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 0 → α

![] is the vector with no entries.

def Matrix.vecCons {α : Type u} {n : ℕ} (h : α) (t : Fin n → α) : Fin (Nat.succ n) → α

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 ...).

def Matrix.vecNotation : Lean.ParserDescr

Construct a vector Fin n → α using Matrix.vecEmpty and Matrix.vecCons.

def Matrix.vecConsUnexpander : Lean.PrettyPrinter.Unexpander

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

def Matrix.vecEmptyUnexpander : Lean.PrettyPrinter.Unexpander

Unexpander for the ![] notation.

def Matrix.vecHead {α : Type u} {n : ℕ} (v : Fin (Nat.succ n) → α) : α

vecHead v gives the first entry of the vector v

def Matrix.vecTail {α : Type u} {n : ℕ} (v : Fin (Nat.succ n) → α) : Fin n → α

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

instance PiFin.hasRepr {α : Type u} {n : ℕ} [Repr α] : Repr (Fin n → α)

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

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

theorem Matrix.empty_eq {α : Type u} (v : Fin 0 → α) : v = ![]

@[simp]

theorem Matrix.head_fin_const {α : Type u} {n : ℕ} (a : α) : (Matrix.vecHead fun x => a) = a

@[simp]

theorem Matrix.cons_val_zero {α : Type u} {m : ℕ} (x : α) (u : Fin m → α) : Matrix.vecCons x u 0 = x

theorem Matrix.cons_val_zero' {α : Type u} {m : ℕ} (h : 0 < Nat.succ m) (x : α) (u : Fin m → α) : Matrix.vecCons x u { val := 0, isLt := h } = x

@[simp]

theorem Matrix.cons_val_succ {α : Type u} {m : ℕ} (x : α) (u : Fin m → α) (i : Fin m) : Matrix.vecCons x u (Fin.succ i) = u i

@[simp]

theorem Matrix.cons_val_succ' {α : Type u} {m : ℕ} {i : ℕ} (h : Nat.succ i < Nat.succ m) (x : α) (u : Fin m → α) : Matrix.vecCons x u { val := Nat.succ i, isLt := h } = u { val := i, isLt := (_ : i < m) }

@[simp]

theorem Matrix.head_cons {α : Type u} {m : ℕ} (x : α) (u : Fin m → α) : Matrix.vecHead (Matrix.vecCons x u) = x

@[simp]

theorem Matrix.tail_cons {α : Type u} {m : ℕ} (x : α) (u : Fin m → α) : Matrix.vecTail (Matrix.vecCons x u) = u

@[simp]

theorem Matrix.empty_val' {α : Type u} {n' : Type u_4} (j : n') : (fun i => ![]) = ![]

@[simp]

theorem Matrix.cons_head_tail {α : Type u} {m : ℕ} (u : Fin (Nat.succ m) → α) : Matrix.vecCons (Matrix.vecHead u) (Matrix.vecTail u) = u

@[simp]

theorem Matrix.range_cons {α : Type u} {n : ℕ} (x : α) (u : Fin n → α) : Set.range (Matrix.vecCons x u) = {x} ∪ Set.range u

@[simp]

theorem Matrix.range_empty {α : Type u} (u : Fin 0 → α) : Set.range u = ∅

theorem Matrix.range_cons_empty {α : Type u} (x : α) (u : Fin 0 → α) : Set.range (Matrix.vecCons x u) = {x}

theorem Matrix.range_cons_cons_empty {α : Type u} (x : α) (y : α) (u : Fin 0 → α) : Set.range (Matrix.vecCons x (Matrix.vecCons y u)) = {x, y}

@[simp]

theorem Matrix.vecCons_const {α : Type u} {n : ℕ} (a : α) : (Matrix.vecCons a fun x => a) = fun x => a

theorem Matrix.vec_single_eq_const {α : Type u} (a : α) : ![a] = fun x => a

@[simp]

theorem Matrix.cons_val_one {α : Type u} {m : ℕ} (x : α) (u : Fin (Nat.succ m) → α) : Matrix.vecCons x u 1 = Matrix.vecHead u

![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 : ℕ} (x : α) (u : Fin (Nat.succ (Nat.succ m)) → α) : Matrix.vecCons x u 2 = Matrix.vecHead (Matrix.vecTail u)

@[simp]

theorem Matrix.cons_val_fin_one {α : Type u} (x : α) (u : Fin 0 → α) (i : Fin 1) : Matrix.vecCons x u i = x

theorem Matrix.cons_fin_one {α : Type u} (x : α) (u : Fin 0 → α) : Matrix.vecCons x u = fun x => x

instance PiFin.toExpr {α : Type u} [Lean.ToLevel.{u}] [Lean.ToExpr α] (n : ℕ) : Lean.ToExpr (Fin n → α)

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} {n : ℕ} {α : Type u_4} {o : ℕ} (ho : o = m + n) (u : Fin m → α) (v : Fin n → α) : Fin o → α

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.

theorem Matrix.vecAppend_eq_ite {m : ℕ} {n : ℕ} {α : Type u_4} {o : ℕ} (ho : o = m + n) (u : Fin m → α) (v : Fin n → α) : Matrix.vecAppend ho u v = fun i => if h : ↑i < m then u { val := ↑i, isLt := h } else v { val := ↑i - m, isLt := (_ : ↑i - m < n) }

@[simp]

theorem Matrix.vecAppend_apply_zero {m : ℕ} {n : ℕ} {α : Type u_4} {o : ℕ} (ho : o + 1 = m + 1 + n) (u : Fin (m + 1) → α) (v : Fin n → α) : Matrix.vecAppend ho u v 0 = u 0

@[simp]

theorem Matrix.empty_vecAppend {α : Type u} {n : ℕ} (v : Fin n → α) : Matrix.vecAppend (_ : n = 0 + n) ![] v = v

@[simp]

theorem Matrix.cons_vecAppend {α : Type u} {m : ℕ} {n : ℕ} {o : ℕ} (ho : o + 1 = m + 1 + n) (x : α) (u : Fin m → α) (v : Fin n → α) : Matrix.vecAppend ho (Matrix.vecCons x u) v = Matrix.vecCons x (Matrix.vecAppend (_ : o = m + n) u v)

def Matrix.vecAlt0 {α : Type u} {m : ℕ} {n : ℕ} (hm : m = n + n) (v : Fin m → α) (k : Fin n) : α

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

def Matrix.vecAlt1 {α : Type u} {m : ℕ} {n : ℕ} (hm : m = n + n) (v : Fin m → α) (k : Fin n) : α

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

theorem Matrix.vecAlt0_vecAppend {α : Type u} {n : ℕ} (v : Fin n → α) : Matrix.vecAlt0 (_ : n + n = n + n) (Matrix.vecAppend (_ : n + n = n + n) v v) = v ∘ bit0

theorem Matrix.vecAlt1_vecAppend {α : Type u} {n : ℕ} (v : Fin (n + 1) → α) : Matrix.vecAlt1 (_ : n + 1 + (n + 1) = n + 1 + (n + 1)) (Matrix.vecAppend (_ : n + 1 + (n + 1) = n + 1 + (n + 1)) v v) = v ∘ bit1

@[simp]

theorem Matrix.vecHead_vecAlt0 {α : Type u} {m : ℕ} {n : ℕ} (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2) → α) : Matrix.vecHead (Matrix.vecAlt0 hm v) = v 0

@[simp]

theorem Matrix.vecHead_vecAlt1 {α : Type u} {m : ℕ} {n : ℕ} (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2) → α) : Matrix.vecHead (Matrix.vecAlt1 hm v) = v 1

@[simp]

theorem Matrix.cons_vec_bit0_eq_alt0 {α : Type u} {n : ℕ} (x : α) (u : Fin n → α) (i : Fin (n + 1)) : Matrix.vecCons x u (bit0 i) = Matrix.vecAlt0 (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (Matrix.vecAppend (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (Matrix.vecCons x u) (Matrix.vecCons x u)) i

@[simp]

theorem Matrix.cons_vec_bit1_eq_alt1 {α : Type u} {n : ℕ} (x : α) (u : Fin n → α) (i : Fin (n + 1)) : Matrix.vecCons x u (bit1 i) = Matrix.vecAlt1 (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (Matrix.vecAppend (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (Matrix.vecCons x u) (Matrix.vecCons x u)) i

@[simp]

theorem Matrix.cons_vecAlt0 {α : Type u} {m : ℕ} {n : ℕ} (h : m + 1 + 1 = n + 1 + (n + 1)) (x : α) (y : α) (u : Fin m → α) : Matrix.vecAlt0 h (Matrix.vecCons x (Matrix.vecCons y u)) = Matrix.vecCons x (Matrix.vecAlt0 (_ : m = n + n) u)

@[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} {n : ℕ} (h : m + 1 + 1 = n + 1 + (n + 1)) (x : α) (y : α) (u : Fin m → α) : Matrix.vecAlt1 h (Matrix.vecCons x (Matrix.vecCons y u)) = Matrix.vecCons y (Matrix.vecAlt1 (_ : m = n + n) u)

@[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 0 → α) : x • v = ![]

@[simp]

theorem Matrix.smul_cons {α : Type u} {n : ℕ} {M : Type u_4} [SMul M α] (x : M) (y : α) (v : Fin n → α) : x • Matrix.vecCons y v = Matrix.vecCons (x • y) (x • v)

@[simp]

theorem Matrix.empty_add_empty {α : Type u} [Add α] (v : Fin 0 → α) (w : Fin 0 → α) : v + w = ![]

@[simp]

theorem Matrix.cons_add {α : Type u} {n : ℕ} [Add α] (x : α) (v : Fin n → α) (w : Fin (Nat.succ n) → α) : Matrix.vecCons x v + w = Matrix.vecCons (x + Matrix.vecHead w) (v + Matrix.vecTail w)

@[simp]

theorem Matrix.add_cons {α : Type u} {n : ℕ} [Add α] (v : Fin (Nat.succ n) → α) (y : α) (w : Fin n → α) : v + Matrix.vecCons y w = Matrix.vecCons (Matrix.vecHead v + y) (Matrix.vecTail v + w)

theorem Matrix.cons_add_cons {α : Type u} {n : ℕ} [Add α] (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) : Matrix.vecCons x v + Matrix.vecCons y w = Matrix.vecCons (x + y) (v + w)

@[simp]

theorem Matrix.head_add {α : Type u} {n : ℕ} [Add α] (a : Fin (Nat.succ n) → α) (b : Fin (Nat.succ n) → α) : Matrix.vecHead (a + b) = Matrix.vecHead a + Matrix.vecHead b

@[simp]

theorem Matrix.tail_add {α : Type u} {n : ℕ} [Add α] (a : Fin (Nat.succ n) → α) (b : Fin (Nat.succ n) → α) : Matrix.vecTail (a + b) = Matrix.vecTail a + Matrix.vecTail b

@[simp]

theorem Matrix.empty_sub_empty {α : Type u} [Sub α] (v : Fin 0 → α) (w : Fin 0 → α) : v - w = ![]

@[simp]

theorem Matrix.cons_sub {α : Type u} {n : ℕ} [Sub α] (x : α) (v : Fin n → α) (w : Fin (Nat.succ n) → α) : Matrix.vecCons x v - w = Matrix.vecCons (x - Matrix.vecHead w) (v - Matrix.vecTail w)

@[simp]

theorem Matrix.sub_cons {α : Type u} {n : ℕ} [Sub α] (v : Fin (Nat.succ n) → α) (y : α) (w : Fin n → α) : v - Matrix.vecCons y w = Matrix.vecCons (Matrix.vecHead v - y) (Matrix.vecTail v - w)

theorem Matrix.cons_sub_cons {α : Type u} {n : ℕ} [Sub α] (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) : Matrix.vecCons x v - Matrix.vecCons y w = Matrix.vecCons (x - y) (v - w)

@[simp]

theorem Matrix.head_sub {α : Type u} {n : ℕ} [Sub α] (a : Fin (Nat.succ n) → α) (b : Fin (Nat.succ n) → α) : Matrix.vecHead (a - b) = Matrix.vecHead a - Matrix.vecHead b

@[simp]

theorem Matrix.tail_sub {α : Type u} {n : ℕ} [Sub α] (a : Fin (Nat.succ n) → α) (b : Fin (Nat.succ n) → α) : Matrix.vecTail (a - b) = Matrix.vecTail a - Matrix.vecTail b

@[simp]

theorem Matrix.zero_empty {α : Type u} [Zero α] : 0 = ![]

@[simp]

theorem Matrix.cons_zero_zero {α : Type u} {n : ℕ} [Zero α] : Matrix.vecCons 0 0 = 0

@[simp]

theorem Matrix.head_zero {α : Type u} {n : ℕ} [Zero α] : Matrix.vecHead 0 = 0

@[simp]

theorem Matrix.tail_zero {α : Type u} {n : ℕ} [Zero α] : Matrix.vecTail 0 = 0

@[simp]

theorem Matrix.cons_eq_zero_iff {α : Type u} {n : ℕ} [Zero α] {v : Fin n → α} {x : α} : Matrix.vecCons x v = 0 ↔ x = 0 ∧ v = 0

theorem Matrix.cons_nonzero_iff {α : Type u} {n : ℕ} [Zero α] {v : Fin n → α} {x : α} : Matrix.vecCons x v ≠ 0 ↔ x ≠ 0 ∨ v ≠ 0

@[simp]

theorem Matrix.neg_empty {α : Type u} [Neg α] (v : Fin 0 → α) : -v = ![]

@[simp]

theorem Matrix.neg_cons {α : Type u} {n : ℕ} [Neg α] (x : α) (v : Fin n → α) : -Matrix.vecCons x v = Matrix.vecCons (-x) (-v)

@[simp]

theorem Matrix.head_neg {α : Type u} {n : ℕ} [Neg α] (a : Fin (Nat.succ n) → α) : Matrix.vecHead (-a) = -Matrix.vecHead a

@[simp]

theorem Matrix.tail_neg {α : Type u} {n : ℕ} [Neg α] (a : Fin (Nat.succ n) → α) : Matrix.vecTail (-a) = -Matrix.vecTail a