# Documentation

## Mathlib.Data.Vector

The type Vector represents lists with fixed length.

def Vector (α : Type u) (n : ) :

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

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Instances For
instance Vector.instDecidableEq {α : Type u} {n : } [] :
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@[match_pattern]
def Vector.nil {α : Type u} :
Vector α 0

The empty vector with elements of type α

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• Vector.nil = [],
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@[match_pattern]
def Vector.cons {α : Type u} {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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• Vector.cons x✝ x = match x✝, x with | a, v, h => a :: v,
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@[reducible]
def Vector.length {α : Type u} {n : } :
Vector α n

The length of a vector.

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• x.length = n
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def Vector.head {α : Type u} {n : } :
Vector α n.succα

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

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• x.head = match x with | a :: tail, property => a
Instances For
theorem Vector.head_cons {α : Type u} {n : } (a : α) (v : Vector α n) :
().head = a

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

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

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

Equations
• x.tail = match x with | [], h => [], | head :: v, h => v,
Instances For
theorem Vector.tail_cons {α : Type u} {n : } (a : α) (v : Vector α n) :
().tail = v

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

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

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

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

The list obtained from a vector.

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• v.toList = v
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def Vector.get {α : Type u} {n : } (l : Vector α n) (i : Fin n) :
α

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

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

Appending a vector to another.

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• x✝.append x = match x✝, x with | l₁, h₁, l₂, h₂ => l₁ ++ l₂,
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def Vector.elim {α : Type u_1} {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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• = match x with | l, h => match n, h with | .(l.length), => H l
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def Vector.map {α : Type u} {β : Type v} {n : } (f : αβ) :
Vector α nVector β n

Map a vector under a function.

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• = match x with | l, h => List.map f l,
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@[simp]
theorem Vector.map_nil {α : Type u} {β : Type v} (f : αβ) :
Vector.map f Vector.nil = Vector.nil

A nil vector maps to a nil vector.

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

map is natural with respect to cons.

def Vector.map₂ {α : Type u} {β : Type v} {φ : Type w} {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 Vector.replicate {α : Type u} (n : ) (a : α) :
Vector α n

Vector obtained by repeating an element.

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• = ⟨,
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def Vector.drop {α : Type u} {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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• = match x with | l, p => ⟨,
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def Vector.take {α : Type u} {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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• = match x with | l, p => ⟨,
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def Vector.eraseIdx {α : Type u} {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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• = match x with | l, p => l.eraseIdx i,
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@[deprecated Vector.eraseIdx]
def Vector.removeNth {α : Type u} {n : } (i : Fin n) :
Vector α nVector α (n - 1)

Alias of Vector.eraseIdx.

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

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

Vector of length n from a function on Fin n.

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

Create a vector from another with a provably equal length.

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• = match x with | x, p => x,
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def Vector.mapAccumr {α : Type u} {β : Type v} {n : } {σ : Type} (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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• Vector.mapAccumr f x✝ x = match x✝, x with | x, px, c => let res := ; (res.1, res.2, )
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def Vector.mapAccumr₂ {n : } {α : Type} {β : Type} {σ : Type} {φ : Type} (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 Vector.shiftLeftFill {α : Type u} {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 Vector.shiftRightFill {α : Type u} {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 Vector.eq {α : Type u} {n : } (a1 : Vector α n) (a2 : Vector α n) :
a1.toList = a2.toLista1 = a2

Vector is determined by the underlying list.

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

A vector of length 0 is a nil vector.

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

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

@[simp]
theorem Vector.toList_nil {α : Type u} :
Vector.nil.toList = []

A nil vector maps to a nil list.

@[simp]
theorem Vector.toList_length {α : Type u} {n : } (v : Vector α n) :
v.toList.length = n

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

@[simp]
theorem Vector.toList_cons {α : Type u} {n : } (a : α) (v : Vector α n) :
().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 Vector.toList_append {α : Type u} {n : } {m : } (v : Vector α n) (w : Vector α m) :
(v.append w).toList = v.toList ++ w.toList

Appending of vectors corresponds under toList to appending of lists.

@[simp]
theorem Vector.toList_drop {α : Type u} {n : } {m : } (v : Vector α m) :
().toList = List.drop n v.toList

drop of vectors corresponds under toList to drop of lists.

@[simp]
theorem Vector.toList_take {α : Type u} {n : } {m : } (v : Vector α m) :
().toList = List.take n v.toList

take of vectors corresponds under toList to take of lists.

instance Vector.instGetElemNatLt {α : Type u} {n : } :
GetElem (Vector α n) α fun (x : Vector α n) (i : ) => i < n
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• One or more equations did not get rendered due to their size.