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refactor(library/algebra/ring): using new structure instance syntax sugar to define instance

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Leonardo de Moura 2015-01-16 17:42:30 -08:00
parent a86661f42c
commit 3d63c0b5dc
1 changed files with 15 additions and 19 deletions
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library/algebra/ring.lean
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@ -153,25 +153,22 @@ end comm_semiring
structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
definition ring.to_semiring [instance] [coercion] [s : ring A] : semiring A := definition ring.to_semiring [instance] [coercion] [s : ring A] : semiring A :=
semiring.mk ring.add ring.add_assoc !ring.zero ring.zero_add ⦃ semiring,
add_zero -- note: we've shown that add_zero follows from zero_add in add_comm_group mul_zero := take a,
ring.add_comm ring.mul ring.mul_assoc !ring.one ring.one_mul ring.mul_one
ring.left_distrib ring.right_distrib
(take a,
have H : 0 * a + 0 = 0 * a + 0 * a, from
calc
0 * a + 0 = 0 * a : add_zero
... = (0 + 0) * a : {(add_zero 0)⁻¹}
... = 0 * a + 0 * a : ring.right_distrib,
show 0 * a = 0, from (add.left_cancel H)⁻¹)
(take a,
have H : a * 0 + 0 = a * 0 + a * 0, from have H : a * 0 + 0 = a * 0 + a * 0, from
calc calc
a * 0 + 0 = a * 0 : add_zero a * 0 + 0 = a * 0 : add_zero
... = a * (0 + 0) : {(add_zero 0)⁻¹} ... = a * (0 + 0) : {(add_zero 0)⁻¹}
... = a * 0 + a * 0 : ring.left_distrib, ... = a * 0 + a * 0 : ring.left_distrib,
show a * 0 = 0, from (add.left_cancel H)⁻¹) show a * 0 = 0, from (add.left_cancel H)⁻¹,
!ring.zero_ne_one zero_mul := take a,
have H : 0 * a + 0 = 0 * a + 0 * a, from
calc
0 * a + 0 = 0 * a : add_zero
... = (0 + 0) * a : {(add_zero 0)⁻¹}
... = 0 * a + 0 * a : ring.right_distrib,
show 0 * a = 0, from (add.left_cancel H)⁻¹,
using s ⦄
section section
variables [s : ring A] (a b c d e : A) variables [s : ring A] (a b c d e : A)
@ -226,11 +223,10 @@ end
structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A
definition comm_ring.to_comm_semiring [instance] [coercion] [s : comm_ring A] : comm_semiring A := definition comm_ring.to_comm_semiring [instance] [coercion] [s : comm_ring A] : comm_semiring A :=
comm_semiring.mk comm_ring.add comm_ring.add_assoc (@comm_ring.zero A s) ⦃ comm_semiring,
comm_ring.zero_add comm_ring.add_zero comm_ring.add_comm comm_ring.mul comm_ring.mul_assoc mul_zero := mul_zero,
(@comm_ring.one A s) comm_ring.one_mul comm_ring.mul_one comm_ring.left_distrib zero_mul := zero_mul,
comm_ring.right_distrib zero_mul mul_zero (@comm_ring.zero_ne_one A s) using s ⦄
comm_ring.mul_comm
section section
variables [s : comm_ring A] (a b c d e : A) variables [s : comm_ring A] (a b c d e : A)