# Documentation

## Tail recursive implementations for definitions from core #

@[inline]
def List.setTR {α : Type u_1} (l : List α) (n : Nat) (a : α) :
List α

Tail recursive version of erase.

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def List.setTR.go {α : Type u_1} (l : List α) (a : α) :
List αNatList α

Auxiliary for setTR: setTR.go l a xs n acc = acc.toList ++ set xs a, unless n ≥ l.length in which case it returns l

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@[csimp]
theorem List.set_eq_setTR.go (α : Type u_1) (l : List α) (a : α) (acc : ) (xs : List α) (n : Nat) :
l = acc.data ++ xsList.setTR.go l a xs n acc = acc.data ++ List.set xs n a
@[inline]
def List.eraseTR {α : Type u_1} [BEq α] (l : List α) (a : α) :
List α

Tail recursive version of erase.

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def List.eraseTR.go {α : Type u_1} [BEq α] (l : List α) (a : α) :
List αList α

Auxiliary for eraseTR: eraseTR.go l a xs acc = acc.toList ++ erase xs a, unless a is not present in which case it returns l

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@[csimp]
@[inline]
def List.eraseIdxTR {α : Type u_1} (l : List α) (n : Nat) :
List α

Tail recursive version of eraseIdx.

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def List.eraseIdxTR.go {α : Type u_1} (l : List α) :
List αNatList α

Auxiliary for eraseIdxTR: eraseIdxTR.go l n xs acc = acc.toList ++ eraseIdx xs a, unless a is not present in which case it returns l

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@[csimp]
@[inline]
def List.bindTR {α : Type u_1} {β : Type u_2} (as : List α) (f : αList β) :
List β

Tail recursive version of bind.

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@[specialize #[]]
def List.bindTR.go {α : Type u_1} {β : Type u_2} (f : αList β) :
List αList β

Auxiliary for bind: bind.go f as = acc.toList ++ bind f as

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@[csimp]
theorem List.bind_eq_bindTR.go (α : Type u_2) (β : Type u_1) (f : αList β) (as : List α) (acc : ) :
List.bindTR.go f as acc = acc.data ++ List.bind as f
@[inline]
def List.joinTR {α : Type u_1} (l : List (List α)) :
List α

Tail recursive version of join.

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@[csimp]
@[inline]
def List.filterMapTR {α : Type u_1} {β : Type u_2} (f : α) (l : List α) :
List β

Tail recursive version of filterMap.

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@[specialize #[]]
def List.filterMapTR.go {α : Type u_1} {β : Type u_2} (f : α) :
List αList β

Auxiliary for filterMap: filterMap.go f l = acc.toList ++ filterMap f l

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@[csimp]
theorem List.filterMap_eq_filterMapTR.go (α : Type u_2) (β : Type u_1) (f : α) (as : List α) (acc : ) :
List.filterMapTR.go f as acc = acc.data ++
@[inline]
def List.replaceTR {α : Type u_1} [BEq α] (l : List α) (b : α) (c : α) :
List α

Tail recursive version of replace.

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@[specialize #[]]
def List.replaceTR.go {α : Type u_1} [BEq α] (l : List α) (b : α) (c : α) :
List αList α

Auxiliary for replace: replace.go l b c xs acc = acc.toList ++ replace xs b c, unless b is not found in xs in which case it returns l.

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@[csimp]
@[inline]
def List.takeTR {α : Type u_1} (n : Nat) (l : List α) :
List α

Tail recursive version of take.

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@[specialize #[]]
def List.takeTR.go {α : Type u_1} (l : List α) :
List αNatList α

Auxiliary for take: take.go l xs n acc = acc.toList ++ take n xs, unless n ≥ xs.length in which case it returns l.

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@[csimp]
@[inline]
def List.takeWhileTR {α : Type u_1} (p : αBool) (l : List α) :
List α

Tail recursive version of takeWhile.

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@[specialize #[]]
def List.takeWhileTR.go {α : Type u_1} (p : αBool) (l : List α) :
List αList α

Auxiliary for takeWhile: takeWhile.go p l xs acc = acc.toList ++ takeWhile p xs, unless no element satisfying p is found in xs in which case it returns l.

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@[csimp]
@[specialize #[]]
def List.foldrTR {α : Type u_1} {β : Type u_2} (f : αββ) (init : β) (l : List α) :
β

Tail recursive version of foldr.

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@[csimp]
@[inline]
def List.zipWithTR {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβγ) (as : List α) (bs : List β) :
List γ

Tail recursive version of zipWith.

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def List.zipWithTR.go {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβγ) :
List αList βList γ

Auxiliary for zipWith: zipWith.go f as bs acc = acc.toList ++ zipWith f as bs

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@[csimp]
theorem List.zipWith_eq_zipWithTR.go (α : Type u_3) (β : Type u_2) (γ : Type u_1) (f : αβγ) (as : List α) (bs : List β) (acc : ) :
List.zipWithTR.go f as bs acc = acc.data ++ List.zipWith f as bs
def List.unzipTR {α : Type u_1} {β : Type u_2} (l : List (α × β)) :
List α × List β

Tail recursive version of unzip.

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• One or more equations did not get rendered due to their size.
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@[csimp]
def List.enumFromTR {α : Type u_1} (n : Nat) (l : List α) :
List (Nat × α)

Tail recursive version of enumFrom.

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• One or more equations did not get rendered due to their size.
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@[csimp]
theorem List.enumFrom_eq_enumFromTR.go (α : Type u_1) (l : List α) (n : Nat) :
let f := fun (a : α) (x : Nat × List (Nat × α)) => match x with | (n, acc) => (n - 1, (n - 1, a) :: acc); List.foldr f (n + , []) l = (n, )
theorem List.replicateTR_loop_eq :
∀ {α : Type u_1} {a : α} {acc : List α} (n : Nat), = ++ acc
@[inline]
def List.dropLastTR {α : Type u_1} (l : List α) :
List α

Tail recursive version of dropLast.

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@[csimp]
def List.intersperseTR {α : Type u_1} (sep : α) :
List αList α

Tail recursive version of intersperse.

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@[csimp]
def List.intercalateTR {α : Type u_1} (sep : List α) :
List (List α)List α

Tail recursive version of intercalate.

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def List.intercalateTR.go {α : Type u_1} (sep : ) :
List αList (List α)List α

Auxiliary for intercalateTR: intercalateTR.go sep x xs acc = acc.toList ++ intercalate sep.toList (x::xs)

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@[csimp]
theorem List.intercalate_eq_intercalateTR.go (α : Type u_1) (sep : List α) {acc : } {x : List α} (xs : List (List α)) :
List.intercalateTR.go (List.toArray sep) x xs acc = acc.data ++ List.join (List.intersperse sep (x :: xs))

## New definitions #

def List.Subset {α : Type u_1} (l₁ : List α) (l₂ : List α) :

l₁ ⊆ l₂ means that every element of l₁ is also an element of l₂, ignoring multiplicity.

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instance List.instHasSubsetList {α : Type u_1} :
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• List.instHasSubsetList = { Subset := List.Subset }
instance List.decidableBEx {α : Type u_1} (p : αProp) [] (l : List α) :
Decidable (∃ (x : α), x l p x)
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• One or more equations did not get rendered due to their size.
• = isFalse (_ : (∃ (x : α), x [] p x)False)
instance List.decidableBAll {α : Type u_1} (p : αProp) [] (l : List α) :
Decidable (∀ (x : α), x lp x)
Equations
• One or more equations did not get rendered due to their size.
• = isTrue (_ : ∀ (a : α), a []p a)
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def List.bagInter {α : Type u_1} [BEq α] :
List αList αList α

Computes the "bag intersection" of l₁ and l₂, that is, the collection of elements of l₁ which are also in l₂. As each element is identified, it is removed from l₂, so elements are counted with multiplicity.

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def List.diff {α : Type u_1} [BEq α] :
List αList αList α

Computes the difference of l₁ and l₂, by removing each element in l₂ from l₁.

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def List.tail {α : Type u_1} :
List αList α

Get the tail of a nonempty list, or return [] for [].

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• = match x with | [] => [] | head :: as => as
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@[simp]
theorem List.tail_nil {α : Type u_1} :
= []
@[simp]
theorem List.tail_cons {α : Type u_1} {a : α} {as : List α} :
List.tail (a :: as) = as
@[inline]
def List.next? {α : Type u_1} :
List αOption (α × List α)

Get the head and tail of a list, if it is nonempty.

Equations
• = match x with | [] => none | a :: l => some (a, l)
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@[inline]
def List.mapIdx {α : Type u_1} {β : Type u_2} (f : Natαβ) (as : List α) :
List β

Given a function f : Nat → α → β and as : list α, as = [a₀, a₁, ...], returns the list [f 0 a₀, f 1 a₁, ...].

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@[specialize #[]]
def List.mapIdx.go {α : Type u_1} {β : Type u_2} (f : Natαβ) :
List αList β

Auxiliary for mapIdx: mapIdx.go [a₀, a₁, ...] acc = acc.toList ++ [f acc.size a₀, f (acc.size + 1) a₁, ...]

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@[inline]
def List.mapIdxM {α : Type u_1} {β : Type v} {m : Type v → Type w} [] (as : List α) (f : Natαm β) :
m (List β)

Monadic variant of mapIdx.

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@[specialize #[]]
def List.mapIdxM.go {α : Type u_1} {β : Type v} {m : Type v → Type w} [] (f : Natαm β) :
List αm (List β)

Auxiliary for mapIdxM: mapIdxM.go as f acc = acc.toList ++ [← f acc.size a₀, ← f (acc.size + 1) a₁, ...]

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@[specialize #[]]
def List.after {α : Type u_1} (p : αBool) :
List αList α

after p xs is the suffix of xs after the first element that satisfies p, not including that element.

after      (· == 1) [0, 1, 2, 3] = [2, 3]
drop_while (· != 1) [0, 1, 2, 3] = [1, 2, 3]

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@[inline]
def List.findIdx {α : Type u_1} (p : αBool) (l : List α) :

Returns the index of the first element satisfying p, or the length of the list otherwise.

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@[specialize #[]]
def List.findIdx.go {α : Type u_1} (p : αBool) :
List α

Auxiliary for findIdx: findIdx.go p l n = findIdx p l + n

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def List.indexOf {α : Type u_1} [BEq α] (a : α) :
List αNat

Returns the index of the first element equal to a, or the length of the list otherwise.

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def List.removeNth {α : Type u_1} :
List αNatList α

Removes the nth element of l, or the original list if n is out of bounds.

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@[inline]
def List.removeNthTR {α : Type u_1} (l : List α) (n : Nat) :
List α

Tail recursive version of removeNth.

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def List.removeNthTR.go {α : Type u_1} (l : List α) :
List αNatList α

Auxiliary for removeNthTR: removeNthTR.go l xs n acc = acc.toList ++ removeNth xs n if n < length xs, else l.

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@[csimp]
def List.replaceF {α : Type u_1} (f : α) :
List αList α

Replaces the first element of the list for which f returns some with the returned value.

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@[inline]
def List.replaceFTR {α : Type u_1} (f : α) (l : List α) :
List α

Tail-recursive version of replaceF.

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@[specialize #[]]
def List.replaceFTR.go {α : Type u_1} (f : α) :
List αList α

Auxiliary for replaceFTR: replaceFTR.go f xs acc = acc.toList ++ replaceF f xs.

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@[csimp]
theorem List.replaceF_eq_replaceFTR.go (α : Type u_1) (p : α) (acc : ) (xs : List α) :
List.replaceFTR.go p xs acc = acc.data ++
@[inline]
def List.insert {α : Type u_1} [BEq α] (a : α) (l : List α) :
List α

Inserts an element into a list without duplication.

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@[inline]
def List.union {α : Type u_1} [] (l₁ : List α) (l₂ : List α) :
List α

Constructs the union of two lists, by inserting the elements of l₁ in reverse order to l₂. As a result, l₂ will always be a suffix, but only the last occurrence of each element in l₁ will be retained (but order will otherwise be preserved).

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instance List.instUnionList {α : Type u_1} [] :
Union (List α)
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• List.instUnionList = { union := List.union }
@[inline]
def List.inter {α : Type u_1} [] (l₁ : List α) (l₂ : List α) :
List α

Constructs the intersection of two lists, by filtering the elements of l₁ that are in l₂. Unlike bagInter this does not preserve multiplicity: [1, 1].inter [1] is [1, 1].

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instance List.instInterList {α : Type u_1} [] :
Inter (List α)
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• List.instInterList = { inter := List.inter }
inductive List.Sublist {α : Type u_1} :
List αList αProp

l₁ <+ l₂, or Sublist l₁ l₂, says that l₁ is a (non-contiguous) subsequence of l₂.

• slnil: ∀ {α : Type u_1}, List.Sublist [] []

the base case: [] is a sublist of []

• cons: ∀ {α : Type u_1} {l₁ l₂ : List α} (a : α), List.Sublist l₁ l₂List.Sublist l₁ (a :: l₂)

If l₁ is a subsequence of l₂, then it is also a subsequence of a :: l₂.

• cons₂: ∀ {α : Type u_1} {l₁ l₂ : List α} (a : α), List.Sublist l₁ l₂List.Sublist (a :: l₁) (a :: l₂)

If l₁ is a subsequence of l₂, then a :: l₁ is a subsequence of a :: l₂.

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l₁ <+ l₂, or Sublist l₁ l₂, says that l₁ is a (non-contiguous) subsequence of l₂.

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def List.isSublist {α : Type u_1} [] :
List αList αBool

True if the first list is a potentially non-contiguous sub-sequence of the second list.

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def List.splitAt {α : Type u_1} (n : Nat) (l : List α) :
List α × List α

Split a list at an index.

splitAt 2 [a, b, c] = ([a, b], [c])

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def List.splitAt.go {α : Type u_1} (l : List α) :
List αNatList α × List α

Auxiliary for splitAt: splitAt.go l n xs acc = (acc.toList ++ take n xs, drop n xs) if n < length xs, else (l, []).

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def List.splitAtD {α : Type u_1} (n : Nat) (l : List α) (dflt : α) :
List α × List α

Split a list at an index. Ensures the left list always has the specified length by right padding with the provided default element.

splitAtD 2 [a, b, c] x = ([a, b], [c])
splitAtD 4 [a, b, c] x = ([a, b, c, x], [])

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def List.splitAtD.go {α : Type u_1} (dflt : α) :
NatList αList α × List α

Auxiliary for splitAtD: splitAtD.go dflt n l acc = (acc.toList ++ left, right) if splitAtD n l dflt = (left, right).

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def List.splitOnP {α : Type u_1} (P : αBool) (l : List α) :
List (List α)

Split a list at every element satisfying a predicate. The separators are not in the result.

[1, 1, 2, 3, 2, 4, 4].splitOnP (· == 2) = [[1, 1], [3], [4, 4]]

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def List.splitOnP.go {α : Type u_1} (P : αBool) :
List αList αList (List α)

Auxiliary for splitOnP: splitOnP.go xs acc = res' where res' is obtained from splitOnP P xs by prepending acc.reverse to the first element.

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@[inline]
def List.splitOnPTR {α : Type u_1} (P : αBool) (l : List α) :
List (List α)

Tail recursive version of removeNth.

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@[specialize #[]]
def List.splitOnPTR.go {α : Type u_1} (P : αBool) :
List αArray (List α)List (List α)

Auxiliary for splitOnP: splitOnP.go xs acc r = r.toList ++ res' where res' is obtained from splitOnP P xs by prepending acc.toList to the first element.

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@[csimp]
@[inline]
def List.splitOn {α : Type u_1} [BEq α] (a : α) (as : List α) :
List (List α)

Split a list at every occurrence of a separator element. The separators are not in the result.

[1, 1, 2, 3, 2, 4, 4].splitOn 2 = [[1, 1], [3], [4, 4]]

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def List.modifyNthTail {α : Type u_1} (f : List αList α) :
NatList αList α

Apply a function to the nth tail of l. Returns the input without using f if the index is larger than the length of the List.

modifyNthTail f 2 [a, b, c] = [a, b] ++ f [c]

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@[inline]
def List.modifyHead {α : Type u_1} (f : αα) :
List αList α

Apply f to the head of the list, if it exists.

Equations
• = match x with | [] => [] | a :: l => f a :: l
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def List.modifyNth {α : Type u_1} (f : αα) :
NatList αList α

Apply f to the nth element of the list, if it exists.

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def List.modifyNthTR {α : Type u_1} (f : αα) (n : Nat) (l : List α) :
List α

Tail-recursive version of modifyNth.

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def List.modifyNthTR.go {α : Type u_1} (f : αα) :
List αNatList α

Auxiliary for modifyNthTR: modifyNthTR.go f l n acc = acc.toList ++ modifyNth f n l.

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theorem List.modifyNthTR_go_eq :
∀ {α : Type u_1} {f : αα} {acc : } (l : List α) (n : Nat), List.modifyNthTR.go f l n acc = acc.data ++
@[csimp]
@[inline]
def List.modifyLast {α : Type u_1} (f : αα) (l : List α) :
List α

Apply f to the last element of l, if it exists.

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@[specialize #[]]
def List.modifyLast.go {α : Type u_1} (f : αα) :
List αList α

Auxiliary for modifyLast: modifyLast.go f l acc = acc.toList ++ modifyLast f l.

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def List.insertNth {α : Type u_1} (n : Nat) (a : α) :
List αList α

insertNth n a l inserts a into the list l after the first n elements of l

insertNth 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]

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@[inline]
def List.insertNthTR {α : Type u_1} (n : Nat) (a : α) (l : List α) :
List α

Tail-recursive version of insertNth.

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def List.insertNthTR.go {α : Type u_1} (a : α) :
NatList αList α

Auxiliary for insertNthTR: insertNthTR.go a n l acc = acc.toList ++ insertNth n a l.

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theorem List.insertNthTR_go_eq :
∀ {α : Type u_1} {a : α} {acc : } (n : Nat) (l : List α), List.insertNthTR.go a n l acc = acc.data ++
@[csimp]
@[simp]
theorem List.headD_eq_head? {α : Type u_1} (l : List α) (a : α) :
=
def List.takeD {α : Type u_1} :
NatList ααList α

Take n elements from a list l. If l has less than n elements, append n - length l elements x.

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@[simp]
theorem List.takeD_zero {α : Type u_1} (l : List α) (a : α) :
List.takeD 0 l a = []
@[simp]
theorem List.takeD_succ {α : Type u_1} {n : Nat} (l : List α) (a : α) :
List.takeD (n + 1) l a = :: List.takeD n () a
@[simp]
theorem List.takeD_nil {α : Type u_1} (n : Nat) (a : α) :
List.takeD n [] a =
def List.takeDTR {α : Type u_1} (n : Nat) (l : List α) (dflt : α) :
List α

Tail-recursive version of takeD.

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def List.takeDTR.go {α : Type u_1} (dflt : α) :
NatList αList α

Auxiliary for takeDTR: takeDTR.go dflt n l acc = acc.toList ++ takeD n l dflt.

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theorem List.takeDTR_go_eq :
∀ {α : Type u_1} {dflt : α} {acc : } (n : Nat) (l : List α), List.takeDTR.go dflt n l acc = acc.data ++ List.takeD n l dflt
@[csimp]
def List.leftpad {α : Type u_1} (n : Nat) (a : α) (l : List α) :
List α

Pads l : List α with repeated occurrences of a : α until it is of length n. If l is initially larger than n, just return l.

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@[inline]
def List.leftpadTR {α : Type u_1} (n : Nat) (a : α) (l : List α) :
List α

Optimized version of leftpad.

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@[csimp]
def List.scanl {α : Type u_1} {β : Type u_2} (f : αβα) (a : α) :
List βList α

Fold a function f over the list from the left, returning the list of partial results.

scanl (+) 0 [1, 2, 3] = [0, 1, 3, 6]

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@[inline]
def List.scanlTR {α : Type u_1} {β : Type u_2} (f : αβα) (a : α) (l : List β) :
List α

Tail-recursive version of scanl.

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@[specialize #[]]
def List.scanlTR.go {α : Type u_1} {β : Type u_2} (f : αβα) :
List βαList α

Auxiliary for scanlTR: scanlTR.go f l a acc = acc.toList ++ scanl f a l.

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theorem List.scanlTR_go_eq :
∀ {α : Type u_1} {α_1 : Type u_2} {f : αα_1α} {a : α} {acc : } (l : List α_1), List.scanlTR.go f l a acc = acc.data ++ List.scanl f a l
@[csimp]
def List.scanr {α : Type u_1} {β : Type u_2} (f : αββ) (b : β) (l : List α) :
List β

Fold a function f over the list from the right, returning the list of partial results.

scanr (+) 0 [1, 2, 3] = [6, 5, 3, 0]

Equations
• List.scanr f b l = match List.foldr (fun (a : α) (x : β × List β) => match x with | (b', l') => (f a b', b' :: l')) (b, []) l with | (b', l') => b' :: l'
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@[inline]
def List.partitionMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβ γ) (l : List α) :
List β × List γ

Given a function f : α → β ⊕ γ, partitionMap f l maps the list by f whilst partitioning the result it into a pair of lists, List β × List γ, partitioning the .inl _ into the left list, and the .inr _ into the right List.

partitionMap (id : Nat ⊕ Nat → Nat ⊕ Nat) [inl 0, inr 1, inl 2] = ([0, 2], [1])

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@[specialize #[]]
def List.partitionMap.go {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβ γ) :
List αList β × List γ

Auxiliary for partitionMap: partitionMap.go f l acc₁ acc₂ = (acc₁.toList ++ left, acc₂.toList ++ right) if partitionMap f l = (left, right).

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@[inline]
def List.partitionM {m : TypeType u_1} {α : Type} [] (p : αm Bool) (l : List α) :
m (List α × List α)

Monadic generalization of List.partition.

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@[specialize #[]]
def List.partitionM.go {m : TypeType u_1} {α : Type} [] (p : αm Bool) :
List αm (List α × List α)

Auxiliary for partitionM: partitionM.go p l acc₁ acc₂ returns (acc₁.toList ++ left, acc₂.toList ++ right) if partitionM p l returns (left, right).

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@[specialize #[]]
def List.foldlIdx {α : Sort u_1} {β : Type u_2} (f : Natαβα) (init : α) :
List βα

Fold a list from left to right as with foldl, but the combining function also receives each element's index.

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@[specialize #[]]
def List.foldrIdx {α : Type u_1} {β : Sort u_2} (f : Natαββ) (init : β) (l : List α) (start : ) :
β

Fold a list from right to left as with foldr, but the combining function also receives each element's index.

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@[inline]
def List.findIdxs {α : Type u_1} (p : αBool) (l : List α) :

findIdxs p l is the list of indexes of elements of l that satisfy p.

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@[inline]
def List.indexesValues {α : Type u_1} (p : αBool) (l : List α) :
List (Nat × α)

Returns the elements of l that satisfy p together with their indexes in l. The returned list is ordered by index.

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@[inline]
def List.indexesOf {α : Type u_1} [BEq α] (a : α) :
List α

indexesOf a l is the list of all indexes of a in l. For example:

indexesOf a [a, b, a, a] = [0, 2, 3]

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def List.findIdx? {α : Type u_1} (p : αBool) :
List α

Return the index of the first occurrence of an element satisfying p.

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@[inline]
def List.indexOf? {α : Type u_1} [BEq α] (a : α) :
List α

Return the index of the first occurrence of a in the list.

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@[inline]
def List.lookmap {α : Type u_1} (f : α) (l : List α) :
List α

lookmap is a combination of lookup and filterMap. lookmap f l will apply f : α → Option α to each element of the list, replacing a → b at the first value a in the list such that f a = some b.

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@[specialize #[]]
def List.lookmap.go {α : Type u_1} (f : α) :
List αList α

Auxiliary for lookmap: lookmap.go f l acc = acc.toList ++ lookmap f l.

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@[inline]
def List.countP {α : Type u_1} (p : αBool) (l : List α) :

countP p l is the number of elements of l that satisfy p.

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@[specialize #[]]
def List.countP.go {α : Type u_1} (p : αBool) :
List α

Auxiliary for countP: countP.go p l acc = countP p l + acc.

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@[inline]
def List.count {α : Type u_1} [BEq α] (a : α) :
List αNat

count a l is the number of occurrences of a in l.

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def List.IsPrefix {α : Type u_1} (l₁ : List α) (l₂ : List α) :

IsPrefix l₁ l₂, or l₁ <+: l₂, means that l₁ is a prefix of l₂, that is, l₂ has the form l₁ ++ t for some t.

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def List.IsSuffix {α : Type u_1} (l₁ : List α) (l₂ : List α) :

IsSuffix l₁ l₂, or l₁ <:+ l₂, means that l₁ is a suffix of l₂, that is, l₂ has the form t ++ l₁ for some t.

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def List.IsInfix {α : Type u_1} (l₁ : List α) (l₂ : List α) :

IsInfix l₁ l₂, or l₁ <:+: l₂, means that l₁ is a contiguous substring of l₂, that is, l₂ has the form s ++ l₁ ++ t for some s, t.

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IsPrefix l₁ l₂, or l₁ <+: l₂, means that l₁ is a prefix of l₂, that is, l₂ has the form l₁ ++ t for some t.

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IsSuffix l₁ l₂, or l₁ <:+ l₂, means that l₁ is a suffix of l₂, that is, l₂ has the form t ++ l₁ for some t.

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IsInfix l₁ l₂, or l₁ <:+: l₂, means that l₁ is a contiguous substring of l₂, that is, l₂ has the form s ++ l₁ ++ t for some s, t.

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def List.inits {α : Type u_1} :
List αList (List α)

inits l is the list of initial segments of l.

inits [1, 2, 3] = [[], [1], [1, 2], [1, 2, 3]]

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

Tail-recursive version of inits.

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@[csimp]
def List.tails {α : Type u_1} :
List αList (List α)

tails l is the list of terminal segments of l.

tails [1, 2, 3] = [[1, 2, 3], [2, 3], [3], []]

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

Tail-recursive version of tails.

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def List.tailsTR.go {α : Type u_1} (l : List α) (acc : Array (List α)) :
List (List α)

Auxiliary for tailsTR: tailsTR.go l acc = acc.toList ++ tails l.

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@[csimp]
def List.sublists' {α : Type u_1} (l : List α) :
List (List α)

sublists' l is the list of all (non-contiguous) sublists of l. It differs from sublists only in the order of appearance of the sublists; sublists' uses the first element of the list as the MSB, sublists uses the first element of the list as the LSB.

sublists' [1, 2, 3] = [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]]

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• One or more equations did not get rendered due to their size.
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@[implemented_by List.sublistsFast]
def List.sublists {α : Type u_1} (l : List α) :
List (List α)

sublists l is the list of all (non-contiguous) sublists of l; cf. sublists' for a different ordering.

sublists [1, 2, 3] = [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]

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

A version of List.sublists that has faster runtime performance but worse kernel performance

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inductive List.Forall₂ {α : Type u_1} {β : Type u_2} (R : αβProp) :
List αList βProp

Forall₂ R l₁ l₂ means that l₁ and l₂ have the same length, and whenever a is the nth element of l₁, and b is the nth element of l₂, then R a b is satisfied.

• nil: ∀ {α : Type u_1} {β : Type u_2} {R : αβProp}, List.Forall₂ R [] []

Two nil lists are Forall₂-related

• cons: ∀ {α : Type u_1} {β : Type u_2} {R : αβProp} {a : α} {b : β} {l₁ : List α} {l₂ : List β}, R a bList.Forall₂ R l₁ l₂List.Forall₂ R (a :: l₁) (b :: l₂)

Two cons lists are related by Forall₂ R if the heads are related by R and the tails are related by Forall₂ R

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

Transpose of a list of lists, treated as a matrix.

transpose [[1, 2], [3, 4], [5, 6]] = [[1, 3, 5], [2, 4, 6]]

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def List.transpose.pop {α : Type u_1} (old : List α) :
List αId (List α × List α)

pop : List α → StateM (List α) (List α) transforms the input list old by taking the head of the current state and pushing it on the head of old. If the state list is empty, then old is left unchanged.

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• = match x with | [] => (old, []) | a :: l => (a :: old, l)
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def List.transpose.go {α : Type u_1} (l : List α) (acc : Array (List α)) :
Array (List α)

go : List α → Array (List α) → Array (List α) handles the insertion of a new list into all the lists in the array: go [a, b, c] #[l₁, l₂, l₃] = #[a::l₁, b::l₂, c::l₃]. If the new list is too short, the later lists are unchanged, and if it is too long the array is extended:

go [a] #[l₁, l₂, l₃] = #[a::l₁, l₂, l₃]
go [a, b, c, d] #[l₁, l₂, l₃] = #[a::l₁, b::l₂, c::l₃, [d]]

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def List.sections {α : Type u_1} :
List (List α)List (List α)

List of all sections through a list of lists. A section of [L₁, L₂, ..., Lₙ] is a list whose first element comes from L₁, whose second element comes from L₂, and so on.

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def List.sectionsTR {α : Type u_1} (L : List (List α)) :
List (List α)

Optimized version of sections.

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def List.sectionsTR.go {α : Type u_1} (l : List α) (acc : Array (List α)) :
Array (List α)

go : List α → Array (List α) → Array (List α) inserts one list into the accumulated list of sections acc: go [a, b] #[l₁, l₂] = [a::l₁, b::l₁, a::l₂, b::l₂].

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theorem List.sections_eq_nil_of_isEmpty {α : Type u_1} {L : List (List α)} :
List.any L List.isEmpty = true = []
@[csimp]
def List.eraseP {α : Type u_1} (p : αBool) :
List αList α

eraseP p l removes the first element of l satisfying the predicate p.

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@[inline]
def List.erasePTR {α : Type u_1} (p : αBool) (l : List α) :
List α

Tail-recursive version of eraseP.

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@[specialize #[]]
def List.erasePTR.go {α : Type u_1} (p : αBool) (l : List α) :
List αList α

Auxiliary for erasePTR: erasePTR.go p l xs acc = acc.toList ++ eraseP p xs, unless xs does not contain any elements satisfying p, where it returns l.

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@[csimp]
theorem List.eraseP_eq_erasePTR.go (α : Type u_1) (p : αBool) (l : List α) (acc : ) (xs : List α) :
l = acc.data ++ xsList.erasePTR.go p l xs acc = acc.data ++ List.eraseP p xs
def List.extractP {α : Type u_1} (p : αBool) (l : List 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).

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def List.extractP.go {α : Type u_1} (p : αBool) (l : List α) :
List α × List α

Auxiliary for extractP: extractP.go p l xs acc = (some a, acc.toList ++ out) if extractP p xs = (some a, out), and extractP.go p l xs acc = (none, l) if extractP p xs = (none, _).

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

revzip l returns a list of pairs of the elements of l paired with the elements of l in reverse order.

revzip [1, 2, 3, 4, 5] = [(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)]

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def List.product {α : Type u_1} {β : Type u_2} (l₁ : List α) (l₂ : List β) :
List (α × β)

product l₁ l₂ is the list of pairs (a, b) where a ∈ l₁ and b ∈ l₂.

product [1, 2] [5, 6] = [(1, 5), (1, 6), (2, 5), (2, 6)]

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def List.productTR {α : Type u_1} {β : Type u_2} (l₁ : List α) (l₂ : List β) :
List (α × β)

Optimized version of product.

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• One or more equations did not get rendered due to their size.
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@[csimp]
def List.sigma {α : Type u_1} {σ : αType u_2} (l₁ : List α) (l₂ : (a : α) → List (σ a)) :
List ((a : α) × σ a)

sigma l₁ l₂ is the list of dependent pairs (a, b) where a ∈ l₁ and b ∈ l₂ a.

sigma [1, 2] (λ_, [(5 : Nat), 6]) = [(1, 5), (1, 6), (2, 5), (2, 6)]

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def List.sigmaTR {α : Type u_1} {σ : αType u_2} (l₁ : List α) (l₂ : (a : α) → List (σ a)) :
List ((a : α) × σ a)

Optimized version of sigma.

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• One or more equations did not get rendered due to their size.
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@[csimp]
def List.ofFn {α : Type u_1} {n : Nat} (f : Fin nα) :
List α

ofFn f with f : fin n → α returns the list whose ith element is f i

ofFn f = [f 0, f 1, ... , f (n - 1)]

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def List.ofFnNthVal {α : Type u_1} {n : Nat} (f : Fin nα) (i : Nat) :

ofFnNthVal f i returns some (f i) if i < n and none otherwise.

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• = if h : i < n then some (f { val := i, isLt := h }) else none
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def List.Disjoint {α : Type u_1} (l₁ : List α) (l₂ : List α) :

disjoint l₁ l₂ means that l₁ and l₂ have no elements in common.

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def List.takeWhile₂ {α : Type u_1} {β : Type u_2} (R : αβBool) :
List αList βList α × List β

Returns the longest initial prefix of two lists such that they are pairwise related by R.

takeWhile₂ (· < ·) [1, 2, 4, 5] [5, 4, 3, 6] = ([1, 2], [5, 4])

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@[inline]
def List.takeWhile₂TR {α : Type u_1} {β : Type u_2} (R : αβBool) (as : List α) (bs : List β) :
List α × List β

Tail-recursive version of takeWhile₂.

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@[specialize #[]]
def List.takeWhile₂TR.go {α : Type u_1} {β : Type u_2} (R : αβBool) :
List αList βList αList βList α × List β

Auxiliary for takeWhile₂TR: takeWhile₂TR.go R as bs acca accb = (acca.reverse ++ as', acca.reverse ++ bs') if takeWhile₂ R as bs = (as', bs').

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@[csimp]
theorem List.takeWhile₂_eq_takeWhile₂TR.go (α : Type u_2) (β : Type u_1) (R : αβBool) (as : List α) (bs : List β) (acca : List α) (accb : List β) :
List.takeWhile₂TR.go R as bs acca accb = (List.reverse acca ++ (List.takeWhile₂ R as bs).fst, List.reverse accb ++ (List.takeWhile₂ R as bs).snd)
inductive List.Pairwise {α : Type u_1} (R : ααProp) :
List αProp

Pairwise R l means that all the elements with earlier indexes are R-related to all the elements with later indexes.

Pairwise R [1, 2, 3] ↔ R 1 2 ∧ R 1 3 ∧ R 2 3


For example if R = (·≠·) then it asserts l has no duplicates, and if R = (·<·) then it asserts that l is (strictly) sorted.

• nil: ∀ {α : Type u_1} {R : ααProp},

All elements of the empty list are vacuously pairwise related.

• cons: ∀ {α : Type u_1} {R : ααProp} {a : α} {l : List α}, (∀ (a' : α), a' lR a a')List.Pairwise R (a :: l)

a :: l is Pairwise R if a R-relates to every element of l, and l is Pairwise R.

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@[simp]
theorem List.pairwise_cons {α : Type u_1} {R : ααProp} {a : α} {l : List α} :
List.Pairwise R (a :: l) (∀ (a' : α), a' lR a a')
instance List.instDecidablePairwise {α : Type u_1} {R : ααProp} [] (l : List α) :
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• One or more equations did not get rendered due to their size.
• = isTrue (_ : )
def List.pwFilter {α : Type u_1} (R : ααProp) [] (l : List α) :
List α

pwFilter R l is a maximal sublist of l which is Pairwise R. pwFilter (·≠·) is the erase duplicates function (cf. eraseDup), and pwFilter (·<·) finds a maximal increasing subsequence in l. For example,

pwFilter (·<·) [0, 1, 5, 2, 6, 3, 4] = [0, 1, 2, 3, 4]

Equations
• = List.foldr (fun (x : α) (IH : List α) => if ∀ (y : α), y IHR x y then x :: IH else IH) [] l
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inductive List.Chain {α : Type u_1} (R : ααProp) :
αList αProp

Chain R a l means that R holds between adjacent elements of a::l.

Chain R a [b, c, d] ↔ R a b ∧ R b c ∧ R c d

• nil: ∀ {α : Type u_1} {R : ααProp} {a : α}, List.Chain R a []

A chain of length 1 is trivially a chain.

• cons: ∀ {α : Type u_1} {R : ααProp} {a b : α} {l : List α}, R a bList.Chain R b lList.Chain R a (b :: l)

If a relates to b and b::l is a chain, then a :: b :: l is also a chain.

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def List.Chain' {α : Type u_1} (R : ααProp) :
List αProp

Chain' R l means that R holds between adjacent elements of l.

Chain' R [a, b, c, d] ↔ R a b ∧ R b c ∧ R c d

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def List.Nodup {α : Type u_1} :
List αProp

Nodup l means that l has no duplicates, that is, any element appears at most once in the List. It is defined as Pairwise (≠).

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instance List.nodupDecidable {α : Type u_1} [] (l : List α) :
Equations
• List.nodupDecidable = List.instDecidablePairwise
@[inline]
def List.eraseDup {α : Type u_1} [] :
List αList α

eraseDup l removes duplicates from l (taking only the first occurrence). Defined as pwFilter (≠).

eraseDup [1, 0, 2, 2, 1] = [0, 2, 1]

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def List.range' (start : Nat) (len : Nat) (step : ) :

range' start len step is the list of numbers [start, start+step, ..., start+(len-1)*step]. It is intended mainly for proving properties of range and iota.

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@[inline]
def List.range'TR (s : Nat) (n : Nat) (step : ) :

Optimized version of range'.

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def List.range'TR.go (step : ) :
NatNat

Auxiliary for range'TR: range'TR.go n e = [e-n, ..., e-1] ++ acc.

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@[csimp]
theorem List.range'_eq_range'TR.go (step : ) (s : Nat) (n : Nat) (m : Nat) :
List.range'TR.go step n (s + step * n) (List.range' (s + step * n) m step) = List.range' s (n + m) step
@[inline]
def List.reduceOption {α : Type u_1} :
List ()List α

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

Equations
• List.reduceOption =
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@[deprecated List.getLastD]
def List.ilast' {α : Type u_1} :
αList αα

ilast' x xs returns the last element of xs if xs is non-empty; it returns x otherwise. Use List.getLastD instead.

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@[deprecated List.getLast?]
def List.last' {α : Type u_1} :
List α

last' xs returns the last element of xs if xs is non-empty; it returns none otherwise. Use List.getLast? instead

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@[inline]
def List.rotate {α : Type u_1} (l : List α) (n : Nat) :
List α

rotate l n rotates the elements of l to the left by n

rotate [0, 1, 2, 3, 4, 5] 2 = [2, 3, 4, 5, 0, 1]

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def List.rotate' {α : Type u_1} :
List αNatList α

rotate' is the same as rotate, but slower. Used for proofs about rotate

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def List.mapDiagM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [] (f : ααm β) (l : List α) :
m (List β)

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.

mapDiagM f [1, 2, 3] =
return [← f 1 1, ← f 1 2, ← f 1 3, ← f 2 2, ← f 2 3, ← f 3 3]

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def List.mapDiagM.go {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [] (f : ααm β) :
List αm (List β)

Auxiliary for mapDiagM: mapDiagM.go as f acc = (acc.toList ++ ·) <\$> mapDiagM f as

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def List.forDiagM {m : Type u_1 → Type u_2} {α : Type u_3} [] (f : αα) :
List α

forDiagM 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.

forDiagM f [1, 2, 3] = do f 1 1; f 1 2; f 1 3; f 2 2; f 2 3; f 3 3

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def List.getRest {α : Type u_1} [] :
List αList αOption (List α)

getRest l l₁ returns some l₂ if l = l₁ ++ l₂. If l₁ is not a prefix of l, returns none

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def List.dropSlice {α : Type u_1} :
NatNatList αList α

List.dropSlice n m xs removes a slice of length m at index n in list xs.

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@[inline]
def List.dropSliceTR {α : Type u_1} (n : Nat) (m : Nat) (l : List α) :
List α

Optimized version of dropSlice.

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def List.dropSliceTR.go {α : Type u_1} (l : List α) (m : Nat) :
List αNatList α

Auxiliary for dropSliceTR: dropSliceTR.go l m xs n acc = acc.toList ++ dropSlice n m xs unless n ≥ length xs, in which case it is l.

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theorem List.dropSlice_zero₂ {α : Type u_1} (n : Nat) (l : List α) :
= l
@[csimp]
theorem List.dropSlice_eq_dropSliceTR.go (α : Type u_1) (l : List α) (m : Nat) (acc : ) (xs : List α) (n : Nat) :
l = acc.data ++ xsList.dropSliceTR.go l m xs n acc = acc.data ++ List.dropSlice n (m + 1) xs
def List.zipWithLeft' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) :
List αList βList γ × List β

Left-biased version of List.zipWith. zipWithLeft' 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.

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

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@[inline]
def List.zipWithLeft'TR {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) (as : List α) (bs : List β) :
List γ × List β

Tail-recursive version of zipWithLeft'.

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def List.zipWithLeft'TR.go {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) :
List αList βList γ × List β

Auxiliary for zipWithLeft'TR: zipWithLeft'TR.go l acc = acc.toList ++ zipWithLeft' l.

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@[csimp]
theorem List.zipWithLeft'_eq_zipWithLeft'TR.go (α : Type u_3) (β : Type u_2) (γ : Type u_1) (f : αγ) (acc : ) (as : List α) (bs : List β) :
List.zipWithLeft'TR.go f as bs acc = match List.zipWithLeft' f as bs with | (l, r) => (Array.toList acc ++ l, r)
@[inline]
def List.zipWithRight' {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : βγ) (as : List α) (bs : List β) :
List γ × List α

Right-biased version of List.zipWith. zipWithRight' 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.

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

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@[inline]
def List.zipLeft' {α : Type u_1} {β : Type u_2} :
List αList βList (α × ) × List β

Left-biased version of List.zip. zipLeft' as bs returns the list of pairs (aᵢ, bᵢ) for aᵢ ∈ as and bᵢ ∈ bs. If bs is shorter than as, the remaining aᵢ are paired with none. Also returns the remaining bs.

zipLeft' [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])
zipLeft' [1] ['a', 'b'] = ([(1, some 'a')], ['b'])
zipLeft' = zipWithLeft' prod.mk

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@[inline]
def List.zipRight' {α : Type u_1} {β : Type u_2} :
List αList βList ( × β) × List α

Right-biased version of List.zip. zipRight' as bs returns the list of pairs (aᵢ, bᵢ) for aᵢ ∈ as and bᵢ ∈ bs. If as is shorter than bs, the remaining bᵢ are paired with none. Also returns the remaining as.

zipRight' [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])
zipRight' [1, 2] ['a'] = ([(some 1, 'a')], [2])
zipRight' = zipWithRight' prod.mk

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def List.zipWithLeft {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) :
List αList βList γ

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

zipWithLeft prod.mk [1, 2] ['a'] = [(1, some 'a'), (2, none)]
zipWithLeft prod.mk [1] ['a', 'b'] = [(1, some 'a')]
zipWithLeft f as bs = (zipWithLeft' f as bs).fst

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@[inline]
def List.zipWithLeftTR {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) (as : List α) (bs : List β) :
List γ

Tail-recursive version of zipWithLeft.

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def List.zipWithLeftTR.go {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) :
List αList βList γ

Auxiliary for zipWithLeftTR: zipWithLeftTR.go l acc = acc.toList ++ zipWithLeft l.

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@[csimp]
theorem List.zipWithLeft_eq_zipWithLeftTR.go (α : Type u_3) (β : Type u_2) (γ : Type u_1) (f : αγ) (acc : ) (as : List α) (bs : List β) :
@[inline]
def List.zipWithRight {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : βγ) (as : List α) (bs : List β) :
List γ

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

zipWithRight prod.mk [1, 2] ['a'] = [(some 1, 'a')]
zipWithRight prod.mk [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]
zipWithRight f as bs = (zipWithRight' f as bs).fst

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@[inline]
def List.zipLeft {α : Type u_1} {β : Type u_2} :
List αList βList (α × )

Left-biased version of List.zip. zipLeft as bs returns the list of pairs (aᵢ, bᵢ) for aᵢ ∈ as and bᵢ ∈ bs. If bs is shorter than as, the remaining aᵢ are paired with none.

zipLeft [1, 2] ['a'] = [(1, some 'a'), (2, none)]
zipLeft [1] ['a', 'b'] = [(1, some 'a')]
zipLeft = zipWithLeft prod.mk

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@[inline]
def List.zipRight {α : Type u_1} {β : Type u_2} :
List αList βList ( × β)

Right-biased version of List.zip. zipRight as bs returns the list of pairs (aᵢ, bᵢ) for aᵢ ∈ as and bᵢ ∈ bs. If as is shorter than bs, the remaining bᵢ are paired with none.

zipRight [1, 2] ['a'] = [(some 1, 'a')]
zipRight [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]
zipRight = zipWithRight prod.mk

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@[inline]
def List.allSome {α : Type u_1} (l : List ()) :

If all elements of xs are some xᵢ, allSome xs returns the xᵢ. Otherwise it returns none.

allSome [some 1, some 2] = some [1, 2]
allSome [some 1, none  ] = none

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def List.fillNones {α : Type u_1} :
List ()List αList α

fillNones xs ys replaces the nones in xs with elements of ys. If there are not enough ys to replace all the nones, the remaining nones are dropped from xs.

fillNones [none, some 1, none, none] [2, 3] = [2, 1, 3]

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@[inline]
def List.fillNonesTR {α : Type u_1} (as : List ()) (as' : List α) :
List α

Tail-recursive version of fillNones.

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def List.fillNonesTR.go {α : Type u_1} :
List ()List αList α

Auxiliary for fillNonesTR: fillNonesTR.go as as' acc = acc.toList ++ fillNones as as'.

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@[csimp]
theorem List.fillNones_eq_fillNonesTR.go (α : Type u_1) (acc : ) (as : List ()) (as' : List α) :
List.fillNonesTR.go as as' acc = acc.data ++ List.fillNones as as'
def List.takeList {α : Type u_1} :
List αList (List α) × List α

takeList as ns extracts successive sublists from as. For ns = n₁ ... nₘ, it first takes the n₁ initial elements from as, then the next n₂ ones, etc. It returns the sublists of as -- one for each nᵢ -- and the remaining elements of as. If as does not have at least as many elements as the sum of the nᵢ, the corresponding sublists will have less than nᵢ elements.

takeList ['a', 'b', 'c', 'd', 'e'] [2, 1, 1] = ([['a', 'b'], ['c'], ['d']], ['e'])
takeList ['a', 'b'] [3, 1] = ([['a', 'b'], []], [])

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@[inline]
def List.takeListTR {α : Type u_1} (xs : List α) (ns : ) :
List (List α) × List α

Tail-recursive version of takeList.

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def List.takeListTR.go {α : Type u_1} :
List αArray (List α)List (List α) × List α

Auxiliary for takeListTR: takeListTR.go as as' acc = acc.toList ++ takeList as as'.

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@[csimp]
theorem List.takeList_eq_takeListTR.go (α : Type u_1) (acc : Array (List α)) (ns : ) (xs : List α) :
List.takeListTR.go ns xs acc = match List.takeList xs ns with | (l, r) => (Array.toList acc ++ l, r)
def List.toChunksAux {α : Type u_1} (n : Nat) :
List αNatList α × List (List α)

Auxliary definition used to define toChunks. toChunksAux n xs i returns (xs.take i, (xs.drop i).toChunks (n+1)), that is, the first i elements of xs, and the remaining elements chunked into sublists of length n+1.

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def List.toChunks {α : Type u_1} :
NatList αList (List α)

xs.toChunks n splits the list into sublists of size at most n, such that (xs.toChunks n).join = xs.

[1, 2, 3, 4, 5, 6, 7, 8].toChunks 10 = [[1, 2, 3, 4, 5, 6, 7, 8]]
[1, 2, 3, 4, 5, 6, 7, 8].toChunks 3 = [[1, 2, 3], [4, 5, 6], [7, 8]]
[1, 2, 3, 4, 5, 6, 7, 8].toChunks 2 = [[1, 2], [3, 4], [5, 6], [7, 8]]
[1, 2, 3, 4, 5, 6, 7, 8].toChunks 0 = [[1, 2, 3, 4, 5, 6, 7, 8]]

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• = match x✝, x with | x, [] => [] | 0, xs => [xs] | n, x :: xs => List.toChunks.go n xs #[x] #[]
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def List.toChunks.go {α : Type u_1} (n : Nat) :
List αArray (List α)List (List α)

Auxliary definition used to define toChunks. toChunks.go xs acc₁ acc₂ pushes elements into acc₁ until it reaches size n, then it pushes the resulting list to acc₂ and continues until xs is exhausted.

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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 : αβγδ) :
List αList βList γList δ

Ternary version of List.zipWith.

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def List.zipWith₄ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} (f : αβγδε) :
List αList βList γList δList ε

Quaternary version of List.zipWith.

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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 : αβγδεζ) :
List αList βList γList δList εList ζ

Quinary version of List.zipWith.

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def List.mapWithPrefixSuffixAux {α : Type u_1} {β : Type u_2} (f : List ααList αβ) :
List αList αList β

An auxiliary function for List.mapWithPrefixSuffix.

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def List.mapWithPrefixSuffix {α : Type u_1} {β : Type u_2} (f : List ααList αβ) (l : List α) :
List β

List.mapWithPrefixSuffix f l maps f across a list l. For each a ∈ l with l = pref ++ [a] ++ suff, a is mapped to f pref a suff. Example: if f : list Nat → Nat → list Nat → β, List.mapWithPrefixSuffix f [1, 2, 3] will produce the list [f [] 1 [2, 3], f [1] 2 [3], f [1, 2] 3 []].

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def List.mapWithComplement {α : Type u_1} {β : Type u_2} (f : αList αβ) :
List αList β

List.mapWithComplement f l is a variant of List.mapWithPrefixSuffix that maps f across a list l. For each a ∈ l with l = pref ++ [a] ++ suff, a is mapped to f a (pref ++ suff), i.e., the list input to f is l with a removed. Example: if f : Nat → list Nat → β, List.mapWithComplement f [1, 2, 3] will produce the list [f 1 [2, 3], f 2 [1, 3], f 3 [1, 2]].

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def List.traverse {F : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [] (f : αF β) :
List αF (List β)

Map each element of a List to an action, evaluate these actions in order, and collect the results.

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inductive List.Perm {α : Type u_1} :
List αList αProp

Perm l₁ l₂ or l₁ ~ l₂ asserts that l₁ and l₂ are permutations of each other. This is defined by induction using pairwise swaps.

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Perm l₁ l₂ or l₁ ~ l₂ asserts that l₁ and l₂ are permutations of each other. This is defined by induction using pairwise swaps.

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def List.isPerm {α : Type u_1} [BEq α] :
List αList αBool

O(|l₁| * |l₂|). Computes whether l₁ is a permutation of l₂. See isPerm_iff for a characterization in terms of List.Perm.

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def List.Subperm {α : Type u_1} (l₁ : List α) (l₂ : List α) :

Subperm l₁ l₂, denoted l₁ <+~ l₂, means that l₁ is a sublist of a permutation of l₂. This is an analogue of l₁ ⊆ l₂ which respects multiplicities of elements, and is used for the ≤ relation on multisets.

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Subperm l₁ l₂, denoted l₁ <+~ l₂, means that l₁ is a sublist of a permutation of l₂. This is an analogue of l₁ ⊆ l₂ which respects multiplicities of elements, and is used for the ≤ relation on multisets.

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def List.isSubperm {α : Type u_1} [BEq α] (l₁ : List α) (l₂ : List α) :

O(|l₁| * (|l₁| + |l₂|)). Computes whether l₁ is a sublist of a permutation of l₂. See isSubperm_iff for a characterization in terms of List.Subperm.

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