7d72c9b6b5
Many theorems for division rings and fields have stronger versions for discrete fields, where we assume x / 0 = 0. Before, we used primes to distinguish the versions, but that has the downside that e.g. for rat and real, all the theorems are equally present. Now, I qualified the weaker ones with division_ring.foo or field.foo. Maybe that is not ideal, but let's try it. I also set implicit arguments with the following convention: an argument to a theorem should be explicit unless it can be inferred from the other arguments and hypotheses.
465 lines
19 KiB
Text
465 lines
19 KiB
Text
/-
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Copyright (c) 2014 Robert Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Robert Lewis
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Structures with multiplicative and additive components, including division rings and fields.
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The development is modeled after Isabelle's library.
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-/
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import logic.eq logic.connectives data.unit data.sigma data.prod
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import algebra.binary algebra.group algebra.ring
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open eq eq.ops
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namespace algebra
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variable {A : Type}
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-- in division rings, 1 / 0 = 0
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structure division_ring [class] (A : Type) extends ring A, has_inv A, zero_ne_one_class A :=
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(mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one)
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(inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one)
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--(inv_zero : inv zero = zero)
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section division_ring
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variables [s : division_ring A] {a b c : A}
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include s
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definition divide (a b : A) : A := a * b⁻¹
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infix [priority algebra.prio] `/` := divide
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-- only in this file
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local attribute divide [reducible]
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theorem mul_inv_cancel (H : a ≠ 0) : a * a⁻¹ = 1 :=
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division_ring.mul_inv_cancel H
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theorem inv_mul_cancel (H : a ≠ 0) : a⁻¹ * a = 1 :=
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division_ring.inv_mul_cancel H
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theorem inv_eq_one_div (a : A) : a⁻¹ = 1 / a := !one_mul⁻¹
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theorem div_eq_mul_one_div (a b : A) : a / b = a * (1 / b) :=
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by rewrite [↑divide, one_mul]
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theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 :=
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by rewrite [-inv_eq_one_div, (mul_inv_cancel H)]
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theorem one_div_mul_cancel (H : a ≠ 0) : (1 / a) * a = 1 :=
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by rewrite [-inv_eq_one_div, (inv_mul_cancel H)]
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theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
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theorem one_div_one : 1 / 1 = (1:A) := div_self (ne.symm zero_ne_one)
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theorem mul_div_assoc (a b : A) : (a * b) / c = a * (b / c) := !mul.assoc
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theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 :=
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assume H2 : 1 / a = 0,
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have C1 : 0 = (1:A), from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
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absurd C1 zero_ne_one
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theorem one_inv_eq : 1⁻¹ = (1:A) :=
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by rewrite [-mul_one, inv_mul_cancel (ne.symm (@zero_ne_one A _))]
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theorem div_one (a : A) : a / 1 = a :=
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by rewrite [↑divide, one_inv_eq, mul_one]
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theorem zero_div (a : A) : 0 / a = 0 := !zero_mul
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-- note: integral domain has a "mul_ne_zero". A commutative division ring is an integral
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-- domain, but let's not define that class for now.
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theorem division_ring.mul_ne_zero (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 :=
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assume H : a * b = 0,
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have C1 : a = 0, by rewrite [-mul_one, -(mul_one_div_cancel Hb), -mul.assoc, H, zero_mul],
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absurd C1 Ha
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theorem mul_ne_zero_comm (H : a * b ≠ 0) : b * a ≠ 0 :=
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have H2 : a ≠ 0 ∧ b ≠ 0, from ne_zero_and_ne_zero_of_mul_ne_zero H,
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division_ring.mul_ne_zero (and.right H2) (and.left H2)
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theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a :=
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have a ≠ 0, from
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(suppose a = 0,
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have 0 = (1:A), by rewrite [-(zero_mul b), -this, H],
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absurd this zero_ne_one),
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show b = 1 / a, from symm (calc
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1 / a = (1 / a) * 1 : mul_one
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... = (1 / a) * (a * b) : H
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... = (1 / a) * a * b : mul.assoc
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... = 1 * b : one_div_mul_cancel this
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... = b : one_mul)
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theorem eq_one_div_of_mul_eq_one_left (H : b * a = 1) : b = 1 / a :=
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have a ≠ 0, from
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(suppose a = 0,
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have 0 = 1, from symm (calc
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1 = b * a : symm H
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... = b * 0 : this
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... = 0 : mul_zero),
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absurd this zero_ne_one),
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show b = 1 / a, from symm (calc
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1 / a = 1 * (1 / a) : one_mul
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... = b * a * (1 / a) : H
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... = b * (a * (1 / a)) : mul.assoc
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... = b * 1 : mul_one_div_cancel this
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... = b : mul_one)
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theorem division_ring.one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) :
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(1 / a) * (1 / b) = 1 / (b * a) :=
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have (b * a) * ((1 / a) * (1 / b)) = 1, by
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rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul,
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(mul_one_div_cancel Hb)],
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eq_one_div_of_mul_eq_one this
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theorem one_div_neg_one_eq_neg_one : (1:A) / (-1) = -1 :=
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have (-1) * (-1) = (1:A), by rewrite [-neg_eq_neg_one_mul, neg_neg],
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symm (eq_one_div_of_mul_eq_one this)
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theorem division_ring.one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) :=
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have -1 ≠ 0, from
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(suppose -1 = 0, absurd (symm (calc
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1 = -(-1) : neg_neg
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... = -0 : this
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... = (0:A) : neg_zero)) zero_ne_one),
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calc
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1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul
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... = (1 / a) * (1 / (- 1)) : division_ring.one_div_mul_one_div H this
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... = (1 / a) * (-1) : one_div_neg_one_eq_neg_one
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... = - (1 / a) : mul_neg_one_eq_neg
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theorem div_neg_eq_neg_div (b : A) (Ha : a ≠ 0) : b / (- a) = - (b / a) :=
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calc
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b / (- a) = b * (1 / (- a)) : inv_eq_one_div
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... = b * -(1 / a) : division_ring.one_div_neg_eq_neg_one_div Ha
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... = -(b * (1 / a)) : neg_mul_eq_mul_neg
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... = - (b * a⁻¹) : inv_eq_one_div
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theorem neg_div (a b : A) : (-b) / a = - (b / a) :=
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by rewrite [neg_eq_neg_one_mul, mul_div_assoc, -neg_eq_neg_one_mul]
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theorem division_ring.neg_div_neg_eq (a : A) {b : A} (Hb : b ≠ 0) : (-a) / (-b) = a / b :=
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by rewrite [(div_neg_eq_neg_div _ Hb), neg_div, neg_neg]
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theorem division_ring.one_div_one_div (H : a ≠ 0) : 1 / (1 / a) = a :=
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symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel H))
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theorem division_ring.eq_of_one_div_eq_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) (H : 1 / a = 1 / b) :
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a = b :=
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by rewrite [-(division_ring.one_div_one_div Ha), H, (division_ring.one_div_one_div Hb)]
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theorem mul_inv_eq (Ha : a ≠ 0) (Hb : b ≠ 0) : (b * a)⁻¹ = a⁻¹ * b⁻¹ :=
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eq.symm (calc
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a⁻¹ * b⁻¹ = (1 / a) * b⁻¹ : inv_eq_one_div
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... = (1 / a) * (1 / b) : inv_eq_one_div
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... = (1 / (b * a)) : division_ring.one_div_mul_one_div Ha Hb
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... = (b * a)⁻¹ : inv_eq_one_div)
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theorem mul_div_cancel (a : A) {b : A} (Hb : b ≠ 0) : a * b / b = a :=
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by rewrite [↑divide, mul.assoc, (mul_inv_cancel Hb), mul_one]
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theorem div_mul_cancel (a : A) {b : A} (Hb : b ≠ 0) : a / b * b = a :=
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by rewrite [↑divide, mul.assoc, (inv_mul_cancel Hb), mul_one]
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theorem div_add_div_same (a b c : A) : a / c + b / c = (a + b) / c := !right_distrib⁻¹
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theorem div_sub_div_same (a b c : A) : (a / c) - (b / c) = (a - b) / c :=
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by rewrite [sub_eq_add_neg, -neg_div, div_add_div_same]
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theorem one_div_mul_add_mul_one_div_eq_one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) :
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(1 / a) * (a + b) * (1 / b) = 1 / a + 1 / b :=
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by rewrite [(left_distrib (1 / a)), (one_div_mul_cancel Ha), right_distrib, one_mul,
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mul.assoc, (mul_one_div_cancel Hb), mul_one, add.comm]
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theorem one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) :
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(1 / a) * (b - a) * (1 / b) = 1 / a - 1 / b :=
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by rewrite [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel Ha), mul_sub_right_distrib,
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one_mul, mul.assoc, (mul_one_div_cancel Hb), mul_one, one_mul]
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theorem div_eq_one_iff_eq (a : A) {b : A} (Hb : b ≠ 0) : a / b = 1 ↔ a = b :=
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iff.intro
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(suppose a / b = 1, symm (calc
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b = 1 * b : one_mul
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... = a / b * b : this
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... = a : div_mul_cancel _ Hb))
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(suppose a = b, calc
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a / b = b / b : this
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... = 1 : div_self Hb)
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theorem eq_of_div_eq_one (a : A) {b : A} (Hb : b ≠ 0) : a / b = 1 → a = b :=
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iff.mp (!div_eq_one_iff_eq Hb)
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theorem eq_div_iff_mul_eq (a : A) {b : A} (Hc : c ≠ 0) : a = b / c ↔ a * c = b :=
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iff.intro
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(suppose a = b / c, by rewrite [this, (!div_mul_cancel Hc)])
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(suppose a * c = b, by rewrite [-(!mul_div_cancel Hc), this])
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theorem eq_div_of_mul_eq (a b : A) {c : A} (Hc : c ≠ 0) : a * c = b → a = b / c :=
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iff.mpr (!eq_div_iff_mul_eq Hc)
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theorem mul_eq_of_eq_div (a b: A) {c : A} (Hc : c ≠ 0) : a = b / c → a * c = b :=
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iff.mp (!eq_div_iff_mul_eq Hc)
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theorem add_div_eq_mul_add_div (a b : A) {c : A} (Hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
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have (a + b / c) * c = a * c + b, by rewrite [right_distrib, (!div_mul_cancel Hc)],
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(iff.elim_right (!eq_div_iff_mul_eq Hc)) this
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theorem mul_mul_div (a : A) {c : A} (Hc : c ≠ 0) : a = a * c * (1 / c) :=
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calc
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a = a * 1 : mul_one
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... = a * (c * (1 / c)) : mul_one_div_cancel Hc
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... = a * c * (1 / c) : mul.assoc
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-- There are many similar rules to these last two in the Isabelle library
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-- that haven't been ported yet. Do as necessary.
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end division_ring
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structure field [class] (A : Type) extends division_ring A, comm_ring A
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section field
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variables [s : field A] {a b c d: A}
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include s
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local attribute divide [reducible]
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theorem field.one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (a * b) :=
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by rewrite [(division_ring.one_div_mul_one_div Ha Hb), mul.comm b]
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theorem field.div_mul_right (Hb : b ≠ 0) (H : a * b ≠ 0) : a / (a * b) = 1 / b :=
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have a ≠ 0, from and.left (ne_zero_and_ne_zero_of_mul_ne_zero H),
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symm (calc
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1 / b = 1 * (1 / b) : one_mul
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... = (a * a⁻¹) * (1 / b) : mul_inv_cancel this
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... = a * (a⁻¹ * (1 / b)) : mul.assoc
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... = a * ((1 / a) * (1 / b)) : inv_eq_one_div
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... = a * (1 / (b * a)) : division_ring.one_div_mul_one_div this Hb
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... = a * (1 / (a * b)) : mul.comm
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... = a * (a * b)⁻¹ : inv_eq_one_div)
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theorem field.div_mul_left (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a :=
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let H1 : b * a ≠ 0 := mul_ne_zero_comm H in
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by rewrite [mul.comm a, (field.div_mul_right Ha H1)]
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theorem mul_div_cancel_left (Ha : a ≠ 0) : a * b / a = b :=
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by rewrite [mul.comm a, (!mul_div_cancel Ha)]
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theorem mul_div_cancel' (Hb : b ≠ 0) : b * (a / b) = a :=
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by rewrite [mul.comm, (!div_mul_cancel Hb)]
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theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) :=
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assert a * b ≠ 0, from (division_ring.mul_ne_zero Ha Hb),
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by rewrite [add.comm, -(field.div_mul_left Ha this), -(field.div_mul_right Hb this), ↑divide,
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-right_distrib]
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theorem field.div_mul_div (a : A) {b : A} (c : A) {d : A} (Hb : b ≠ 0) (Hd : d ≠ 0) :
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(a / b) * (c / d) = (a * c) / (b * d) :=
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by rewrite [↑divide, 2 mul.assoc, (mul.comm b⁻¹), mul.assoc, (mul_inv_eq Hd Hb)]
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theorem mul_div_mul_left (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) :
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(c * a) / (c * b) = a / b :=
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by rewrite [-(!field.div_mul_div Hc Hb), (div_self Hc), one_mul]
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theorem mul_div_mul_right (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) :
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(a * c) / (b * c) = a / b :=
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by rewrite [(mul.comm a), (mul.comm b), (!mul_div_mul_left Hb Hc)]
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theorem div_mul_eq_mul_div (a b c : A) : (b / c) * a = (b * a) / c :=
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by rewrite [↑divide, mul.assoc, (mul.comm c⁻¹), -mul.assoc]
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theorem field.div_mul_eq_mul_div_comm (a b : A) {c : A} (Hc : c ≠ 0) :
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(b / c) * a = b * (a / c) :=
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by rewrite [(div_mul_eq_mul_div), -(one_mul c), -(!field.div_mul_div (ne.symm zero_ne_one) Hc),
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div_one, one_mul]
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theorem div_add_div (a : A) {b : A} (c : A) {d : A} (Hb : b ≠ 0) (Hd : d ≠ 0) :
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(a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) :=
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by rewrite [-(!mul_div_mul_right Hb Hd), -(!mul_div_mul_left Hd Hb), div_add_div_same]
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theorem div_sub_div (a : A) {b : A} (c : A) {d : A} (Hb : b ≠ 0) (Hd : d ≠ 0) :
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(a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) :=
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by rewrite [↑sub, neg_eq_neg_one_mul, -mul_div_assoc, (!div_add_div Hb Hd),
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-mul.assoc, (mul.comm b), mul.assoc, -neg_eq_neg_one_mul]
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theorem mul_eq_mul_of_div_eq_div (a : A) {b : A} (c : A) {d : A} (Hb : b ≠ 0)
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(Hd : d ≠ 0) (H : a / b = c / d) : a * d = c * b :=
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by rewrite [-mul_one, mul.assoc, (mul.comm d), -mul.assoc, -(div_self Hb),
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-(!field.div_mul_eq_mul_div_comm Hb), H, (div_mul_eq_mul_div), (!div_mul_cancel Hd)]
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theorem field.one_div_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / (a / b) = b / a :=
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have (a / b) * (b / a) = 1, from calc
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(a / b) * (b / a) = (a * b) / (b * a) : !field.div_mul_div Hb Ha
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... = (a * b) / (a * b) : mul.comm
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... = 1 : div_self (division_ring.mul_ne_zero Ha Hb),
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symm (eq_one_div_of_mul_eq_one this)
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theorem field.div_div_eq_mul_div (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) :
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a / (b / c) = (a * c) / b :=
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by rewrite [div_eq_mul_one_div, (field.one_div_div Hb Hc), -mul_div_assoc]
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theorem field.div_div_eq_div_mul (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) :
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(a / b) / c = a / (b * c) :=
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by rewrite [div_eq_mul_one_div, (!field.div_mul_div Hb Hc), mul_one]
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theorem field.div_div_div_div_eq (a : A) {b c d : A} (Hb : b ≠ 0) (Hc : c ≠ 0) (Hd : d ≠ 0) :
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(a / b) / (c / d) = (a * d) / (b * c) :=
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by rewrite [(!field.div_div_eq_mul_div Hc Hd), (div_mul_eq_mul_div),
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(!field.div_div_eq_div_mul Hb Hc)]
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theorem field.div_mul_eq_div_mul_one_div (a : A) {b c : A} (Hb : b ≠ 0) (Hc : c ≠ 0) :
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a / (b * c) = (a / b) * (1 / c) :=
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by rewrite [-!field.div_div_eq_div_mul Hb Hc, -div_eq_mul_one_div]
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end field
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structure discrete_field [class] (A : Type) extends field A :=
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(has_decidable_eq : decidable_eq A)
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(inv_zero : inv zero = zero)
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attribute discrete_field.has_decidable_eq [instance]
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section discrete_field
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variable [s : discrete_field A]
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include s
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variables {a b c d : A}
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-- many of the theorems in discrete_field are the same as theorems in field or division ring,
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-- but with fewer hypotheses since 0⁻¹ = 0 and equality is decidable.
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theorem discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero
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(x y : A) (H : x * y = 0) : x = 0 ∨ y = 0 :=
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decidable.by_cases
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(suppose x = 0, or.inl this)
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(suppose x ≠ 0,
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or.inr (by rewrite [-one_mul, -(inv_mul_cancel this), mul.assoc, H, mul_zero]))
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definition discrete_field.to_integral_domain [trans-instance] [reducible] [coercion] :
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integral_domain A :=
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⦃ integral_domain, s,
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eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero⦄
|
||
|
||
theorem inv_zero : 0⁻¹ = (0:A) := !discrete_field.inv_zero
|
||
|
||
theorem one_div_zero : 1 / 0 = (0:A) :=
|
||
calc
|
||
1 / 0 = 1 * 0⁻¹ : refl
|
||
... = 1 * 0 : discrete_field.inv_zero A
|
||
... = 0 : mul_zero
|
||
|
||
theorem div_zero (a : A) : a / 0 = 0 := by rewrite [div_eq_mul_one_div, one_div_zero, mul_zero]
|
||
|
||
theorem ne_zero_of_one_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 :=
|
||
assume Ha : a = 0, absurd (Ha⁻¹ ▸ one_div_zero) H
|
||
|
||
theorem eq_zero_of_one_div_eq_zero (H : 1 / a = 0) : a = 0 :=
|
||
decidable.by_cases
|
||
(assume Ha, Ha)
|
||
(assume Ha, false.elim ((one_div_ne_zero Ha) H))
|
||
|
||
variables (a b)
|
||
theorem one_div_mul_one_div' : (1 / a) * (1 / b) = 1 / (b * a) :=
|
||
decidable.by_cases
|
||
(suppose a = 0,
|
||
by rewrite [this, div_zero, zero_mul, -(@div_zero A s 1), mul_zero b])
|
||
(assume Ha : a ≠ 0,
|
||
decidable.by_cases
|
||
(suppose b = 0,
|
||
by rewrite [this, div_zero, mul_zero, -(@div_zero A s 1), zero_mul a])
|
||
(suppose b ≠ 0, division_ring.one_div_mul_one_div Ha this))
|
||
|
||
theorem one_div_neg_eq_neg_one_div : 1 / (- a) = - (1 / a) :=
|
||
decidable.by_cases
|
||
(suppose a = 0, by rewrite [this, neg_zero, 2 div_zero, neg_zero])
|
||
(suppose a ≠ 0, division_ring.one_div_neg_eq_neg_one_div this)
|
||
|
||
theorem neg_div_neg_eq : (-a) / (-b) = a / b :=
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, neg_zero, 2 div_zero])
|
||
(assume Hb : b ≠ 0, !division_ring.neg_div_neg_eq Hb)
|
||
|
||
theorem one_div_one_div : 1 / (1 / a) = a :=
|
||
decidable.by_cases
|
||
(assume Ha : a = 0, by rewrite [Ha, 2 div_zero])
|
||
(assume Ha : a ≠ 0, division_ring.one_div_one_div Ha)
|
||
|
||
variables {a b}
|
||
theorem eq_of_one_div_eq_one_div (H : 1 / a = 1 / b) : a = b :=
|
||
decidable.by_cases
|
||
(assume Ha : a = 0,
|
||
have Hb : b = 0, from eq_zero_of_one_div_eq_zero (by rewrite [-H, Ha, div_zero]),
|
||
Hb⁻¹ ▸ Ha)
|
||
(assume Ha : a ≠ 0,
|
||
have Hb : b ≠ 0, from ne_zero_of_one_div_ne_zero (H ▸ (one_div_ne_zero Ha)),
|
||
division_ring.eq_of_one_div_eq_one_div Ha Hb H)
|
||
|
||
variables (a b)
|
||
theorem mul_inv' : (b * a)⁻¹ = a⁻¹ * b⁻¹ :=
|
||
decidable.by_cases
|
||
(assume Ha : a = 0, by rewrite [Ha, mul_zero, 2 inv_zero, zero_mul])
|
||
(assume Ha : a ≠ 0,
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, zero_mul, 2 inv_zero, mul_zero])
|
||
(assume Hb : b ≠ 0, mul_inv_eq Ha Hb))
|
||
|
||
-- the following are specifically for fields
|
||
theorem one_div_mul_one_div : (1 / a) * (1 / b) = 1 / (a * b) :=
|
||
by rewrite [one_div_mul_one_div', mul.comm b]
|
||
|
||
variable {a}
|
||
theorem div_mul_right (Ha : a ≠ 0) : a / (a * b) = 1 / b :=
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, mul_zero, 2 div_zero])
|
||
(assume Hb : b ≠ 0, field.div_mul_right Hb (mul_ne_zero Ha Hb))
|
||
|
||
variables (a) {b}
|
||
theorem div_mul_left (Hb : b ≠ 0) : b / (a * b) = 1 / a :=
|
||
by rewrite [mul.comm a, div_mul_right _ Hb]
|
||
|
||
variables (a b c)
|
||
theorem div_mul_div : (a / b) * (c / d) = (a * c) / (b * d) :=
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, div_zero, zero_mul, -(@div_zero A s (a * c)), zero_mul])
|
||
(assume Hb : b ≠ 0,
|
||
decidable.by_cases
|
||
(assume Hd : d = 0, by rewrite [Hd, div_zero, mul_zero, -(@div_zero A s (a * c)),
|
||
mul_zero])
|
||
(assume Hd : d ≠ 0, !field.div_mul_div Hb Hd))
|
||
|
||
variable {c}
|
||
theorem mul_div_mul_left' (Hc : c ≠ 0) : (c * a) / (c * b) = a / b :=
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, mul_zero, 2 div_zero])
|
||
(assume Hb : b ≠ 0, !mul_div_mul_left Hb Hc)
|
||
|
||
theorem mul_div_mul_right' (Hc : c ≠ 0) : (a * c) / (b * c) = a / b :=
|
||
by rewrite [(mul.comm a), (mul.comm b), (!mul_div_mul_left' Hc)]
|
||
|
||
variables (a b c d)
|
||
theorem div_mul_eq_mul_div_comm : (b / c) * a = b * (a / c) :=
|
||
decidable.by_cases
|
||
(assume Hc : c = 0, by rewrite [Hc, div_zero, zero_mul, -(mul_zero b), -(@div_zero A s a)])
|
||
(assume Hc : c ≠ 0, !field.div_mul_eq_mul_div_comm Hc)
|
||
|
||
theorem one_div_div : 1 / (a / b) = b / a :=
|
||
decidable.by_cases
|
||
(assume Ha : a = 0, by rewrite [Ha, zero_div, 2 div_zero])
|
||
(assume Ha : a ≠ 0,
|
||
decidable.by_cases
|
||
(assume Hb : b = 0, by rewrite [Hb, 2 div_zero, zero_div])
|
||
(assume Hb : b ≠ 0, field.one_div_div Ha Hb))
|
||
|
||
theorem div_div_eq_mul_div : a / (b / c) = (a * c) / b :=
|
||
by rewrite [div_eq_mul_one_div, one_div_div, -mul_div_assoc]
|
||
|
||
theorem div_div_eq_div_mul : (a / b) / c = a / (b * c) :=
|
||
by rewrite [div_eq_mul_one_div, div_mul_div, mul_one]
|
||
|
||
theorem div_div_div_div_eq : (a / b) / (c / d) = (a * d) / (b * c) :=
|
||
by rewrite [div_div_eq_mul_div, div_mul_eq_mul_div, div_div_eq_div_mul]
|
||
|
||
variable {a}
|
||
theorem div_helper (H : a ≠ 0) : (1 / (a * b)) * a = 1 / b :=
|
||
by rewrite [div_mul_eq_mul_div, one_mul, !div_mul_right H]
|
||
|
||
variable (a)
|
||
theorem div_mul_eq_div_mul_one_div : a / (b * c) = (a / b) * (1 / c) :=
|
||
by rewrite [-div_div_eq_div_mul, -div_eq_mul_one_div]
|
||
end discrete_field
|
||
|
||
end algebra
|