lean2/library/algebra/ring.lean

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/-
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.
-/
2014-12-01 04:34:12 +00:00
import logic.eq 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] (A : Type) extends has_mul A, has_zero A :=
(zero_mul_eq : ∀a, mul zero a = zero)
(mul_zero_eq : ∀a, mul a zero = zero)
theorem zero_mul_eq [s : mul_zero A] (a : A) : 0 * a = 0 := !mul_zero.zero_mul_eq
theorem mul_zero_eq [s : mul_zero A] (a : A) : a * 0 = 0 := !mul_zero.mul_zero_eq
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 A,
zero_ne_one_class A
structure comm_semiring [class] (A : Type) extends semiring A, comm_semigroup A
/- abstract divisibility -/
structure has_dvd [class] (A : Type) extends has_mul A :=
(dvd : A → A → Prop)
(dvd_intro : ∀a b c, mul a b = c → dvd a c)
(dvd_ex : ∀ a b, dvd a b → ∃c, mul a c = b)
definition dvd [s : has_dvd A] (a b : A) : Prop := has_dvd.dvd a b
infix `|` := dvd
theorem dvd.intro [s : has_dvd A] {a b c : A} : a * b = c → a | c := !has_dvd.dvd_intro
theorem dvd.ex [s : has_dvd A] {a b : A} : a | b → ∃c, a * c = b := !has_dvd.dvd_ex
theorem dvd.elim [s : has_dvd A] {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
exists_elim (dvd.ex H₁) H₂
structure comm_semiring_dvd [class] (A : Type) extends comm_semiring A, has_dvd A
section comm_semiring_dvd
variables [s : comm_semiring_dvd A] (a b c : A)
include s
theorem dvd.refl : a | a := dvd.intro (!mul.right_id)
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 {H : 0 | a} : a = 0 :=
dvd.elim H (take c, assume H' : 0 * c = a, (H')⁻¹ ⬝ !zero_mul_eq)
theorem dvd_zero : a | 0 := dvd.intro !mul_zero_eq
theorem one_dvd : 1 | a := dvd.intro !mul.left_id
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_dvd
/- ring -/
structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
definition ring.to_semiring [instance] [s : ring A] : semiring A :=
semiring.mk ring.add ring.add_assoc !ring.zero ring.add_left_id
add.right_id -- note: we've shown that add_right_id follows from add_left_id in add_comm_group
ring.add_comm ring.mul ring.mul_assoc !ring.one ring.mul_left_id ring.mul_right_id
ring.left_distrib ring.right_distrib
(take a,
have H : 0 * a + 0 = 0 * a + 0 * a, from
calc
0 * a + 0 = 0 * a : add.right_id
... = (0 + 0) * a : {(add.right_id 0)⁻¹}
... = 0 * a + 0 * a : ring.right_distrib,
show 0 * a = 0, from (add.left_cancel H)⁻¹)
(take a,
have H : a * 0 + 0 = a * 0 + a * 0, from
calc
a * 0 + 0 = a * 0 : add.right_id
... = a * (0 + 0) : {(add.right_id 0)⁻¹}
... = a * 0 + a * 0 : ring.left_distrib,
show a * 0 = 0, from (add.left_cancel H)⁻¹)
!ring.zero_ne_one
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_eq)
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_eq)
theorem neg_mul_neg_eq : -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_eq
theorem neg_mul_comm : -a * b = a * -b := !neg_mul_eq_neg_mul⁻¹ ⬝ !neg_mul_eq_mul_neg
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] [s : comm_ring A] : comm_semiring A :=
comm_semiring.mk comm_ring.add comm_ring.add_assoc (@comm_ring.zero A s)
comm_ring.add_left_id comm_ring.add_right_id comm_ring.add_comm comm_ring.mul comm_ring.mul_assoc
(@comm_ring.one A s) comm_ring.mul_left_id comm_ring.mul_right_id comm_ring.left_distrib
comm_ring.right_distrib zero_mul_eq mul_zero_eq (@comm_ring.zero_ne_one A s)
comm_ring.mul_comm
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.right_id 1 ▸ mul_self_sub_mul_self_eq a 1
end
structure comm_ring_dvd [class] (A : Type) extends comm_ring A, has_dvd A
definition comm_ring_dvd.to_comm_semiring_dvd [instance] [s : comm_ring_dvd A] :
comm_semiring_dvd A :=
comm_semiring_dvd.mk comm_ring_dvd.add comm_ring_dvd.add_assoc (@comm_ring_dvd.zero A s)
comm_ring_dvd.add_left_id comm_ring_dvd.add_right_id comm_ring_dvd.add_comm
comm_ring_dvd.mul comm_ring_dvd.mul_assoc (@comm_ring_dvd.one A s) comm_ring_dvd.mul_left_id
comm_ring_dvd.mul_right_id comm_ring_dvd.left_distrib comm_ring_dvd.right_distrib
zero_mul_eq mul_zero_eq (@comm_ring_dvd.zero_ne_one A s) comm_ring_dvd.mul_comm
comm_ring_dvd.dvd (@comm_ring_dvd.dvd_intro A s) (@comm_ring_dvd.dvd_ex A s)
section
variables [s : comm_ring_dvd A] (a b c d e : A)
include s
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_eq)))
(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_eq
... = 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 -/
-- TODO: some properties here may extend to cancellative semirings. It is worth the effort?
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_dvd A, no_zero_divisors A
section
variables [s : integral_domain A] (a b c d e : A)
include s
theorem mul_ne_zero_of_ne_zero_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