Documentation

Mathlib.Data.Vector.Defs

The type List.Vector represents lists with fixed length.

TODO: The API of List.Vector is quite incomplete relative to Vector, and in particular does not use x[i] (that is GetElem notation) as the preferred accessor. Any combination of reducing the use of List.Vector in Mathlib, or modernising its API, would be welcome.

def List.Vector (α : Type u) (n : ) :

List.Vector α n is the type of lists of length n with elements of type α.

Note that there is also Vector α n in the root namespace, which is the type of arrays of length n with elements of type α.

Typically, if you are doing programming or verification, you will primarily use Vector α n, and if you are doing mathematics, you may want to use List.Vector α n instead.

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Instances For
@[match_pattern]
def List.Vector.nil {α : Type u_1} :
Vector α 0

The empty vector with elements of type α

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@[match_pattern]
def List.Vector.cons {α : Type u_1} {n : } :
αVector α nVector α n.succ

If a : α and l : Vector α n, then cons a l, is the vector of length n + 1 whose first element is a and with l as the rest of the list.

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@[reducible]
def List.Vector.length {α : Type u_1} {n : } :
Vector α n

The length of a vector.

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def List.Vector.head {α : Type u_1} {n : } :
Vector α n.succα

The first element of a vector with length at least 1.

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theorem List.Vector.head_cons {α : Type u_1} {n : } (a : α) (v : Vector α n) :
(cons a v).head = a

The head of a vector obtained by prepending is the element prepended.

def List.Vector.tail {α : Type u_1} {n : } :
Vector α nVector α (n - 1)

The tail of a vector, with an empty vector having empty tail.

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theorem List.Vector.tail_cons {α : Type u_1} {n : } (a : α) (v : Vector α n) :
(cons a v).tail = v

The tail of a vector obtained by prepending is the vector prepended. to

@[simp]
theorem List.Vector.cons_head_tail {α : Type u_1} {n : } (v : Vector α n.succ) :
cons v.head v.tail = v

Prepending the head of a vector to its tail gives the vector.

def List.Vector.toList {α : Type u_1} {n : } (v : Vector α n) :
List α

The list obtained from a vector.

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def List.Vector.get {α : Type u_1} {n : } (l : Vector α n) (i : Fin n) :
α

nth element of a vector, indexed by a Fin type.

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def List.Vector.append {α : Type u_1} {n m : } :
Vector α nVector α mVector α (n + m)

Appending a vector to another.

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def List.Vector.elim {α : Type u_5} {C : {n : } → Vector α nSort u} (H : (l : List α) → C l, ) {n : } (v : Vector α n) :
C v

Elimination rule for Vector.

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def List.Vector.map {α : Type u_1} {β : Type u_2} {n : } (f : αβ) :
Vector α nVector β n

Map a vector under a function.

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@[simp]
theorem List.Vector.map_nil {α : Type u_1} {β : Type u_2} (f : αβ) :

A nil vector maps to a nil vector.

@[simp]
theorem List.Vector.map_cons {α : Type u_1} {β : Type u_2} {n : } (f : αβ) (a : α) (v : Vector α n) :
map f (cons a v) = cons (f a) (map f v)

map is natural with respect to cons.

def List.Vector.pmap {α : Type u_1} {β : Type u_2} {n : } {p : αProp} (f : (a : α) → p aβ) (v : Vector α n) :
(∀ (x : α), x v.toListp x)Vector β n

Map a vector under a partial function.

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@[simp]
theorem List.Vector.pmap_nil {α : Type u_1} {β : Type u_2} {p : αProp} (f : (a : α) → p aβ) (hp : ∀ (x : α), x nil.toListp x) :
pmap f nil hp = nil
def List.Vector.map₂ {α : Type u_1} {β : Type u_2} {φ : Type u_4} {n : } (f : αβφ) :
Vector α nVector β nVector φ n

Mapping two vectors under a curried function of two variables.

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def List.Vector.replicate {α : Type u_1} (n : ) (a : α) :
Vector α n

Vector obtained by repeating an element.

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def List.Vector.drop {α : Type u_1} {n : } (i : ) :
Vector α nVector α (n - i)

Drop i elements from a vector of length n; we can have i > n.

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def List.Vector.take {α : Type u_1} {n : } (i : ) :
Vector α nVector α (min i n)

Take i elements from a vector of length n; we can have i > n.

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def List.Vector.eraseIdx {α : Type u_1} {n : } (i : Fin n) :
Vector α nVector α (n - 1)

Remove the element at position i from a vector of length n.

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def List.Vector.ofFn {α : Type u_1} {n : } :
(Fin nα)Vector α n

Vector of length n from a function on Fin n.

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def List.Vector.congr {α : Type u_1} {n m : } (h : n = m) :
Vector α nVector α m

Create a vector from another with a provably equal length.

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def List.Vector.mapAccumr {α : Type u_1} {β : Type u_2} {σ : Type u_3} {n : } (f : ασσ × β) :
Vector α nσσ × Vector β n

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

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def List.Vector.mapAccumr₂ {α : Type u_1} {β : Type u_2} {σ : Type u_3} {φ : Type u_4} {n : } (f : αβσσ × φ) :
Vector α nVector β nσσ × Vector φ n

Runs a function over a pair of vectors returning the intermediate results and a final result.

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Shift Primitives #

def List.Vector.shiftLeftFill {α : Type u_1} {n : } (v : Vector α n) (i : ) (fill : α) :
Vector α n

shiftLeftFill v i is the vector obtained by left-shifting v i times and padding with the fill argument. If v.length < i then this will return replicate n fill.

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def List.Vector.shiftRightFill {α : Type u_1} {n : } (v : Vector α n) (i : ) (fill : α) :
Vector α n

shiftRightFill v i is the vector obtained by right-shifting v i times and padding with the fill argument. If v.length < i then this will return replicate n fill.

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Basic Theorems #

theorem List.Vector.eq {α : Type u_1} {n : } (a1 a2 : Vector α n) :
a1.toList = a2.toLista1 = a2

Vector is determined by the underlying list.

theorem List.Vector.eq_nil {α : Type u_1} (v : Vector α 0) :
v = nil

A vector of length 0 is a nil vector.

@[simp]
theorem List.Vector.toList_mk {α : Type u_1} {n : } (v : List α) (P : v.length = n) :
toList v, P = v

Vector of length from a list v with witness that v has length n maps to v under toList.

@[simp]

A nil vector maps to a nil list.

@[simp]
theorem List.Vector.toList_length {α : Type u_1} {n : } (v : Vector α n) :

The length of the list to which a vector of length n maps is n.

@[simp]
theorem List.Vector.toList_cons {α : Type u_1} {n : } (a : α) (v : Vector α n) :
(cons a v).toList = a :: v.toList

toList of cons of a vector and an element is the cons of the list obtained by toList and the element

@[simp]
theorem List.Vector.toList_append {α : Type u_1} {n m : } (v : Vector α n) (w : Vector α m) :

Appending of vectors corresponds under toList to appending of lists.

@[simp]
theorem List.Vector.toList_drop {α : Type u_1} {n m : } (v : Vector α m) :

drop of vectors corresponds under toList to drop of lists.

@[simp]
theorem List.Vector.toList_take {α : Type u_1} {n m : } (v : Vector α m) :

take of vectors corresponds under toList to take of lists.

instance List.Vector.instGetElemNatLt {α : Type u_1} {n : } :
GetElem (Vector α n) α fun (x : Vector α n) (i : ) => i < n
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theorem List.Vector.getElem_def {α : Type u_1} {n : } (v : Vector α n) (i : ) {hi : i < n} :
theorem List.Vector.toList_getElem {α : Type u_1} {n : } (v : Vector α n) (i : ) {hi : i < v.toList.length} :