• a.uget i h = a[i.toNat]

@[extern lean_array_uset]

def Array.uset {α : Type u} (xs : Array α) (i : USize) (v : α) (h : i.toNat < xs.size) : Array α

Low-level modification operator which is as fast as a C array write. The modification is performed in-place when the reference to the array is unique.

This avoids overhead due to unboxing a Nat used as an index.

Equations

• xs.uset i v h = xs.set i.toNat v h

@[extern lean_array_pop]

def Array.pop {α : Type u} (xs : Array α) : Array α

Removes the last element of an array. If the array is empty, then it is returned unmodified. The modification is performed in-place when the reference to the array is unique.

Examples:

• #[1, 2, 3].pop = #[1, 2]
• #["orange", "yellow"].pop = #["orange"]
• (#[] : Array String).pop = #[]

Equations

• xs.pop = { toList := xs.toList.dropLast }

@[simp]

theorem Array.size_pop {α : Type u} {xs : Array α} : xs.pop.size = xs.size - 1

@[extern lean_mk_array]

def Array.replicate {α : Type u} (n : Nat) (v : α) : Array α

Creates an array that contains n repetitions of v.

The corresponding List function is List.replicate.

Examples:

• Array.replicate 2 true = #[true, true]
• Array.replicate 3 () = #[(), (), ()]
• Array.replicate 0 "anything" = #[]

Equations

• Array.replicate n v = { toList := List.replicate n v }

@[extern lean_mk_array, deprecated Array.replicate (since := "2025-03-18")]

def Array.mkArray {α : Type u} (n : Nat) (v : α) : Array α

Creates an array that contains n repetitions of v.

The corresponding List function is List.replicate.

Examples:

• Array.mkArray 2 true = #[true, true]
• Array.mkArray 3 () = #[(), (), ()]
• Array.mkArray 0 "anything" = #[]

Equations

• mkArray n v = { toList := List.replicate n v }

@[extern lean_array_fswap]

def Array.swap {α : Type u} (xs : Array α) (i j : Nat) (hi : i < xs.size := by get_elem_tactic) (hj : j < xs.size := by get_elem_tactic) : Array α

Swaps two elements of an array. The modification is performed in-place when the reference to the array is unique.

Examples:

• #["red", "green", "blue", "brown"].swap 0 3 = #["brown", "green", "blue", "red"]
• #["red", "green", "blue", "brown"].swap 0 2 = #["blue", "green", "red", "brown"]
• #["red", "green", "blue", "brown"].swap 1 2 = #["red", "blue", "green", "brown"]
• #["red", "green", "blue", "brown"].swap 3 0 = #["brown", "green", "blue", "red"]

Equations

• xs.swap i j hi hj = (xs.set i xs[j] hi).set j xs[i] ⋯

@[simp]

theorem Array.size_swap {α : Type u} {xs : Array α} {i j : Nat} {hi : i < xs.size} {hj : j < xs.size} : (xs.swap i j hi hj).size = xs.size

@[extern lean_array_swap]

def Array.swapIfInBounds {α : Type u} (xs : Array α) (i j : Nat) : Array α

Swaps two elements of an array, returning the array unchanged if either index is out of bounds. The modification is performed in-place when the reference to the array is unique.

Examples:

• #["red", "green", "blue", "brown"].swapIfInBounds 0 3 = #["brown", "green", "blue", "red"]
• #["red", "green", "blue", "brown"].swapIfInBounds 0 2 = #["blue", "green", "red", "brown"]
• #["red", "green", "blue", "brown"].swapIfInBounds 1 2 = #["red", "blue", "green", "brown"]
• #["red", "green", "blue", "brown"].swapIfInBounds 0 4 = #["red", "green", "blue", "brown"]
• #["red", "green", "blue", "brown"].swapIfInBounds 9 2 = #["red", "green", "blue", "brown"]

Equations

• xs.swapIfInBounds i j = if h₁ : i < xs.size then if h₂ : j < xs.size then xs.swap i j h₁ h₂ else xs else xs

GetElem instance for USize, backed by uget

instance Array.instGetElemUSizeLtNatToNatSize {α : Type u} : GetElem (Array α) USize α fun (xs : Array α) (i : USize) => i.toNat < xs.size

Equations

• Array.instGetElemUSizeLtNatToNatSize = { getElem := fun (xs : Array α) (i : USize) (h : i.toNat < xs.size) => xs.uget i h }

Definitions

instance Array.instEmptyCollection {α : Type u} : EmptyCollection (Array α)

Equations

• Array.instEmptyCollection = { emptyCollection := #[] }

instance Array.instInhabited {α : Type u} : Inhabited (Array α)

Equations

• Array.instInhabited = { default := #[] }

def Array.isEmpty {α : Type u} (xs : Array α) : Bool

Checks whether an array is empty.

An array is empty if its size is 0.

Examples:

• (#[] : Array String).isEmpty = true
• #[1, 2].isEmpty = false
• #[()].isEmpty = false

Equations

• xs.isEmpty = decide (xs.size = 0)

@[specialize #[]]

def Array.isEqvAux {α : Type u} (xs ys : Array α) (hsz : xs.size = ys.size) (p : α → α → Bool) (i : Nat) : i ≤ xs.size → Bool

Equations

• xs.isEqvAux ys hsz p 0 x_2 = true
• xs.isEqvAux ys hsz p i.succ h = (p xs[i] ys[i] && xs.isEqvAux ys hsz p i ⋯)

@[inline]

def Array.isEqv {α : Type u} (xs ys : Array α) (p : α → α → Bool) : Bool

Returns true if as and bs have the same length and they are pairwise related by eqv.

Short-circuits at the first non-related pair of elements.

Examples:

• #[1, 2, 3].isEqv #[2, 3, 4] (· < ·) = true
• #[1, 2, 3].isEqv #[2, 2, 4] (· < ·) = false
• #[1, 2, 3].isEqv #[2, 3] (· < ·) = false

Equations

• xs.isEqv ys p = if h : xs.size = ys.size then xs.isEqvAux ys h p xs.size ⋯ else false

instance Array.instBEq {α : Type u} [BEq α] : BEq (Array α)

Equations

• Array.instBEq = { beq := fun (xs ys : Array α) => xs.isEqv ys BEq.beq }

def Array.ofFn {α : Type u} {n : Nat} (f : Fin n → α) : Array α

Creates an array by applying f to each potential index in order, starting at 0.

Examples:

• Array.ofFn (n := 3) toString = #["0", "1", "2"]
• Array.ofFn (fun i => #["red", "green", "blue"].get i.val i.isLt) = #["red", "green", "blue"]

Equations

• Array.ofFn f = Array.ofFn.go f (Array.emptyWithCapacity n) n ⋯

def Array.ofFn.go {α : Type u} {n : Nat} (f : Fin n → α) (acc : Array α) (i : Nat) : i ≤ n → Array α

Auxiliary for ofFn. ofFn.go f acc i h = acc ++ #[f (n - i), ..., f(n - 1)]

Equations

• Array.ofFn.go f acc i.succ h = Array.ofFn.go f (acc.push (f ⟨n - i - 1, ⋯⟩)) i ⋯
• Array.ofFn.go f acc 0 x_2 = acc

def Array.range (n : Nat) : Array Nat

Constructs an array that contains all the numbers from 0 to n, exclusive.

Examples:

• Array.range 5 := #[0, 1, 2, 3, 4]
• Array.range 0 := #[]
• Array.range 1 := #[0]

Equations

• Array.range n = Array.ofFn fun (i : Fin n) => ↑i

def Array.range' (start size : Nat) (step : Nat := 1) : Array Nat

Constructs an array of numbers of size size, starting at start and increasing by step at each element.

In other words, Array.range' start size step is #[start, start+step, ..., start+(len-1)*step].

Examples:

• Array.range' 0 3 (step := 1) = #[0, 1, 2]
• Array.range' 0 3 (step := 2) = #[0, 2, 4]
• Array.range' 0 4 (step := 2) = #[0, 2, 4, 6]
• Array.range' 3 4 (step := 2) = #[3, 5, 7, 9]

Equations

• Array.range' start size step = Array.ofFn fun (i : Fin size) => start + step * ↑i

@[inline]

def Array.singleton {α : Type u} (v : α) : Array α

Constructs a single-element array that contains v.

Examples:

• Array.singleton 5 = #[5]
• Array.singleton "one" = #["one"]

Equations

• Array.singleton v = #[v]

def Array.back! {α : Type u} [Inhabited α] (xs : Array α) : α

Returns the last element of an array, or panics if the array is empty.

Safer alternatives include Array.back, which requires a proof the array is non-empty, and Array.back?, which returns an Option.

Equations

• xs.back! = xs[xs.size - 1]!

def Array.back {α : Type u} (xs : Array α) (h : 0 < xs.size := by get_elem_tactic) : α

Returns the last element of an array, given a proof that the array is not empty.

See Array.back! for the version that panics if the array is empty, or Array.back? for the version that returns an option.

Equations

• xs.back h = xs[xs.size - 1]

def Array.back? {α : Type u} (xs : Array α) : Option α

Returns the last element of an array, or none if the array is empty.

See Array.back! for the version that panics if the array is empty, or Array.back for the version that requires a proof the array is non-empty.

Equations

• xs.back? = xs[xs.size - 1]?

@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]

def Array.get? {α : Type u} (xs : Array α) (i : Nat) : Option α

Equations

• xs.get? i = if h : i < xs.size then some xs[i] else none

@[inline]

def Array.swapAt {α : Type u} (xs : Array α) (i : Nat) (v : α) (hi : i < xs.size := by get_elem_tactic) : α × Array α

Swaps a new element with the element at the given index.

Returns the value formerly found at i, paired with an array in which the value at i has been replaced with v.

Examples:

• #["spinach", "broccoli", "carrot"].swapAt 1 "pepper" = ("broccoli", #["spinach", "pepper", "carrot"])
• #["spinach", "broccoli", "carrot"].swapAt 2 "pepper" = ("carrot", #["spinach", "broccoli", "pepper"])

Equations

• xs.swapAt i v hi = (xs[i], xs.set i v hi)

@[inline]

def Array.swapAt! {α : Type u} (xs : Array α) (i : Nat) (v : α) : α × Array α

Swaps a new element with the element at the given index. Panics if the index is out of bounds.

Returns the value formerly found at i, paired with an array in which the value at i has been replaced with v.

Examples:

• #["spinach", "broccoli", "carrot"].swapAt! 1 "pepper" = (#["spinach", "pepper", "carrot"], "broccoli")
• #["spinach", "broccoli", "carrot"].swapAt! 2 "pepper" = (#["spinach", "broccoli", "pepper"], "carrot")

Equations

• One or more equations did not get rendered due to their size.

def Array.shrink {α : Type u} (xs : Array α) (n : Nat) : Array α

Returns the first n elements of an array. The resulting array is produced by repeatedly calling Array.pop. If n is greater than the size of the array, it is returned unmodified.

If the reference to the array is unique, then this function uses in-place modification.

Examples:

• #[0, 1, 2, 3, 4].shrink 2 = #[0, 1]
• #[0, 1, 2, 3, 4].shrink 0 = #[]
• #[0, 1, 2, 3, 4].shrink 10 = #[0, 1, 2, 3, 4]

Equations

• xs.shrink n = Array.shrink.loop (xs.size - n) xs

def Array.shrink.loop {α : Type u} : Nat → Array α → Array α

Equations

• Array.shrink.loop 0 x✝ = x✝
• Array.shrink.loop n.succ x✝ = Array.shrink.loop n x✝.pop

@[reducible, inline]

abbrev Array.take {α : Type u} (xs : Array α) (i : Nat) : Array α

Returns a new array that contains the first i elements of xs. If xs has fewer than i elements, the new array contains all the elements of xs.

The returned array is always a new array, even if it contains the same elements as the input array.

Examples:

• #["red", "green", "blue"].take 1 = #["red"]
• #["red", "green", "blue"].take 2 = #["red", "green"]
• #["red", "green", "blue"].take 5 = #["red", "green", "blue"]

Equations

• xs.take i = xs.extract 0 i

@[simp]

theorem Array.take_eq_extract {α : Type u} {xs : Array α} {i : Nat} : xs.take i = xs.extract 0 i

@[reducible, inline]

abbrev Array.drop {α : Type u} (xs : Array α) (i : Nat) : Array α

Removes the first i elements of xs. If xs has fewer than i elements, the new array is empty.

The returned array is always a new array, even if it contains the same elements as the input array.

Examples:

• #["red", "green", "blue"].drop 1 = #["green", "blue"]
• #["red", "green", "blue"].drop 2 = #["blue"]
• #["red", "green", "blue"].drop 5 = #[]

Equations

• xs.drop i = xs.extract i

@[simp]

theorem Array.drop_eq_extract {α : Type u} {xs : Array α} {i : Nat} : xs.drop i = xs.extract i

@[inline]

unsafe def Array.modifyMUnsafe {α : Type u} {m : Type u → Type u_1} [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α)

Equations

• xs.modifyMUnsafe i f = if h : i < xs.size then let v := xs[i]; let xs' := xs.set i (unsafeCast ()) h; do let v ← f v pure (xs'.set i v ⋯) else pure xs

@[implemented_by Array.modifyMUnsafe]

def Array.modifyM {α : Type u} {m : Type u → Type u_1} [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α)

Replaces the element at the given index, if it exists, with the result of applying the monadic function f to it. If the index is invalid, the array is returned unmodified and f is not called.

Examples:

#eval #[1, 2, 3, 4].modifyM 2 fun x => do
  IO.println s!"It was {x}"
  return x * 10

It was 3

#[1, 2, 30, 4]

#eval #[1, 2, 3, 4].modifyM 6 fun x => do
  IO.println s!"It was {x}"
  return x * 10

#[1, 2, 3, 4]

Equations

• xs.modifyM i f = if h : i < xs.size then let v := xs[i]; do let v ← f v pure (xs.set i v h) else pure xs

@[inline]

def Array.modify {α : Type u} (xs : Array α) (i : Nat) (f : α → α) : Array α

Replaces the element at the given index, if it exists, with the result of applying f to it. If the index is invalid, the array is returned unmodified.

Examples:

• #[1, 2, 3].modify 0 (· * 10) = #[10, 2, 3]
• #[1, 2, 3].modify 2 (· * 10) = #[1, 2, 30]
• #[1, 2, 3].modify 3 (· * 10) = #[1, 2, 3]

Equations

• xs.modify i f = (xs.modifyM i fun (x : α) => pure (f x)).run

@[inline]

def Array.modifyOp {α : Type u} (xs : Array α) (idx : Nat) (f : α → α) : Array α

Replaces the element at the given index, if it exists, with the result of applying f to it. If the index is invalid, the array is returned unmodified.

Examples:

• #[1, 2, 3].modifyOp 0 (· * 10) = #[10, 2, 3]
• #[1, 2, 3].modifyOp 2 (· * 10) = #[1, 2, 30]
• #[1, 2, 3].modifyOp 3 (· * 10) = #[1, 2, 3]

Equations

• xs.modifyOp idx f = xs.modify idx f

@[inline]

unsafe def Array.forIn'Unsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : (a : α) → a ∈ as → β → m (ForInStep β)) : m β

We claim this unsafe implementation is correct because an array cannot have more than usizeSz elements in our runtime.

This kind of low level trick can be removed with a little bit of compiler support. For example, if the compiler simplifies as.size < usizeSz to true.

Equations

• as.forIn'Unsafe b f = Array.forIn'Unsafe.loop as f as.usize 0 b

@[specialize #[]]

unsafe def Array.forIn'Unsafe.loop {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : (a : α) → a ∈ as → β → m (ForInStep β)) (sz i : USize) (b : β) : m β

Equations

• One or more equations did not get rendered due to their size.

@[implemented_by Array.forIn'Unsafe]

def Array.forIn' {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : (a : α) → a ∈ as → β → m (ForInStep β)) : m β

Reference implementation for forIn'

Equations

• as.forIn' b f = Array.forIn'.loop as f as.size ⋯ b

def Array.forIn'.loop {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : (a : α) → a ∈ as → β → m (ForInStep β)) (i : Nat) (h : i ≤ as.size) (b : β) : m β

Equations

• One or more equations did not get rendered due to their size.
• Array.forIn'.loop as f 0 x_2 b = pure b

instance Array.instForIn'InferInstanceMembership {α : Type u} {m : Type u_1 → Type u_2} : ForIn' m (Array α) α inferInstance

Equations

• Array.instForIn'InferInstanceMembership = { forIn' := fun {β : Type ?u.14} [Monad m] => Array.forIn' }

@[simp]

theorem Array.forIn'_eq_forIn' {α : Type u} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] : Array.forIn' = forIn'

@[inline]

unsafe def Array.foldlMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m β

See comment at forIn'Unsafe

Equations

• Array.foldlMUnsafe f init as start stop = if start < stop then if stop ≤ as.size then Array.foldlMUnsafe.fold f as (USize.ofNat start) (USize.ofNat stop) init else pure init else pure init

@[specialize #[]]

unsafe def Array.foldlMUnsafe.fold {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (as : Array α) (i stop : USize) (b : β) : m β

Equations

• Array.foldlMUnsafe.fold f as i stop b = if (i == stop) = true then pure b else do let __do_lift ← f b (as.uget i ⋯) Array.foldlMUnsafe.fold f as (i + 1) stop __do_lift

@[implemented_by Array.foldlMUnsafe]

def Array.foldlM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m β

Folds a monadic function over a list from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

The optional parameters start and stop control the region of the array to be folded. Folding proceeds from start (inclusive) to stop (exclusive), so no folding occurs unless start < stop. By default, the entire array is folded.

Examples:

example [Monad m] (f : α → β → m α) :
    Array.foldlM (m := m) f x₀ #[a, b, c] = (do
      let x₁ ← f x₀ a
      let x₂ ← f x₁ b
      let x₃ ← f x₂ c
      pure x₃)
  := by rfl

example [Monad m] (f : α → β → m α) :
    Array.foldlM (m := m) f x₀ #[a, b, c] (start := 1) = (do
      let x₁ ← f x₀ b
      let x₂ ← f x₁ c
      pure x₂)
  := by rfl

Equations

• One or more equations did not get rendered due to their size.

def Array.foldlM.loop {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (as : Array α) (stop : Nat) (h : stop ≤ as.size) (i j : Nat) (b : β) : m β

Equations

• One or more equations did not get rendered due to their size.

@[inline]

unsafe def Array.foldrMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start : Nat := as.size) (stop : Nat := 0) : m β

See comment at forIn'Unsafe

Equations

• One or more equations did not get rendered due to their size.

@[specialize #[]]

unsafe def Array.foldrMUnsafe.fold {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (as : Array α) (i stop : USize) (b : β) : m β

Equations

• Array.foldrMUnsafe.fold f as i stop b = if (i == stop) = true then pure b else do let __do_lift ← f (as.uget (i - 1) ⋯) b Array.foldrMUnsafe.fold f as (i - 1) stop __do_lift

@[implemented_by Array.foldrMUnsafe]

def Array.foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start : Nat := as.size) (stop : Nat := 0) : m β

Folds a monadic function over an array from the right, accumulating a value starting with init. The accumulated value is combined with the each element of the list in reverse order, using f.

The optional parameters start and stop control the region of the array to be folded. Folding proceeds from start (exclusive) to stop (inclusive), so no folding occurs unless start > stop. By default, the entire array is folded.

Examples:

example [Monad m] (f : α → β → m β) :
  Array.foldrM (m := m) f x₀ #[a, b, c] = (do
    let x₁ ← f c x₀
    let x₂ ← f b x₁
    let x₃ ← f a x₂
    pure x₃)
  := by rfl

example [Monad m] (f : α → β → m β) :
  Array.foldrM (m := m) f x₀ #[a, b, c] (start := 2) = (do
    let x₁ ← f b x₀
    let x₂ ← f a x₁
    pure x₂)
  := by rfl

Equations

• One or more equations did not get rendered due to their size.

def Array.foldrM.fold {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (as : Array α) (stop : Nat := 0) (i : Nat) (h : i ≤ as.size) (b : β) : m β

Equations

• One or more equations did not get rendered due to their size.
• Array.foldrM.fold f as stop 0 x_2 b = if (0 == stop) = true then pure b else pure b

@[inline]

unsafe def Array.mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β)

See comment at forIn'Unsafe

Equations

• Array.mapMUnsafe f as = unsafeCast (Array.mapMUnsafe.map f as.usize 0 (unsafeCast as))

@[specialize #[]]

unsafe def Array.mapMUnsafe.map {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (sz i : USize) (bs : Array NonScalar) : m (Array PNonScalar)

Equations

• One or more equations did not get rendered due to their size.

@[implemented_by Array.mapMUnsafe]

def Array.mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β)

Applies the monadic action f to every element in the array, left-to-right, and returns the array of results.

Equations

• Array.mapM f as = Array.mapM.map f as 0 (Array.emptyWithCapacity as.size)

def Array.mapM.map {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) (i : Nat) (bs : Array β) : m (Array β)

Equations

• Array.mapM.map f as i bs = if hlt : i < as.size then do let __do_lift ← f as[i] Array.mapM.map f as (i + 1) (bs.push __do_lift) else pure bs

@[inline]

def Array.mapFinIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : (i : Nat) → α → i < as.size → m β) : m (Array β)

Applies the monadic action f to every element in the array, along with the element's index and a proof that the index is in bounds, from left to right. Returns the array of results.

Equations

• as.mapFinIdxM f = Array.mapFinIdxM.map as f as.size 0 ⋯ (Array.emptyWithCapacity as.size)

@[specialize #[]]

def Array.mapFinIdxM.map {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : (i : Nat) → α → i < as.size → m β) (i j : Nat) (inv : i + j = as.size) (bs : Array β) : m (Array β)

Equations

• Array.mapFinIdxM.map as f 0 j x bs = pure bs
• Array.mapFinIdxM.map as f i_2.succ j inv_2 bs = do let __do_lift ← f j as[j] ⋯ Array.mapFinIdxM.map as f i_2 (j + 1) ⋯ (bs.push __do_lift)

@[inline]

def Array.mapIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : Nat → α → m β) (as : Array α) : m (Array β)

Applies the monadic action f to every element in the array, along with the element's index, from left to right. Returns the array of results.

Equations

• Array.mapIdxM f as = as.mapFinIdxM fun (i : Nat) (a : α) (x : i < as.size) => f i a

@[inline]

def Array.firstM {β : Type v} {α : Type u} {m : Type v → Type w} [Alternative m] (f : α → m β) (as : Array α) : m β

Maps f over the array and collects the results with <|>. The result for the end of the array is failure.

Examples:

• #[[], [1, 2], [], [2]].firstM List.head? = some 1
• #[[], [], []].firstM List.head? = none
• #[].firstM List.head? = none

Equations

• Array.firstM f as = Array.firstM.go f as 0

@[irreducible]

def Array.firstM.go {β : Type v} {α : Type u} {m : Type v → Type w} [Alternative m] (f : α → m β) (as : Array α) (i : Nat) : m β

Equations

• Array.firstM.go f as i = if hlt : i < as.size then f as[i] <|> Array.firstM.go f as (i + 1) else failure

@[inline]

def Array.findSomeM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m (Option β)) (as : Array α) : m (Option β)

Returns the first non-none result of applying the monadic function f to each element of the array, in order. Returns none if f returns none for all elements.

Example:

#eval #[7, 6, 5, 8, 1, 2, 6].findSomeM? fun i => do
  if i < 5 then
    return some (i * 10)
  if i ≤ 6 then
    IO.println s!"Almost! {i}"
  return none

Almost! 6
Almost! 5

some 10

Equations

• One or more equations did not get rendered due to their size.

@[inline]

def Array.findM? {m : Type → Type u_1} {α : Type} [Monad m] (p : α → m Bool) (as : Array α) : m (Option α)

Returns the first element of the array for which the monadic predicate p returns true, or none if no such element is found. Elements of the array are checked in order.

The monad m is restricted to Type → Type to avoid needing to use ULift Bool in p's type.

Example:

#eval #[7, 6, 5, 8, 1, 2, 6].findM? fun i => do
  if i < 5 then
    return true
  if i ≤ 6 then
    IO.println s!"Almost! {i}"
  return false

Almost! 6
Almost! 5

some 1

Equations

• One or more equations did not get rendered due to their size.

@[inline]

def Array.findIdxM? {α : Type u} {m : Type → Type u_1} [Monad m] (p : α → m Bool) (as : Array α) : m (Option Nat)

Finds the index of the first element of an array for which the monadic predicate p returns true. Elements are examined in order from left to right, and the search is terminated when an element that satisfies p is found. If no such element exists in the array, then none is returned.

Equations

• One or more equations did not get rendered due to their size.

@[inline]

unsafe def Array.anyMUnsafe {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m Bool

Equations

• One or more equations did not get rendered due to their size.

@[specialize #[]]

unsafe def Array.anyMUnsafe.any {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (i stop : USize) : m Bool

Equations

• One or more equations did not get rendered due to their size.

@[implemented_by Array.anyMUnsafe]

def Array.anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m Bool

Returns true if the monadic predicate p returns true for any element of as.

Short-circuits upon encountering the first true. The elements in as are examined in order from left to right.

The optional parameters start and stop control the region of the array to be checked. Only the elements with indices from start (inclusive) to stop (exclusive) are checked. By default, the entire array is checked.

Equations

• One or more equations did not get rendered due to their size.

def Array.anyM.loop {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (stop : Nat) (h : stop ≤ as.size) (j : Nat) : m Bool

Equations

• One or more equations did not get rendered due to their size.

@[inline]

def Array.allM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m Bool

Returns true if the monadic predicate p returns true for every element of as.

Short-circuits upon encountering the first false. The elements in as are examined in order from left to right.

The optional parameters start and stop control the region of the array to be checked. Only the elements with indices from start (inclusive) to stop (exclusive) are checked. By default, the entire array is checked.

Equations

• Array.allM p as start stop = do let __do_lift ← Array.anyM (fun (v : α) => do let __do_lift ← p v pure !__do_lift) as start stop pure !__do_lift

@[inline]

def Array.findSomeRevM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m (Option β)) (as : Array α) : m (Option β)

Returns the first non-none result of applying the monadic function f to each element of the array in reverse order, from right to left. Once a non-none result is found, no further elements are checked. Returns none if f returns none for all elements of the array.

Examples:

#eval #[1, 2, 0, -4, 1].findSomeRevM? (m := Except String) fun x => do
  if x = 0 then throw "Zero!"
  else if x < 0 then return (some x)
  else return none

Except.ok (some (-4))

#eval #[1, 2, 0, 4, 1].findSomeRevM? (m := Except String) fun x => do
  if x = 0 then throw "Zero!"
  else if x < 0 then return (some x)
  else return none

Except.error "Zero!"

Equations

• Array.findSomeRevM? f as = Array.findSomeRevM?.find f as as.size ⋯

@[specialize #[]]

def Array.findSomeRevM?.find {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m (Option β)) (as : Array α) (i : Nat) : i ≤ as.size → m (Option β)

Equations

• Array.findSomeRevM?.find f as 0 x_2 = pure none
• Array.findSomeRevM?.find f as i.succ h = do let r ← f as[i] match r with | some val => pure r | none => have this := ⋯; Array.findSomeRevM?.find f as i this

@[inline]

def Array.findRevM? {α : Type} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) : m (Option α)

Returns the last element of the array for which the monadic predicate p returns true, or none if no such element is found. Elements of the array are checked in reverse, from right to left..

The monad m is restricted to Type → Type to avoid needing to use ULift Bool in p's type.

Example:

#eval #[7, 5, 8, 1, 2, 6, 5, 8].findRevM? fun i => do
  if i < 5 then
    return true
  if i ≤ 6 then
    IO.println s!"Almost! {i}"
  return false

Almost! 5
Almost! 6

some 2

Equations

• Array.findRevM? p as = Array.findSomeRevM? (fun (a : α) => do let __do_lift ← p a pure (if __do_lift = true then some a else none)) as

@[inline]

def Array.forM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m PUnit

Applies the monadic action f to each element of an array, in order.

The optional parameters start and stop control the region of the array to which f should be applied. Iteration proceeds from start (inclusive) to stop (exclusive), so f is not invoked unless start < stop. By default, the entire array is used.

Equations

• Array.forM f as start stop = Array.foldlM (fun (x : PUnit) => f) PUnit.unit as start stop

instance Array.instForM {α : Type u} {m : Type u_1 → Type u_2} : ForM m (Array α) α

Equations

• Array.instForM = { forM := fun [Monad m] (xs : Array α) (f : α → m PUnit) => Array.forM f xs }

@[simp]

theorem Array.forM_eq_forM {α : Type u} {m : Type u_1 → Type u_2} {as : Array α} [Monad m] {f : α → m PUnit} : Array.forM f as = forM as f

@[inline]

def Array.forRevM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start : Nat := as.size) (stop : Nat := 0) : m PUnit

Applies the monadic action f to each element of an array from right to left, in reverse order.

The optional parameters start and stop control the region of the array to which f should be applied. Iteration proceeds from start (exclusive) to stop (inclusive), so no f is not invoked unless start > stop. By default, the entire array is used.

Equations

• Array.forRevM f as start stop = Array.foldrM (fun (a : α) (x : PUnit) => f a) PUnit.unit as start stop

@[inline]

def Array.foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : β

Folds a function over an array from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the array in order, using f.

The optional parameters start and stop control the region of the array to be folded. Folding proceeds from start (inclusive) to stop (exclusive), so no folding occurs unless start < stop. By default, the entire array is used.

Examples:

• #[a, b, c].foldl f z = f (f (f z a) b) c
• #[1, 2, 3].foldl (· ++ toString ·) "" = "123"
• #[1, 2, 3].foldl (s!"({·} {·})") "" = "((( 1) 2) 3)"

Equations

• Array.foldl f init as start stop = (Array.foldlM (fun (x1 : β) (x2 : α) => pure (f x1 x2)) init as start stop).run

@[inline]

def Array.foldr {α : Type u} {β : Type v} (f : α → β → β) (init : β) (as : Array α) (start : Nat := as.size) (stop : Nat := 0) : β

Folds a function over an array from the right, accumulating a value starting with init. The accumulated value is combined with the each element of the array in reverse order, using f.

The optional parameters start and stop control the region of the array to be folded. Folding proceeds from start (exclusive) to stop (inclusive), so no folding occurs unless start > stop. By default, the entire array is used.

Examples:

• #[a, b, c].foldr f init = f a (f b (f c init))
• #[1, 2, 3].foldr (toString · ++ ·) "" = "123"
• #[1, 2, 3].foldr (s!"({·} {·})") "!" = "(1 (2 (3 !)))"

Equations

• Array.foldr f init as start stop = (Array.foldrM (fun (x1 : α) (x2 : β) => pure (f x1 x2)) init as start stop).run

@[inline]

def Array.sum {α : Type u_1} [Add α] [Zero α] : Array α → α

Computes the sum of the elements of an array.

Examples:

• #[a, b, c].sum = a + (b + (c + 0))
• #[1, 2, 5].sum = 8

Equations

• as.sum = Array.foldr (fun (x1 x2 : α) => x1 + x2) 0 as

@[inline]

def Array.countP {α : Type u} (p : α → Bool) (as : Array α) : Nat

Counts the number of elements in the array as that satisfy the Boolean predicate p.

Examples:

• #[1, 2, 3, 4, 5].countP (· % 2 == 0) = 2
• #[1, 2, 3, 4, 5].countP (· < 5) = 4
• #[1, 2, 3, 4, 5].countP (· > 5) = 0

Equations

• Array.countP p as = Array.foldr (fun (a : α) (acc : Nat) => bif p a then acc + 1 else acc) 0 as

@[inline]

def Array.count {α : Type u} [BEq α] (a : α) (as : Array α) : Nat

Counts the number of times an element occurs in an array.

Examples:

• #[1, 1, 2, 3, 5].count 1 = 2
• #[1, 1, 2, 3, 5].count 5 = 1
• #[1, 1, 2, 3, 5].count 4 = 0

Equations

• Array.count a as = Array.countP (fun (x : α) => x == a) as

@[inline]

def Array.map {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β

Applies a function to each element of the array, returning the resulting array of values.

Examples:

• #[a, b, c].map f = #[f a, f b, f c]
• #[].map Nat.succ = #[]
• #["one", "two", "three"].map (·.length) = #[3, 3, 5]
• #["one", "two", "three"].map (·.reverse) = #["eno", "owt", "eerht"]

Equations

• Array.map f as = (Array.mapM (fun (x : α) => pure (f x)) as).run

instance Array.instFunctor : Functor Array

Equations

• Array.instFunctor = { map := fun {α β : Type ?u.9} => Array.map }

@[inline]

def Array.mapFinIdx {α : Type u} {β : Type v} (as : Array α) (f : (i : Nat) → α → i < as.size → β) : Array β

Applies a function to each element of the array along with the index at which that element is found, returning the array of results. In addition to the index, the function is also provided with a proof that the index is valid.

Array.mapIdx is a variant that does not provide the function with evidence that the index is valid.

Equations

• as.mapFinIdx f = (as.mapFinIdxM fun (x1 : Nat) (x2 : α) (x3 : x1 < as.size) => pure (f x1 x2 x3)).run

@[inline]

def Array.mapIdx {α : Type u} {β : Type v} (f : Nat → α → β) (as : Array α) : Array β

Applies a function to each element of the array along with the index at which that element is found, returning the array of results.

Array.mapFinIdx is a variant that additionally provides the function with a proof that the index is valid.

Equations

• Array.mapIdx f as = (Array.mapIdxM (fun (x1 : Nat) (x2 : α) => pure (f x1 x2)) as).run

def Array.zipIdx {α : Type u} (xs : Array α) (start : Nat := 0) : Array (α × Nat)

Pairs each element of an array with its index, optionally starting from an index other than 0.

Examples:

• #[a, b, c].zipIdx = #[(a, 0), (b, 1), (c, 2)]
• #[a, b, c].zipIdx 5 = #[(a, 5), (b, 6), (c, 7)]

Equations

• xs.zipIdx start = Array.mapIdx (fun (i : Nat) (a : α) => (a, start + i)) xs

@[reducible, inline, deprecated Array.zipIdx (since := "2025-01-21")]

abbrev Array.zipWithIndex {α : Type u_1} (xs : Array α) (start : Nat := 0) : Array (α × Nat)

Equations

• @Array.zipWithIndex = @Array.zipIdx

@[inline]

def Array.find? {α : Type u} (p : α → Bool) (as : Array α) : Option α

Returns the first element of the array for which the predicate p returns true, or none if no such element is found.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].find? (· < 5) = some 1
• #[7, 6, 5, 8, 1, 2, 6].find? (· < 1) = none

Equations

• One or more equations did not get rendered due to their size.

@[inline]

def Array.findSome? {α : Type u} {β : Type v} (f : α → Option β) (as : Array α) : Option β

Returns the first non-none result of applying the function f to each element of the array, in order. Returns none if f returns none for all elements.

Example:

#eval #[7, 6, 5, 8, 1, 2, 6].findSome? fun i =>
  if i < 5 then
    some (i * 10)
  else
    none

some 10

Equations

• Array.findSome? f as = (Array.findSomeM? (fun (x : α) => pure (f x)) as).run

@[inline]

def Array.findSome! {α : Type u} {β : Type v} [Inhabited β] (f : α → Option β) (xs : Array α) : β

Returns the first non-none result of applying the function f to each element of the array, in order. Panics if f returns none for all elements.

Example:

#eval #[7, 6, 5, 8, 1, 2, 6].findSome? fun i =>
  if i < 5 then
    some (i * 10)
  else
    none

some 10

Equations

• Array.findSome! f xs = match Array.findSome? f xs with | some b => b | none => panicWithPosWithDecl "Init.Data.Array.Basic" "Array.findSome!" 1224 14 "failed to find element"

@[inline]

def Array.findSomeRev? {α : Type u} {β : Type v} (f : α → Option β) (as : Array α) : Option β

Returns the first non-none result of applying f to each element of the array in reverse order, from right to left. Returns none if f returns none for all elements of the array.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].findSome? (fun x => if x < 5 then some (10 * x) else none) = some 10
• #[7, 6, 5, 8, 1, 2, 6].findSome? (fun x => if x < 1 then some (10 * x) else none) = none

Equations

• Array.findSomeRev? f as = (Array.findSomeRevM? (fun (x : α) => pure (f x)) as).run

@[inline]

def Array.findRev? {α : Type} (p : α → Bool) (as : Array α) : Option α

Returns the last element of the array for which the predicate p returns true, or none if no such element is found.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].findRev? (· < 5) = some 2
• #[7, 6, 5, 8, 1, 2, 6].findRev? (· < 1) = none

Equations

• Array.findRev? p as = (Array.findRevM? (fun (x : α) => pure (p x)) as).run

@[inline]

def Array.findIdx? {α : Type u} (p : α → Bool) (as : Array α) : Option Nat

Returns the index of the first element for which p returns true, or none if there is no such element.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].findIdx (· < 5) = some 4
• #[7, 6, 5, 8, 1, 2, 6].findIdx (· < 1) = none

Equations

• Array.findIdx? p as = Array.findIdx?.loop p as 0

def Array.findIdx?.loop {α : Type u} (p : α → Bool) (as : Array α) (j : Nat) : Option Nat

Equations

• Array.findIdx?.loop p as j = if h : j < as.size then if p as[j] = true then some j else Array.findIdx?.loop p as (j + 1) else none

@[inline]

def Array.findFinIdx? {α : Type u} (p : α → Bool) (as : Array α) : Option (Fin as.size)

Returns the index of the first element for which p returns true, or none if there is no such element. The index is returned as a Fin, which guarantees that it is in bounds.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].findFinIdx? (· < 5) = some (4 : Fin 7)
• #[7, 6, 5, 8, 1, 2, 6].findFinIdx? (· < 1) = none

Equations

• Array.findFinIdx? p as = Array.findFinIdx?.loop p as 0

def Array.findFinIdx?.loop {α : Type u} (p : α → Bool) (as : Array α) (j : Nat) : Option (Fin as.size)

Equations

• Array.findFinIdx?.loop p as j = if h : j < as.size then if p as[j] = true then some ⟨j, h⟩ else Array.findFinIdx?.loop p as (j + 1) else none

@[irreducible]

theorem Array.findIdx?_loop_eq_map_findFinIdx?_loop_val {α : Type u} {xs : Array α} {p : α → Bool} {j : Nat} : findIdx?.loop p xs j = Option.map (fun (x : Fin xs.size) => ↑x) (findFinIdx?.loop p xs j)

theorem Array.findIdx?_eq_map_findFinIdx?_val {α : Type u} {xs : Array α} {p : α → Bool} : findIdx? p xs = Option.map (fun (x : Fin xs.size) => ↑x) (findFinIdx? p xs)

@[inline]

def Array.findIdx {α : Type u} (p : α → Bool) (as : Array α) : Nat

Returns the index of the first element for which p returns true, or the size of the array if there is no such element.

Examples:

• #[7, 6, 5, 8, 1, 2, 6].findIdx (· < 5) = 4
• #[7, 6, 5, 8, 1, 2, 6].findIdx (· < 1) = 7

Equations

• Array.findIdx p as = (Array.findIdx? p as).getD as.size

def Array.idxOfAux {α : Type u} [BEq α] (xs : Array α) (v : α) (i : Nat) : Option (Fin xs.size)

Equations

• xs.idxOfAux v i = if h : i < xs.size then if (xs[i] == v) = true then some ⟨i, h⟩ else xs.idxOfAux v (i + 1) else none

@[reducible, inline, deprecated Array.idxOfAux (since := "2025-01-29")]

abbrev Array.indexOfAux {α : Type u_1} [BEq α] (xs : Array α) (v : α) (i : Nat) : Option (Fin xs.size)

Equations

• @Array.indexOfAux = @Array.idxOfAux

def Array.finIdxOf? {α : Type u} [BEq α] (xs : Array α) (v : α) : Option (Fin xs.size)

Returns the index of the first element equal to a, or the size of the array if no element is equal to a. The index is returned as a Fin, which guarantees that it is in bounds.

Examples:

• #["carrot", "potato", "broccoli"].finIdxOf? "carrot" = some 0
• #["carrot", "potato", "broccoli"].finIdxOf? "broccoli" = some 2
• #["carrot", "potato", "broccoli"].finIdxOf? "tomato" = none
• #["carrot", "potato", "broccoli"].finIdxOf? "anything else" = none

Equations

• xs.finIdxOf? v = xs.idxOfAux v 0

@[reducible, inline, deprecated "`Array.indexOf?` has been deprecated, use `idxOf?` or `finIdxOf?` instead." (since := "2025-01-29")]

abbrev Array.indexOf? {α : Type u_1} [BEq α] (xs : Array α) (v : α) : Option (Fin xs.size)

Equations

• @Array.indexOf? = @Array.finIdxOf?

def Array.idxOf {α : Type u} [BEq α] (a : α) : Array α → Nat

Returns the index of the first element equal to a, or the size of the array if no element is equal to a.

Examples:

• #["carrot", "potato", "broccoli"].idxOf "carrot" = 0
• #["carrot", "potato", "broccoli"].idxOf "broccoli" = 2
• #["carrot", "potato", "broccoli"].idxOf "tomato" = 3
• #["carrot", "potato", "broccoli"].idxOf "anything else" = 3

Equations

• Array.idxOf a = Array.findIdx fun (x : α) => x == a

def Array.idxOf? {α : Type u} [BEq α] (xs : Array α) (v : α) : Option Nat

Returns the index of the first element equal to a, or none if no element is equal to a.

Examples:

• #["carrot", "potato", "broccoli"].idxOf? "carrot" = some 0
• #["carrot", "potato", "broccoli"].idxOf? "broccoli" = some 2
• #["carrot", "potato", "broccoli"].idxOf? "tomato" = none
• #["carrot", "potato", "broccoli"].idxOf? "anything else" = none

Equations

• xs.idxOf? v = Option.map (fun (x : Fin xs.size) => ↑x) (xs.finIdxOf? v)

@[inline]

def Array.any {α : Type u} (as : Array α) (p : α → Bool) (start : Nat := 0) (stop : Nat := as.size) : Bool

Returns true if p returns true for any element of as.

Short-circuits upon encountering the first true.

The optional parameters start and stop control the region of the array to be checked. Only the elements with indices from start (inclusive) to stop (exclusive) are checked. By default, the entire array is checked.

Examples:

• #[2, 4, 6].any (· % 2 = 0) = true
• #[2, 4, 6].any (· % 2 = 1) = false
• #[2, 4, 5, 6].any (· % 2 = 0) = true
• #[2, 4, 5, 6].any (· % 2 = 1) = true

Equations

• as.any p start stop = (Array.anyM (fun (x : α) => pure (p x)) as start stop).run

@[inline]

def Array.all {α : Type u} (as : Array α) (p : α → Bool) (start : Nat := 0) (stop : Nat := as.size) : Bool

Returns true if p returns true for every element of as.

Short-circuits upon encountering the first false.

The optional parameters start and stop control the region of the array to be checked. Only the elements with indices from start (inclusive) to stop (exclusive) are checked. By default, the entire array is checked.

Examples:

• #[a, b, c].all p = (p a && (p b && p c))
• #[2, 4, 6].all (· % 2 = 0) = true
• #[2, 4, 5, 6].all (· % 2 = 0) = false

Equations

• as.all p start stop = (Array.allM (fun (x : α) => pure (p x)) as start stop).run

def Array.contains {α : Type u} [BEq α] (as : Array α) (a : α) : Bool

Checks whether a is an element of as, using == to compare elements.

Array.elem is a synonym that takes the element before the array.

Examples:

• #[1, 4, 2, 3, 3, 7].contains 3 = true
• Array.contains #[1, 4, 2, 3, 3, 7] 5 = false

Equations

• as.contains a = as.any fun (x : α) => a == x

def Array.elem {α : Type u} [BEq α] (a : α) (as : Array α) : Bool

Checks whether a is an element of as, using == to compare elements.

Array.contains is a synonym that takes the array before the element.

For verification purposes, Array.elem is simplified to Array.contains.

Example:

• Array.elem 3 #[1, 4, 2, 3, 3, 7] = true
• Array.elem 5 #[1, 4, 2, 3, 3, 7] = false

Equations

• Array.elem a as = as.contains a

@[export lean_array_to_list_impl]

def Array.toListImpl {α : Type u} (as : Array α) : List α

Convert a Array α into an List α. This is O(n) in the size of the array.

Equations

• as.toListImpl = Array.foldr List.cons [] as

@[inline]

def Array.toListAppend {α : Type u} (as : Array α) (l : List α) : List α

Prepends an array to a list. The elements of the array are at the beginning of the resulting list.

Equivalent to as.toList ++ l.

Examples:

• #[1, 2].toListAppend [3, 4] = [1, 2, 3, 4]
• #[1, 2].toListAppend [] = [1, 2]
• #[].toListAppend [3, 4, 5] = [3, 4, 5]

Equations

• as.toListAppend l = Array.foldr List.cons l as

def Array.append {α : Type u} (as bs : Array α) : Array α

Appends two arrays. Normally used via the ++ operator.

Appending arrays takes time proportional to the length of the second array.

Examples:

• #[1, 2, 3] ++ #[4, 5] = #[1, 2, 3, 4, 5].
• #[] ++ #[4, 5] = #[4, 5].
• #[1, 2, 3] ++ #[] = #[1, 2, 3].

Equations

• as.append bs = Array.foldl (fun (xs : Array α) (v : α) => xs.push v) as bs

instance Array.instAppend {α : Type u} : Append (Array α)

Equations

• Array.instAppend = { append := Array.append }

def Array.appendList {α : Type u} (as : Array α) (bs : List α) : Array α

Appends an array and a list.

Takes time proportional to the length of the list..

Examples:

• #[1, 2, 3].appendList [4, 5] = #[1, 2, 3, 4, 5].
• #[].appendList [4, 5] = #[4, 5].
• #[1, 2, 3].appendList [] = #[1, 2, 3].

Equations

• as.appendList bs = List.foldl (fun (xs : Array α) (v : α) => xs.push v) as bs

instance Array.instHAppendList {α : Type u} : HAppend (Array α) (List α) (Array α)

Equations

• Array.instHAppendList = { hAppend := Array.appendList }

@[inline]

def Array.flatMapM {α : Type u} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (f : α → m (Array β)) (as : Array α) : m (Array β)

Applies a monadic function that returns an array to each element of an array, from left to right. The resulting arrays are appended.

Equations

• Array.flatMapM f as = Array.foldlM (fun (bs : Array β) (a : α) => do let __do_lift ← f a pure (bs ++ __do_lift)) #[] as

@[inline]

def Array.flatMap {α : Type u} {β : Type u_1} (f : α → Array β) (as : Array α) : Array β

Applies a function that returns an array to each element of an array. The resulting arrays are appended.

Examples:

• #[2, 3, 2].flatMap Array.range = #[0, 1, 0, 1, 2, 0, 1]
• #[['a', 'b'], ['c', 'd', 'e']].flatMap List.toArray = #['a', 'b', 'c', 'd', 'e']

Equations

• Array.flatMap f as = Array.foldl (fun (bs : Array β) (a : α) => bs ++ f a) #[] as

@[inline]

def Array.flatten {α : Type u} (xss : Array (Array α)) : Array α

Appends the contents of array of arrays into a single array. The resulting array contains the same elements as the nested arrays in the same order.

Examples:

• #[#[5], #[4], #[3, 2]].flatten = #[5, 4, 3, 2]
• #[#[0, 1], #[], #[2], #[1, 0, 1]].flatten = #[0, 1, 2, 1, 0, 1]
• (#[] : Array Nat).flatten = #[]

Equations

• xss.flatten = Array.foldl (fun (acc xs : Array α) => acc ++ xs) #[] xss

def Array.reverse {α : Type u} (as : Array α) : Array α

Reverses an array by repeatedly swapping elements.

The original array is modified in place if there are no other references to it.

Examples:

• (#[] : Array Nat).reverse = #[]
• #[0, 1].reverse = #[1, 0]
• #[0, 1, 2].reverse = #[2, 1, 0]

Equations

• as.reverse = if h : as.size ≤ 1 then as else Array.reverse.loop as 0 ⟨as.size - 1, ⋯⟩

theorem Array.reverse.termination {i j : Nat} (h : i < j) : j - 1 - (i + 1) < j - i

@[irreducible]

def Array.reverse.loop {α : Type u} (as : Array α) (i : Nat) (j : Fin as.size) : Array α

Equations

• Array.reverse.loop as i j = if h : i < ↑j then have this := ⋯; let as_1 := as.swap i ↑j ⋯ ⋯; have this := ⋯; Array.reverse.loop as_1 (i + 1) ⟨↑j - 1, this⟩ else as

@[inline]

def Array.filter {α : Type u} (p : α → Bool) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : Array α

Returns the array of elements in as for which p returns true.

Only elements from start (inclusive) to stop (exclusive) are considered. Elements outside that range are discarded. By default, the entire array is considered.

Examples:

• #[1, 2, 5, 2, 7, 7].filter (· > 2) = #[5, 7, 7]
• #[1, 2, 5, 2, 7, 7].filter (fun _ => false) = #[]
• #[1, 2, 5, 2, 7, 7].filter (fun _ => true) = #[1, 2, 5, 2, 7, 7]
• #[1, 2, 5, 2, 7, 7].filter (· > 2) (start := 3) = #[7, 7]
• #[1, 2, 5, 2, 7, 7].filter (fun _ => true) (start := 3) = #[2, 7, 7]
• #[1, 2, 5, 2, 7, 7].filter (fun _ => true) (stop := 3) = #[1, 2, 5]

Equations

• Array.filter p as start stop = Array.foldl (fun (acc : Array α) (a : α) => if p a = true then acc.push a else acc) #[] as start stop

@[inline]

def Array.filterM {m : Type → Type u_1} {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m (Array α)

Applies the monadic predicate p to every element in the array, in order from left to right, and returns the array of elements for which p returns true.

Only elements from start (inclusive) to stop (exclusive) are considered. Elements outside that range are discarded. By default, the entire array is checked.

Example:

#eval #[1, 2, 5, 2, 7, 7].filterM fun x => do
  IO.println s!"Checking {x}"
  return x < 3

Checking 1
Checking 2
Checking 5
Checking 2
Checking 7
Checking 7

#[1, 2, 2]

Equations

• Array.filterM p as start stop = Array.foldlM (fun (acc : Array α) (a : α) => do let __do_lift ← p a if __do_lift = true then pure (acc.push a) else pure acc) #[] as start stop

@[inline]

def Array.filterRevM {m : Type → Type u_1} {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start : Nat := as.size) (stop : Nat := 0) : m (Array α)

Applies the monadic predicate p on every element in the array in reverse order, from right to left, and returns those elements for which p returns true. The elements of the returned list are in the same order as in the input list.

Only elements from start (exclusive) to stop (inclusive) are considered. Elements outside that range are discarded. Because the array is examined in reverse order, elements are only examined when start > stop. By default, the entire array is considered.

Example:

#eval #[1, 2, 5, 2, 7, 7].filterRevM fun x => do
  IO.println s!"Checking {x}"
  return x < 3

Checking 7
Checking 7
Checking 2
Checking 5
Checking 2
Checking 1

#[1, 2, 2]

Equations

• One or more equations did not get rendered due to their size.

@[specialize #[]]

def Array.filterMapM {α : Type u} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (f : α → m (Option β)) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : m (Array β)

Applies a monadic function that returns an Option to each element of an array, collecting the non-none values.

Only elements from start (inclusive) to stop (exclusive) are considered. Elements outside that range are discarded. By default, the entire array is considered.

Example:

#eval #[1, 2, 5, 2, 7, 7].filterMapM fun x => do
  IO.println s!"Examining {x}"
  if x > 2 then return some (2 * x)
  else return none

Examining 1
Examining 2
Examining 5
Examining 2
Examining 7
Examining 7

#[10, 14, 14]

Equations

• One or more equations did not get rendered due to their size.

@[inline]

def Array.filterMap {α : Type u} {β : Type u_1} (f : α → Option β) (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : Array β

Applies a function that returns an Option to each element of an array, collecting the non-none values.

Example:

#eval #[1, 2, 5, 2, 7, 7].filterMap fun x =>
  if x > 2 then some (2 * x) else none

#[10, 14, 14]

Equations

• Array.filterMap f as start stop = (Array.filterMapM (fun (x : α) => pure (f x)) as start stop).run

@[specialize #[]]

def Array.getMax? {α : Type u} (as : Array α) (lt : α → α → Bool) : Option α

Returns the largest element of the array, as determined by the comparison lt, or none if the array is empty.

Examples:

• (#[] : Array Nat).getMax? (· < ·) = none
• #["red", "green", "blue"].getMax? (·.length < ·.length) = some "green"
• #["red", "green", "blue"].getMax? (· < ·) = some "red"

Equations

• as.getMax? lt = if h : 0 < as.size then let a0 := as[0]; some (Array.foldl (fun (best a : α) => if lt best a = true then a else best) a0 as 1) else none

@[inline]

def Array.partition {α : Type u} (p : α → Bool) (as : Array α) : Array α × Array α

Returns a pair of arrays that together contain all the elements of as. The first array contains those elements for which p returns true, and the second contains those for which p returns false.

as.partition p is equivalent to (as.filter p, as.filter (not ∘ p)), but it is more efficient since it only has to do one pass over the array.

Examples:

• #[1, 2, 5, 2, 7, 7].partition (· > 2) = (#[5, 7, 7], #[1, 2, 2])
• #[1, 2, 5, 2, 7, 7].partition (fun _ => false) = (#[], #[1, 2, 5, 2, 7, 7])
• #[1, 2, 5, 2, 7, 7].partition (fun _ => true) = (#[1, 2, 5, 2, 7, 7], #[])

Equations

• One or more equations did not get rendered due to their size.

def Array.popWhile {α : Type u} (p : α → Bool) (as : Array α) : Array α

Removes all the elements that satisfy a predicate from the end of an array.

The longest contiguous sequence of elements that all satisfy the predicate is removed.

Examples:

• #[0, 1, 2, 3, 4].popWhile (· > 2) = #[0, 1, 2]
• #[3, 2, 3, 4].popWhile (· > 2) = #[3, 2]
• (#[] : Array Nat).popWhile (· > 2) = #[]

Equations

• Array.popWhile p as = if h : as.size > 0 then if p as[as.size - 1] = true then Array.popWhile p as.pop else as else as

@[simp]

theorem Array.popWhile_empty {α : Type u} {p : α → Bool} : popWhile p #[] = #[]

def Array.takeWhile {α : Type u} (p : α → Bool) (as : Array α) : Array α

Returns a new array that contains the longest prefix of elements that satisfy the predicate p from an array.

Examples:

• #[0, 1, 2, 3, 2, 1].takeWhile (· < 2) = #[0, 1]
• #[0, 1, 2, 3, 2, 1].takeWhile (· < 20) = #[0, 1, 2, 3, 2, 1]
• #[0, 1, 2, 3, 2, 1].takeWhile (· < 0) = #[]

Equations

• Array.takeWhile p as = Array.takeWhile.go p as 0 #[]

def Array.takeWhile.go {α : Type u} (p : α → Bool) (as : Array α) (i : Nat) (acc : Array α) : Array α

Equations

• Array.takeWhile.go p as i acc = if h : i < as.size then let a := as[i]; if p a = true then Array.takeWhile.go p as (i + 1) (acc.push a) else acc else acc

def Array.eraseIdx {α : Type u} (xs : Array α) (i : Nat) (h : i < xs.size := by get_elem_tactic) : Array α

Removes the element at a given index from an array without a run-time bounds check.

This function takes worst-case O(n) time because it back-shifts all elements at positions greater than i.

Examples:

• #["apple", "pear", "orange"].eraseIdx 0 = #["pear", "orange"]
• #["apple", "pear", "orange"].eraseIdx 1 = #["apple", "orange"]
• #["apple", "pear", "orange"].eraseIdx 2 = #["apple", "pear"]

Equations

• xs.eraseIdx i h = if h' : i + 1 < xs.size then let xs' := xs.swap (i + 1) i h' h; xs'.eraseIdx (i + 1) ⋯ else xs.pop

@[simp]

theorem Array.size_eraseIdx {α : Type u} {xs : Array α} (i : Nat) (h : i < xs.size) : (xs.eraseIdx i h).size = xs.size - 1

def Array.eraseIdxIfInBounds {α : Type u} (xs : Array α) (i : Nat) : Array α

Removes the element at a given index from an array. Does nothing if the index is out of bounds.

This function takes worst-case O(n) time because it back-shifts all elements at positions greater than i.

Examples:

• #["apple", "pear", "orange"].eraseIdxIfInBounds 0 = #["pear", "orange"]
• #["apple", "pear", "orange"].eraseIdxIfInBounds 1 = #["apple", "orange"]
• #["apple", "pear", "orange"].eraseIdxIfInBounds 2 = #["apple", "pear"]
• #["apple", "pear", "orange"].eraseIdxIfInBounds 3 = #["apple", "pear", "orange"]
• #["apple", "pear", "orange"].eraseIdxIfInBounds 5 = #["apple", "pear", "orange"]

Equations

• xs.eraseIdxIfInBounds i = if h : i < xs.size then xs.eraseIdx i h else xs

def Array.eraseIdx! {α : Type u} (xs : Array α) (i : Nat) : Array α

Removes the element at a given index from an array. Panics if the index is out of bounds.

This function takes worst-case O(n) time because it back-shifts all elements at positions greater than i.

Equations

• xs.eraseIdx! i = if h : i < xs.size then xs.eraseIdx i h else panicWithPosWithDecl "Init.Data.Array.Basic" "Array.eraseIdx!" 1819 47 "invalid index"

def Array.erase {α : Type u} [BEq α] (as : Array α) (a : α) : Array α

Removes the first occurrence of a specified element from an array, or does nothing if it is not present.

This function takes worst-case O(n) time because it back-shifts all later elements.

Examples:

• #[1, 2, 3].erase 2 = #[1, 3]
• #[1, 2, 3].erase 5 = #[1, 2, 3]
• #[1, 2, 3, 2, 1].erase 2 = #[1, 3, 2, 1]
• (#[] : List Nat).erase 2 = #[]

Equations

• as.erase a = match as.finIdxOf? a with | none => as | some i => as.eraseIdx ↑i ⋯

def Array.eraseP {α : Type u} (as : Array α) (p : α → Bool) : Array α

Removes the first element that satisfies the predicate p. If no element satisfies p, the array is returned unmodified.

This function takes worst-case O(n) time because it back-shifts all later elements.

Examples:

• #["red", "green", "", "blue"].eraseP (·.isEmpty) = #["red", "green", "blue"]
• #["red", "green", "", "blue", ""].eraseP (·.isEmpty) = #["red", "green", "blue", ""]
• #["red", "green", "blue"].eraseP (·.length % 2 == 0) = #["red", "green"]
• #["red", "green", "blue"].eraseP (fun _ => true) = #["green", "blue"]
• (#[] : Array String).eraseP (fun _ => true) = #[]

Equations

• as.eraseP p = match Array.findFinIdx? p as with | none => as | some i => as.eraseIdx ↑i ⋯

@[inline]

def Array.insertIdx {α : Type u} (as : Array α) (i : Nat) (a : α) : autoParam (i ≤ as.size) _auto✝ → Array α

Inserts an element into an array at the specified index. If the index is greater than the size of the array, then the array is returned unmodified.

In other words, the new element is inserted into the array as after the first i elements of as.

This function takes worst case O(n) time because it has to swap the inserted element into place.

Examples:

• #["tues", "thur", "sat"].insertIdx 1 "wed" = #["tues", "wed", "thur", "sat"]
• #["tues", "thur", "sat"].insertIdx 2 "wed" = #["tues", "thur", "wed", "sat"]
• #["tues", "thur", "sat"].insertIdx 3 "wed" = #["tues", "thur", "sat", "wed"]

Equations

• as.insertIdx i a x✝ = Array.insertIdx.loop i (as.push a) ⟨as.size, ⋯⟩

def Array.insertIdx.loop {α : Type u} (i : Nat) (as : Array α) (j : Fin as.size) : Array α

Equations

• Array.insertIdx.loop i as j = if i < ↑j then let j' := ⟨↑j - 1, ⋯⟩; let as_1 := as.swap ↑j' ↑j ⋯ ⋯; Array.insertIdx.loop i as_1 ⟨↑j', ⋯⟩ else as

def Array.insertIdx! {α : Type u} (as : Array α) (i : Nat) (a : α) : Array α

Inserts an element into an array at the specified index. Panics if the index is greater than the size of the array.

In other words, the new element is inserted into the array as after the first i elements of as.

This function takes worst case O(n) time because it has to swap the inserted element into place. Array.insertIdx and Array.insertIdxIfInBounds are safer alternatives.

Examples:

• #["tues", "thur", "sat"].insertIdx! 1 "wed" = #["tues", "wed", "thur", "sat"]
• #["tues", "thur", "sat"].insertIdx! 2 "wed" = #["tues", "thur", "wed", "sat"]
• #["tues", "thur", "sat"].insertIdx! 3 "wed" = #["tues", "thur", "sat", "wed"]

Equations

• as.insertIdx! i a = if h : i ≤ as.size then as.insertIdx i a h else panicWithPosWithDecl "Init.Data.Array.Basic" "Array.insertIdx!" 1902 7 "invalid index"

def Array.insertIdxIfInBounds {α : Type u} (as : Array α) (i : Nat) (a : α) : Array α

Inserts an element into an array at the specified index. The array is returned unmodified if the index is greater than the size of the array.

In other words, the new element is inserted into the array as after the first i elements of as.

This function takes worst case O(n) time because it has to swap the inserted element into place.

Examples:

• #["tues", "thur", "sat"].insertIdxIfInBounds 1 "wed" = #["tues", "wed", "thur", "sat"]
• #["tues", "thur", "sat"].insertIdxIfInBounds 2 "wed" = #["tues", "thur", "wed", "sat"]
• #["tues", "thur", "sat"].insertIdxIfInBounds 3 "wed" = #["tues", "thur", "sat", "wed"]
• #["tues", "thur", "sat"].insertIdxIfInBounds 4 "wed" = #["tues", "thur", "sat"]

Equations

• as.insertIdxIfInBounds i a = if h : i ≤ as.size then as.insertIdx i a h else as

def Array.isPrefixOfAux {α : Type u} [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : Nat) : Bool

Equations

• as.isPrefixOfAux bs hle i = if h : i < as.size then let a := as[i]; have this := ⋯; let b := bs[i]; if (a == b) = true then as.isPrefixOfAux bs hle (i + 1) else false else true

def Array.isPrefixOf {α : Type u} [BEq α] (as bs : Array α) : Bool

Return true if as is a prefix of bs, or false otherwise.

Examples:

• #[0, 1, 2].isPrefixOf #[0, 1, 2, 3] = true
• #[0, 1, 2].isPrefixOf #[0, 1, 2] = true
• #[0, 1, 2].isPrefixOf #[0, 1] = false
• #[].isPrefixOf #[0, 1] = true

Equations

• as.isPrefixOf bs = if h : as.size ≤ bs.size then as.isPrefixOfAux bs h 0 else false

@[specialize #[]]

def Array.zipWithAux {α : Type u} {β : Type u_1} {γ : Type u_2} (as : Array α) (bs : Array β) (f : α → β → γ) (i : Nat) (cs : Array γ) : Array γ

Equations

• as.zipWithAux bs f i cs = if h : i < as.size then let a := as[i]; if h : i < bs.size then let b := bs[i]; as.zipWithAux bs f (i + 1) (cs.push (f a b)) else cs else cs

@[inline]

def Array.zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (as : Array α) (bs : Array β) : Array γ

Applies a function to the corresponding elements of two arrays, stopping at the end of the shorter array.

Examples:

• #[1, 2].zipWith (· + ·) #[5, 6] = #[6, 8]
• #[1, 2, 3].zipWith (· + ·) #[5, 6, 10] = #[6, 8, 13]
• #[].zipWith (· + ·) #[5, 6] = #[]
• #[x₁, x₂, x₃].zipWith f #[y₁, y₂, y₃, y₄] = #[f x₁ y₁, f x₂ y₂, f x₃ y₃]

Equations

• Array.zipWith f as bs = as.zipWithAux bs f 0 #[]

def Array.zip {α : Type u} {β : Type u_1} (as : Array α) (bs : Array β) : Array (α × β)

Combines two arrays into an array of pairs in which the first and second components are the corresponding elements of each input array. The resulting array is the length of the shorter of the input arrays.

Examples:

• #["Mon", "Tue", "Wed"].zip #[1, 2, 3] = #[("Mon", 1), ("Tue", 2), ("Wed", 3)]
• #["Mon", "Tue", "Wed"].zip #[1, 2] = #[("Mon", 1), ("Tue", 2)]
• #[x₁, x₂, x₃].zip #[y₁, y₂, y₃, y₄] = #[(x₁, y₁), (x₂, y₂), (x₃, y₃)]

Equations

• as.zip bs = Array.zipWith Prod.mk as bs

def Array.zipWithAll {α : Type u} {β : Type u_1} {γ : Type u_2} (f : Option α → Option β → γ) (as : Array α) (bs : Array β) : Array γ

Applies a function to the corresponding elements of both arrays, stopping when there are no more elements in either array. If one array is shorter than the other, the function is passed none for the missing elements.

Examples:

• #[1, 6].zipWithAll min #[5, 2] = #[some 1, some 2]
• #[1, 2, 3].zipWithAll Prod.mk #[5, 6] = #[(some 1, some 5), (some 2, some 6), (some 3, none)]
• #[x₁, x₂].zipWithAll f #[y] = #[f (some x₁) (some y), f (some x₂) none]

Equations

• Array.zipWithAll f as bs = Array.zipWithAll.go f as bs 0 #[]

@[irreducible]

def Array.zipWithAll.go {α : Type u} {β : Type u_1} {γ : Type u_2} (f : Option α → Option β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ

Equations

• Array.zipWithAll.go f as bs i cs = if i < max as.size bs.size then let a := as[i]?; let b := bs[i]?; Array.zipWithAll.go f as bs (i + 1) (cs.push (f a b)) else cs

def Array.unzip {α : Type u} {β : Type u_1} (as : Array (α × β)) : Array α × Array β

Separates an array of pairs into two arrays that contain the respective first and second components.

Examples:

• #[("Monday", 1), ("Tuesday", 2)].unzip = (#["Monday", "Tuesday"], #[1, 2])
• #[(x₁, y₁), (x₂, y₂), (x₃, y₃)].unzip = (#[x₁, x₂, x₃], #[y₁, y₂, y₃])
• (#[] : Array (Nat × String)).unzip = ((#[], #[]) : List Nat × List String)

Equations

• as.unzip = Array.foldl (fun (x : Array α × Array β) (x_1 : α × β) => match x with | (as, bs) => match x_1 with | (a, b) => (as.push a, bs.push b)) (#[], #[]) as

def Array.replace {α : Type u} [BEq α] (xs : Array α) (a b : α) : Array α

Replaces the first occurrence of a with b in an array. The modification is performed in-place when the reference to the array is unique. Returns the array unmodified when a is not present.

Examples:

• #[1, 2, 3, 2, 1].replace 2 5 = #[1, 5, 3, 2, 1]
• #[1, 2, 3, 2, 1].replace 0 5 = #[1, 2, 3, 2, 1]
• #[].replace 2 5 = #[]

Equations

• xs.replace a b = match xs.finIdxOf? a with | none => xs | some i => xs.set (↑i) b ⋯

Lexicographic ordering

instance Array.instLT {α : Type u} [LT α] : LT (Array α)

Equations

• Array.instLT = { lt := fun (as bs : Array α) => as.toList < bs.toList }

instance Array.instLE {α : Type u} [LT α] : LE (Array α)

Equations

• Array.instLE = { le := fun (as bs : Array α) => as.toList ≤ bs.toList }

Auxiliary functions used in metaprogramming.

We do not currently intend to provide verification theorems for these functions.

leftpad and rightpad

def Array.leftpad {α : Type u} (n : Nat) (a : α) (xs : Array α) : Array α

Pads xs : Array α on the left with repeated occurrences of a : α until it is of size n. If xs already has at least n elements, it is returned unmodified.

Examples:

• #[1, 2, 3].leftpad 5 0 = #[0, 0, 1, 2, 3]
• #["red", "green", "blue"].leftpad 4 "blank" = #["blank", "red", "green", "blue"]
• #["red", "green", "blue"].leftpad 3 "blank" = #["red", "green", "blue"]
• #["red", "green", "blue"].leftpad 1 "blank" = #["red", "green", "blue"]

Equations

• Array.leftpad n a xs = Array.replicate (n - xs.size) a ++ xs

def Array.rightpad {α : Type u} (n : Nat) (a : α) (xs : Array α) : Array α

Pads xs : Array α on the right with repeated occurrences of a : α until it is of length n. If l already has at least n elements, it is returned unmodified.

Examples:

• #[1, 2, 3].rightpad 5 0 = #[1, 2, 3, 0, 0]
• #["red", "green", "blue"].rightpad 4 "blank" = #["red", "green", "blue", "blank"]
• #["red", "green", "blue"].rightpad 3 "blank" = #["red", "green", "blue"]
• #["red", "green", "blue"].rightpad 1 "blank" = #["red", "green", "blue"]

Equations

• Array.rightpad n a xs = xs ++ Array.replicate (n - xs.size) a

@[inline]

def Array.reduceOption {α : Type u} (as : Array (Option α)) : Array α

Drop nones from a Array, and replace each remaining some a with a.

Equations

• as.reduceOption = Array.filterMap id as

eraseReps

def Array.eraseReps {α : Type u_1} [BEq α] (as : Array α) : Array α

Erases repeated elements, keeping the first element of each run.

O(|as|).

Example:

• #[1, 3, 2, 2, 2, 3, 3, 5].eraseReps = #[1, 3, 2, 3, 5]

Equations

• One or more equations did not get rendered due to their size.

allDiff

def Array.allDiff {α : Type u} [BEq α] (as : Array α) : Bool

Returns true if no two elements of as are equal according to the == operator.

Examples:

• #["red", "green", "blue"].allDiff = true
• #["red", "green", "red"].allDiff = false
• (#[] : Array Nat).allDiff = true

Equations

• as.allDiff = Array.allDiffAux✝ as 0

getEvenElems

@[inline]

def Array.getEvenElems {α : Type u} (as : Array α) : Array α

Returns a new array that contains the elements at even indices in as, starting with the element at index 0.

Examples:

• #[0, 1, 2, 3, 4].getEvenElems = #[0, 2, 4]
• #[1, 2, 3, 4].getEvenElems = #[1, 3]
• #["red", "green", "blue"].getEvenElems = #["red", "blue"]
• (#[] : Array String).getEvenElems = #[]

Equations

• One or more equations did not get rendered due to their size.

Repr and ToString

def Array.Array.repr {α : Type u} [Repr α] (xs : Array α) : Std.Format

Equations

• Array.Array.repr xs = if (xs.size == 0) = true then Std.Format.text "#[]" else Std.Format.bracketFill "#[" (Std.Format.joinSep xs.toList (Std.Format.text "," ++ Std.Format.line)) "]"

instance Array.instRepr {α : Type u} [Repr α] : Repr (Array α)

Equations

• Array.instRepr = { reprPrec := fun (xs : Array α) (x : Nat) => Array.Array.repr xs }

instance Array.instToString {α : Type u} [ToString α] : ToString (Array α)

Equations

• Array.instToString = { toString := fun (xs : Array α) => "#" ++ toString xs.toList }