342 lines
11 KiB
Text
342 lines
11 KiB
Text
/-
|
||
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: algebra.ring
|
||
Authors: Jeremy Avigad, Leonardo de Moura
|
||
|
||
Structures with multiplicative and additive components, including semirings, rings, and fields.
|
||
The development is modeled after Isabelle's library.
|
||
-/
|
||
|
||
import logic.eq logic.connectives data.unit data.sigma data.prod
|
||
import algebra.function algebra.binary algebra.group
|
||
open eq eq.ops
|
||
|
||
namespace algebra
|
||
|
||
variable {A : Type}
|
||
|
||
/- auxiliary classes -/
|
||
|
||
structure distrib [class] (A : Type) extends has_mul A, has_add A :=
|
||
(left_distrib : ∀a b c, mul a (add b c) = add (mul a b) (mul a c))
|
||
(right_distrib : ∀a b c, mul (add a b) c = add (mul a c) (mul b c))
|
||
|
||
theorem left_distrib [s : distrib A] (a b c : A) : a * (b + c) = a * b + a * c :=
|
||
!distrib.left_distrib
|
||
|
||
theorem right_distrib [s: distrib A] (a b c : A) : (a + b) * c = a * c + b * c :=
|
||
!distrib.right_distrib
|
||
|
||
structure mul_zero_class [class] (A : Type) extends has_mul A, has_zero A :=
|
||
(zero_mul : ∀a, mul zero a = zero)
|
||
(mul_zero : ∀a, mul a zero = zero)
|
||
|
||
theorem zero_mul [s : mul_zero_class A] (a : A) : 0 * a = 0 := !mul_zero_class.zero_mul
|
||
theorem mul_zero [s : mul_zero_class A] (a : A) : a * 0 = 0 := !mul_zero_class.mul_zero
|
||
|
||
structure zero_ne_one_class [class] (A : Type) extends has_zero A, has_one A :=
|
||
(zero_ne_one : zero ≠ one)
|
||
|
||
theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ 1 := @zero_ne_one_class.zero_ne_one A s
|
||
|
||
/- semiring -/
|
||
|
||
structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A,
|
||
mul_zero_class A, zero_ne_one_class A
|
||
|
||
section semiring
|
||
variables [s : semiring A] (a b c : A)
|
||
include s
|
||
|
||
theorem ne_zero_of_mul_ne_zero_right {a b : A} (H : a * b ≠ 0) : a ≠ 0 :=
|
||
assume H1 : a = 0,
|
||
have H2 : a * b = 0, from H1⁻¹ ▸ zero_mul b,
|
||
H H2
|
||
|
||
theorem ne_zero_of_mul_ne_zero_left {a b : A} (H : a * b ≠ 0) : b ≠ 0 :=
|
||
assume H1 : b = 0,
|
||
have H2 : a * b = 0, from H1⁻¹ ▸ mul_zero a,
|
||
H H2
|
||
end semiring
|
||
|
||
/- comm semiring -/
|
||
|
||
structure comm_semiring [class] (A : Type) extends semiring A, comm_semigroup A
|
||
-- TODO: we could also define a cancelative comm_semiring, i.e. satisfying
|
||
-- c ≠ 0 → c * a = c * b → a = b.
|
||
|
||
section comm_semiring
|
||
variables [s : comm_semiring A] (a b c : A)
|
||
include s
|
||
|
||
definition dvd (a b : A) : Prop := ∃c, a * c = b
|
||
infix `|` := dvd
|
||
|
||
theorem dvd.intro {a b c : A} (H : a * b = c) : a | c :=
|
||
exists.intro _ H
|
||
|
||
theorem dvd.intro_right {a b c : A} (H : a * b = c) : b | c :=
|
||
dvd.intro (!mul.comm ▸ H)
|
||
|
||
theorem dvd.ex {a b : A} (H : a | b) : ∃c, a * c = b := H
|
||
|
||
theorem dvd.elim {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
|
||
exists.elim H₁ H₂
|
||
|
||
theorem dvd.refl : a | a := dvd.intro !mul_one
|
||
|
||
theorem dvd.trans {a b c : A} (H₁ : a | b) (H₂ : b | c) : a | c :=
|
||
dvd.elim H₁
|
||
(take d, assume H₃ : a * d = b,
|
||
dvd.elim H₂
|
||
(take e, assume H₄ : b * e = c,
|
||
@dvd.intro _ _ _ (d * e) _
|
||
(calc
|
||
a * (d * e) = (a * d) * e : mul.assoc
|
||
... = b * e : H₃
|
||
... = c : H₄)))
|
||
|
||
theorem eq_zero_of_zero_dvd {a : A} (H : 0 | a) : a = 0 :=
|
||
dvd.elim H (take c, assume H' : 0 * c = a, (H')⁻¹ ⬝ !zero_mul)
|
||
|
||
theorem dvd_zero : a | 0 := dvd.intro !mul_zero
|
||
|
||
theorem one_dvd : 1 | a := dvd.intro !one_mul
|
||
|
||
theorem dvd_mul_right : a | a * b := dvd.intro rfl
|
||
|
||
theorem dvd_mul_left : a | b * a := mul.comm a b ▸ dvd_mul_right a b
|
||
|
||
theorem dvd_mul_of_dvd_left {a b : A} (H : a | b) (c : A) : a | b * c :=
|
||
dvd.elim H
|
||
(take d,
|
||
assume H₁ : a * d = b,
|
||
dvd.intro
|
||
(calc
|
||
a * (d * c) = a * d * c : (!mul.assoc)⁻¹
|
||
... = b * c : H₁))
|
||
|
||
theorem dvd_mul_of_dvd_right {a b : A} (H : a | b) (c : A) : a | c * b :=
|
||
!mul.comm ▸ (dvd_mul_of_dvd_left H _)
|
||
|
||
theorem mul_dvd_mul {a b c d : A} (dvd_ab : a | b) (dvd_cd : c | d) : a * c | b * d :=
|
||
dvd.elim dvd_ab
|
||
(take e, assume Haeb : a * e = b,
|
||
dvd.elim dvd_cd
|
||
(take f, assume Hcfd : c * f = d,
|
||
dvd.intro
|
||
(calc
|
||
a * c * (e * f) = a * (c * (e * f)) : mul.assoc
|
||
... = a * (e * (c * f)) : mul.left_comm
|
||
... = a * e * (c * f) : (!mul.assoc)⁻¹
|
||
... = b * (c * f) : Haeb
|
||
... = b * d : Hcfd)))
|
||
|
||
theorem dvd_of_mul_right_dvd {a b c : A} (H : a * b | c) : a | c :=
|
||
dvd.elim H (take d, assume Habdc : a * b * d = c, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc))
|
||
|
||
theorem dvd_of_mul_left_dvd {a b c : A} (H : a * b | c) : b | c :=
|
||
dvd_of_mul_right_dvd (mul.comm a b ▸ H)
|
||
|
||
theorem dvd_add {a b c : A} (Hab : a | b) (Hac : a | c) : a | b + c :=
|
||
dvd.elim Hab
|
||
(take d, assume Hadb : a * d = b,
|
||
dvd.elim Hac
|
||
(take e, assume Haec : a * e = c,
|
||
dvd.intro (show a * (d + e) = b + c, from Hadb ▸ Haec ▸ left_distrib a d e)))
|
||
end comm_semiring
|
||
|
||
/- ring -/
|
||
|
||
structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
|
||
|
||
theorem ring.mul_zero [s : ring A] (a : A) : a * 0 = 0 :=
|
||
have H : a * 0 + 0 = a * 0 + a * 0, from calc
|
||
a * 0 + 0 = a * 0 : add_zero
|
||
... = a * (0 + 0) : {(add_zero 0)⁻¹}
|
||
... = a * 0 + a * 0 : ring.left_distrib,
|
||
show a * 0 = 0, from (add.left_cancel H)⁻¹
|
||
|
||
theorem ring.zero_mul [s : ring A] (a : A) : 0 * a = 0 :=
|
||
have H : 0 * a + 0 = 0 * a + 0 * a, from calc
|
||
0 * a + 0 = 0 * a : add_zero
|
||
... = (0 + 0) * a : {(add_zero 0)⁻¹}
|
||
... = 0 * a + 0 * a : ring.right_distrib,
|
||
show 0 * a = 0, from (add.left_cancel H)⁻¹
|
||
|
||
definition ring.to_semiring [instance] [coercion] [reducible] [s : ring A] : semiring A :=
|
||
⦃ semiring, s,
|
||
mul_zero := ring.mul_zero,
|
||
zero_mul := ring.zero_mul ⦄
|
||
|
||
section
|
||
variables [s : ring A] (a b c d e : A)
|
||
include s
|
||
|
||
theorem neg_mul_eq_neg_mul : -(a * b) = -a * b :=
|
||
neg_eq_of_add_eq_zero
|
||
(calc
|
||
a * b + -a * b = (a + -a) * b : right_distrib
|
||
... = 0 * b : add.right_inv
|
||
... = 0 : zero_mul)
|
||
|
||
theorem neg_mul_eq_mul_neg : -(a * b) = a * -b :=
|
||
neg_eq_of_add_eq_zero
|
||
(calc
|
||
a * b + a * -b = a * (b + -b) : left_distrib
|
||
... = a * 0 : add.right_inv
|
||
... = 0 : mul_zero)
|
||
|
||
theorem neg_mul_neg : -a * -b = a * b :=
|
||
calc
|
||
-a * -b = -(a * -b) : !neg_mul_eq_neg_mul⁻¹
|
||
... = - -(a * b) : neg_mul_eq_mul_neg
|
||
... = a * b : neg_neg
|
||
|
||
theorem neg_mul_comm : -a * b = a * -b := !neg_mul_eq_neg_mul⁻¹ ⬝ !neg_mul_eq_mul_neg
|
||
|
||
theorem neg_eq_neg_one_mul : -a = -1 * a :=
|
||
calc
|
||
-a = -(1 * a) : one_mul
|
||
... = -1 * a : neg_mul_eq_neg_mul
|
||
|
||
theorem mul_sub_left_distrib : a * (b - c) = a * b - a * c :=
|
||
calc
|
||
a * (b - c) = a * b + a * -c : left_distrib
|
||
... = a * b + - (a * c) : {!neg_mul_eq_mul_neg⁻¹}
|
||
... = a * b - a * c : rfl
|
||
|
||
theorem mul_sub_right_distrib : (a - b) * c = a * c - b * c :=
|
||
calc
|
||
(a - b) * c = a * c + -b * c : right_distrib
|
||
... = a * c + - (b * c) : {!neg_mul_eq_neg_mul⁻¹}
|
||
... = a * c - b * c : rfl
|
||
|
||
-- TODO: can calc mode be improved to make this easier?
|
||
-- TODO: there is also the other direction. It will be easier when we
|
||
-- have the simplifier.
|
||
|
||
theorem mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d :=
|
||
calc
|
||
a * e + c = b * e + d ↔ a * e + c = d + b * e : !add.comm ▸ !iff.refl
|
||
... ↔ a * e + c - b * e = d : iff.symm !sub_eq_iff_eq_add
|
||
... ↔ a * e - b * e + c = d : !sub_add_eq_add_sub ▸ !iff.refl
|
||
... ↔ (a - b) * e + c = d : !mul_sub_right_distrib ▸ !iff.refl
|
||
end
|
||
|
||
structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
|
||
|
||
definition comm_ring.to_comm_semiring [instance] [coercion] [reducible] [s : comm_ring A] : comm_semiring A :=
|
||
⦃ comm_semiring, s,
|
||
mul_zero := mul_zero,
|
||
zero_mul := zero_mul ⦄
|
||
|
||
section
|
||
variables [s : comm_ring A] (a b c d e : A)
|
||
include s
|
||
|
||
-- TODO: wait for the simplifier
|
||
theorem mul_self_sub_mul_self_eq : a * a - b * b = (a + b) * (a - b) := sorry
|
||
|
||
theorem mul_self_sub_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
|
||
mul_one 1 ▸ mul_self_sub_mul_self_eq a 1
|
||
|
||
theorem dvd_neg_iff_dvd : a | -b ↔ a | b :=
|
||
iff.intro
|
||
(assume H : a | -b,
|
||
dvd.elim H
|
||
(take c, assume H' : a * c = -b,
|
||
dvd.intro
|
||
(calc
|
||
a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
|
||
... = -(-b) : {H'}
|
||
... = b : neg_neg)))
|
||
(assume H : a | b,
|
||
dvd.elim H
|
||
(take c, assume H' : a * c = b,
|
||
dvd.intro
|
||
(calc
|
||
a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
|
||
... = -b : {H'})))
|
||
|
||
theorem neg_dvd_iff_dvd : -a | b ↔ a | b :=
|
||
iff.intro
|
||
(assume H : -a | b,
|
||
dvd.elim H
|
||
(take c, assume H' : -a * c = b,
|
||
dvd.intro
|
||
(calc
|
||
a * -c = -a * c : !neg_mul_comm⁻¹
|
||
... = b : H')))
|
||
(assume H : a | b,
|
||
dvd.elim H
|
||
(take c, assume H' : a * c = b,
|
||
dvd.intro
|
||
(calc
|
||
-a * -c = a * c : neg_mul_neg
|
||
... = b : H')))
|
||
|
||
theorem dvd_sub (H₁ : a | b) (H₂ : a | c) : a | (b - c) :=
|
||
dvd_add H₁ (iff.elim_right !dvd_neg_iff_dvd H₂)
|
||
end
|
||
|
||
/- integral domains -/
|
||
|
||
structure no_zero_divisors [class] (A : Type) extends has_mul A, has_zero A :=
|
||
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀a b, mul a b = zero → a = zero ∨ b = zero)
|
||
|
||
theorem eq_zero_or_eq_zero_of_mul_eq_zero {A : Type} [s : no_zero_divisors A] {a b : A}
|
||
(H : a * b = 0) :
|
||
a = 0 ∨ b = 0 := !no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero H
|
||
|
||
structure integral_domain [class] (A : Type) extends comm_ring A, no_zero_divisors A
|
||
|
||
section
|
||
|
||
variables [s : integral_domain A] (a b c d e : A)
|
||
include s
|
||
|
||
theorem mul_ne_zero {a b : A} (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
|
||
assume H : a * b = 0,
|
||
or.elim (eq_zero_or_eq_zero_of_mul_eq_zero H) (assume H3, H1 H3) (assume H4, H2 H4)
|
||
|
||
theorem mul.cancel_right {a b c : A} (Ha : a ≠ 0) (H : b * a = c * a) : b = c :=
|
||
have H1 : b * a - c * a = 0, from iff.mp !eq_iff_sub_eq_zero H,
|
||
have H2 : (b - c) * a = 0, from eq.trans !mul_sub_right_distrib H1,
|
||
have H3 : b - c = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
|
||
iff.elim_right !eq_iff_sub_eq_zero H3
|
||
|
||
theorem mul.cancel_left {a b c : A} (Ha : a ≠ 0) (H : a * b = a * c) : b = c :=
|
||
have H1 : a * b - a * c = 0, from iff.mp !eq_iff_sub_eq_zero H,
|
||
have H2 : a * (b - c) = 0, from eq.trans !mul_sub_left_distrib H1,
|
||
have H3 : b - c = 0, from or_resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
|
||
iff.elim_right !eq_iff_sub_eq_zero H3
|
||
|
||
-- TODO: do we want the iff versions?
|
||
|
||
-- TODO: wait for simplifier
|
||
theorem mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b := sorry
|
||
|
||
theorem mul_self_eq_one_iff (a : A) : a * a = 1 ↔ a = 1 ∨ a = -1 := sorry
|
||
|
||
-- TODO: c - b * c → c = 0 ∨ b = 1 and variants
|
||
|
||
theorem dvd_of_mul_dvd_mul_left {a b c : A} (Ha : a ≠ 0) (Hdvd : a * b | a * c) : b | c :=
|
||
dvd.elim Hdvd
|
||
(take d,
|
||
assume H : a * b * d = a * c,
|
||
have H1 : b * d = c, from mul.cancel_left Ha (mul.assoc a b d ▸ H),
|
||
dvd.intro H1)
|
||
|
||
theorem dvd_of_mul_dvd_mul_right {a b c : A} (Ha : a ≠ 0) (Hdvd : b * a | c * a) : b | c :=
|
||
dvd.elim Hdvd
|
||
(take d,
|
||
assume H : b * a * d = c * a,
|
||
have H1 : b * d * a = c * a, from eq.trans !mul.right_comm H,
|
||
have H2 : b * d = c, from mul.cancel_right Ha H1,
|
||
dvd.intro H2)
|
||
end
|
||
|
||
end algebra
|