lean2/library/algebra/ring.lean
2014-11-17 18:32:14 -08:00

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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.
-/
import logic.eq logic.connectives
import 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 :=
(distrib_left : ∀a b c, mul a (add b c) = add (mul a b) (mul a c))
(distrib_right : ∀a b c, mul (add a b) c = add (mul a c) (mul b c))
theorem distrib_left [s : distrib A] (a b c : A) : a * (b + c) = a * b + a * c := !distrib.distrib_left
theorem distrib_right [s: distrib A] (a b c : A) : (a + b) * c = a * c + b * c := !distrib.distrib_right
structure mul_zero [class] (A : Type) extends has_mul A, has_zero A :=
(mul_zero_left : ∀a, mul zero a = zero)
(mul_zero_right : ∀a, mul a zero = zero)
theorem mul_zero_left [s : mul_zero A] (a : A) : 0 * a = 0 := !mul_zero.mul_zero_left
theorem mul_zero_right [s : mul_zero A] (a : A) : a * 0 = 0 := !mul_zero.mul_zero_right
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
/- semiring -/
structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A, mul_zero A,
zero_ne_one_class A
section
variable [s : semiring A]
include s
/- -/
end
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_imp_exists : ∀ 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_imp_exists [s : has_dvd A] {a b : A} : a | b → ∃c, a * c = b := !has_dvd.dvd_imp_exists
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_imp_exists 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 zero_dvd {H : 0 | a} : a = 0 :=
dvd_elim H (take c, assume H' : 0 * c = a, (H')⁻¹ ⬝ !mul_zero_left)
theorem dvd_zero : a | 0 := dvd_intro !mul_zero_right
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 ▸ !dvd_mul_right
theorem dvd_imp_dvd_mul_right {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
... = b * c : H₁))
theorem dvd_imp_dvd_mul_left {a b : A} (H : a | b) (c : A) : a | c * b :=
!mul_comm ▸ (dvd_imp_dvd_mul_right H _)
theorem mul_dvd_mono {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
... = b * (c * f) : Haeb
... = b * d : Hcfd)))
theorem mul_dvd_imp_dvd_left {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 mul_dvd_imp_dvd_right {a b c : A} (H : a * b | c) : b | c :=
mul_dvd_imp_dvd_left (!mul_comm ▸ 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 ▸ !distrib_left)))
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.distrib_left ring.distrib_right
(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 * a + 0 * a : ring.distrib_right,
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
... = a * 0 + a * 0 : ring.distrib_left,
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_left : -(a * b) = -a * b :=
neg_unique
(calc
a * b + -a * b = (a + -a) * b : distrib_right
... = 0 * b : add_right_inv
... = 0 : mul_zero_left)
theorem neg_mul_right : -(a * b) = a * -b :=
neg_unique
(calc
a * b + a * -b = a * (b + -b) : distrib_left
... = a * 0 : add_right_inv
... = 0 : mul_zero_right)
theorem neg_mul_neg : -a * -b = a * b :=
calc
-a * -b = -(a * -b) : neg_mul_left
... = - -(a * b) : neg_mul_right
... = a * b : neg_neg
theorem neg_mul_comm : -a * b = a * -b := !neg_mul_left⁻¹ ⬝ !neg_mul_right
theorem minus_distrib_left : a * (b - c) = a * b - a * c :=
calc
a * (b - c) = a * b + a * -c : distrib_left
... = a * b + - (a * c) : neg_mul_right
... = a * b - a * c : rfl
theorem minus_distrib_right : (a - b) * c = a * c - b * c :=
calc
(a - b) * c = a * c + -b * c : distrib_right
... = a * c + - (b * c) : neg_mul_left
... = 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 eq_add_iff1 : 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 !minus_eq_iff_eq_add
... ↔ a * e - b * e + c = d : !minus_add_right_comm ▸ !iff.refl
... ↔ (a - b) * e + c = d : !minus_distrib_right ▸ !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 has_add.add add_assoc has_zero.zero add_left_id add_right_id add_comm
has_mul.mul mul_assoc has_one.one mul_left_id mul_right_id distrib_left distrib_right
mul_zero_left mul_zero_right zero_ne_one mul_comm
section
variables [s : comm_ring A] (a b c d e : A)
include s
-- TODO: wait for the simplifier
theorem square_minus_square_eq : a * a - b * b = (a + b) * (a - b) := sorry
theorem square_minus_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
!mul_right_id ▸ !square_minus_square_eq
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 has_add.add add_assoc has_zero.zero add_left_id add_right_id add_comm
has_mul.mul mul_assoc has_one.one mul_left_id mul_right_id distrib_left distrib_right
mul_zero_left mul_zero_right zero_ne_one mul_comm dvd (@dvd_intro A s) (@dvd_imp_exists A s)
section
variables [s : comm_ring_dvd A] (a b c d e : A)
include s
theorem dvd_neg_iff : 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_right
... = -(-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_right
... = -b : H')))
theorem neg_dvd_iff : -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
... = 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_diff (H₁ : a | b) (H₂ : a | c) : a | (b - c) :=
dvd_add H₁ (iff.elim_right !dvd_neg_iff H₂)
end
/- ring no_zero_divisors -/
end algebra