refactor(library/algebra/ordered_ring): add workarounds to improve performance
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Leonardo de Moura
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6f775be1b6
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628faa10eb
1 changed files with 42 additions and 55 deletions
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@ -34,6 +34,14 @@ section
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variables (a b c d e : A)
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include s
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-- TODO: remove after we short-circuit class-graph
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definition ordered_semiring.to_mul [instance] [priority 100000] : has_mul A :=
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has_mul.mk (@ordered_semiring.mul A s)
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definition ordered_semiring.to_lt [instance] [priority 100000] : has_lt A :=
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has_lt.mk (@ordered_semiring.lt A s)
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definition ordered_semiring.to_zero [instance] [priority 100000] : has_zero A :=
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has_zero.mk (@ordered_semiring.zero A s)
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theorem mul_le_mul_of_nonneg_left {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) :
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c * a ≤ c * b := !ordered_semiring.mul_le_mul_left Hab Hc
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@ -95,6 +103,14 @@ section
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variables (a b c : A)
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include s
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-- TODO: remove after we short-circuit class-graph
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definition linear_ordered_semiring.to_mul [instance] [priority 100000] : has_mul A :=
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has_mul.mk (@linear_ordered_semiring.mul A s)
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definition linear_ordered_semiring.to_lt [instance] [priority 100000] : has_lt A :=
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has_lt.mk (@linear_ordered_semiring.lt A s)
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definition linear_ordered_semiring.to_zero [instance] [priority 100000] : has_zero A :=
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has_zero.mk (@linear_ordered_semiring.zero A s)
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theorem lt_of_mul_lt_mul_left {a b c : A} (H : c * a < c * b) (Hc : c ≥ 0) : a < b :=
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lt_of_not_le
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(assume H1 : b ≤ a,
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@ -190,6 +206,14 @@ section
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variables (a b c : A)
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include s
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-- TODO: remove after we short-circuit class-graph
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definition ordered_ring.to_mul [instance] [priority 100000] : has_mul A :=
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has_mul.mk (@ordered_ring.mul A s)
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definition ordered_ring.to_lt [instance] [priority 100000] : has_lt A :=
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has_lt.mk (@ordered_ring.lt A s)
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definition ordered_ring.to_zero [instance] [priority 100000] : has_zero A :=
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has_zero.mk (@ordered_ring.zero A s)
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theorem mul_le_mul_of_nonpos_left {a b c : A} (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b :=
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have Hc' : -c ≥ 0, from iff.mp' !neg_nonneg_iff_nonpos Hc,
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have H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
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@ -249,6 +273,14 @@ section
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theorem le_add_of_nonneg_right {a b : A} (H : b ≥ 0) : a ≤ a + b := !add.right_id ▸ add_le_add_left H a
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theorem le_add_of_nonneg_left {a b : A} (H : b ≥ 0) : a ≤ b + a := !add.left_id ▸ add_le_add_right H a
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-- TODO: remove after we short-circuit class-graph
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definition linear_ordered_ring.to_mul [instance] [priority 100000] : has_mul A :=
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has_mul.mk (@linear_ordered_ring.mul A s)
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definition linear_ordered_ring.to_lt [instance] [priority 100000] : has_lt A :=
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has_lt.mk (@linear_ordered_ring.lt A s)
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definition linear_ordered_ring.to_zero [instance] [priority 100000] : has_zero A :=
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has_zero.mk (@linear_ordered_ring.zero A s)
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/- TODO: a good example of a performance bottleneck.
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Without any of the "proof ... qed" pairs, it exceeds the unifier maximum number of steps.
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@ -265,72 +297,27 @@ section
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theorem pos_and_pos_or_neg_and_neg_of_mul_pos (Hab : a * b > 0) :
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(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=
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lt_or_eq_or_lt_cases
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-- @lt_or_eq_or_lt_cases A s 0 a ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0))
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(assume Ha : 0 < a,
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proof
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lt_or_eq_or_lt_cases
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-- @lt_or_eq_or_lt_cases A s 0 b ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0))
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(assume Hb : 0 < b, or.inl (and.intro Ha Hb))
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(assume Hb : 0 = b,
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have H : 0 > 0, from !mul_zero_eq ▸ Hb⁻¹ ▸ Hab,
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absurd H (lt.irrefl 0))
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absurd (!mul_zero_eq ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0))
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(assume Hb : b < 0,
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have Hab' : a * b < 0, from mul_neg_of_pos_of_neg Ha Hb,
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absurd Hab (not_lt_of_lt Hab'))
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qed)
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absurd Hab (not_lt_of_lt (mul_neg_of_pos_of_neg Ha Hb))))
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(assume Ha : 0 = a,
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proof
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have H : 0 > 0, from !zero_mul_eq ▸ Ha⁻¹ ▸ Hab,
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absurd H (lt.irrefl 0)
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qed)
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absurd (!zero_mul_eq ▸ Ha⁻¹ ▸ Hab) (lt.irrefl 0))
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(assume Ha : a < 0,
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proof
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lt_or_eq_or_lt_cases
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-- @lt_or_eq_or_lt_cases A s 0 b ((a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0))
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(assume Hb : 0 < b,
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have Hab' : a * b < 0, from mul_neg_of_neg_of_pos Ha Hb,
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absurd Hab (not_lt_of_lt Hab'))
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absurd Hab (not_lt_of_lt (mul_neg_of_neg_of_pos Ha Hb)))
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(assume Hb : 0 = b,
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have H : 0 > 0, from !mul_zero_eq ▸ Hb⁻¹ ▸ Hab,
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absurd H (lt.irrefl 0))
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(assume Hb : b < 0, or.inr (and.intro Ha Hb))
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qed)
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absurd (!mul_zero_eq ▸ Hb⁻¹ ▸ Hab) (lt.irrefl 0))
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(assume Hb : b < 0, or.inr (and.intro Ha Hb)))
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/-
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This version, with tactics, is also slow.
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Also, I do not understand why I cannot succesfully return to "proof mode"
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in the proof below. I think "s" is not visible there. Can I make it visible?
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Here is the really strange thing: compiling the file with both proofs seems faster than
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compiling the file with just one of them.
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-/
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theorem pos_and_pos_or_neg_and_neg_of_mul_pos' (Hab : a * b > 0) :
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(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=
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begin
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apply (@lt_or_eq_or_lt_cases _ _ 0 a),
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intro Ha,
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apply (@lt_or_eq_or_lt_cases _ _ 0 b),
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intro Hb, apply (or.inl (and.intro Ha Hb)),
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intro Hb,
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/-
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proof -- gives invalid local context
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have H : 0 > 0, from !mul_zero_eq ▸ (eq.symm Hb) ▸ Hab,
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absurd H (lt.irrefl 0)
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qed
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-/
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apply (absurd (!mul_zero_eq ▸ (eq.symm Hb) ▸ Hab) (lt.irrefl 0)),
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intro Hb,
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apply (absurd Hab (not_lt_of_lt (mul_neg_of_pos_of_neg Ha Hb))),
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intro Ha,
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apply (absurd (!zero_mul_eq ▸ Ha⁻¹ ▸ Hab) (lt.irrefl 0)),
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intro Ha,
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apply (@lt_or_eq_or_lt_cases _ _ 0 b),
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intro Hb,
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apply (absurd Hab (not_lt_of_lt (mul_neg_of_neg_of_pos Ha Hb))),
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intro Hb,
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apply (absurd (!mul_zero_eq ▸ (eq.symm Hb) ▸ Hab) (lt.irrefl 0)),
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intro Hb, apply (or.inr (and.intro Ha Hb))
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end
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set_option pp.coercions true
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set_option pp.implicit true
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set_option pp.notation false
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-- print definition pos_and_pos_or_neg_and_neg_of_mul_pos
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-- TODO: use previous and integral domain
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theorem noneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonneg (Hab : a * b ≥ 0) :
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