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refactor(library/algebra/ordered_ring): remove 0 ~= 1 from ordered_semiring, add 0 < 1 to linear_ordered_semiring

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Jeremy Avigad 2015-07-08 10:13:56 +10:00 committed by Leonardo de Moura
parent e35f05ad47
commit 31aeff95d5
2 changed files with 6 additions and 5 deletions
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8
library/algebra/ordered_ring.lean
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@ -19,7 +19,7 @@ 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 semiring A, ordered_cancel_comm_monoid A, zero_ne_one_class A :=
extends semiring A, ordered_cancel_comm_monoid A :=
(mul_le_mul_of_nonneg_left: ∀a b c, le a b → le zero c → le (mul c a) (mul c b))
(mul_le_mul_of_nonneg_right: ∀a b c, le a b → le zero c → le (mul a c) (mul b c))
(mul_lt_mul_of_pos_left: ∀a b c, lt a b → lt zero c → lt (mul c a) (mul c b))
@ -100,13 +100,16 @@ section
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 :=
(zero_lt_one : lt zero one)
section
variable [s : linear_ordered_semiring A]
variables {a b c : A}
include s
theorem zero_lt_one : 0 < (1:A) := linear_ordered_semiring.zero_lt_one A
theorem lt_of_mul_lt_mul_left (H : c * a < c * b) (Hc : c ≥ 0) : a < b :=
lt_of_not_ge
(assume H1 : b ≤ a,
@ -378,7 +381,6 @@ section
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
theorem zero_le_one : 0 ≤ (1:A) := one_mul 1 ▸ mul_self_nonneg 1
theorem zero_lt_one : 0 < (1:A) := linear_ordered_ring.zero_lt_one A
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=

3
library/data/nat/order.lean
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@ -165,7 +165,7 @@ section migrate_algebra
add_lt_add_left := @add_lt_add_left,
add_le_add_left := @add_le_add_left,
le_of_add_le_add_left := @le_of_add_le_add_left,
zero_ne_one := ne.symm (succ_ne_zero zero),
zero_lt_one := zero_lt_succ 0,
mul_le_mul_of_nonneg_left := (take a b c H1 H2, mul_le_mul_left c H1),
mul_le_mul_of_nonneg_right := (take a b c H1 H2, mul_le_mul_right c H1),
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left,
@ -223,7 +223,6 @@ section migrate_algebra
end migrate_algebra
theorem zero_le_one : 0 ≤ 1 := dec_trivial
theorem zero_lt_one : 0 < 1 := dec_trivial
/- properties specific to nat -/