359 lines
12 KiB
Text
359 lines
12 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.ring
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Authors: Jeremy Avigad, Leonardo de Moura
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Structures with multiplicative and additive components, including semirings, 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.function algebra.binary algebra.group
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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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/- auxiliary classes -/
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structure distrib [class] (A : Type) extends has_mul A, has_add A :=
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(left_distrib : ∀a b c, mul a (add b c) = add (mul a b) (mul a c))
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(right_distrib : ∀a b c, mul (add a b) c = add (mul a c) (mul b c))
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theorem left_distrib [s : distrib A] (a b c : A) : a * (b + c) = a * b + a * c :=
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!distrib.left_distrib
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theorem right_distrib [s: distrib A] (a b c : A) : (a + b) * c = a * c + b * c :=
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!distrib.right_distrib
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structure mul_zero [class] (A : Type) extends has_mul A, has_zero A :=
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(zero_mul_eq : ∀a, mul zero a = zero)
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(mul_zero_eq : ∀a, mul a zero = zero)
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theorem zero_mul_eq [s : mul_zero A] (a : A) : 0 * a = 0 := !mul_zero.zero_mul_eq
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theorem mul_zero_eq [s : mul_zero A] (a : A) : a * 0 = 0 := !mul_zero.mul_zero_eq
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structure zero_ne_one_class [class] (A : Type) extends has_zero A, has_one A :=
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(zero_ne_one : zero ≠ one)
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theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ 1 := @zero_ne_one_class.zero_ne_one A s
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/- semiring -/
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structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A, mul_zero A,
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zero_ne_one_class A
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structure comm_semiring [class] (A : Type) extends semiring A, comm_semigroup A
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-- TODO: we could also define a cancelative comm_semiring, i.e. satisfying c ≠ 0 → c * a = c * b → a = b.
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/- abstract divisibility -/
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structure has_dvd [class] (A : Type) extends has_mul A :=
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(dvd : A → A → Prop)
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(dvd_intro : ∀a b c, mul a b = c → dvd a c)
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(dvd_ex : ∀ a b, dvd a b → ∃c, mul a c = b)
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definition dvd [s : has_dvd A] (a b : A) : Prop := has_dvd.dvd a b
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infix `|` := dvd
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theorem dvd.intro [s : has_dvd A] {a b c : A} : a * b = c → a | c := !has_dvd.dvd_intro
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theorem dvd.ex [s : has_dvd A] {a b : A} : a | b → ∃c, a * c = b := !has_dvd.dvd_ex
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theorem dvd.elim [s : has_dvd A] {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
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exists.elim (dvd.ex H₁) H₂
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structure comm_semiring_dvd [class] (A : Type) extends comm_semiring A, has_dvd A
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section comm_semiring_dvd
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variables [s : comm_semiring_dvd A] (a b c : A)
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include s
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theorem dvd.refl : a | a := dvd.intro (!mul.right_id)
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theorem dvd.trans {a b c : A} (H₁ : a | b) (H₂ : b | c) : a | c :=
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dvd.elim H₁
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(take d, assume H₃ : a * d = b,
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dvd.elim H₂
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(take e, assume H₄ : b * e = c,
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@dvd.intro _ _ _ (d * e) _
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(calc
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a * (d * e) = (a * d) * e : mul.assoc
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... = b * e : H₃
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... = c : H₄)))
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theorem eq_zero_of_zero_dvd {H : 0 | a} : a = 0 :=
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dvd.elim H (take c, assume H' : 0 * c = a, (H')⁻¹ ⬝ !zero_mul_eq)
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theorem dvd_zero : a | 0 := dvd.intro !mul_zero_eq
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theorem one_dvd : 1 | a := dvd.intro !mul.left_id
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theorem dvd_mul_right : a | a * b := dvd.intro rfl
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theorem dvd_mul_left : a | b * a := mul.comm a b ▸ dvd_mul_right a b
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theorem dvd_mul_of_dvd_left {a b : A} (H : a | b) (c : A) : a | b * c :=
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dvd.elim H
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(take d,
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assume H₁ : a * d = b,
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dvd.intro
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(calc
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a * (d * c) = a * d * c : (!mul.assoc)⁻¹
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... = b * c : H₁))
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theorem dvd_mul_of_dvd_right {a b : A} (H : a | b) (c : A) : a | c * b :=
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!mul.comm ▸ (dvd_mul_of_dvd_left H _)
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theorem mul_dvd_mul {a b c d : A} (dvd_ab : a | b) (dvd_cd : c | d) : a * c | b * d :=
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dvd.elim dvd_ab
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(take e, assume Haeb : a * e = b,
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dvd.elim dvd_cd
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(take f, assume Hcfd : c * f = d,
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dvd.intro
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(calc
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a * c * (e * f) = a * (c * (e * f)) : mul.assoc
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... = a * (e * (c * f)) : mul.left_comm
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... = a * e * (c * f) : (!mul.assoc)⁻¹
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... = b * (c * f) : Haeb
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... = b * d : Hcfd)))
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theorem dvd_of_mul_right_dvd {a b c : A} (H : a * b | c) : a | c :=
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dvd.elim H (take d, assume Habdc : a * b * d = c, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc))
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theorem dvd_of_mul_left_dvd {a b c : A} (H : a * b | c) : b | c :=
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dvd_of_mul_right_dvd (mul.comm a b ▸ H)
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theorem dvd_add {a b c : A} (Hab : a | b) (Hac : a | c) : a | b + c :=
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dvd.elim Hab
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(take d, assume Hadb : a * d = b,
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dvd.elim Hac
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(take e, assume Haec : a * e = c,
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dvd.intro (show a * (d + e) = b + c, from Hadb ▸ Haec ▸ left_distrib a d e)))
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end comm_semiring_dvd
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/- ring -/
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structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
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definition ring.to_semiring [instance] [s : ring A] : semiring A :=
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semiring.mk ring.add ring.add_assoc !ring.zero ring.add_left_id
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add.right_id -- note: we've shown that add_right_id follows from add_left_id in add_comm_group
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ring.add_comm ring.mul ring.mul_assoc !ring.one ring.mul_left_id ring.mul_right_id
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ring.left_distrib ring.right_distrib
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(take a,
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have H : 0 * a + 0 = 0 * a + 0 * a, from
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calc
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0 * a + 0 = 0 * a : add.right_id
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... = (0 + 0) * a : {(add.right_id 0)⁻¹}
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... = 0 * a + 0 * a : ring.right_distrib,
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show 0 * a = 0, from (add.left_cancel H)⁻¹)
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(take a,
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have H : a * 0 + 0 = a * 0 + a * 0, from
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calc
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a * 0 + 0 = a * 0 : add.right_id
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... = a * (0 + 0) : {(add.right_id 0)⁻¹}
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... = a * 0 + a * 0 : ring.left_distrib,
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show a * 0 = 0, from (add.left_cancel H)⁻¹)
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!ring.zero_ne_one
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section
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variables [s : ring A] (a b c d e : A)
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include s
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theorem neg_mul_eq_neg_mul : -(a * b) = -a * b :=
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neg_eq_of_add_eq_zero
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(calc
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a * b + -a * b = (a + -a) * b : right_distrib
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... = 0 * b : add.right_inv
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... = 0 : zero_mul_eq)
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theorem neg_mul_eq_mul_neg : -(a * b) = a * -b :=
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neg_eq_of_add_eq_zero
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(calc
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a * b + a * -b = a * (b + -b) : left_distrib
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... = a * 0 : add.right_inv
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... = 0 : mul_zero_eq)
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theorem neg_mul_neg_eq : -a * -b = a * b :=
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calc
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-a * -b = -(a * -b) : !neg_mul_eq_neg_mul⁻¹
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... = - -(a * b) : neg_mul_eq_mul_neg
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... = a * b : neg_neg_eq
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theorem neg_mul_comm : -a * b = a * -b := !neg_mul_eq_neg_mul⁻¹ ⬝ !neg_mul_eq_mul_neg
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theorem mul_sub_left_distrib : a * (b - c) = a * b - a * c :=
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calc
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a * (b - c) = a * b + a * -c : left_distrib
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... = a * b + - (a * c) : {!neg_mul_eq_mul_neg⁻¹}
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... = a * b - a * c : rfl
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theorem mul_sub_right_distrib : (a - b) * c = a * c - b * c :=
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calc
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(a - b) * c = a * c + -b * c : right_distrib
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... = a * c + - (b * c) : {!neg_mul_eq_neg_mul⁻¹}
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... = a * c - b * c : rfl
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-- TODO: can calc mode be improved to make this easier?
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-- TODO: there is also the other direction. It will be easier when we
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-- have the simplifier.
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theorem mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d :=
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calc
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a * e + c = b * e + d ↔ a * e + c = d + b * e : !add.comm ▸ !iff.refl
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... ↔ a * e + c - b * e = d : iff.symm !sub_eq_iff_eq_add
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... ↔ a * e - b * e + c = d : !sub_add_eq_add_sub ▸ !iff.refl
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... ↔ (a - b) * e + c = d : !mul_sub_right_distrib ▸ !iff.refl
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end
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structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
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definition comm_ring.to_comm_semiring [instance] [s : comm_ring A] : comm_semiring A :=
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comm_semiring.mk comm_ring.add comm_ring.add_assoc (@comm_ring.zero A s)
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comm_ring.add_left_id comm_ring.add_right_id comm_ring.add_comm comm_ring.mul comm_ring.mul_assoc
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(@comm_ring.one A s) comm_ring.mul_left_id comm_ring.mul_right_id comm_ring.left_distrib
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comm_ring.right_distrib zero_mul_eq mul_zero_eq (@comm_ring.zero_ne_one A s)
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comm_ring.mul_comm
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section
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variables [s : comm_ring A] (a b c d e : A)
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include s
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-- TODO: wait for the simplifier
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theorem mul_self_sub_mul_self_eq : a * a - b * b = (a + b) * (a - b) := sorry
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theorem mul_self_sub_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
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mul.right_id 1 ▸ mul_self_sub_mul_self_eq a 1
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end
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structure comm_ring_dvd [class] (A : Type) extends comm_ring A, has_dvd A
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definition comm_ring_dvd.to_comm_semiring_dvd [instance] [s : comm_ring_dvd A] :
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comm_semiring_dvd A :=
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comm_semiring_dvd.mk comm_ring_dvd.add comm_ring_dvd.add_assoc (@comm_ring_dvd.zero A s)
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comm_ring_dvd.add_left_id comm_ring_dvd.add_right_id comm_ring_dvd.add_comm
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comm_ring_dvd.mul comm_ring_dvd.mul_assoc (@comm_ring_dvd.one A s) comm_ring_dvd.mul_left_id
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comm_ring_dvd.mul_right_id comm_ring_dvd.left_distrib comm_ring_dvd.right_distrib
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zero_mul_eq mul_zero_eq (@comm_ring_dvd.zero_ne_one A s) comm_ring_dvd.mul_comm
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comm_ring_dvd.dvd (@comm_ring_dvd.dvd_intro A s) (@comm_ring_dvd.dvd_ex A s)
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section
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variables [s : comm_ring_dvd A] (a b c d e : A)
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include s
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theorem dvd_neg_iff_dvd : a | -b ↔ a | b :=
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iff.intro
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(assume H : a | -b,
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dvd.elim H
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(take c, assume H' : a * c = -b,
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dvd.intro
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(calc
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a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
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... = -(-b) : H'
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... = b : neg_neg_eq)))
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(assume H : a | b,
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dvd.elim H
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(take c, assume H' : a * c = b,
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dvd.intro
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(calc
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a * -c = -(a * c) : {!neg_mul_eq_mul_neg⁻¹}
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... = -b : H')))
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theorem neg_dvd_iff_dvd : -a | b ↔ a | b :=
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iff.intro
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(assume H : -a | b,
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dvd.elim H
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(take c, assume H' : -a * c = b,
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dvd.intro
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(calc
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a * -c = -a * c : !neg_mul_comm⁻¹
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... = b : H')))
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(assume H : a | b,
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dvd.elim H
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(take c, assume H' : a * c = b,
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dvd.intro
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(calc
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-a * -c = a * c : neg_mul_neg_eq
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... = b : H')))
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theorem dvd_sub (H₁ : a | b) (H₂ : a | c) : a | (b - c) :=
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dvd_add H₁ (iff.elim_right !dvd_neg_iff_dvd H₂)
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end
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/- integral domains -/
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-- TODO: some properties here may extend to cancellative semirings. It is worth the effort?
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structure no_zero_divisors [class] (A : Type) extends has_mul A, has_zero A :=
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(eq_zero_or_eq_zero_of_mul_eq_zero : ∀a b, mul a b = zero → a = zero ∨ b = zero)
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theorem eq_zero_or_eq_zero_of_mul_eq_zero {A : Type} [s : no_zero_divisors A] {a b : A}
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(H : a * b = 0) :
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a = 0 ∨ b = 0 := !no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero H
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structure integral_domain [class] (A : Type) extends comm_ring_dvd A, no_zero_divisors A
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section
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variables [s : integral_domain A] (a b c d e : A)
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include s
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theorem mul_ne_zero_of_ne_zero_ne_zero {a b : A} (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
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assume H : a * b = 0,
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or.elim (eq_zero_or_eq_zero_of_mul_eq_zero H) (assume H3, H1 H3) (assume H4, H2 H4)
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theorem mul.cancel_right {a b c : A} (Ha : a ≠ 0) (H : b * a = c * a) : b = c :=
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have H1 : b * a - c * a = 0, from iff.mp !eq_iff_sub_eq_zero H,
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have H2 : (b - c) * a = 0, from eq.trans !mul_sub_right_distrib H1,
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have H3 : b - c = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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iff.elim_right !eq_iff_sub_eq_zero H3
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theorem mul.cancel_left {a b c : A} (Ha : a ≠ 0) (H : a * b = a * c) : b = c :=
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have H1 : a * b - a * c = 0, from iff.mp !eq_iff_sub_eq_zero H,
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have H2 : a * (b - c) = 0, from eq.trans !mul_sub_left_distrib H1,
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have H3 : b - c = 0, from or_resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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iff.elim_right !eq_iff_sub_eq_zero H3
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-- TODO: do we want the iff versions?
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-- TODO: wait for simplifier
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theorem mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b := sorry
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theorem mul_self_eq_one_iff (a : A) : a * a = 1 ↔ a = 1 ∨ a = -1 := sorry
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-- TODO: c - b * c → c = 0 ∨ b = 1 and variants
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theorem dvd_of_mul_dvd_mul_left {a b c : A} (Ha : a ≠ 0) (Hdvd : a * b | a * c) : b | c :=
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dvd.elim Hdvd
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(take d,
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assume H : a * b * d = a * c,
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have H1 : b * d = c, from mul.cancel_left Ha (mul.assoc a b d ▸ H),
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dvd.intro H1)
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theorem dvd_of_mul_dvd_mul_right {a b c : A} (Ha : a ≠ 0) (Hdvd : b * a | c * a) : b | c :=
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dvd.elim Hdvd
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(take d,
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assume H : b * a * d = c * a,
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have H1 : b * d * a = c * a, from eq.trans !mul.right_comm H,
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have H2 : b * d = c, from mul.cancel_right Ha H1,
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dvd.intro H2)
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end
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end algebra
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