Documentation

Mathlib.Data.Vector.Basic

Additional theorems and definitions about the Vector type #

This file introduces the infix notation ::ᵥ for Vector.cons.

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.

Equations
instance Vector.instInhabitedVector {n : } {α : Type u_1} [Inhabited α] :
Equations
  • Vector.instInhabitedVector = { default := Vector.ofFn default }
theorem Vector.toList_injective {n : } {α : Type u_1} :
Function.Injective Vector.toList
theorem Vector.ext {n : } {α : Type u_1} {v : Vector α n} {w : Vector α n} :
(∀ (m : Fin n), Vector.get v m = Vector.get w m)v = w

Two v w : Vector α n are equal iff they are equal at every single index.

instance Vector.zero_subsingleton {α : Type u_1} :

The empty Vector is a Subsingleton.

Equations
  • =
@[simp]
theorem Vector.cons_val {n : } {α : Type u_1} (a : α) (v : Vector α n) :
(a ::ᵥ v) = a :: v
theorem Vector.eq_cons_iff {n : } {α : Type u_1} (a : α) (v : Vector α (Nat.succ n)) (v' : Vector α n) :
theorem Vector.ne_cons_iff {n : } {α : Type u_1} (a : α) (v : Vector α (Nat.succ n)) (v' : Vector α n) :
theorem Vector.exists_eq_cons {n : } {α : Type u_1} (v : Vector α (Nat.succ n)) :
∃ (a : α) (as : Vector α n), v = a ::ᵥ as
@[simp]
theorem Vector.toList_ofFn {α : Type u_1} {n : } (f : Fin nα) :
@[simp]
theorem Vector.mk_toList {n : } {α : Type u_1} (v : Vector α n) (h : List.length (Vector.toList v) = n) :
{ val := Vector.toList v, property := h } = v
@[simp]
theorem Vector.length_val {n : } {α : Type u_1} (v : Vector α n) :
List.length v = n
@[simp]
theorem Vector.toList_map {n : } {α : Type u_1} {β : Type u_2} (v : Vector α n) (f : αβ) :
@[simp]
theorem Vector.head_map {n : } {α : Type u_1} {β : Type u_2} (v : Vector α (n + 1)) (f : αβ) :
@[simp]
theorem Vector.tail_map {n : } {α : Type u_1} {β : Type u_2} (v : Vector α (n + 1)) (f : αβ) :
theorem Vector.get_eq_get {n : } {α : Type u_1} (v : Vector α n) (i : Fin n) :
@[deprecated Vector.get_eq_get]
theorem Vector.nth_eq_nthLe {n : } {α : Type u_1} (v : Vector α n) (i : Fin n) :
@[simp]
theorem Vector.get_replicate {n : } {α : Type u_1} (a : α) (i : Fin n) :
@[simp]
theorem Vector.get_map {n : } {α : Type u_1} {β : Type u_2} (v : Vector α n) (f : αβ) (i : Fin n) :
@[simp]
theorem Vector.map₂_nil {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβγ) :
Vector.map₂ f Vector.nil Vector.nil = Vector.nil
@[simp]
theorem Vector.map₂_cons {n : } {α : Type u_1} {β : Type u_2} {γ : Type u_3} (hd₁ : α) (tl₁ : Vector α n) (hd₂ : β) (tl₂ : Vector β n) (f : αβγ) :
Vector.map₂ f (hd₁ ::ᵥ tl₁) (hd₂ ::ᵥ tl₂) = f hd₁ hd₂ ::ᵥ Vector.map₂ f tl₁ tl₂
@[simp]
theorem Vector.get_ofFn {α : Type u_1} {n : } (f : Fin nα) (i : Fin n) :
@[simp]
theorem Vector.ofFn_get {n : } {α : Type u_1} (v : Vector α n) :
def Equiv.vectorEquivFin (α : Type u_2) (n : ) :
Vector α n (Fin nα)

The natural equivalence between length-n vectors and functions from Fin n.

Equations
  • Equiv.vectorEquivFin α n = { toFun := Vector.get, invFun := Vector.ofFn, left_inv := , right_inv := }
theorem Vector.get_tail {n : } {α : Type u_1} (x : Vector α n) (i : Fin (n - 1)) :
Vector.get (Vector.tail x) i = Vector.get x { val := i + 1, isLt := }
@[simp]
theorem Vector.get_tail_succ {n : } {α : Type u_1} (v : Vector α (Nat.succ n)) (i : Fin n) :
@[simp]
theorem Vector.tail_val {n : } {α : Type u_1} (v : Vector α (Nat.succ n)) :
(Vector.tail v) = List.tail v
@[simp]
theorem Vector.tail_nil {α : Type u_1} :
Vector.tail Vector.nil = Vector.nil

The tail of a nil vector is nil.

@[simp]
theorem Vector.singleton_tail {α : Type u_1} (v : Vector α 1) :
Vector.tail v = Vector.nil

The tail of a vector made up of one element is nil.

@[simp]
theorem Vector.tail_ofFn {α : Type u_1} {n : } (f : Fin (Nat.succ n)α) :
Vector.tail (Vector.ofFn f) = Vector.ofFn fun (i : Fin n) => f (Fin.succ i)
@[simp]
theorem Vector.toList_empty {α : Type u_1} (v : Vector α 0) :
@[simp]
theorem Vector.toList_singleton {α : Type u_1} (v : Vector α 1) :

The list that makes up a Vector made up of a single element, retrieved via toList, is equal to the list of that single element.

@[simp]
theorem Vector.empty_toList_eq_ff {n : } {α : Type u_1} (v : Vector α (n + 1)) :
theorem Vector.not_empty_toList {n : } {α : Type u_1} (v : Vector α (n + 1)) :
@[simp]
theorem Vector.map_id {α : Type u_1} {n : } (v : Vector α n) :
Vector.map id v = v

Mapping under id does not change a vector.

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

Reverse a vector.

Equations

The List of a vector after a reverse, retrieved by toList is equal to the List.reverse after retrieving a vector's toList.

@[simp]
theorem Vector.reverse_reverse {n : } {α : Type u_1} {v : Vector α n} :
@[simp]
theorem Vector.get_zero {n : } {α : Type u_1} (v : Vector α (Nat.succ n)) :
@[simp]
theorem Vector.head_ofFn {α : Type u_1} {n : } (f : Fin (Nat.succ n)α) :
theorem Vector.get_cons_zero {n : } {α : Type u_1} (a : α) (v : Vector α n) :
Vector.get (a ::ᵥ v) 0 = a
@[simp]
theorem Vector.get_cons_nil {α : Type u_1} {ix : Fin 1} (x : α) :
Vector.get (x ::ᵥ Vector.nil) ix = x

Accessing the nth element of a vector made up of one element x : α is x itself.

@[simp]
theorem Vector.get_cons_succ {n : } {α : Type u_1} (a : α) (v : Vector α n) (i : Fin n) :
def Vector.last {n : } {α : Type u_1} (v : Vector α (n + 1)) :
α

The last element of a Vector, given that the vector is at least one element.

Equations
theorem Vector.last_def {n : } {α : Type u_1} {v : Vector α (n + 1)} :

The last element of a Vector, given that the vector is at least one element.

theorem Vector.reverse_get_zero {n : } {α : Type u_1} {v : Vector α (n + 1)} :

The last element of a vector is the head of the reverse vector.

def Vector.scanl {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α n) :
Vector β (n + 1)

Construct a Vector β (n + 1) from a Vector α n by scanning f : β → α → β from the "left", that is, from 0 to Fin.last n, using b : β as the starting value.

Equations
@[simp]
theorem Vector.scanl_nil {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) :
Vector.scanl f b Vector.nil = b ::ᵥ Vector.nil

Providing an empty vector to scanl gives the starting value b : β.

@[simp]
theorem Vector.scanl_cons {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α n) (x : α) :
Vector.scanl f b (x ::ᵥ v) = b ::ᵥ Vector.scanl f (f b x) v

The recursive step of scanl splits a vector x ::ᵥ v : Vector α (n + 1) into the provided starting value b : β and the recursed scanl f b x : β as the starting value.

This lemma is the cons version of scanl_get.

@[simp]
theorem Vector.scanl_val {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) {v : Vector α n} :
(Vector.scanl f b v) = List.scanl f b v

The underlying List of a Vector after a scanl is the List.scanl of the underlying List of the original Vector.

@[simp]
theorem Vector.toList_scanl {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α n) :

The toList of a Vector after a scanl is the List.scanl of the toList of the original Vector.

@[simp]
theorem Vector.scanl_singleton {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α 1) :
Vector.scanl f b v = b ::ᵥ f b (Vector.head v) ::ᵥ Vector.nil

The recursive step of scanl splits a vector made up of a single element x ::ᵥ nil : Vector α 1 into a Vector of the provided starting value b : β and the mapped f b x : β as the last value.

@[simp]
theorem Vector.scanl_head {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α n) :

The first element of scanl of a vector v : Vector α n, retrieved via head, is the starting value b : β.

@[simp]
theorem Vector.scanl_get {n : } {α : Type u_1} {β : Type u_2} (f : βαβ) (b : β) (v : Vector α n) (i : Fin n) :

For an index i : Fin n, the nth element of scanl of a vector v : Vector α n at i.succ, is equal to the application function f : β → α → β of the castSucc i element of scanl f b v and get v i.

This lemma is the get version of scanl_cons.

def Vector.mOfFn {m : Type u → Type u_2} [Monad m] {α : Type u} {n : } :
(Fin nm α)m (Vector α n)

Monadic analog of Vector.ofFn. Given a monadic function on Fin n, return a Vector α n inside the monad.

Equations
theorem Vector.mOfFn_pure {m : Type u_2 → Type u_3} [Monad m] [LawfulMonad m] {α : Type u_2} {n : } (f : Fin nα) :
(Vector.mOfFn fun (i : Fin n) => pure (f i)) = pure (Vector.ofFn f)
def Vector.mmap {m : Type u → Type u_2} [Monad m] {α : Type u_3} {β : Type u} (f : αm β) {n : } :
Vector α nm (Vector β n)

Apply a monadic function to each component of a vector, returning a vector inside the monad.

Equations
@[simp]
theorem Vector.mmap_nil {m : Type u_2 → Type u_3} [Monad m] {α : Type u_4} {β : Type u_2} (f : αm β) :
Vector.mmap f Vector.nil = pure Vector.nil
@[simp]
theorem Vector.mmap_cons {m : Type u_2 → Type u_3} [Monad m] {α : Type u_4} {β : Type u_2} (f : αm β) (a : α) {n : } (v : Vector α n) :
Vector.mmap f (a ::ᵥ v) = do let h' ← f a let t' ← Vector.mmap f v pure (h' ::ᵥ t')
def Vector.inductionOn {α : Type u_1} {C : {n : } → Vector α nSort u_2} {n : } (v : Vector α n) (h_nil : C Vector.nil) (h_cons : {n : } → {x : α} → {w : Vector α n} → C wC (x ::ᵥ w)) :
C v

Define C v by induction on v : Vector α n.

This function has two arguments: h_nil handles the base case on C nil, and h_cons defines the inductive step using ∀ x : α, C w → C (x ::ᵥ w).

This can be used as induction v using Vector.inductionOn.

Equations
  • One or more equations did not get rendered due to their size.
def Vector.inductionOn₂ {n : } {α : Type u_1} {β : Type u_2} {C : {n : } → Vector α nVector β nSort u_4} (v : Vector α n) (w : Vector β n) (nil : C Vector.nil Vector.nil) (cons : {n : } → {a : α} → {b : β} → {x : Vector α n} → {y : Vector β n} → C x yC (a ::ᵥ x) (b ::ᵥ y)) :
C v w

Define C v w by induction on a pair of vectors v : Vector α n and w : Vector β n.

Equations
  • One or more equations did not get rendered due to their size.
def Vector.inductionOn₃ {n : } {α : Type u_1} {β : Type u_2} {γ : Type u_3} {C : {n : } → Vector α nVector β nVector γ nSort u_4} (u : Vector α n) (v : Vector β n) (w : Vector γ n) (nil : C Vector.nil Vector.nil Vector.nil) (cons : {n : } → {a : α} → {b : β} → {c : γ} → {x : Vector α n} → {y : Vector β n} → {z : Vector γ n} → C x y zC (a ::ᵥ x) (b ::ᵥ y) (c ::ᵥ z)) :
C u v w

Define C u v w by induction on a triplet of vectors u : Vector α n, v : Vector β n, and w : Vector γ b.

Equations
  • One or more equations did not get rendered due to their size.
def Vector.casesOn {α : Type u_1} {m : } {motive : {n : } → Vector α nSort u_4} (v : Vector α m) (nil : motive Vector.nil) (cons : {n : } → (hd : α) → (tl : Vector α n) → motive (hd ::ᵥ tl)) :
motive v

Define motive v by case-analysis on v : Vector α n

Equations
def Vector.casesOn₂ {α : Type u_1} {β : Type u_2} {m : } {motive : {n : } → Vector α nVector β nSort u_4} (v₁ : Vector α m) (v₂ : Vector β m) (nil : motive Vector.nil Vector.nil) (cons : {n : } → (x : α) → (y : β) → (xs : Vector α n) → (ys : Vector β n) → motive (x ::ᵥ xs) (y ::ᵥ ys)) :
motive v₁ v₂

Define motive v₁ v₂ by case-analysis on v₁ : Vector α n and v₂ : Vector β n

Equations
def Vector.casesOn₃ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {motive : {n : } → Vector α nVector β nVector γ nSort u_4} (v₁ : Vector α m) (v₂ : Vector β m) (v₃ : Vector γ m) (nil : motive Vector.nil Vector.nil Vector.nil) (cons : {n : } → (x : α) → (y : β) → (z : γ) → (xs : Vector α n) → (ys : Vector β n) → (zs : Vector γ n) → motive (x ::ᵥ xs) (y ::ᵥ ys) (z ::ᵥ zs)) :
motive v₁ v₂ v₃

Define motive v₁ v₂ v₃ by case-analysis on v₁ : Vector α n, v₂ : Vector β n, and v₃ : Vector γ n

Equations
  • One or more equations did not get rendered due to their size.
def Vector.toArray {n : } {α : Type u_1} :
Vector α nArray α

Cast a vector to an array.

Equations
def Vector.insertNth {n : } {α : Type u_1} (a : α) (i : Fin (n + 1)) (v : Vector α n) :
Vector α (n + 1)

v.insertNth a i inserts a into the vector v at position i (and shifting later components to the right).

Equations
theorem Vector.insertNth_val {n : } {α : Type u_1} {a : α} {i : Fin (n + 1)} {v : Vector α n} :
(Vector.insertNth a i v) = List.insertNth (i) a v
@[simp]
theorem Vector.removeNth_val {n : } {α : Type u_1} {i : Fin n} {v : Vector α n} :
(Vector.removeNth i v) = List.removeNth v i
theorem Vector.removeNth_insertNth {n : } {α : Type u_1} {a : α} {v : Vector α n} {i : Fin (n + 1)} :
theorem Vector.removeNth_insertNth' {n : } {α : Type u_1} {a : α} {v : Vector α (n + 1)} {i : Fin (n + 1)} {j : Fin (n + 2)} :
theorem Vector.insertNth_comm {n : } {α : Type u_1} (a : α) (b : α) (i : Fin (n + 1)) (j : Fin (n + 1)) (h : i j) (v : Vector α n) :
def Vector.set {n : } {α : Type u_1} (v : Vector α n) (i : Fin n) (a : α) :
Vector α n

set v n a replaces the nth element of v with a

Equations
@[simp]
theorem Vector.toList_set {n : } {α : Type u_1} (v : Vector α n) (i : Fin n) (a : α) :
@[simp]
theorem Vector.get_set_same {n : } {α : Type u_1} (v : Vector α n) (i : Fin n) (a : α) :
Vector.get (Vector.set v i a) i = a
theorem Vector.get_set_of_ne {n : } {α : Type u_1} {v : Vector α n} {i : Fin n} {j : Fin n} (h : i j) (a : α) :
theorem Vector.get_set_eq_if {n : } {α : Type u_1} {v : Vector α n} {i : Fin n} {j : Fin n} (a : α) :
Vector.get (Vector.set v i a) j = if i = j then a else Vector.get v j
theorem Vector.sum_set {n : } {α : Type u_1} [AddMonoid α] (v : Vector α n) (i : Fin n) (a : α) :
theorem Vector.prod_set {n : } {α : Type u_1} [Monoid α] (v : Vector α n) (i : Fin n) (a : α) :
theorem Vector.sum_set' {n : } {α : Type u_1} [AddCommGroup α] (v : Vector α n) (i : Fin n) (a : α) :
theorem Vector.prod_set' {n : } {α : Type u_1} [CommGroup α] (v : Vector α n) (i : Fin n) (a : α) :
def Vector.traverse {n : } {F : Type u → Type u} [Applicative F] {α : Type u} {β : Type u} (f : αF β) :
Vector α nF (Vector β n)

Apply an applicative function to each component of a vector.

Equations
@[simp]
theorem Vector.traverse_def {n : } {F : Type u → Type u} [Applicative F] {α : Type u} {β : Type u} (f : αF β) (x : α) (xs : Vector α n) :
Vector.traverse f (x ::ᵥ xs) = Seq.seq (Vector.cons <$> f x) fun (x : Unit) => Vector.traverse f xs
theorem Vector.id_traverse {n : } {α : Type u} (x : Vector α n) :
theorem Vector.comp_traverse {n : } {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [LawfulApplicative G] {α : Type u} {β : Type u} {γ : Type u} (f : βF γ) (g : αG β) (x : Vector α n) :
theorem Vector.traverse_eq_map_id {n : } {α : Type u_1} {β : Type u_1} (f : αβ) (x : Vector α n) :
theorem Vector.naturality {n : } {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] (η : ApplicativeTransformation F G) {α : Type u} {β : Type u} (f : αF β) (x : Vector α n) :
(fun {α : Type u} => η.app α) (Vector.traverse f x) = Vector.traverse ((fun {α : Type u} => η.app α) f) x
Equations
@[simp]
theorem Vector.replicate_succ {α : Type u_1} {n : } (val : α) :
@[simp]
theorem Vector.get_append_cons_zero {α : Type u_1} {m : } {n : } (xs : Vector α n) (ys : Vector α m) {x : α} :
Vector.get (Vector.append (x ::ᵥ xs) ys) { val := 0, isLt := } = x
@[simp]
theorem Vector.get_append_cons_succ {α : Type u_1} {m : } {n : } (xs : Vector α n) (ys : Vector α m) {x : α} {i : Fin (n + m)} {h : i + 1 < Nat.succ n + m} :
Vector.get (Vector.append (x ::ᵥ xs) ys) { val := i + 1, isLt := h } = Vector.get (Vector.append xs ys) i
@[simp]
theorem Vector.append_nil {α : Type u_1} {n : } (xs : Vector α n) :
Vector.append xs Vector.nil = xs
@[simp]
theorem Vector.get_map₂ {α : Type u_2} {β : Type u_3} {n : } {γ : Type u_1} (v₁ : Vector α n) (v₂ : Vector β n) (f : αβγ) (i : Fin n) :
Vector.get (Vector.map₂ f v₁ v₂) i = f (Vector.get v₁ i) (Vector.get v₂ i)
@[simp]
theorem Vector.mapAccumr_cons {α : Type u_2} {n : } (xs : Vector α n) :
∀ {α_1 : Type} {β : Type u_1} {f : αα_1α_1 × β} {x : α} {s : α_1}, Vector.mapAccumr f (x ::ᵥ xs) s = let r := Vector.mapAccumr f xs s; let q := f x r.1; (q.1, q.2 ::ᵥ r.2)
@[simp]
theorem Vector.mapAccumr₂_cons {α : Type} {β : Type} {n : } (xs : Vector α n) (ys : Vector β n) :
∀ {α_1 β_1 : Type} {f : αβα_1α_1 × β_1} {x : α} {y : β} {s : α_1}, Vector.mapAccumr₂ f (x ::ᵥ xs) (y ::ᵥ ys) s = let r := Vector.mapAccumr₂ f xs ys s; let q := f x y r.1; (q.1, q.2 ::ᵥ r.2)