Documentation

Definitions on lists #

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

instance List.instSDiffOfDecidableEq_mathlib {α : Type u_1} [] :
SDiff (List α)
Equations
• List.instSDiffOfDecidableEq_mathlib = { sdiff := List.diff }
def List.getI {α : Type u_1} [] (l : List α) (n : ) :
α

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

Equations
• l.getI n = l.getD n default
Instances For
def List.headI {α : Type u_1} [] :
List αα

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

Equations
• (a :: tail).headI = a
Instances For
@[simp]
theorem List.headI_nil {α : Type u_1} [] :
@[simp]
theorem List.headI_cons {α : Type u_1} [] {h : α} {t : List α} :
def List.getLastI {α : Type u_1} [] :
List αα

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

Equations
• [].getLastI = default
• [a].getLastI = a
Instances For
@[inline, deprecated List.pure]
def List.ret {α : Type u} (a : α) :
List α

List with a single given element.

Equations
• = [a]
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def List.takeI {α : Type u_1} [] (n : ) (l : List α) :
List α

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

Equations
Instances For
def List.findM {α : Type u} {m : Type u → Type v} [] (tac : α) :
List αm α

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

Equations
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def List.findM?' {m : Type u → Type v} [] {α : Type u} (p : αm ) :
List αm (Option α)

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

Equations
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def List.orM {m : TypeType v} [] :
List (m Bool)m Bool

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

Equations
• List.orM =
Instances For
def List.andM {m : TypeType v} [] :
List (m Bool)m Bool

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

Equations
• List.andM =
Instances For
def List.foldlIdxM {m : Type v → Type w} [] {α : Type u_7} {β : Type v} (f : βαm β) (b : β) (as : List α) :
m β

Monadic variant of foldlIdx.

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

Monadic variant of foldrIdx.

Equations
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def List.mapIdxMAux' {m : Type v → Type w} [] {α : Type u_7} (f : α) :
List α

Auxiliary definition for mapIdxM'.

Equations
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def List.mapIdxM' {m : Type v → Type w} [] {α : Type u_7} (f : α) (as : List α) :

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

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

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

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

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

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

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

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

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

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

Equations
• One or more equations did not get rendered due to their size.
Instances For
def List.permutations {α : Type u_1} (l : List α) :
List (List α)

List of all permutations of l.

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

Equations
• l.permutations = l :: l.permutationsAux []
Instances For
def List.permutations'Aux {α : Type u_1} (t : α) :
List αList (List α)

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

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

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

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

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

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

Equations
• [].permutations' = [[]]
• (a :: tail).permutations' = tail.permutations'.bind
Instances For
def List.extractp {α : Type u_1} (p : αProp) [] :
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).

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

Notation for calculating the product of a List

Equations
• List.instSProd = { sprod := List.product }
instance List.decidableChain {α : Type u_1} {R : ααProp} [] (a : α) (l : List α) :
Equations
• One or more equations did not get rendered due to their size.
instance List.decidableChain' {α : Type u_1} {R : ααProp} [] (l : List α) :
Equations
• One or more equations did not get rendered due to their size.
def List.dedup {α : Type u_1} [] :
List αList α

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

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

Equations
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def List.destutter' {α : Type u_1} (R : ααProp) [] :
αList αList α

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

Equations
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def List.destutter {α : Type u_1} (R : ααProp) [] :
List αList α

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

Equations
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def List.chooseX {α : Type u_1} (p : αProp) [] (l : List α) :
(∃ (a : α), a l p a){ a : α // a l p a }

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

Equations
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def List.choose {α : Type u_1} (p : αProp) [] (l : List α) (hp : ∃ (a : α), a l p a) :
α

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

Equations
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def List.mapDiagM' {m : TypeType u_7} [] {α : Type u_8} (f : ααm Unit) :
List αm Unit

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

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

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

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

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

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

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

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

Equations
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def List.map₂Left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγ) :
List αList βList γ

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

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

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

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

Equations
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def List.map₂Right {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.map₂. map₂Right f as bs applies f to each pair aᵢ ∈ as and bᵢ ∈ bs. If as is shorter than bs, f is applied to none for the remaining bᵢ.

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

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

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

Equations
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def List.mapAsyncChunked {α : Type u_7} {β : Type u_8} (f : αβ) (xs : List α) (chunk_size : optParam 1024) :
List β

Asynchronous version of List.map.

Equations
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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.

Equations
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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.

Equations
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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.

Equations
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def List.replaceIf {α : Type u_1} :
List αList αList α

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

Equations
• x✝.replaceIf x [] = x✝
• [].replaceIf x✝ x = []
• x✝.replaceIf [] x = x✝
• (n :: ns).replaceIf (tf :: bs) (c :: cs) = if tf = true then c :: ns.replaceIf bs cs else n :: ns.replaceIf bs (c :: cs)
Instances For
def List.iterate {α : Type u_1} (f : αα) (a : α) (n : ) :
List α

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

Equations
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@[inline]
def List.iterateTR {α : Type u_1} (f : αα) (a : α) (n : ) :
List α

Tail-recursive version of List.iterate.

Equations
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@[specialize #[]]
def List.iterateTR.loop {α : Type u_1} (f : αα) (a : α) (n : ) (l : List α) :
List α

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

Equations
Instances For
theorem List.iterateTR_loop_eq {α : Type u_1} (f : αα) (a : α) (n : ) (l : List α) :
= l.reverse ++ List.iterate f a n
@[csimp]
def List.mapAccumr {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγγ × β) :
List αγγ × List β

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

Equations
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@[simp]
theorem List.length_mapAccumr {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αγγ × β) (x : List α) (s : γ) :
(List.mapAccumr f x s).snd.length = x.length

Length of the list obtained by mapAccumr.

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

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

Equations
Instances For
@[simp]
theorem List.length_mapAccumr₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αβγγ × δ) (x : List α) (y : List β) (c : γ) :
(List.mapAccumr₂ f x y c).snd.length = min x.length y.length

Length of a list obtained using mapAccumr₂.

@[deprecated List.mem_cons]
theorem List.mem_cons_eq {α : Type u_1} (a : α) (y : α) (l : List α) :
(a y :: l) = (a = y a l)
theorem List.eq_or_mem_of_mem_cons :
∀ {α : Type u_1} {b : α} {l : List α} {a : α}, a b :: la = b a l

Alias of the forward direction of List.mem_cons.

@[deprecated List.not_mem_nil]
theorem List.not_exists_mem_nil {α : Type u_1} (p : αProp) :
¬∃ (x : α), x [] p x
@[deprecated List.not_exists_mem_nil]
theorem List.not_bex_nil {α : Type u_1} (p : αProp) :
¬∃ (x : α), x [] p x

Alias of List.not_exists_mem_nil.

@[deprecated List.exists_mem_cons]
theorem List.bex_cons {α : Type u_1} {p : αProp} {a : α} {l : List α} :
(∃ (x : α), ∃ (x_1 : x a :: l), p x) p a ∃ (x : α), ∃ (x_1 : x l), p x

Alias of List.exists_mem_cons.

@[deprecated List.Sublist.length_le]
theorem List.length_le_of_sublist :
∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Sublist l₂l₁.length l₂.length

Alias of List.Sublist.length_le.