## New definitions #

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.

## Equations

## Instances For

Auxiliary for `mapIdxM`

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

## Equations

- List.mapIdxM.go f [] x = pure x.toList
- List.mapIdxM.go f (a :: as) x = do let __do_lift ← f x.size a List.mapIdxM.go f as (x.push __do_lift)

## Instances For

`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]
```

## Equations

- List.after p [] = []
- List.after p (a :: l) = bif p a then l else List.after p l

## Instances For

**Alias** of `List.eraseIdx`

.

`O(i)`

. `eraseIdx l i`

removes the `i`

'th element of the list `l`

.

`erase [a, b, c, d, e] 0 = [b, c, d, e]`

`erase [a, b, c, d, e] 1 = [a, c, d, e]`

`erase [a, b, c, d, e] 5 = [a, b, c, d, e]`

## Equations

## Instances For

**Alias** of `List.eraseIdxTR`

.

Tail recursive version of `List.eraseIdx`

.

## Equations

## Instances For

**Alias** of `List.eraseIdx_eq_eraseIdxTR`

.

Replaces the first element of the list for which `f`

returns `some`

with the returned value.

## Equations

- List.replaceF f [] = []
- List.replaceF f (a :: l) = match f a with | none => a :: List.replaceF f l | some a => a :: l

## Instances For

Tail-recursive version of `replaceF`

.

## Equations

- List.replaceFTR f l = List.replaceFTR.go f l #[]

## Instances For

Auxiliary for `replaceFTR`

: `replaceFTR.go f xs acc = acc.toList ++ replaceF f xs`

.

## Equations

- List.replaceFTR.go f [] x = x.toList
- List.replaceFTR.go f (x_2 :: xs) x = match f x_2 with | none => List.replaceFTR.go f xs (x.push x_2) | some a' => x.toListAppend (a' :: xs)

## Instances For

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).

## Equations

- l₁.union l₂ = List.foldr List.insert l₂ l₁

## Instances For

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]`

.

## Equations

- l₁.inter l₂ = List.filter (fun (x : α) => List.elem x l₂) l₁

## Instances For

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], [])
```

## Equations

- List.splitAtD n l dflt = List.splitAtD.go dflt n l []

## Instances For

Auxiliary for `splitAtD`

: `splitAtD.go dflt n l acc = (acc.reverse ++ left, right)`

if `splitAtD n l dflt = (left, right)`

.

## Equations

- List.splitAtD.go dflt n.succ (x_3 :: xs) x = List.splitAtD.go dflt n xs (x_3 :: x)
- List.splitAtD.go dflt 0 x✝ x = (x.reverse, x✝)
- List.splitAtD.go dflt x✝ [] x = (x.reverseAux (List.replicate x✝ dflt), [])

## Instances For

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]]
```

## Equations

- List.splitOnP P l = List.splitOnP.go P l []

## Instances For

Auxiliary for `splitOnP`

: `splitOnP.go xs acc = res'`

where `res'`

is obtained from `splitOnP P xs`

by prepending `acc.reverse`

to the first element.

## Equations

- List.splitOnP.go P [] x = [x.reverse]
- List.splitOnP.go P (a :: t) x = if P a = true then x.reverse :: List.splitOnP.go P t [] else List.splitOnP.go P t (a :: x)

## Instances For

Tail recursive version of `splitOnP`

.

## Equations

- List.splitOnPTR P l = List.splitOnPTR.go P l #[] #[]

## Instances For

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.

## Equations

- List.splitOnPTR.go P [] x✝ x = x.toListAppend [x✝.toList]
- List.splitOnPTR.go P (a :: t) x✝ x = bif P a then List.splitOnPTR.go P t #[] (x.push x✝.toList) else List.splitOnPTR.go P t (x✝.push a) x

## Instances For

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]]
```

## Equations

- List.splitOn a as = List.splitOnP (fun (x : α) => x == a) as

## Instances For

**Alias** of `List.modifyTailIdx`

.

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.

```
modifyTailIdx f 2 [a, b, c] = [a, b] ++ f [c]
```

## Equations

## Instances For

**Alias** of `List.modify`

.

Apply `f`

to the nth element of the list, if it exists, replacing that element with the result.

## Equations

## Instances For

Apply `f`

to the last element of `l`

, if it exists.

## Equations

- List.modifyLast f l = List.modifyLast.go f l #[]

## Instances For

Auxiliary for `modifyLast`

: `modifyLast.go f l acc = acc.toList ++ modifyLast f l`

.

## Equations

- List.modifyLast.go f [] x = []
- List.modifyLast.go f [x_2] x = x.toListAppend [f x_2]
- List.modifyLast.go f (x_2 :: xs) x = List.modifyLast.go f xs (x.push x_2)

## Instances For

`insertIdx n a l`

inserts `a`

into the list `l`

after the first `n`

elements of `l`

```
insertIdx 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]
```

## Equations

- List.insertIdx n a = List.modifyTailIdx (List.cons a) n

## Instances For

**Alias** of `List.insertIdx`

.

`insertIdx n a l`

inserts `a`

into the list `l`

after the first `n`

elements of `l`

```
insertIdx 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]
```

## Equations

## Instances For

Tail-recursive version of `insertIdx`

.

## Equations

- List.insertIdxTR n a l = List.insertIdxTR.go a n l #[]

## Instances For

Auxiliary for `insertIdxTR`

: `insertIdxTR.go a n l acc = acc.toList ++ insertIdx n a l`

.

## Equations

- List.insertIdxTR.go a 0 x✝ x = x.toListAppend (a :: x✝)
- List.insertIdxTR.go a x✝ [] x = x.toList
- List.insertIdxTR.go a n.succ (a_1 :: l) x = List.insertIdxTR.go a n l (x.push a_1)

## Instances For

Take `n`

elements from a list `l`

. If `l`

has less than `n`

elements, append `n - length l`

elements `x`

.

## Equations

- List.takeD 0 x✝ x = []
- List.takeD n.succ x✝ x = x✝.headD x :: List.takeD n x✝.tail x

## Instances For

Tail-recursive version of `takeD`

.

## Equations

- List.takeDTR n l dflt = List.takeDTR.go dflt n l #[]

## Instances For

Auxiliary for `takeDTR`

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

.

## Equations

- List.takeDTR.go dflt n.succ (x_3 :: xs) x = List.takeDTR.go dflt n xs (x.push x_3)
- List.takeDTR.go dflt 0 x✝ x = x.toList
- List.takeDTR.go dflt x✝ [] x = x.toListAppend (List.replicate x✝ dflt)

## Instances For

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]
```

## Equations

- List.scanl f a [] = [a]
- List.scanl f a (a_1 :: l) = a :: List.scanl f (f a a_1) l

## Instances For

Tail-recursive version of `scanl`

.

## Equations

- List.scanlTR f a l = List.scanlTR.go f l a #[]

## Instances For

Auxiliary for `scanlTR`

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

.

## Equations

- List.scanlTR.go f [] x✝ x = x.toListAppend [x✝]
- List.scanlTR.go f (b :: l) x✝ x = List.scanlTR.go f l (f x✝ b) (x.push x✝)

## Instances For

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'

## Instances For

Fold a list from left to right as with `foldl`

, but the combining function
also receives each element's index.

## Equations

- List.foldlIdx f init [] x = init
- List.foldlIdx f init (b :: l) x = List.foldlIdx f (f x init b) l (x + 1)

## Instances For

Fold a list from right to left as with `foldr`

, but the combining function
also receives each element's index.

## Equations

- List.foldrIdx f init [] x = init
- List.foldrIdx f init (b :: l) x = f x b (List.foldrIdx f init l (x + 1))

## Instances For

Returns the elements of `l`

that satisfy `p`

together with their indexes in
`l`

. The returned list is ordered by index.

## Equations

- List.indexesValues p l = List.foldrIdx (fun (i : Nat) (a : α) (l : List (Nat × α)) => if p a = true then (i, a) :: l else l) [] l

## Instances For

`indexesOf a l`

is the list of all indexes of `a`

in `l`

. For example:

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

## Equations

- List.indexesOf a = List.findIdxs fun (x : α) => x == a

## Instances For

`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`

.

## Equations

- List.lookmap f l = List.lookmap.go f l #[]

## Instances For

Auxiliary for `lookmap`

: `lookmap.go f l acc = acc.toList ++ lookmap f l`

.

## Equations

- List.lookmap.go f [] x = x.toList
- List.lookmap.go f (x_2 :: xs) x = match f x_2 with | some b => x.toListAppend (b :: xs) | none => List.lookmap.go f xs (x.push x_2)

## Instances For

Auxiliary for `tailsTR`

: `tailsTR.go l acc = acc.toList ++ tails l`

.

## Equations

- List.tailsTR.go [] acc = acc.toListAppend [[]]
- List.tailsTR.go (a :: l_1) acc = List.tailsTR.go l_1 (acc.push (a :: l_1))

## Instances For

`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]]
```

## Equations

- l.sublists' = (List.foldr (fun (a : α) (arr : Array (List α)) => Array.foldl (fun (r : Array (List α)) (l : List α) => r.push (a :: l)) arr arr) #[[]] l).toList

## Instances For

A version of `List.sublists`

that has faster runtime performance but worse kernel performance

## Equations

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

## Instances For

`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 : α → β → Prop}, List.Forall₂ R [] []
Two nil lists are

`Forall₂`

-related - cons: ∀ {α : Type u_1} {β : Type u_2} {R : α → β → Prop} {a : α} {b : β} {l₁ : List α} {l₂ : List β}, R a b → List.Forall₂ R l₁ l₂ → List.Forall₂ R (a :: l₁) (b :: l₂)

## Instances For

Check for all elements `a`

, `b`

, where `a`

and `b`

are the nth element of the first and second
List respectively, that `r a b = true`

.

## Equations

## Instances For

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

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

## Equations

- l.transpose = (List.foldr List.transpose.go #[] l).toList

## Instances For

`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.

## Equations

- List.transpose.pop old [] = (old, [])
- List.transpose.pop old (a :: l) = (a :: old, l)

## Instances For

`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]]
```

## Equations

- List.transpose.go l acc = match Array.mapM List.transpose.pop acc l with | (acc, l) => List.foldl (fun (arr : Array (List α)) (a : α) => arr.push [a]) acc l

## Instances For

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.

## Equations

## Instances For

Optimized version of `sections`

.

## Equations

- L.sectionsTR = bif L.any List.isEmpty then [] else (List.foldr List.sectionsTR.go #[[]] L).toList

## Instances For

`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]`

.

## Equations

- List.sectionsTR.go l acc = Array.foldl (fun (acc' : Array (List α)) (l' : List α) => List.foldl (fun (acc' : Array (List α)) (a : α) => acc'.push (a :: l')) acc' l) #[] acc

## Instances For

`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)`

.

## Equations

- List.extractP p l = List.extractP.go p l l #[]

## Instances For

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, _)`

.

## Equations

- List.extractP.go p l [] x = (none, l)
- List.extractP.go p l (x_2 :: xs) x = bif p x_2 then (some x_2, x.toListAppend xs) else List.extractP.go p l xs (x.push x_2)

## Instances For

Optimized version of `product`

.

## Equations

- l₁.productTR l₂ = (List.foldl (fun (acc : Array (α × β)) (a : α) => List.foldl (fun (acc : Array (α × β)) (b : β) => acc.push (a, b)) acc l₂) #[] l₁).toList

## Instances For

Optimized version of `sigma`

.

## Equations

- l₁.sigmaTR l₂ = (List.foldl (fun (acc : Array ((a : α) × σ a)) (a : α) => List.foldl (fun (acc : Array ((a : α) × σ a)) (b : σ a) => acc.push ⟨a, b⟩) acc (l₂ a)) #[] l₁).toList

## Instances For

`ofFnNthVal f i`

returns `some (f i)`

if `i < n`

and `none`

otherwise.

## Equations

- List.ofFnNthVal f i = if h : i < n then some (f ⟨i, h⟩) else none

## Instances For

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])
```

## Equations

- List.takeWhile₂ R (a :: as) (b :: bs) = if R a b = true then match List.takeWhile₂ R as bs with | (as', bs') => (a :: as', b :: bs') else ([], [])
- List.takeWhile₂ R x✝ x = ([], [])

## Instances For

Tail-recursive version of `takeWhile₂`

.

## Equations

- List.takeWhile₂TR R as bs = List.takeWhile₂TR.go R as bs [] []

## Instances For

Auxiliary for `takeWhile₂TR`

:
`takeWhile₂TR.go R as bs acca accb = (acca.reverse ++ as', acca.reverse ++ bs')`

if `takeWhile₂ R as bs = (as', bs')`

.

## Equations

- List.takeWhile₂TR.go R (a :: as) (b :: bs) x✝ x = bif R a b then List.takeWhile₂TR.go R as bs (a :: x✝) (b :: x) else (x✝.reverse, x.reverse)
- List.takeWhile₂TR.go R x✝² x✝¹ x✝ x = (x✝.reverse, x.reverse)

## Instances For

`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.pwFilter R l = List.foldr (fun (x : α) (IH : List α) => if ∀ (y : α), y ∈ IH → R x y then x :: IH else IH) [] l

## Instances For

`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 : α → α → Prop} {a : α}, List.Chain R a []
A chain of length 1 is trivially a chain.

- cons: ∀ {α : Type u_1} {R : α → α → Prop} {a b : α} {l : List α}, R a b → List.Chain R b l → List.Chain R a (b :: l)
If

`a`

relates to`b`

and`b::l`

is a chain, then`a :: b :: l`

is also a chain.

## Instances For

`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
```

## Equations

- List.Chain' R [] = True
- List.Chain' R (a :: l) = List.Chain R a l

## Instances For

`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]
```

## Equations

- List.mapDiagM f l = List.mapDiagM.go f l #[]

## Instances For

Auxiliary for `mapDiagM`

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

## Equations

- List.mapDiagM.go f [] x = pure x.toList
- List.mapDiagM.go f (a :: as) x = do let b ← f a a let acc ← List.foldlM (fun (x1 : Array β) (x2 : α) => x1.push <$> f a x2) (x.push b) as List.mapDiagM.go f as acc

## Instances For

`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
```

## Equations

- List.forDiagM f [] = pure PUnit.unit
- List.forDiagM f (a :: l) = do f a a l.forM (f a) List.forDiagM f l

## Instances For

`getRest l l₁`

returns `some l₂`

if `l = l₁ ++ l₂`

.
If `l₁`

is not a prefix of `l`

, returns `none`

## Equations

## Instances For

`List.dropSlice n m xs`

removes a slice of length `m`

at index `n`

in list `xs`

.

## Equations

- List.dropSlice x✝ x [] = []
- List.dropSlice 0 x✝ x = List.drop x✝ x
- List.dropSlice n.succ x (x_3 :: xs) = x_3 :: List.dropSlice n x xs

## Instances For

Optimized version of `dropSlice`

.

## Equations

- List.dropSliceTR n 0 l = l
- List.dropSliceTR n m_2.succ l = List.dropSliceTR.go l m_2 l n #[]

## Instances For

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`

.

## Equations

- List.dropSliceTR.go l m [] x✝ x = l
- List.dropSliceTR.go l m (head :: xs) 0 x = x.toListAppend (List.drop m xs)
- List.dropSliceTR.go l m (x_3 :: xs) n.succ x = List.dropSliceTR.go l m xs n (x.push x_3)

## Instances For

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'])
```

## Equations

- List.zipWithLeft' f [] x = ([], x)
- List.zipWithLeft' f (a :: as) [] = (List.map (fun (a : α) => f a none) (a :: as), [])
- List.zipWithLeft' f (a :: as) (b :: bs) = (f a (some b) :: (List.zipWithLeft' f as bs).fst, (List.zipWithLeft' f as bs).snd)

## Instances For

Tail-recursive version of `zipWithLeft'`

.

## Equations

- List.zipWithLeft'TR f as bs = List.zipWithLeft'TR.go f as bs #[]

## Instances For

Auxiliary for `zipWithLeft'TR`

: `zipWithLeft'TR.go l acc = acc.toList ++ zipWithLeft' l`

.

## Equations

- List.zipWithLeft'TR.go f [] x✝ x = (x.toList, x✝)
- List.zipWithLeft'TR.go f x✝ [] x = ((List.foldl (fun (acc : Array γ) (a : α) => acc.push (f a none)) x x✝).toList, [])
- List.zipWithLeft'TR.go f (a :: as) (b :: bs) x = List.zipWithLeft'TR.go f as bs (x.push (f a (some b)))

## Instances For

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])
```

## Equations

- List.zipWithRight' f as bs = List.zipWithLeft' (flip f) bs as

## Instances For

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
```

## Equations

- List.zipLeft' = List.zipWithLeft' Prod.mk

## Instances For

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
```

## Equations

- List.zipRight' = List.zipWithRight' Prod.mk

## Instances For

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
```

## Equations

- List.zipWithLeft f [] x = []
- List.zipWithLeft f (a :: as) [] = List.map (fun (a : α) => f a none) (a :: as)
- List.zipWithLeft f (a :: as) (b :: bs) = f a (some b) :: List.zipWithLeft f as bs

## Instances For

Tail-recursive version of `zipWithLeft`

.

## Equations

- List.zipWithLeftTR f as bs = List.zipWithLeftTR.go f as bs #[]

## Instances For

Auxiliary for `zipWithLeftTR`

: `zipWithLeftTR.go l acc = acc.toList ++ zipWithLeft l`

.

## Equations

- List.zipWithLeftTR.go f [] x✝ x = x.toList
- List.zipWithLeftTR.go f x✝ [] x = (List.foldl (fun (acc : Array γ) (a : α) => acc.push (f a none)) x x✝).toList
- List.zipWithLeftTR.go f (a :: as) (b :: bs) x = List.zipWithLeftTR.go f as bs (x.push (f a (some b)))

## Instances For

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
```

## Equations

- List.zipWithRight f as bs = List.zipWithLeft (flip f) bs as

## Instances For

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
```

## Equations

- List.zipLeft = List.zipWithLeft Prod.mk

## Instances For

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
```

## Equations

- List.zipRight = List.zipWithRight Prod.mk

## Instances For

Tail-recursive version of `fillNones`

.

## Equations

- as.fillNonesTR as' = List.fillNonesTR.go as as' #[]

## Instances For

Auxiliary for `fillNonesTR`

: `fillNonesTR.go as as' acc = acc.toList ++ fillNones as as'`

.

## Equations

- List.fillNonesTR.go [] x✝ x = x.toList
- List.fillNonesTR.go (some a :: as) x✝ x = List.fillNonesTR.go as x✝ (x.push a)
- List.fillNonesTR.go (none :: as) [] x = List.filterMapTR.go id as x
- List.fillNonesTR.go (none :: as) (a :: as') x = List.fillNonesTR.go as as' (x.push a)

## Instances For

`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'], []], [])
```

## Equations

- x.takeList [] = ([], x)
- x.takeList (n :: ns) = match List.splitAt n x with | (xs₁, xs₂) => match xs₂.takeList ns with | (xss, rest) => (xs₁ :: xss, rest)

## Instances For

Auxiliary for `takeListTR`

: `takeListTR.go as as' acc = acc.toList ++ takeList as as'`

.

## Equations

- List.takeListTR.go [] x✝ x = (x.toList, x✝)
- List.takeListTR.go (n :: ns) x✝ x = match List.splitAt n x✝ with | (xs₁, xs₂) => List.takeListTR.go ns xs₂ (x.push xs₁)

## Instances For

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`

.

## Equations

- List.toChunksAux n [] x = ([], [])
- List.toChunksAux n (x_2 :: xs) 0 = match List.toChunksAux n xs n with | (l, L) => ([], (x_2 :: l) :: L)
- List.toChunksAux n (x_2 :: xs) i.succ = match List.toChunksAux n xs i with | (l, L) => (x_2 :: l, L)

## Instances For

`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]]
```

## Equations

- List.toChunks x [] = []
- List.toChunks 0 x = [x]
- List.toChunks x (x_2 :: xs) = List.toChunks.go x xs #[x_2] #[]

## Instances For

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.

## Equations

- List.toChunks.go n [] x✝ x = (x.push x✝.toList).toList
- List.toChunks.go n (a :: t) x✝ x = if (x✝.size == n) = true then List.toChunks.go n t ((Array.mkEmpty n).push a) (x.push x✝.toList) else List.toChunks.go n t (x✝.push a) x

## Instances For

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`

.

Ternary version of `List.zipWith`

.

## Equations

- List.zipWith₃ f (x_3 :: xs) (y :: ys) (z :: zs) = f x_3 y z :: List.zipWith₃ f xs ys zs
- List.zipWith₃ f x✝¹ x✝ x = []

## Instances For

Quaternary version of `List.zipWith`

.

## Equations

- List.zipWith₄ f (x_4 :: xs) (y :: ys) (z :: zs) (u :: us) = f x_4 y z u :: List.zipWith₄ f xs ys zs us
- List.zipWith₄ f x✝² x✝¹ x✝ x = []

## Instances For

Quinary version of `List.zipWith`

.

## Equations

- List.zipWith₅ f (x_5 :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) = f x_5 y z u v :: List.zipWith₅ f xs ys zs us vs
- List.zipWith₅ f x✝³ x✝² x✝¹ x✝ x = []

## Instances For

An auxiliary function for `List.mapWithPrefixSuffix`

.

## Equations

- List.mapWithPrefixSuffixAux f x [] = []
- List.mapWithPrefixSuffixAux f x (a :: l₂) = f x a l₂ :: List.mapWithPrefixSuffixAux f (x.concat a) l₂

## Instances For

`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 []]`

.

## Equations

- List.mapWithPrefixSuffix f l = List.mapWithPrefixSuffixAux f [] l

## Instances For

`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]]`

.

## Equations

- List.mapWithComplement f = List.mapWithPrefixSuffix fun (pref : List α) (a : α) (suff : List α) => f a (pref ++ suff)

## Instances For

Map each element of a `List`

to an action, evaluate these actions in order,
and collect the results.

## Equations

- List.traverse f [] = pure []
- List.traverse f (a :: l) = List.cons <$> f a <*> List.traverse f l

## Instances For

`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.

## Instances For

`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.

## Equations

- List.«term_<+~_» = Lean.ParserDescr.trailingNode `List.«term_<+~_» 50 50 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " <+~ ") (Lean.ParserDescr.cat `term 51))

## Instances For

`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`

.

## Equations

- l₁.isSubperm l₂ = decide (∀ (x : α), x ∈ l₁ → List.count x l₁ ≤ List.count x l₂)

## Instances For

`O(|l|)`

. Inserts `a`

in `l`

right before the first element such that `p`

is true, or at the end of
the list if `p`

always false on `l`

.

## Equations

- List.insertP p a l = List.insertP.loop p a l []

## Instances For

Inner loop for `insertP`

. Tail recursive.

## Equations

- List.insertP.loop p a [] x = (a :: x).reverseAux []
- List.insertP.loop p a (a_1 :: t) x = bif p a_1 then (a :: x).reverseAux (a_1 :: t) else List.insertP.loop p a t (a_1 :: x)