# Matrix and vector notation #

This file defines notation for vectors and matrices. Given a b c d : α, the notation allows us to write ![a, b, c, d] : Fin 4 → α. Nesting vectors gives coefficients of a matrix, so ![![a, b], ![c, d]] : Fin 2 → Fin 2 → α. In later files we introduce !![a, b; c, d] as notation for Matrix.of ![![a, b], ![c, d]].

## Main definitions #

• vecEmpty is the empty vector (or 0 by n matrix) ![]
• vecCons prepends an entry to a vector, so ![a, b] is vecCons a (vecCons b vecEmpty)

## Implementation notes #

The simp lemmas require that one of the arguments is of the form vecCons _ _. This ensures simp works with entries only when (some) entries are already given. In other words, this notation will only appear in the output of simp if it already appears in the input.

## Notations #

The main new notation is ![a, b], which gets expanded to vecCons a (vecCons b vecEmpty).

## Examples #

Examples of usage can be found in the test/matrix.lean file.

def Matrix.vecEmpty {α : Type u} :
Fin 0α

![] is the vector with no entries.

Equations
• ![] = Fin.elim0
Instances For
def Matrix.vecCons {α : Type u} {n : } (h : α) (t : Fin nα) :
Fin n.succα

vecCons h t prepends an entry h to a vector t.

The inverse functions are vecHead and vecTail. The notation ![a, b, ...] expands to vecCons a (vecCons b ...).

Equations
Instances For

![...] notation is used to construct a vector Fin n → α using Matrix.vecEmpty and Matrix.vecCons.

For instance, ![a, b, c] : Fin 3 is syntax for vecCons a (vecCons b (vecCons c vecEmpty)).

Note that this should not be used as syntax for Matrix as it generates a term with the wrong type. The !![a, b; c, d] syntax (provided by Matrix.matrixNotation) should be used instead.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Unexpander for the ![x, y, ...] notation.

Equations
• One or more equations did not get rendered due to their size.
Instances For

Unexpander for the ![] notation.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def Matrix.vecHead {α : Type u} {n : } (v : Fin n.succα) :
α

vecHead v gives the first entry of the vector v

Equations
• = v 0
Instances For
def Matrix.vecTail {α : Type u} {n : } (v : Fin n.succα) :
Fin nα

vecTail v gives a vector consisting of all entries of v except the first

Equations
Instances For
instance PiFin.hasRepr {α : Type u} {n : } [Repr α] :
Repr (Fin nα)

Use ![...] notation for displaying a vector Fin n → α, for example:

#eval ![1, 2] + ![3, 4] -- ![4, 6]

Equations
• One or more equations did not get rendered due to their size.
theorem Matrix.empty_eq {α : Type u} (v : Fin 0α) :
v = ![]
@[simp]
theorem Matrix.head_fin_const {α : Type u} {n : } (a : α) :
(Matrix.vecHead fun (x : Fin (n + 1)) => a) = a
@[simp]
theorem Matrix.cons_val_zero {α : Type u} {m : } (x : α) (u : Fin mα) :
= x
theorem Matrix.cons_val_zero' {α : Type u} {m : } (h : 0 < m.succ) (x : α) (u : Fin mα) :
Matrix.vecCons x u 0, h = x
@[simp]
theorem Matrix.cons_val_succ {α : Type u} {m : } (x : α) (u : Fin mα) (i : Fin m) :
Matrix.vecCons x u i.succ = u i
@[simp]
theorem Matrix.cons_val_succ' {α : Type u} {m : } {i : } (h : i.succ < m.succ) (x : α) (u : Fin mα) :
Matrix.vecCons x u i.succ, h = u i,
@[simp]
theorem Matrix.head_cons {α : Type u} {m : } (x : α) (u : Fin mα) :
= x
@[simp]
theorem Matrix.tail_cons {α : Type u} {m : } (x : α) (u : Fin mα) :
= u
@[simp]
theorem Matrix.empty_val' {α : Type u} {n' : Type u_4} (j : n') :
(fun (i : Fin 0) => ![] i j) = ![]
@[simp]
theorem Matrix.cons_head_tail {α : Type u} {m : } (u : Fin m.succα) :
= u
@[simp]
theorem Matrix.range_cons {α : Type u} {n : } (x : α) (u : Fin nα) :
Set.range () = {x}
@[simp]
theorem Matrix.range_empty {α : Type u} (u : Fin 0α) :
theorem Matrix.range_cons_empty {α : Type u} (x : α) (u : Fin 0α) :
Set.range () = {x}
theorem Matrix.range_cons_cons_empty {α : Type u} (x : α) (y : α) (u : Fin 0α) :
Set.range (Matrix.vecCons x ()) = {x, y}
@[simp]
theorem Matrix.vecCons_const {α : Type u} {n : } (a : α) :
(Matrix.vecCons a fun (x : Fin n) => a) = fun (x : Fin n.succ) => a
theorem Matrix.vec_single_eq_const {α : Type u} (a : α) :
![a] = fun (x : Fin ()) => a
@[simp]
theorem Matrix.cons_val_one {α : Type u} {m : } (x : α) (u : Fin m.succα) :
=

![a, b, ...] 1 is equal to b.

The simplifier needs a special lemma for length ≥ 2, in addition to cons_val_succ, because 1 : Fin 1 = 0 : Fin 1.

@[simp]
theorem Matrix.cons_val_two {α : Type u} {m : } (x : α) (u : Fin m.succ.succα) :
=
@[simp]
theorem Matrix.cons_val_three {α : Type u} {m : } (x : α) (u : Fin m.succ.succ.succα) :
@[simp]
theorem Matrix.cons_val_four {α : Type u} {m : } (x : α) (u : Fin m.succ.succ.succ.succα) :
@[simp]
theorem Matrix.cons_val_fin_one {α : Type u} (x : α) (u : Fin 0α) (i : Fin 1) :
= x
theorem Matrix.cons_fin_one {α : Type u} (x : α) (u : Fin 0α) :
= fun (x_1 : Fin ()) => x
instance PiFin.toExpr {α : Type u} [] (n : ) :
Lean.ToExpr (Fin nα)

### bit0 and bit1 indices #

The following definitions and simp lemmas are used to allow numeral-indexed element of a vector given with matrix notation to be extracted by simp in Lean 3 (even when the numeral is larger than the number of elements in the vector, which is taken modulo that number of elements by virtue of the semantics of bit0 and bit1 and of addition on Fin n).

def Matrix.vecAppend {m : } {n : } {α : Type u_4} {o : } (ho : o = m + n) (u : Fin mα) (v : Fin nα) :
Fin oα

vecAppend ho u v appends two vectors of lengths m and n to produce one of length o = m + n. This is a variant of Fin.append with an additional ho argument, which provides control of definitional equality for the vector length.

This turns out to be helpful when providing simp lemmas to reduce ![a, b, c] n, and also means that vecAppend ho u v 0 is valid. Fin.append u v 0 is not valid in this case because there is no Zero (Fin (m + n)) instance.

Equations
Instances For
theorem Matrix.vecAppend_eq_ite {m : } {n : } {α : Type u_4} {o : } (ho : o = m + n) (u : Fin mα) (v : Fin nα) :
Matrix.vecAppend ho u v = fun (i : Fin o) => if h : i < m then u i, h else v i - m,
@[simp]
theorem Matrix.vecAppend_apply_zero {m : } {n : } {α : Type u_4} {o : } (ho : o + 1 = m + 1 + n) (u : Fin (m + 1)α) (v : Fin nα) :
Matrix.vecAppend ho u v 0 = u 0
@[simp]
theorem Matrix.empty_vecAppend {α : Type u} {n : } (v : Fin nα) :
Matrix.vecAppend ![] v = v
@[simp]
theorem Matrix.cons_vecAppend {α : Type u} {m : } {n : } {o : } (ho : o + 1 = m + 1 + n) (x : α) (u : Fin mα) (v : Fin nα) :
def Matrix.vecAlt0 {α : Type u} {m : } {n : } (hm : m = n + n) (v : Fin mα) (k : Fin n) :
α

vecAlt0 v gives a vector with half the length of v, with only alternate elements (even-numbered).

Equations
Instances For
def Matrix.vecAlt1 {α : Type u} {m : } {n : } (hm : m = n + n) (v : Fin mα) (k : Fin n) :
α

vecAlt1 v gives a vector with half the length of v, with only alternate elements (odd-numbered).

Equations
Instances For
theorem Matrix.vecAlt0_vecAppend {α : Type u} {n : } (v : Fin nα) :
Matrix.vecAlt0 () = v bit0
theorem Matrix.vecAlt1_vecAppend {α : Type u} {n : } (v : Fin (n + 1)α) :
Matrix.vecAlt1 () = v bit1
@[simp]
theorem Matrix.vecHead_vecAlt0 {α : Type u} {m : } {n : } (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2)α) :
@[simp]
theorem Matrix.vecHead_vecAlt1 {α : Type u} {m : } {n : } (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2)α) :
@[simp]
theorem Matrix.cons_vec_bit0_eq_alt0 {α : Type u} {n : } (x : α) (u : Fin nα) (i : Fin (n + 1)) :
@[simp]
theorem Matrix.cons_vec_bit1_eq_alt1 {α : Type u} {n : } (x : α) (u : Fin nα) (i : Fin (n + 1)) :
@[simp]
theorem Matrix.cons_vecAlt0 {α : Type u} {m : } {n : } (h : m + 1 + 1 = n + 1 + (n + 1)) (x : α) (y : α) (u : Fin mα) :
@[simp]
theorem Matrix.empty_vecAlt0 (α : Type u_4) {h : 0 = 0 + 0} :
Matrix.vecAlt0 h ![] = ![]
@[simp]
theorem Matrix.cons_vecAlt1 {α : Type u} {m : } {n : } (h : m + 1 + 1 = n + 1 + (n + 1)) (x : α) (y : α) (u : Fin mα) :
@[simp]
theorem Matrix.empty_vecAlt1 (α : Type u_4) {h : 0 = 0 + 0} :
Matrix.vecAlt1 h ![] = ![]
@[simp]
theorem Matrix.smul_empty {α : Type u} {M : Type u_4} [SMul M α] (x : M) (v : Fin 0α) :
x v = ![]
@[simp]
theorem Matrix.smul_cons {α : Type u} {n : } {M : Type u_4} [SMul M α] (x : M) (y : α) (v : Fin nα) :
x = Matrix.vecCons (x y) (x v)
@[simp]
theorem Matrix.empty_add_empty {α : Type u} [Add α] (v : Fin 0α) (w : Fin 0α) :
v + w = ![]
@[simp]
theorem Matrix.cons_add {α : Type u} {n : } [Add α] (x : α) (v : Fin nα) (w : Fin n.succα) :
+ w = Matrix.vecCons () ()
@[simp]
theorem Matrix.add_cons {α : Type u} {n : } [Add α] (v : Fin n.succα) (y : α) (w : Fin nα) :
theorem Matrix.cons_add_cons {α : Type u} {n : } [Add α] (x : α) (v : Fin nα) (y : α) (w : Fin nα) :
+ = Matrix.vecCons (x + y) (v + w)
@[simp]
theorem Matrix.head_add {α : Type u} {n : } [Add α] (a : Fin n.succα) (b : Fin n.succα) :
@[simp]
theorem Matrix.tail_add {α : Type u} {n : } [Add α] (a : Fin n.succα) (b : Fin n.succα) :
@[simp]
theorem Matrix.empty_sub_empty {α : Type u} [Sub α] (v : Fin 0α) (w : Fin 0α) :
v - w = ![]
@[simp]
theorem Matrix.cons_sub {α : Type u} {n : } [Sub α] (x : α) (v : Fin nα) (w : Fin n.succα) :
- w = Matrix.vecCons () ()
@[simp]
theorem Matrix.sub_cons {α : Type u} {n : } [Sub α] (v : Fin n.succα) (y : α) (w : Fin nα) :
theorem Matrix.cons_sub_cons {α : Type u} {n : } [Sub α] (x : α) (v : Fin nα) (y : α) (w : Fin nα) :
- = Matrix.vecCons (x - y) (v - w)
@[simp]
theorem Matrix.head_sub {α : Type u} {n : } [Sub α] (a : Fin n.succα) (b : Fin n.succα) :
@[simp]
theorem Matrix.tail_sub {α : Type u} {n : } [Sub α] (a : Fin n.succα) (b : Fin n.succα) :
@[simp]
theorem Matrix.zero_empty {α : Type u} [Zero α] :
0 = ![]
@[simp]
theorem Matrix.cons_zero_zero {α : Type u} {n : } [Zero α] :
= 0
@[simp]
theorem Matrix.head_zero {α : Type u} {n : } [Zero α] :
@[simp]
theorem Matrix.tail_zero {α : Type u} {n : } [Zero α] :
@[simp]
theorem Matrix.cons_eq_zero_iff {α : Type u} {n : } [Zero α] {v : Fin nα} {x : α} :
= 0 x = 0 v = 0
theorem Matrix.cons_nonzero_iff {α : Type u} {n : } [Zero α] {v : Fin nα} {x : α} :
0 x 0 v 0
@[simp]
theorem Matrix.neg_empty {α : Type u} [Neg α] (v : Fin 0α) :
-v = ![]
@[simp]
theorem Matrix.neg_cons {α : Type u} {n : } [Neg α] (x : α) (v : Fin nα) :
@[simp]
theorem Matrix.head_neg {α : Type u} {n : } [Neg α] (a : Fin n.succα) :
@[simp]
theorem Matrix.tail_neg {α : Type u} {n : } [Neg α] (a : Fin n.succα) :
theorem Matrix.const_fin1_eq {α : Type u} (x : α) :
(fun (x_1 : Fin 1) => x) = ![x]