363 lines
13 KiB
Text
363 lines
13 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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Ported from the standard library
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-/
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import algebra.group
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open core
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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 : Πa b c, mul a (add b c) = add (mul a b) (mul a c))
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(right_distrib : Πa b c, mul (add a b) c = add (mul a c) (mul b c))
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definition 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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definition 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 [class] (A : Type) extends has_mul A, has_zero A :=
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(zero_mul : Πa, mul zero a = zero)
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(mul_zero : Πa, mul a zero = zero)
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definition zero_mul [s : mul_zero_class A] (a : A) : 0 * a = 0 := !mul_zero_class.zero_mul
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definition mul_zero [s : mul_zero_class A] (a : A) : a * 0 = 0 := !mul_zero_class.mul_zero
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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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definition 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,
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mul_zero_class A
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section semiring
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variables [s : semiring A] (a b c : A)
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include s
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definition ne_zero_of_mul_ne_zero_right {a b : A} (H : a * b ≠ 0) : a ≠ 0 :=
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assume H1 : a = 0,
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have H2 : a * b = 0, from H1⁻¹ ▸ zero_mul b,
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H H2
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definition ne_zero_of_mul_ne_zero_left {a b : A} (H : a * b ≠ 0) : b ≠ 0 :=
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assume H1 : b = 0,
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have H2 : a * b = 0, from H1⁻¹ ▸ mul_zero a,
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H H2
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end semiring
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/- comm semiring -/
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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
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-- c ≠ 0 → c * a = c * b → a = b.
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section comm_semiring
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variables [s : comm_semiring A] (a b c : A)
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include s
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definition dvd (a b : A) : Type := Σc, b = a * c
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notation a ∣ b := dvd a b
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definition dvd.intro {a b c : A} (H : a * c = b) : a ∣ b :=
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sigma.mk _ H⁻¹
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definition dvd.intro_left {a b c : A} (H : c * a = b) : a ∣ b :=
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dvd.intro (!mul.comm ▸ H)
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definition exists_eq_mul_right_of_dvd {a b : A} (H : a ∣ b) : Σc, b = a * c := H
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definition dvd.elim {P : Type} {a b : A} (H₁ : a ∣ b) (H₂ : Πc, b = a * c → P) : P :=
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sigma.rec_on H₁ H₂
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definition exists_eq_mul_left_of_dvd {a b : A} (H : a ∣ b) : Σc, b = c * a :=
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dvd.elim H (take c, assume H1 : b = a * c, sigma.mk c (H1 ⬝ !mul.comm))
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definition dvd.elim_left {P : Type} {a b : A} (H₁ : a ∣ b) (H₂ : Πc, b = c * a → P) : P :=
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sigma.rec_on (exists_eq_mul_left_of_dvd H₁) (take c, assume H₃ : b = c * a, H₂ c H₃)
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definition dvd.refl : a ∣ a := dvd.intro !mul_one
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definition 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 : b = a * d,
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dvd.elim H₂
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(take e, assume H₄ : c = b * e,
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dvd.intro
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(show a * (d * e) = c, by rewrite [-mul.assoc, -H₃, H₄])))
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definition eq_zero_of_zero_dvd {a : A} (H : 0 ∣ a) : a = 0 :=
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dvd.elim H (take c, assume H' : a = 0 * c, H' ⬝ !zero_mul)
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definition dvd_zero : a ∣ 0 := dvd.intro !mul_zero
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definition one_dvd : 1 ∣ a := dvd.intro !one_mul
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definition dvd_mul_right : a ∣ a * b := dvd.intro rfl
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definition dvd_mul_left : a ∣ b * a := mul.comm a b ▸ dvd_mul_right a b
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definition 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 : b = a * d,
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dvd.intro
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(show a * (d * c) = b * c, from by rewrite [-mul.assoc, H₁]))
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definition 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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definition 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 : b = a * e,
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dvd.elim dvd_cd
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(take f, assume Hcfd : d = c * f,
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dvd.intro
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(show a * c * (e * f) = b * d,
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by rewrite [mul.assoc, {c*_}mul.left_comm, -mul.assoc, Haeb, Hcfd])))
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definition 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 : c = a * b * d, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc⁻¹))
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definition 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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definition 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 : b = a * d,
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dvd.elim Hac
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(take e, assume Haec : c = a * e,
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dvd.intro (show a * (d + e) = b + c,
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by rewrite [left_distrib, -Hadb, -Haec])))
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end comm_semiring
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/- ring -/
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structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A
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definition ring.mul_zero [s : ring A] (a : A) : a * 0 = 0 :=
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have H : a * 0 + 0 = a * 0 + a * 0, from calc
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a * 0 + 0 = a * 0 : by rewrite add_zero
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... = a * (0 + 0) : by rewrite add_zero
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... = a * 0 + a * 0 : by rewrite {a*_}ring.left_distrib,
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show a * 0 = 0, from (add.left_cancel H)⁻¹
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definition ring.zero_mul [s : ring A] (a : A) : 0 * a = 0 :=
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have H : 0 * a + 0 = 0 * a + 0 * a, from calc
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0 * a + 0 = 0 * a : by rewrite add_zero
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... = (0 + 0) * a : by rewrite add_zero
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... = 0 * a + 0 * a : by rewrite {_*a}ring.right_distrib,
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show 0 * a = 0, from (add.left_cancel H)⁻¹
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definition ring.to_semiring [instance] [coercion] [reducible] [s : ring A] : semiring A :=
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⦃ semiring, s,
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mul_zero := ring.mul_zero,
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zero_mul := ring.zero_mul ⦄
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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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definition neg_mul_eq_neg_mul : -(a * b) = -a * b :=
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neg_eq_of_add_eq_zero
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begin
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rewrite [-right_distrib, add.right_inv, zero_mul]
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end
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definition neg_mul_eq_mul_neg : -(a * b) = a * -b :=
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neg_eq_of_add_eq_zero
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begin
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rewrite [-left_distrib, add.right_inv, mul_zero]
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end
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definition neg_mul_neg : -a * -b = a * b :=
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calc
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-a * -b = -(a * -b) : by rewrite -neg_mul_eq_neg_mul
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... = - -(a * b) : by rewrite -neg_mul_eq_mul_neg
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... = a * b : by rewrite neg_neg
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definition neg_mul_comm : -a * b = a * -b := !neg_mul_eq_neg_mul⁻¹ ⬝ !neg_mul_eq_mul_neg
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definition neg_eq_neg_one_mul : -a = -1 * a :=
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calc
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-a = -(1 * a) : by rewrite one_mul
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... = -1 * a : by rewrite neg_mul_eq_neg_mul
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definition 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) : by rewrite -neg_mul_eq_mul_neg
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... = a * b - a * c : rfl
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definition 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) : by rewrite neg_mul_eq_neg_mul
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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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definition 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 : by rewrite {b*e+_}add.comm
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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 : by rewrite sub_add_eq_add_sub
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... ↔ (a - b) * e + c = d : by rewrite mul_sub_right_distrib
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definition mul_neg_one_eq_neg : a * (-1) = -a :=
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have H : a + a * -1 = 0, from calc
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a + a * -1 = a * 1 + a * -1 : mul_one
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... = a * (1 + -1) : left_distrib
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... = a * 0 : add.right_inv
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... = 0 : mul_zero,
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inverse (neg_eq_of_add_eq_zero H)
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definition ne_zero_and_ne_zero_of_mul_ne_zero {a b : A} (H : a * b ≠ 0) : a ≠ 0 × b ≠ 0 :=
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have Ha : a ≠ 0, from
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(assume Ha1 : a = 0,
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have H1 : a * b = 0, by rewrite [Ha1, zero_mul],
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absurd H1 H),
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have Hb : b ≠ 0, from
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(assume Hb1 : b = 0,
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have H1 : a * b = 0, by rewrite [Hb1, mul_zero],
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absurd H1 H),
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pair Ha Hb
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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] [coercion] [reducible] [s : comm_ring A] : comm_semiring A :=
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⦃ comm_semiring, s,
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mul_zero := mul_zero,
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zero_mul := zero_mul ⦄
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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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definition mul_self_sub_mul_self_eq : a * a - b * b = (a + b) * (a - b) :=
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by rewrite [left_distrib, *right_distrib, add.assoc, -{b*a + _}add.assoc,
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-*neg_mul_eq_mul_neg, {a*b}mul.comm, add.right_inv, zero_add]
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definition mul_self_sub_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
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mul_one 1 ▸ mul_self_sub_mul_self_eq a 1
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definition 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' : -b = a * c,
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dvd.intro
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(show a * -c = b,
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by rewrite [-neg_mul_eq_mul_neg, -H', neg_neg])))
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(assume H : (a ∣ b),
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dvd.elim H
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(take c, assume H' : b = a * c,
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dvd.intro
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(show a * -c = -b,
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by rewrite [-neg_mul_eq_mul_neg, -H'])))
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definition 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' : b = -a * c,
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dvd.intro
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(show a * -c = b, by rewrite [-neg_mul_comm, H'])))
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(assume H : (a ∣ b),
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dvd.elim H
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(take c, assume H' : b = a * c,
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dvd.intro
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(show -a * -c = b, by rewrite [neg_mul_neg, H'])))
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definition 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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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 : Πa b, mul a b = zero → a = zero ⊎ b = zero)
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definition 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 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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definition mul_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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sum.rec_on (eq_zero_or_eq_zero_of_mul_eq_zero H) (assume H3, H1 H3) (assume H4, H2 H4)
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definition 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, using H1, by rewrite [mul_sub_right_distrib, H1],
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have H3 : b - c = 0, from sum_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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definition 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, using H1, by rewrite [mul_sub_left_distrib, H1],
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have H3 : b - c = 0, from sum_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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definition mul_self_eq_mul_self_iff (a b : A) : a * a = b * b ↔ a = b ⊎ a = -b :=
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iff.intro
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(λ H : a * a = b * b,
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have aux₁ : (a - b) * (a + b) = 0,
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by rewrite [mul.comm, -mul_self_sub_mul_self_eq, H, sub_self],
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assert aux₂ : a - b = 0 ⊎ a + b = 0, from !eq_zero_or_eq_zero_of_mul_eq_zero aux₁,
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sum.rec_on aux₂
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(λ H : a - b = 0, sum.inl (eq_of_sub_eq_zero H))
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(λ H : a + b = 0, sum.inr (eq_neg_of_add_eq_zero H)))
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(λ H : a = b ⊎ a = -b, sum.rec_on H
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(λ a_eq_b, by rewrite a_eq_b)
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(λ a_eq_mb, by rewrite [a_eq_mb, neg_mul_neg]))
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definition mul_self_eq_one_iff (a : A) : a * a = 1 ↔ a = 1 ⊎ a = -1 :=
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assert aux : a * a = 1 * 1 ↔ a = 1 ⊎ a = -1, from mul_self_eq_mul_self_iff a 1,
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by rewrite mul_one at aux; exact aux
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-- TODO: c - b * c → c = 0 ⊎ b = 1 and variants
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definition 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 * c = a * b * d,
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have H1 : b * d = c, from mul.cancel_left Ha (mul.assoc a b d ▸ H⁻¹),
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dvd.intro H1)
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definition 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 : c * a = b * a * d,
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have H1 : b * d * a = c * a, from by rewrite [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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