refactor(library/algebra/ring): minor cleanup
This commit is contained in:
Leonardo de Moura
parent
3557bd36e7
commit
726867a8fb
1 changed files with 71 additions and 75 deletions
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@ -29,8 +29,8 @@ structure mul_zero_class [class] (A : Type) extends has_mul A, has_zero A :=
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(zero_mul : ∀a, mul zero a = zero)
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(mul_zero : ∀a, mul a zero = zero)
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theorem zero_mul [mul_zero_class A] (a : A) : 0 * a = 0 := !mul_zero_class.zero_mul
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theorem mul_zero [mul_zero_class A] (a : A) : a * 0 = 0 := !mul_zero_class.mul_zero
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theorem zero_mul [simp] [mul_zero_class A] (a : A) : 0 * a = 0 := !mul_zero_class.zero_mul
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theorem mul_zero [simp] [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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@ -59,8 +59,10 @@ section semiring
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have a * b = 0, from this⁻¹ ▸ mul_zero a,
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H this
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local attribute right_distrib [simp]
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theorem distrib_three_right (a b c d : A) : (a + b + c) * d = a * d + b * d + c * d :=
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by rewrite *right_distrib
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by simp
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end semiring
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/- comm semiring -/
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@ -99,7 +101,8 @@ section comm_semiring
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theorem dvd.elim_left {P : Prop} {a b : A} (H₁ : a ∣ b) (H₂ : ∀c, b = c * a → P) : P :=
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exists.elim (exists_eq_mul_left_of_dvd H₁) (take c, assume H₃ : b = c * a, H₂ c H₃)
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theorem dvd.refl : a ∣ a := dvd.intro !mul_one
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theorem dvd.refl [simp] : a ∣ a :=
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dvd.intro !mul_one
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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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@ -112,20 +115,21 @@ section comm_semiring
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theorem 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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theorem dvd_zero : a ∣ 0 := dvd.intro !mul_zero
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theorem dvd_zero [simp] : a ∣ 0 := dvd.intro !mul_zero
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theorem one_dvd : 1 ∣ a := dvd.intro !one_mul
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theorem one_dvd [simp] : 1 ∣ a := dvd.intro !one_mul
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theorem dvd_mul_right : a ∣ a * b := dvd.intro rfl
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theorem dvd_mul_right [simp] : 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_left [simp] : a ∣ b * a :=
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by simp
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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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suppose b = a * d,
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dvd.intro
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(show a * (d * c) = b * c, from by rewrite [-mul.assoc]; substvars))
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(show a * (d * c) = b * c, by simp))
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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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@ -137,7 +141,7 @@ section comm_semiring
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(take f, suppose 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]; substvars)))
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by simp)))
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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 : c = a * b * d, dvd.intro (!mul.assoc⁻¹ ⬝ Habdc⁻¹))
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@ -158,18 +162,16 @@ end comm_semiring
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structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A
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theorem ring.mul_zero [ring A] (a : A) : a * 0 = 0 :=
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theorem ring.mul_zero [simp] [ring A] (a : A) : a * 0 = 0 :=
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have 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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a * 0 + 0 = a * (0 + 0) : by simp
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... = a * 0 + a * 0 : by rewrite left_distrib,
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show a * 0 = 0, from (add.left_cancel this)⁻¹
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theorem ring.zero_mul [ring A] (a : A) : 0 * a = 0 :=
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theorem ring.zero_mul [simp] [ring A] (a : A) : 0 * a = 0 :=
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have 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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0 * a + 0 = (0 + 0) * a : by simp
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... = 0 * a + 0 * a : by rewrite right_distrib,
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show 0 * a = 0, from (add.left_cancel this)⁻¹
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definition ring.to_semiring [trans_instance] [reducible] [s : ring A] : semiring A :=
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@ -193,33 +195,27 @@ section
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rewrite [-left_distrib, add.right_inv, mul_zero]
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end
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theorem neg_mul_eq_neg_mul_symm : - a * b = - (a * b) := eq.symm !neg_mul_eq_neg_mul
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theorem mul_neg_eq_neg_mul_symm : a * - b = - (a * b) := eq.symm !neg_mul_eq_mul_neg
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theorem neg_mul_eq_neg_mul_symm [simp] : - a * b = - (a * b) := eq.symm !neg_mul_eq_neg_mul
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theorem mul_neg_eq_neg_mul_symm [simp] : a * - b = - (a * b) := eq.symm !neg_mul_eq_mul_neg
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theorem 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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by simp
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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 neg_mul_comm : -a * b = a * -b :=
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by simp
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theorem 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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by simp
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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) : by rewrite -neg_mul_eq_mul_neg
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... = a * b - a * c : rfl
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... = a * b - a * c : by simp
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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) : by rewrite neg_mul_eq_neg_mul
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... = a * c - b * c : rfl
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... = a * c - b * c : by simp
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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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@ -233,17 +229,16 @@ section
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... ↔ (a - b) * e + c = d : by rewrite mul_sub_right_distrib
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theorem mul_add_eq_mul_add_of_sub_mul_add_eq : (a - b) * e + c = d → a * e + c = b * e + d :=
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iff.mpr !mul_add_eq_mul_add_iff_sub_mul_add_eq
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iff.mpr !mul_add_eq_mul_add_iff_sub_mul_add_eq
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theorem sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d :=
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iff.mp !mul_add_eq_mul_add_iff_sub_mul_add_eq
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iff.mp !mul_add_eq_mul_add_iff_sub_mul_add_eq
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theorem mul_neg_one_eq_neg : a * (-1) = -a :=
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have a + a * -1 = 0, from calc
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a + a * -1 = a * 1 + a * -1 : mul_one
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a + a * -1 = a * 1 + a * -1 : by simp
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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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... = 0 : by simp,
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symm (neg_eq_of_add_eq_zero this)
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theorem ne_zero_and_ne_zero_of_mul_ne_zero {a b : A} (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 :=
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@ -269,15 +264,13 @@ 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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local attribute left_distrib right_distrib [simp]
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theorem mul_self_sub_mul_self_eq : a * a - b * b = (a + b) * (a - b) :=
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begin
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krewrite [left_distrib, *right_distrib, add.assoc],
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rewrite [-{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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end
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by simp
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theorem mul_self_sub_one_eq : a * a - 1 = (a + 1) * (a - 1) :=
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by rewrite [-mul_self_sub_mul_self_eq, mul_one]
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by simp
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theorem dvd_neg_iff_dvd : (a ∣ -b) ↔ (a ∣ b) :=
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iff.intro
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@ -295,10 +288,10 @@ section
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by rewrite [-neg_mul_eq_mul_neg, -this])))
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theorem dvd_neg_of_dvd : (a ∣ b) → (a ∣ -b) :=
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iff.mpr !dvd_neg_iff_dvd
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iff.mpr !dvd_neg_iff_dvd
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theorem dvd_of_dvd_neg : (a ∣ -b) → (a ∣ b) :=
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iff.mp !dvd_neg_iff_dvd
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iff.mp !dvd_neg_iff_dvd
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theorem neg_dvd_iff_dvd : (-a ∣ b) ↔ (a ∣ b) :=
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iff.intro
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@ -314,10 +307,10 @@ section
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(show -a * -c = b, by rewrite [neg_mul_neg, this])))
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theorem neg_dvd_of_dvd : (a ∣ b) → (-a ∣ b) :=
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iff.mpr !neg_dvd_iff_dvd
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iff.mpr !neg_dvd_iff_dvd
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theorem dvd_of_neg_dvd : (-a ∣ b) → (a ∣ b) :=
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iff.mp !neg_dvd_iff_dvd
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iff.mp !neg_dvd_iff_dvd
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theorem dvd_sub (H₁ : (a ∣ b)) (H₂ : (a ∣ c)) : (a ∣ b - c) :=
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dvd_add H₁ (!dvd_neg_of_dvd H₂)
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@ -341,7 +334,7 @@ section
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theorem mul_ne_zero {a b : A} (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
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suppose a * b = 0,
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or.elim (eq_zero_or_eq_zero_of_mul_eq_zero this) (assume H3, H1 H3) (assume H4, H2 H4)
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or.elim (eq_zero_or_eq_zero_of_mul_eq_zero this) (assume H3, H1 H3) (assume H4, H2 H4)
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theorem eq_of_mul_eq_mul_right {a b c : A} (Ha : a ≠ 0) (H : b * a = c * a) : b = c :=
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have b * a - c * a = 0, from iff.mp !eq_iff_sub_eq_zero H,
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@ -360,7 +353,7 @@ section
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theorem eq_zero_of_mul_eq_self_right {a b : A} (H₁ : b ≠ 1) (H₂ : a * b = a) : a = 0 :=
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have b - 1 ≠ 0, from
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suppose b - 1 = 0, H₁ (!zero_add ▸ eq_add_of_sub_eq this),
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have a * b - a = 0, by rewrite H₂; apply sub_self,
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have a * b - a = 0, by simp,
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have a * (b - 1) = 0, by+ rewrite [mul_sub_left_distrib, mul_one]; apply this,
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show a = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero this) `b - 1 ≠ 0`
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@ -404,81 +397,84 @@ end
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namespace norm_num
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local attribute bit0 bit1 add1 [reducible]
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local attribute right_distrib left_distrib [simp]
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theorem mul_zero [mul_zero_class A] (a : A) : a * zero = zero :=
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by rewrite [↑zero, mul_zero]
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by simp
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theorem zero_mul [mul_zero_class A] (a : A) : zero * a = zero :=
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by rewrite [↑zero, zero_mul]
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by simp
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theorem mul_one [monoid A] (a : A) : a * one = a :=
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by rewrite [↑one, mul_one]
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by simp
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theorem mul_bit0 [distrib A] (a b : A) : a * (bit0 b) = bit0 (a * b) :=
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by rewrite [↑bit0, left_distrib]
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by simp
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theorem mul_bit0_helper [distrib A] (a b t : A) (H : a * b = t) : a * (bit0 b) = bit0 t :=
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by rewrite -H; apply mul_bit0
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by rewrite -H; simp
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theorem mul_bit1 [semiring A] (a b : A) : a * (bit1 b) = bit0 (a * b) + a :=
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by rewrite [↑bit1, ↑bit0, +left_distrib, ↑one, mul_one]
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by simp
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theorem mul_bit1_helper [semiring A] (a b s t : A) (Hs : a * b = s) (Ht : bit0 s + a = t) :
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a * (bit1 b) = t :=
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begin rewrite [-Ht, -Hs, mul_bit1] end
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by simp
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theorem subst_into_prod [has_mul A] (l r tl tr t : A) (prl : l = tl) (prr : r = tr)
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(prt : tl * tr = t) :
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l * r = t :=
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by rewrite [prl, prr, prt]
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by simp
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theorem mk_cong (op : A → A) (a b : A) (H : a = b) : op a = op b :=
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by congruence; exact H
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by simp
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theorem neg_add_neg_eq_of_add_add_eq_zero [add_comm_group A] (a b c : A) (H : c + a + b = 0) :
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-a + -b = c :=
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begin
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apply add_neg_eq_of_eq_add,
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apply neg_eq_of_add_eq_zero,
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rewrite [add.comm, add.assoc, add.comm b, -add.assoc, H]
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end
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begin
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apply add_neg_eq_of_eq_add,
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apply neg_eq_of_add_eq_zero,
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simp
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end
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theorem neg_add_neg_helper [add_comm_group A] (a b c : A) (H : a + b = c) : -a + -b = -c :=
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begin apply iff.mp !neg_eq_neg_iff_eq, rewrite [neg_add, *neg_neg, H] end
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begin apply iff.mp !neg_eq_neg_iff_eq, simp end
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theorem neg_add_pos_eq_of_eq_add [add_comm_group A] (a b c : A) (H : b = c + a) : -a + b = c :=
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begin apply neg_add_eq_of_eq_add, rewrite add.comm, exact H end
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begin apply neg_add_eq_of_eq_add, simp end
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theorem neg_add_pos_helper1 [add_comm_group A] (a b c : A) (H : b + c = a) : -a + b = -c :=
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begin apply neg_add_eq_of_eq_add, apply eq_add_neg_of_add_eq H end
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begin apply neg_add_eq_of_eq_add, apply eq_add_neg_of_add_eq H end
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theorem neg_add_pos_helper2 [add_comm_group A] (a b c : A) (H : a + c = b) : -a + b = c :=
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begin apply neg_add_eq_of_eq_add, rewrite H end
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begin apply neg_add_eq_of_eq_add, rewrite H end
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theorem pos_add_neg_helper [add_comm_group A] (a b c : A) (H : b + a = c) : a + b = c :=
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by rewrite [add.comm, H]
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by simp
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theorem sub_eq_add_neg_helper [add_comm_group A] (t₁ t₂ e w₁ w₂: A) (H₁ : t₁ = w₁)
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(H₂ : t₂ = w₂) (H : w₁ + -w₂ = e) : t₁ - t₂ = e :=
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by rewrite [sub_eq_add_neg, H₁, H₂, H]
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by simp
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theorem pos_add_pos_helper [add_comm_group A] (a b c h₁ h₂ : A) (H₁ : a = h₁) (H₂ : b = h₂)
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(H : h₁ + h₂ = c) : a + b = c :=
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by rewrite [H₁, H₂, H]
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by simp
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theorem subst_into_subtr [add_group A] (l r t : A) (prt : l + -r = t) : l - r = t :=
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by rewrite [sub_eq_add_neg, prt]
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by simp
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theorem neg_neg_helper [add_group A] (a b : A) (H : a = -b) : -a = b :=
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by rewrite [H, neg_neg]
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by simp
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theorem neg_mul_neg_helper [ring A] (a b c : A) (H : a * b = c) : (-a) * (-b) = c :=
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begin rewrite [neg_mul_neg, H] end
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by simp
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theorem neg_mul_pos_helper [ring A] (a b c : A) (H : a * b = c) : (-a) * b = -c :=
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begin rewrite [-neg_mul_eq_neg_mul, H] end
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by simp
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theorem pos_mul_neg_helper [ring A] (a b c : A) (H : a * b = c) : a * (-b) = -c :=
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begin rewrite [-neg_mul_comm, -neg_mul_eq_neg_mul, H] end
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by simp
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end norm_num
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