Documentation

Mathlib.Data.Fin.VecNotation

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→ α. Nesting vectors gives coefficients of a matrix, so ![![a, b], ![c, d]] : fin 2 → fin 2 → α→ fin 2 → α→ α. In later files we introduce !![a, b; c, d] as notation for matrix.of ![![a, b], ![c, d]].

Main definitions #

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

Implementation notes #

The simp lemmas require that one of the arguments is of the form vec_cons _ _. 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 vec_cons a (vec_cons b vec_empty).

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.

Equations

Matrix.vecEmpty = Fin.elim0'

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 (vec_cons b ...).

Equations

Matrix.vecCons h t = Fin.cons h t

def Matrix.«![_,]» :

Lean.ParserDescr

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

Equations

One or more equations did not get rendered due to their size.

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

Equations

Matrix.vecHead v = v 0

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

Equations

Matrix.vecTail v = v ∘ Fin.succ

instance PiFin.hasRepr {α : Type u} {n : ℕ} [inst : Repr α] :

Repr (Fin n → α)

Use ![...] notation for displaying a vector fin n → α→ 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 0 → α) :

v = Matrix.vecEmpty

@[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_1} (j : n') :

(fun i => Matrix.vecEmpty (n' → α) i j) = Matrix.vecEmpty

@[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) :

Matrix.vecCons a Matrix.vecEmpty = 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≥ 2, in addition to cons_val_succ, because 1 : fin 1 = 0 : fin 1.

@[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

Numeral (`bit0` and `bit1`) indices #

The following definitions and simp lemmas are to allow any numeral-indexed element of a vector given with matrix notation to be extracted by simp (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_1} {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.

Equations

Matrix.vecAppend ho u v = Fin.append u v ∘ ↑(RelIso.toRelEmbedding (Fin.cast ho)).toEmbedding

theorem Matrix.vecAppend_eq_ite {m : ℕ} {n : ℕ} {α : Type u_1} {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_1} {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) Matrix.vecEmpty 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).

Equations

Matrix.vecAlt0 hm v k = v { val := ↑k + ↑k, isLt := (_ : ↑k + ↑k < m) }

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).

Equations

Matrix.vecAlt1 hm v k = v { val := ↑k + ↑k + 1, isLt := (_ : ↑k + ↑k + 1 < m) }

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_1) {h : 0 = 0 + 0} :

Matrix.vecAlt0 h Matrix.vecEmpty = Matrix.vecEmpty

@[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_1) {h : 0 = 0 + 0} :

Matrix.vecAlt1 h Matrix.vecEmpty = Matrix.vecEmpty

@[simp]

theorem Matrix.smul_empty {α : Type u} {M : Type u_1} [inst : SMul M α] (x : M) (v : Fin 0 → α) :

x • v = Matrix.vecEmpty

@[simp]

theorem Matrix.smul_cons {α : Type u} {n : ℕ} {M : Type u_1} [inst : 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} [inst : Add α] (v : Fin 0 → α) (w : Fin 0 → α) :

v + w = Matrix.vecEmpty

@[simp]

theorem Matrix.cons_add {α : Type u} {n : ℕ} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : Sub α] (v : Fin 0 → α) (w : Fin 0 → α) :

v - w = Matrix.vecEmpty

@[simp]

theorem Matrix.cons_sub {α : Type u} {n : ℕ} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : Zero α] :

0 = Matrix.vecEmpty

@[simp]

theorem Matrix.cons_zero_zero {α : Type u} {n : ℕ} [inst : Zero α] :

Matrix.vecCons 0 0 = 0

@[simp]

theorem Matrix.head_zero {α : Type u} {n : ℕ} [inst : Zero α] :

Matrix.vecHead 0 = 0

@[simp]

theorem Matrix.tail_zero {α : Type u} {n : ℕ} [inst : Zero α] :

Matrix.vecTail 0 = 0

@[simp]

theorem Matrix.cons_eq_zero_iff {α : Type u} {n : ℕ} [inst : 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} [inst : 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} [inst : Neg α] (v : Fin 0 → α) :

-v = Matrix.vecEmpty

@[simp]

theorem Matrix.neg_cons {α : Type u} {n : ℕ} [inst : 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} [inst : Neg α] (a : Fin (Nat.succ n) → α) :

Matrix.vecHead (-a) = -Matrix.vecHead a

@[simp]

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

Matrix.vecTail (-a) = -Matrix.vecTail a