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feat(library/algebra/field): fix broken theorems

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Rob Lewis 2015-02-27 15:44:36 -05:00 committed by Leonardo de Moura
parent 8ef2849b67
commit 10f1232296
1 changed files with 111 additions and 183 deletions
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library/algebra/field.lean
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@ -39,63 +39,48 @@ section division_ring
theorem inv_eq_one_div : a⁻¹ = 1 / a := !one_mul⁻¹
theorem div_eq_mul_one_div : a / b = a * (1 / b) :=
by rewrite [↑divide, one_mul]
theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 :=
by rewrite [-inv_eq_one_div, (mul_inv_cancel H)]
by rewrite [-inv_eq_one_div, (mul_inv_cancel H)]
theorem one_div_mul_cancel (H : a ≠ 0) : (1 / a) * a = 1 :=
calc
(1 / a) * a = a⁻¹ * a : inv_eq_one_div
... = 1 : inv_mul_cancel H
by rewrite [-inv_eq_one_div, (inv_mul_cancel H)]
theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
theorem mul_div_assoc : (a * b) / c = a * (b / c) := !mul.assoc
theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 :=
assume H2 : 1 / a = 0,
have C1 : 0 = 1, from symm (calc
1 = a * (1 / a) : mul_one_div_cancel H
... = a * 0 : H2
... = 0 : mul_zero),
absurd C1 zero_ne_one
have C1 : 0 = 1, from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
absurd C1 zero_ne_one
-- the analogue in group is called inv_one
theorem inv_one_is_one : 1⁻¹ = 1 :=
calc
1⁻¹ = 1⁻¹ * 1 : mul_one
... = 1 : inv_mul_cancel (ne.symm zero_ne_one)
by rewrite [-mul_one, (inv_mul_cancel (ne.symm zero_ne_one))]
theorem div_one : a / 1 = a :=
calc
a / 1 = a * 1 : inv_one_is_one
... = a : mul_one
by rewrite [↑divide, inv_one_is_one, mul_one]
theorem zero_div : 0 / a = 0 := !zero_mul
-- note: integral domain has a "mul_ne_zero". Discrete fields are int domains.
theorem mul_ne_zero' (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 :=
assume H : a * b = 0,
have C1 : a = 0, from (calc
a = a * 1 : mul_one
... = a * (b * (1 / b)) : mul_one_div_cancel Hb
... = (a * b) * (1 / b) : mul.assoc
... = 0 * (1 / b) : H
... = 0 : zero_mul),
have C1 : a = 0, by rewrite [-mul_one, -(mul_one_div_cancel Hb), -mul.assoc, H, zero_mul],
absurd C1 Ha
-- this belongs in ring?
theorem mul_ne_zero_imp_ne_zero (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 :=
have Ha : a ≠ 0, from
(assume Ha1 : a = 0,
have H1 : a * b = 0, from (calc
a * b = 0 * b : Ha1
... = 0 : zero_mul),
have H1 : a * b = 0, by rewrite [Ha1, zero_mul],
absurd H1 H),
have Hb : b ≠ 0, from
(assume Hb1 : b = 0,
have H1 : a * b = 0, from (calc
a * b = a * 0 : Hb1
... = 0 : mul_zero),
have H1 : a * b = 0, by rewrite [Hb1, mul_zero],
absurd H1 H),
and.intro Ha Hb
@ -110,17 +95,14 @@ section division_ring
theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a :=
have H2 : a ≠ 0, from
(assume A : a = 0,
have B : 0 = 1, from symm (calc
1 = a * b : symm H
... = 0 * b : A
... = 0 : zero_mul),
have B : 0 = 1, by rewrite [-(zero_mul b), -A, H],
absurd B zero_ne_one),
show b = 1 / a, from symm (calc
1 / a = (1 / a) * 1 : mul_one
... = (1 / a) * (a * b) : H
... = (1 / a) * a * b : mul.assoc
... = 1 * b : one_div_mul_cancel H2
... = b : one_mul)
1 / a = (1 / a) * 1 : mul_one
... = (1 / a) * (a * b) : H
... = (1 / a) * a * b : mul.assoc
... = 1 * b : one_div_mul_cancel H2
... = b : one_mul)
-- which one is left and which is right?
theorem eq_one_div_of_mul_eq_one_left (H : b * a = 1) : b = 1 / a :=
@ -129,78 +111,63 @@ section division_ring
have B : 0 = 1, from symm (calc
1 = b * a : symm H
... = b * 0 : A
... = 0 : mul_zero),
... = 0 : mul_zero),
absurd B zero_ne_one),
show b = 1 / a, from symm (calc
1 / a = 1 * (1 / a) : one_mul
... = b * a * (1 / a) : H
... = b * (a * (1 / a)) : mul.assoc
... = b * 1 : mul_one_div_cancel H2
... = b : mul_one)
1 / a = 1 * (1 / a) : one_mul
... = b * a * (1 / a) : H
... = b * (a * (1 / a)) : mul.assoc
... = b * 1 : mul_one_div_cancel H2
... = b : mul_one)
theorem one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (b * a) :=
have H : (b * a) * ((1 / a) * (1 / b)) = 1, from (calc
(b * a) * ((1 / a) * (1 / b)) = b * (a * ((1 / a) * (1 / b))) : mul.assoc
... = b * ((a * (1 / a)) * (1 / b)) : mul.assoc
... = b * (1 * (1 / b)) : mul_one_div_cancel Ha
... = b * (1 / b) : one_mul
... = 1 : mul_one_div_cancel Hb),
have H : (b * a) * ((1 / a) * (1 / b)) = 1, by
rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul, (mul_one_div_cancel Hb)],
eq_one_div_of_mul_eq_one H
theorem one_div_neg_one_eq_neg_one : 1 / (-1) = -1 :=
have H : (-1) * (-1) = 1, from calc
(-1) * (-1) = - (-1) : neg_eq_neg_one_mul
... = 1 : neg_neg,
have H : (-1) * (-1) = 1, by rewrite [-neg_eq_neg_one_mul, neg_neg],
symm (eq_one_div_of_mul_eq_one H)
-- this should be in ring
theorem mul_neg_one_eq_neg : a * (-1) = -a :=
have H : a + a * -1 = 0, from calc
a + a * -1 = a * 1 + a * -1 : mul_one
... = a * (1 + -1) : left_distrib
... = a * 0 : add.right_inv
... = 0 : mul_zero,
... = a * (1 + -1) : left_distrib
... = a * 0 : add.right_inv
... = 0 : mul_zero,
symm (neg_eq_of_add_eq_zero H)
theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) :=
have H1 : -1 ≠ 0, from
(assume H2 : -1 = 0, absurd (symm (calc
1 = -(-1) : neg_neg
... = -0 : H2
... = 0 : neg_zero)) zero_ne_one),
1 = -(-1) : neg_neg
... = -0 : H2
... = 0 : neg_zero)) zero_ne_one),
calc
1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1
... = (1 / a) * (-1) : one_div_neg_one_eq_neg_one
... = - (1 / a) : mul_neg_one_eq_neg
1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1
... = (1 / a) * (-1) : one_div_neg_one_eq_neg_one
... = - (1 / a) : mul_neg_one_eq_neg
theorem div_neg_eq_neg_div (Ha : a ≠ 0) : b / (- a) = - (b / a) :=
calc
b / (- a) = b * (1 / (- a)) : inv_eq_one_div
... = b * -(1 / a) : one_div_neg_eq_neg_one_div Ha
... = -(b * (1 / a)) : neg_mul_eq_mul_neg
... = - (b * a⁻¹) : inv_eq_one_div
... = b * -(1 / a) : one_div_neg_eq_neg_one_div Ha
... = -(b * (1 / a)) : neg_mul_eq_mul_neg
... = - (b * a⁻¹) : inv_eq_one_div
theorem neg_div (Ha : a ≠ 0) : (-b) / a = - (b / a) :=
calc
(-b) / a = (-1 * b) / a : neg_eq_neg_one_mul
... = (-1) * (b / a) : mul_div_assoc
... = - (b / a) : neg_eq_neg_one_mul
by rewrite [neg_eq_neg_one_mul, mul_div_assoc, -neg_eq_neg_one_mul]
theorem neg_div_neg_eq_div (Hb : b ≠ 0) : (-a) / (-b) = a / b :=
calc
(-a) / (-b) = - ((-a) / b) : div_neg_eq_neg_div Hb
... = - -(a / b) : neg_div Hb
... = a / b : neg_neg
by rewrite [(div_neg_eq_neg_div Hb), (neg_div Hb), neg_neg]
theorem div_div (H : a ≠ 0) : 1 / (1 / a) = a :=
symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel H))
theorem eq_of_invs_eq (Ha : a ≠ 0) (Hb : b ≠ 0) (H : 1 / a = 1 / b) : a = b :=
calc
a = 1 / (1 / a) : div_div Ha
... = 1 / (1 / b) : H
... = b : div_div Hb
by rewrite [-(div_div Ha), H, (div_div Hb)]
-- oops, the analogous theorem in group is called inv_mul, but it *should* be called
-- mul_inv, in which case, we will have to rename this one
@ -213,63 +180,45 @@ section division_ring
... = (b * a)⁻¹ : inv_eq_one_div)
theorem mul_div_cancel (Hb : b ≠ 0) : a * b / b = a :=
calc
(a * b) / b = a * (b * b⁻¹) : mul.assoc
... = a * 1 : mul_inv_cancel Hb
... = a : mul_one
by rewrite [↑divide, mul.assoc, (mul_inv_cancel Hb), mul_one]
theorem div_mul_cancel (Hb : b ≠ 0) : a / b * b = a :=
calc
(a / b) * b = a * (b⁻¹ * b) : mul.assoc
... = a * 1 : inv_mul_cancel Hb
... = a : mul_one
by rewrite [↑divide, mul.assoc, (inv_mul_cancel Hb), mul_one]
theorem div_add_div_same : a / c + b / c = (a + b) / c := !right_distrib⁻¹
theorem inv_mul_add_mul_inv_eq_inv_add_inv (Ha : a ≠ 0) (Hb : b ≠ 0) :
(1 / a) * (a + b) * (1 / b) = 1 / a + 1 / b :=
by rewrite [(left_distrib (1 / a)), (one_div_mul_cancel Ha), right_distrib, one_mul,
mul.assoc, (mul_one_div_cancel Hb), mul_one, add.comm]
/-calc
(1 / a) * (a + b) * (1 / b) = ((1 / a) * a + (1 / a) * b) * (1 / b) : left_distrib
... = (1 + (1 / a) * b) * (1 / b) : one_div_mul_cancel Ha
... = 1 * (1 / b) + (1 / a) * b * (1 / b) : right_distrib
... = 1 / b + (1 / a) * b * (1 / b) : one_mul
... = 1 / b + (1 / a) * (b * (1 / b)) : mul.assoc
... = 1 / b + (1 / a) * 1 : mul_one_div_cancel Hb
... = 1 / b + (1 / a) : mul_one
... = 1 / a + 1 / b : add.comm-/
theorem inv_mul_sub_mul_inv_eq_inv_add_inv (Ha : a ≠ 0) (Hb : b ≠ 0) :
(1 / a) * (b - a) * (1 / b) = 1 / a - 1 / b :=
by rewrite [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel Ha), mul_sub_right_distrib,
one_mul, mul.assoc, (mul_one_div_cancel Hb), mul_one]
one_mul, mul.assoc, (mul_one_div_cancel Hb), mul_one, one_mul]
theorem div_eq_one_iff_eq (Hb : b ≠ 0) : a / b = 1 ↔ a = b :=
iff.intro
(assume H1 : a / b = 1, symm (calc
b = 1 * b : one_mul
b = 1 * b : one_mul
... = a / b * b : H1
... = a : div_mul_cancel Hb))
... = a : div_mul_cancel Hb))
(assume H2 : a = b, calc
a / b = b / b : H2
... = 1 : div_self Hb)
... = 1 : div_self Hb)
theorem eq_div_iff_mul_eq (Hc : c ≠ 0) : a = b / c ↔ a * c = b :=
iff.intro
(assume H : a = b / c, calc
a * c = b / c * c : H
... = b : div_mul_cancel Hc)
(assume H : a * c = b, symm (calc
b / c = a * c / c : H
... = a : mul_div_cancel Hc))
(assume H : a = b / c, by rewrite [H, (div_mul_cancel Hc)])
(assume H : a * c = b, by rewrite [-(mul_div_cancel Hc), H])
theorem add_div_eq_mul_add_div (Hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
have H : (a + b / c) * c = a * c + b, from calc
(a + b / c) * c = a * c + (b / c) * c : right_distrib
... = a * c + b : div_mul_cancel Hc,
have H : (a + b / c) * c = a * c + b, by rewrite [right_distrib, (div_mul_cancel Hc)],
(iff.elim_right (eq_div_iff_mul_eq Hc)) H
-- There are many similar rules to these last two in the Isabelle library
-- that haven't been ported yet. Do as necessary.
end division_ring
structure field [class] (A : Type) extends division_ring A, comm_ring A
@ -279,103 +228,82 @@ section field
include s
local attribute divide [reducible]
-- I think of "div_cancel" has being something like a * b / b = a or a / b * b = a. The name
-- I chose is clunky, but it has the right prefix
theorem one_div_mul_one_div' (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (a * b) :=
by rewrite [(one_div_mul_one_div Ha Hb), mul.comm b]
theorem div_mul_right (Hb : b ≠ 0) (H : a * b ≠ 0) : a / (a * b) = 1 / b :=
have Ha : a ≠ 0, from and.left (mul_ne_zero_imp_ne_zero H),
let Ha : a ≠ 0 := and.left (mul_ne_zero_imp_ne_zero H) in
symm (calc
1 / b = 1 * (1 / b) : one_mul
... = (a * a⁻¹) * (1 / b) : mul_inv_cancel Ha
... = a * (a⁻¹ * (1 / b)) : mul.assoc
... = a * ((1 / a) * (1 / b)) :inv_eq_one_div
... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb
... = a * (1 / (a * b)) : mul.comm
... = a * (a * b)⁻¹ : inv_eq_one_div)
1 / b = 1 * (1 / b) : one_mul
... = (a * a⁻¹) * (1 / b) : mul_inv_cancel Ha
... = a * (a⁻¹ * (1 / b)) : mul.assoc
... = a * ((1 / a) * (1 / b)) :inv_eq_one_div
... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb
... = a * (1 / (a * b)) : mul.comm
... = a * (a * b)⁻¹ : inv_eq_one_div)
theorem div_mul_left (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a :=
have H1 : b * a ≠ 0, from mul_ne_zero_comm H,
calc
(b / (a * b)) = (b / (b * a)) : mul.comm
... = 1 / a : div_mul_right Ha H1
let H1 : b * a ≠ 0 := mul_ne_zero_comm H in
by rewrite [mul.comm a, (div_mul_right Ha H1)]
theorem mul_div_cancel_left (Ha : a ≠ 0) : a * b / a = b :=
calc
(a * b) / a = (b * a) / a : mul.comm
... = b : mul_div_cancel Ha
by rewrite [mul.comm a, (mul_div_cancel Ha)]
theorem mul_div_cancel' (Hb : b ≠ 0) : b * (a / b) = a :=
calc
b * (a / b) = a / b * b : mul.comm
... = a : div_mul_cancel Hb
by rewrite [mul.comm, (div_mul_cancel Hb)]
theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) :=
have H : a * b ≠ 0, from (mul_ne_zero' Ha Hb),
symm (calc
(a + b) / (a * b) = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib
... = a / (a * b) + b * (a * b)⁻¹ : rfl
... = 1 / b + b * (a * b)⁻¹ : div_mul_right Hb H
... = 1 / b + 1 / a : div_mul_left Ha H
... = 1 / a + 1 / b : add.comm)
have H [visible] : a * b ≠ 0, from (mul_ne_zero' Ha Hb),
by rewrite [add.comm, -(div_mul_left Ha H), -(div_mul_right Hb H), ↑divide, -right_distrib]
theorem div_mul_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) * (c / d) = (a * c) / (b * d) :=
calc
(a / b) * (c / d) = (a * b⁻¹) * (c * d⁻¹) : rfl
... = (a * c) * (d⁻¹ * b⁻¹) : by rewrite [2 mul.assoc, (mul.comm b⁻¹), mul.assoc]
... = (a * c) * (b * d)⁻¹ : mul_inv Hd Hb
by rewrite [↑divide, 2 mul.assoc, (mul.comm b⁻¹), mul.assoc, (mul_inv Hd Hb)]
theorem mul_div_mul_left (Hb : b ≠ 0) (Hc : c ≠ 0) : (c * a) / (c * b) = a / b :=
have H : c * b ≠ 0, from mul_ne_zero' Hc Hb,
calc
(c * a) / (c * b) = (c / c) * (a / b) : div_mul_div Hc Hb
... = 1 * (a / b) : div_self Hc
... = a / b : one_mul
have H [visible] : c * b ≠ 0, from mul_ne_zero' Hc Hb,
by rewrite [-(div_mul_div Hc Hb), (div_self Hc), one_mul]
theorem mul_div_mul_right (Hb : b ≠ 0) (Hc : c ≠ 0) : (a * c) / (b * c) = a / b :=
calc
(a * c) / (b * c) = (c * a) / (b * c) : mul.comm
... = (c * a) / (c * b) : mul.comm
... = a / b : mul_div_mul_left Hb Hc
by rewrite [(mul.comm a), (mul.comm b), (mul_div_mul_left Hb Hc)]
theorem div_mul_eq_mul_div (Hc : c ≠ 0) : (b / c) * a = (b * a) / c :=
calc
(b / c) * a = (b * c⁻¹) * a : rfl
... = (b * a) * c⁻¹ : by rewrite [mul.assoc, (mul.comm c⁻¹), -mul.assoc]
by rewrite [↑divide, mul.assoc, (mul.comm c⁻¹), -mul.assoc]
-- this one is odd -- I am not sure what to call it, but again, the prefix is right
theorem div_mul_eq_mul_div_comm (Hc : c ≠ 0) : (b / c) * a = b * (a / c) :=
calc
(b / c) * a = (b * a) / c : div_mul_eq_mul_div Hc
... = (b * a) / (1 * c) : one_mul
... = (b / 1) * (a / c) : div_mul_div (ne.symm zero_ne_one) Hc
... = b * (a / c) : div_one
by rewrite [(div_mul_eq_mul_div Hc), -(one_mul c), -(div_mul_div (ne.symm zero_ne_one) Hc), div_one, one_mul]
theorem div_add_div (Hb : b ≠ 0) (Hd : d ≠ 0) :
(a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) :=
have H : b * d ≠ 0, from mul_ne_zero' Hb Hd,
calc
a / b + c / d = (a * d) / (b * d) + c / d : mul_div_mul_right Hb Hd
... = (a * d) / (b * d) + (b * c) / (b * d) : mul_div_mul_left Hd Hb
... = ((a * d) + (b * c)) / (b * d) : div_add_div_same
have H [visible] : b * d ≠ 0, from mul_ne_zero' Hb Hd,
by rewrite [-(mul_div_mul_right Hb Hd), -(mul_div_mul_left Hd Hb), div_add_div_same]
theorem div_sub_div (Hb : b ≠ 0) (Hd : d ≠ 0) :
(a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) :=
calc
(a / b) - (c / d) = (a / b) + -1 * (c / d) : neg_eq_neg_one_mul
... = (a / b) + ((-1 * c) / d) : mul_div_assoc
... = ((a * d) + (b * (-1 * c))) / (b * d) : div_add_div Hb Hd
... = ((a * d) + -1 * (b * c)) / (b * d) : by rewrite [-mul.assoc, (mul.comm b), mul.assoc]
... = ((a * d) + -(b * c)) / (b * d) : neg_eq_neg_one_mul
by rewrite [↑sub, neg_eq_neg_one_mul, -mul_div_assoc, (div_add_div Hb Hd),
-mul.assoc, (mul.comm b), mul.assoc, -neg_eq_neg_one_mul]
theorem mul_eq_mul_of_div_eq_div (Hb : b ≠ 0) (Hd : d ≠ 0) (H : a / b = c / d) : a * d = c * b :=
calc
a * d = a * 1 * d : by rewrite [-mul_one, mul.assoc, (mul.comm d), -mul.assoc]
... = (a * (b / b)) * d : div_self Hb
... = ((a / b) * b) * d : div_mul_eq_mul_div_comm Hb
... = ((c / d) * b) * d : H
... = ((c * b) / d) * d : div_mul_eq_mul_div Hd
... = c * b : div_mul_cancel Hd
by rewrite [-mul_one, mul.assoc, (mul.comm d), -mul.assoc, -(div_self Hb),
-(div_mul_eq_mul_div_comm Hb), H, (div_mul_eq_mul_div Hd), (div_mul_cancel Hd)]
theorem one_div_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / (a / b) = b / a :=
have H : (a / b) * (b / a) = 1, from calc
(a / b) * (b / a) = (a * b) / (b * a) : div_mul_div Hb Ha
... = (a * b) / (a * b) : mul.comm
... = 1 : div_self (mul_ne_zero' Ha Hb),
symm (eq_one_div_of_mul_eq_one H)
theorem div_div_eq_mul_div (Hb : b ≠ 0) (Hc : c ≠ 0) : a / (b / c) = (a * c) / b :=
by rewrite [div_eq_mul_one_div, (one_div_div Hb Hc), -mul_div_assoc]
theorem div_div_eq_div_mul (Hb : b ≠ 0) (Hc : c ≠ 0) : (a / b) / c = a / (b * c) :=
by rewrite [div_eq_mul_one_div, (div_mul_div Hb Hc), mul_one]
theorem div_div_div_div (Hb : b ≠ 0) (Hc : c ≠ 0) (Hd : d ≠ 0) : (a / b) / (c / d) = (a * d) / (b * c) :=
by rewrite [(div_div_eq_mul_div Hc Hd), (div_mul_eq_mul_div Hb), (div_div_eq_div_mul Hb Hc)]
-- remaining to transfer from Isabelle fields: ordered fields
end field
@ -395,12 +323,7 @@ section discrete_field
decidable.by_cases
(assume H : x = 0, or.inl H)
(assume H1 : x ≠ 0,
or.inr (calc
y = 1 * y : one_mul
... = x⁻¹ * x * y : inv_mul_cancel H1
... = x⁻¹ * (x * y) : mul.assoc
... = x⁻¹ * 0 : H
... = 0 : mul_zero))
or.inr (by rewrite [-one_mul, -(inv_mul_cancel H1), mul.assoc, H, mul_zero]))
definition discrete_field.to_integral_domain [instance] [reducible] [coercion] :
integral_domain A :=
@ -410,6 +333,11 @@ section discrete_field
example (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
@mul_ne_zero A s a b H1 H2
theorem inv_zero_imp_zero (H : 1 / a = 0) : a = 0 :=
decidable.by_cases
(assume Ha : a = 0, Ha)
(assume Ha: a ≠ 0, false.elim ((one_div_ne_zero Ha) H))
end discrete_field
end algebra