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feat(library/algebra/ordered_ring): improve performance using rewrite tactic

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Leonardo de Moura 2015-03-01 10:10:28 -08:00
parent 9c0375b6e2
commit 18ab9ce4e1
1 changed files with 185 additions and 93 deletions
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278
library/algebra/ordered_ring.lean
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@ -17,6 +17,9 @@ namespace algebra
variable {A : Type}
definition absurd_a_lt_a {B : Type} {a : A} [s : strict_order A] (H : a < a) : B :=
absurd H (lt.irrefl a)
structure ordered_semiring [class] (A : Type)
extends has_mul A, has_zero A, has_lt A, -- TODO: remove hack for improving performance
semiring A, ordered_cancel_comm_monoid A :=
@ -44,16 +47,25 @@ section
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_nonneg {a b : A} (Ha : a ≥ 0) (Hb : b ≥ 0) : a * b ≥ 0 :=
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonneg_right Ha Hb,
!zero_mul ▸ H
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_nonpos_of_nonneg_of_nonpos {a b : A} (Ha : a ≥ 0) (Hb : b ≤ 0) : a * b ≤ 0 :=
have H : a * b ≤ a * 0, from mul_le_mul_of_nonneg_left Hb Ha,
!mul_zero ▸ H
begin
have H : a * b ≤ a * 0, from mul_le_mul_of_nonneg_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_nonpos_of_nonpos_of_nonneg {a b : A} (Ha : a ≤ 0) (Hb : b ≥ 0) : a * b ≤ 0 :=
have H : a * b ≤ 0 * b, from mul_le_mul_of_nonneg_right Ha Hb,
!zero_mul ▸ H
begin
have H : a * b ≤ 0 * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_pos_left {a b c : A} (Hab : a < b) (Hc : 0 < c) :
c * a < c * b := !ordered_semiring.mul_lt_mul_of_pos_left Hab Hc
@ -69,20 +81,29 @@ section
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_pos {a b : A} (Ha : a > 0) (Hb : b > 0) : a * b > 0 :=
have H : 0 * b < a * b, from mul_lt_mul_of_pos_right Ha Hb,
!zero_mul ▸ H
begin
have H : 0 * b < a * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_neg_of_pos_of_neg {a b : A} (Ha : a > 0) (Hb : b < 0) : a * b < 0 :=
have H : a * b < a * 0, from mul_lt_mul_of_pos_left Hb Ha,
!mul_zero ▸ H
begin
have H : a * b < a * 0, from mul_lt_mul_of_pos_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_neg_of_neg_of_pos {a b : A} (Ha : a < 0) (Hb : b > 0) : a * b < 0 :=
have H : a * b < 0 * b, from mul_lt_mul_of_pos_right Ha Hb,
!zero_mul ▸ H
begin
have H : a * b < 0 * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
structure linear_ordered_semiring [class] (A : Type)
extends ordered_semiring A, linear_strong_order_pair A
extends ordered_semiring A, linear_strong_order_pair A
section
variable [s : linear_ordered_semiring A]
@ -133,26 +154,38 @@ structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A :
theorem ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
have H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
iff.mp !sub_nonneg_iff_le (!mul_sub_left_distrib ▸ H2)
assert H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : a * c ≤ b * c :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
have H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
iff.mp !sub_nonneg_iff_le (!mul_sub_right_distrib ▸ H2)
assert H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : c * a < c * b :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
have H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
iff.mp !sub_pos_iff_lt (!mul_sub_left_distrib ▸ H2)
assert H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : a * c < b * c :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
have H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
iff.mp !sub_pos_iff_lt (!mul_sub_right_distrib ▸ H2)
assert H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
definition ordered_ring.to_ordered_semiring [instance] [coercion] [reducible] [s : ordered_ring A] :
ordered_semiring A :=
@ -174,33 +207,57 @@ section
theorem mul_le_mul_of_nonpos_left (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
have H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
have H2 : -(c * b) ≤ -(c * a), from !neg_mul_eq_neg_mul⁻¹ ▸ !neg_mul_eq_neg_mul⁻¹ ▸ H1,
assert H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
have H2 : -(c * b) ≤ -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_le_mul_of_nonpos_right (H : b ≤ a) (Hc : c ≤ 0) : a * c ≤ b * c :=
have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
have H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc',
have H2 : -(b * c) ≤ -(a * c), from !neg_mul_eq_mul_neg⁻¹ ▸ !neg_mul_eq_mul_neg⁻¹ ▸ H1,
assert H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc',
have H2 : -(b * c) ≤ -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_nonneg_of_nonpos_of_nonpos (Ha : a ≤ 0) (Hb : b ≤ 0) : 0 ≤ a * b :=
!zero_mul ▸ mul_le_mul_of_nonpos_right Ha Hb
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonpos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_neg_left (H : b < a) (Hc : c < 0) : c * a < c * b :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
have H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc',
have H2 : -(c * b) < -(c * a), from !neg_mul_eq_neg_mul⁻¹ ▸ !neg_mul_eq_neg_mul⁻¹ ▸ H1,
assert H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc',
have H2 : -(c * b) < -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_lt_mul_of_neg_right (H : b < a) (Hc : c < 0) : a * c < b * c :=
have Hc' : -c > 0, from iff.mp' !neg_pos_iff_neg Hc,
have H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc',
have H2 : -(b * c) < -(a * c), from !neg_mul_eq_mul_neg⁻¹ ▸ !neg_mul_eq_mul_neg⁻¹ ▸ H1,
assert H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc',
have H2 : -(b * c) < -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a * b :=
!zero_mul ▸ mul_lt_mul_of_neg_right Ha Hb
begin
have H : 0 * b < a * b, from mul_lt_mul_of_neg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
-- TODO: we can eliminate mul_pos_of_pos, but now it is not worth the effort to redeclare the
@ -231,18 +288,35 @@ theorem linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero [s : linear_o
lt.by_cases
(assume Ha : 0 < a,
lt.by_cases
(assume Hb : 0 < b, absurd (H ▸ show 0 < a * b, from mul_pos Ha Hb) (lt.irrefl 0))
(assume Hb : 0 < b,
begin
have H1 : 0 < a * b, from mul_pos Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
absurd (H ▸ show 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb) (lt.irrefl 0)))
begin
have H1 : 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end))
(assume Ha : 0 = a, or.inl (Ha⁻¹))
(assume Ha : 0 > a,
lt.by_cases
(assume Hb : 0 < b,
absurd (H ▸ show 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb) (lt.irrefl 0))
begin
have H1 : 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
absurd (H ▸ show 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb) (lt.irrefl 0)))
begin
have H1 : 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb,
rewrite H at H1,
apply (absurd_a_lt_a H1)
end))
-- Linearity implies no zero divisors. Doesn't need commutativity.
definition linear_ordered_comm_ring.to_integral_domain [instance] [coercion] [reducible]
@ -271,17 +345,26 @@ section
lt.by_cases
(assume Hb : 0 < b, or.inl (and.intro Ha Hb))
(assume Hb : 0 = b,
absurd (!mul_zero ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0))
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Hb : b < 0,
absurd Hab (lt.asymm (mul_neg_of_pos_of_neg Ha Hb))))
(assume Ha : 0 = a,
absurd (!zero_mul ▸ Ha⁻¹ ▸ Hab) (lt.irrefl 0))
begin
rewrite [-Ha at Hab, zero_mul at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Ha : a < 0,
lt.by_cases
(assume Hb : 0 < b,
absurd Hab (lt.asymm (mul_neg_of_neg_of_pos Ha Hb)))
(assume Hb : 0 = b,
absurd (!mul_zero ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0))
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply (absurd_a_lt_a Hab)
end)
(assume Hb : b < 0, or.inr (and.intro Ha Hb)))
end
@ -312,31 +395,36 @@ section
lt.by_cases
(assume H : a > 0,
calc
sign (sign a) = sign 1 : {sign_of_pos H}
... = 1 : sign_one
... = sign a : sign_of_pos H)
sign (sign a) = sign 1 : by rewrite (sign_of_pos H)
... = 1 : by rewrite sign_one
... = sign a : by rewrite (sign_of_pos H))
(assume H : 0 = a,
calc
sign (sign a) = sign (sign 0) : H
... = sign 0 : sign_zero
... = sign a : H)
sign (sign a) = sign (sign 0) : by rewrite H
... = sign 0 : by rewrite sign_zero at {1}
... = sign a : by rewrite -H)
(assume H : a < 0,
calc
sign (sign a) = sign (-1) : {sign_of_neg H}
... = -1 : sign_neg_one
... = sign a : sign_of_neg H)
sign (sign a) = sign (-1) : by rewrite (sign_of_neg H)
... = -1 : by rewrite sign_neg_one
... = sign a : by rewrite (sign_of_neg H))
theorem pos_of_sign_eq_one (H : sign a = 1) : a > 0 :=
lt.by_cases
(assume H1 : 0 < a, H1)
(assume H1 : 0 = a, absurd (sign_zero⁻¹ ⬝ (H1⁻¹ ▸ H)) zero_ne_one)
(assume H1 : 0 = a,
begin
rewrite [-H1 at H, sign_zero at H],
apply (absurd H zero_ne_one)
end)
(assume H1 : 0 > a,
have H2 : -1 = 1, from (sign_of_neg H1)⁻¹ ⬝ H,
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
theorem eq_zero_of_sign_eq_zero (H : sign a = 0) : a = 0 :=
lt.by_cases
(assume H1 : 0 < a, absurd (H⁻¹ ⬝ sign_of_pos H1) zero_ne_one)
(assume H1 : 0 < a,
absurd (H⁻¹ ⬝ sign_of_pos H1) zero_ne_one)
(assume H1 : 0 = a, H1⁻¹)
(assume H1 : 0 > a,
have H2 : 0 = -1, from H⁻¹ ⬝ sign_of_neg H1,
@ -349,7 +437,11 @@ section
have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
(assume H1 : 0 = a,
have H2 : 0 = -1, from (H1 ▸ sign_zero)⁻¹ ⬝ H,
have H2 : 0 = -1,
begin
rewrite [-H1 at H, sign_zero at H],
exact H
end,
have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
absurd (H3⁻¹) zero_ne_one)
(assume H1 : 0 > a, H1)
@ -359,19 +451,19 @@ section
(assume H1 : 0 < a,
calc
sign (-a) = -1 : sign_of_neg (neg_neg_of_pos H1)
... = -(sign a) : sign_of_pos H1)
... = -(sign a) : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
sign (-a) = sign (-0) : H1
... = sign 0 : neg_zero
... = 0 : sign_zero
... = -0 : neg_zero
... = -(sign 0) : sign_zero
... = -(sign a) : H1)
sign (-a) = sign (-0) : by rewrite H1
... = sign 0 : by rewrite neg_zero
... = 0 : by rewrite sign_zero
... = -0 : by rewrite neg_zero
... = -(sign 0) : by rewrite sign_zero
... = -(sign a) : by rewrite -H1)
(assume H1 : 0 > a,
calc
sign (-a) = 1 : sign_of_pos (neg_pos_of_neg H1)
... = -(-1) : neg_neg
... = -(-1) : by rewrite neg_neg
... = -(sign a) : sign_of_neg H1)
-- hopefully, will be quick with the simplifier
@ -382,40 +474,40 @@ section
(assume H1 : 0 < a,
calc
abs a = a : abs_of_pos H1
... = 1 * a : one_mul
... = sign a * a : {(sign_of_pos H1)⁻¹})
... = 1 * a : by rewrite one_mul
... = sign a * a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
abs a = abs 0 : H1
... = 0 : abs_zero
... = 0 * a : zero_mul
... = sign 0 * a : sign_zero
... = sign a * a : H1)
abs a = abs 0 : by rewrite H1
... = 0 : by rewrite abs_zero
... = 0 * a : by rewrite zero_mul
... = sign 0 * a : by rewrite sign_zero
... = sign a * a : by rewrite H1)
(assume H1 : a < 0,
calc
abs a = -a : abs_of_neg H1
... = -1 * a : neg_eq_neg_one_mul
... = sign a * a : {(sign_of_neg H1)⁻¹})
abs a = -a : abs_of_neg H1
... = -1 * a : by rewrite neg_eq_neg_one_mul
... = sign a * a : by rewrite (sign_of_neg H1))
theorem eq_sign_mul_abs (a : A) : a = sign a * abs a :=
lt.by_cases
(assume H1 : 0 < a,
calc
a = abs a : abs_of_pos H1
... = 1 * abs a : one_mul
... = sign a * abs a : {(sign_of_pos H1)⁻¹})
... = 1 * abs a : by rewrite one_mul
... = sign a * abs a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
a = 0 : H1
... = 0 * abs a : zero_mul
... = sign 0 * abs a : sign_zero
... = sign a * abs a : H1)
a = 0 : H1⁻¹
... = 0 * abs a : by rewrite zero_mul
... = sign 0 * abs a : by rewrite sign_zero
... = sign a * abs a : by rewrite H1)
(assume H1 : a < 0,
calc
a = -(-a) : neg_neg
... = -abs a : {(abs_of_neg H1)⁻¹}
... = -1 * abs a : neg_eq_neg_one_mul
... = sign a * abs a : {(sign_of_neg H1)⁻¹})
a = -(-a) : by rewrite neg_neg
... = -abs a : by rewrite (abs_of_neg H1)
... = -1 * abs a : by rewrite neg_eq_neg_one_mul
... = sign a * abs a : by rewrite (sign_of_neg H1))
theorem abs_dvd_iff_dvd (a b : A) : (abs a | b) ↔ (a | b) :=
abs.by_cases !iff.refl !neg_dvd_iff_dvd
@ -429,29 +521,29 @@ section
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg H1 H2)
... = abs a * b : {(abs_of_nonneg H1)⁻¹}
... = abs a * abs b : {(abs_of_nonneg H2)⁻¹})
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg H1 H2)
... = abs a * b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonneg_of_nonpos H1 H2)
... = a * -b : neg_mul_eq_mul_neg
... = abs a * -b : {(abs_of_nonneg H1)⁻¹}
... = abs a * abs b : {(abs_of_nonpos H2)⁻¹}))
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonneg_of_nonpos H1 H2)
... = a * -b : by rewrite neg_mul_eq_mul_neg
... = abs a * -b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
(assume H1 : a ≤ 0,
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg H1 H2)
... = -a * b : neg_mul_eq_neg_mul
... = abs a * b : {(abs_of_nonpos H1)⁻¹}
... = abs a * abs b : {(abs_of_nonneg H2)⁻¹})
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg H1 H2)
... = -a * b : by rewrite neg_mul_eq_neg_mul
... = abs a * b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg_of_nonpos_of_nonpos H1 H2)
... = -a * -b : neg_mul_neg
... = abs a * -b : {(abs_of_nonpos H1)⁻¹}
... = abs a * abs b : {(abs_of_nonpos H2)⁻¹}))
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg_of_nonpos_of_nonpos H1 H2)
... = -a * -b : by rewrite neg_mul_neg
... = abs a * -b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
theorem abs_mul_self (a : A) : abs a * abs a = a * a :=
abs.by_cases rfl !neg_mul_neg