refactor(library/data/{nat,int,rat}/{basic.lean,order.lean}: make algebra instance declarations local
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
f25c301c98
commit
42616f766f
6 changed files with 33 additions and 20 deletions
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@ -1,8 +1,6 @@
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/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.int.basic
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Authors: Floris van Doorn, Jeremy Avigad
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The integers, with addition, multiplication, and subtraction. The representation of the integers is
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@ -609,10 +607,10 @@ or_of_or_of_imp_of_imp H3
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(assume H : (nat_abs a) = nat.zero, nat_abs_eq_zero H)
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(assume H : (nat_abs b) = nat.zero, nat_abs_eq_zero H)
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section
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section migrate_algebra
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open [classes] algebra
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protected definition integral_domain [instance] [reducible] : algebra.integral_domain int :=
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protected definition integral_domain [reducible] : algebra.integral_domain int :=
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⦃algebra.integral_domain,
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add := add,
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add_assoc := add.assoc,
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@ -632,6 +630,7 @@ section
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mul_comm := mul.comm,
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eq_zero_or_eq_zero_of_mul_eq_zero := @eq_zero_or_eq_zero_of_mul_eq_zero⦄
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local attribute int.integral_domain [instance]
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definition sub (a b : ℤ) : ℤ := algebra.sub a b
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infix - := int.sub
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definition dvd (a b : ℤ) : Prop := algebra.dvd a b
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@ -639,7 +638,7 @@ section
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migrate from algebra with int
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replacing sub → sub, dvd → dvd
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end
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end migrate_algebra
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/- additional properties -/
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@ -1,8 +1,6 @@
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/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.int.order
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Authors: Floris van Doorn, Jeremy Avigad
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The order relation on the integers. We show that int is an instance of linear_comm_ordered_ring
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@ -218,10 +216,10 @@ lt.intro
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... = of_nat (succ (succ n * m + n)) : nat.add_succ
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... = 0 + succ (succ n * m + n) : zero_add))
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section
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section migrate_algebra
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open [classes] algebra
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protected definition linear_ordered_comm_ring [instance] [reducible] :
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protected definition linear_ordered_comm_ring [reducible] :
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algebra.linear_ordered_comm_ring int :=
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⦃algebra.linear_ordered_comm_ring, int.integral_domain,
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le := le,
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@ -237,12 +235,16 @@ section
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le_total := le.total,
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zero_ne_one := zero_ne_one⦄
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protected definition decidable_linear_ordered_comm_ring [instance] [reducible] :
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protected definition decidable_linear_ordered_comm_ring [reducible] :
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algebra.decidable_linear_ordered_comm_ring int :=
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⦃algebra.decidable_linear_ordered_comm_ring,
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int.linear_ordered_comm_ring,
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decidable_lt := decidable_lt⦄
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local attribute int.integral_domain [instance]
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local attribute int.linear_ordered_comm_ring [instance]
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local attribute int.decidable_linear_ordered_comm_ring [instance]
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definition ge [reducible] (a b : ℤ) := algebra.has_le.ge a b
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definition gt [reducible] (a b : ℤ) := algebra.has_lt.gt a b
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infix >= := int.ge
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@ -257,7 +259,7 @@ section
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migrate from algebra with int
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replacing has_le.ge → ge, has_lt.gt → gt, sign → sign, abs → abs, dvd → dvd, sub → sub
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end
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end migrate_algebra
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/- more facts specific to int -/
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@ -290,7 +290,7 @@ nat.cases_on n
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section
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open [classes] algebra
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protected definition comm_semiring [instance] [reducible] : algebra.comm_semiring nat :=
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protected definition comm_semiring [reducible] : algebra.comm_semiring nat :=
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⦃algebra.comm_semiring,
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add := add,
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add_assoc := add.assoc,
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@ -312,6 +312,7 @@ end
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section port_algebra
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open [classes] algebra
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local attribute comm_semiring [instance]
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theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
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theorem mul.right_comm : ∀a b c : ℕ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
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@ -143,8 +143,9 @@ le.intro !zero_add
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section
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open [classes] algebra
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local attribute nat.comm_semiring [instance]
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protected definition linear_ordered_semiring [instance] [reducible] :
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protected definition linear_ordered_semiring [reducible] :
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algebra.linear_ordered_semiring nat :=
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⦃ algebra.linear_ordered_semiring, nat.comm_semiring,
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add_left_cancel := @add.cancel_left,
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@ -164,6 +165,11 @@ section
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mul_le_mul_of_nonneg_right := (take a b c H1 H2, mul_le_mul_right H1 c),
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mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left,
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mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right ⦄
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end
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section
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open [classes] algebra
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local attribute nat.linear_ordered_semiring [instance]
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migrate from algebra with nat
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replacing has_le.ge → ge, has_lt.gt → gt
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@ -172,6 +178,7 @@ end
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section port_algebra
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open [classes] algebra
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local attribute nat.linear_ordered_semiring [instance]
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theorem add_pos_left : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < a + b :=
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take a H b, @algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le
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theorem add_pos_right : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < b + a :=
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@ -365,10 +365,10 @@ take a b, quot.rec_on_subsingleton₂ a b
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theorem inv_zero : inv 0 = 0 :=
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quot.sound (prerat.inv_zero' ▸ !prerat.equiv.refl)
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section
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section migrate_algebra
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open [classes] algebra
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protected definition discrete_field [instance] [reducible] : algebra.discrete_field rat :=
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protected definition discrete_field [reducible] : algebra.discrete_field rat :=
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⦃algebra.discrete_field,
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add := add,
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add_assoc := add.assoc,
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@ -392,10 +392,12 @@ section
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inv_zero := inv_zero,
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has_decidable_eq := has_decidable_eq⦄
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local attribute rat.discrete_field [instance]
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definition divide (a b : rat) := algebra.divide a b
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definition dvd (a b : rat) := algebra.dvd a b
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migrate from algebra with rat
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replacing sub → rat.sub, divide → divide, dvd → dvd
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end
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end migrate_algebra
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end rat
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@ -202,10 +202,10 @@ have H : pos (a * b), from pos_mul (!sub_zero ▸ H1) (!sub_zero ▸ H2),
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definition decidable_lt [instance] : decidable_rel rat.lt :=
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take a b, decidable_pos (b - a)
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section
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section migrate_algebra
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open [classes] algebra
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protected definition discrete_linear_ordered_field [instance] [reducible] :
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protected definition discrete_linear_ordered_field [reducible] :
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algebra.discrete_linear_ordered_field rat :=
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⦃algebra.discrete_linear_ordered_field,
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rat.discrete_field,
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@ -220,10 +220,12 @@ section
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mul_pos := @mul_pos,
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decidable_lt := @decidable_lt⦄
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local attribute rat.discrete_field [instance]
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local attribute rat.discrete_linear_ordered_field [instance]
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definition abs (n : rat) : rat := algebra.abs n
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definition sign (n : rat) : rat := algebra.sign n
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migrate from algebra with rat
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replacing has_le.ge → ge, has_lt.gt → gt, sub → sub, abs → abs, sign → sign, dvd → dvd, divide → divide
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end
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replacing has_le.ge → ge, has_lt.gt → gt, sub → sub, abs → abs, sign → sign, dvd → dvd, divide → divide
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end migrate_algebra
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end rat
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