diff --git a/library/algebra/ordered_ring.lean b/library/algebra/ordered_ring.lean index 005132433..7d9b4069b 100644 --- a/library/algebra/ordered_ring.lean +++ b/library/algebra/ordered_ring.lean @@ -34,6 +34,14 @@ section variables (a b c d e : A) include s + -- TODO: remove after we short-circuit class-graph + definition ordered_semiring.to_mul [instance] [priority 100000] : has_mul A := + has_mul.mk (@ordered_semiring.mul A s) + definition ordered_semiring.to_lt [instance] [priority 100000] : has_lt A := + has_lt.mk (@ordered_semiring.lt A s) + definition ordered_semiring.to_zero [instance] [priority 100000] : has_zero A := + has_zero.mk (@ordered_semiring.zero A s) + theorem mul_le_mul_of_nonneg_left {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b := !ordered_semiring.mul_le_mul_left Hab Hc @@ -95,6 +103,14 @@ section variables (a b c : A) include s + -- TODO: remove after we short-circuit class-graph + definition linear_ordered_semiring.to_mul [instance] [priority 100000] : has_mul A := + has_mul.mk (@linear_ordered_semiring.mul A s) + definition linear_ordered_semiring.to_lt [instance] [priority 100000] : has_lt A := + has_lt.mk (@linear_ordered_semiring.lt A s) + definition linear_ordered_semiring.to_zero [instance] [priority 100000] : has_zero A := + has_zero.mk (@linear_ordered_semiring.zero A s) + theorem lt_of_mul_lt_mul_left {a b c : A} (H : c * a < c * b) (Hc : c ≥ 0) : a < b := lt_of_not_le (assume H1 : b ≤ a, @@ -190,6 +206,14 @@ section variables (a b c : A) include s + -- TODO: remove after we short-circuit class-graph + definition ordered_ring.to_mul [instance] [priority 100000] : has_mul A := + has_mul.mk (@ordered_ring.mul A s) + definition ordered_ring.to_lt [instance] [priority 100000] : has_lt A := + has_lt.mk (@ordered_ring.lt A s) + definition ordered_ring.to_zero [instance] [priority 100000] : has_zero A := + has_zero.mk (@ordered_ring.zero A s) + theorem mul_le_mul_of_nonpos_left {a b c : A} (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', @@ -249,6 +273,14 @@ section theorem le_add_of_nonneg_right {a b : A} (H : b ≥ 0) : a ≤ a + b := !add.right_id ▸ add_le_add_left H a theorem le_add_of_nonneg_left {a b : A} (H : b ≥ 0) : a ≤ b + a := !add.left_id ▸ add_le_add_right H a + -- TODO: remove after we short-circuit class-graph + definition linear_ordered_ring.to_mul [instance] [priority 100000] : has_mul A := + has_mul.mk (@linear_ordered_ring.mul A s) + definition linear_ordered_ring.to_lt [instance] [priority 100000] : has_lt A := + has_lt.mk (@linear_ordered_ring.lt A s) + definition linear_ordered_ring.to_zero [instance] [priority 100000] : has_zero A := + has_zero.mk (@linear_ordered_ring.zero A s) + /- TODO: a good example of a performance bottleneck. Without any of the "proof ... qed" pairs, it exceeds the unifier maximum number of steps. @@ -265,72 +297,27 @@ section theorem pos_and_pos_or_neg_and_neg_of_mul_pos (Hab : a * b > 0) : (a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) := lt_or_eq_or_lt_cases - -- @lt_or_eq_or_lt_cases A s 0 a ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0)) (assume Ha : 0 < a, - proof lt_or_eq_or_lt_cases - -- @lt_or_eq_or_lt_cases A s 0 b ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0)) (assume Hb : 0 < b, or.inl (and.intro Ha Hb)) (assume Hb : 0 = b, - have H : 0 > 0, from !mul_zero_eq ▸ Hb⁻¹ ▸ Hab, - absurd H (lt.irrefl 0)) + absurd (!mul_zero_eq ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0)) (assume Hb : b < 0, - have Hab' : a * b < 0, from mul_neg_of_pos_of_neg Ha Hb, - absurd Hab (not_lt_of_lt Hab')) - qed) + absurd Hab (not_lt_of_lt (mul_neg_of_pos_of_neg Ha Hb)))) (assume Ha : 0 = a, - proof - have H : 0 > 0, from !zero_mul_eq ▸ Ha⁻¹ ▸ Hab, - absurd H (lt.irrefl 0) - qed) + absurd (!zero_mul_eq ▸ Ha⁻¹ ▸ Hab) (lt.irrefl 0)) (assume Ha : a < 0, - proof lt_or_eq_or_lt_cases - -- @lt_or_eq_or_lt_cases A s 0 b ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0)) (assume Hb : 0 < b, - have Hab' : a * b < 0, from mul_neg_of_neg_of_pos Ha Hb, - absurd Hab (not_lt_of_lt Hab')) + absurd Hab (not_lt_of_lt (mul_neg_of_neg_of_pos Ha Hb))) (assume Hb : 0 = b, - have H : 0 > 0, from !mul_zero_eq ▸ Hb⁻¹ ▸ Hab, - absurd H (lt.irrefl 0)) - (assume Hb : b < 0, or.inr (and.intro Ha Hb)) - qed) + absurd (!mul_zero_eq ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0)) + (assume Hb : b < 0, or.inr (and.intro Ha Hb))) - /- - This version, with tactics, is also slow. - Also, I do not understand why I cannot succesfully return to "proof mode" - in the proof below. I think "s" is not visible there. Can I make it visible? - - Here is the really strange thing: compiling the file with both proofs seems faster than - compiling the file with just one of them. - -/ - theorem pos_and_pos_or_neg_and_neg_of_mul_pos' (Hab : a * b > 0) : - (a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) := - begin - apply (@lt_or_eq_or_lt_cases _ _ 0 a), - intro Ha, - apply (@lt_or_eq_or_lt_cases _ _ 0 b), - intro Hb, apply (or.inl (and.intro Ha Hb)), - intro Hb, - /- - proof -- gives invalid local context - have H : 0 > 0, from !mul_zero_eq ▸ (eq.symm Hb) ▸ Hab, - absurd H (lt.irrefl 0) - qed - -/ - apply (absurd (!mul_zero_eq ▸ (eq.symm Hb) ▸ Hab) (lt.irrefl 0)), - intro Hb, - apply (absurd Hab (not_lt_of_lt (mul_neg_of_pos_of_neg Ha Hb))), - intro Ha, - apply (absurd (!zero_mul_eq ▸ Ha⁻¹ ▸ Hab) (lt.irrefl 0)), - intro Ha, - apply (@lt_or_eq_or_lt_cases _ _ 0 b), - intro Hb, - apply (absurd Hab (not_lt_of_lt (mul_neg_of_neg_of_pos Ha Hb))), - intro Hb, - apply (absurd (!mul_zero_eq ▸ (eq.symm Hb) ▸ Hab) (lt.irrefl 0)), - intro Hb, apply (or.inr (and.intro Ha Hb)) - end + set_option pp.coercions true + set_option pp.implicit true + set_option pp.notation false + -- print definition pos_and_pos_or_neg_and_neg_of_mul_pos -- TODO: use previous and integral domain theorem noneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonneg (Hab : a * b ≥ 0) :