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Cauchy-completion, first parts (#52)

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Patrick Stevens
2019-10-22 07:51:09 +01:00
committed by GitHub
parent 959071214e
commit 6eaa181104
15 changed files with 567 additions and 518 deletions
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14
Rings/Definition.agda
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@@ -34,13 +34,6 @@ module Rings.Definition where
timesZero : {a : A} → a * 0R ∼ 0R
timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
record OrderedRing {n m p} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) : Set (lsuc n ⊔ m ⊔ p) where
open Ring R
open Group additiveGroup
open Setoid S
field
orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
--directSumRing : {m n : _} → {A : Set m} {B : Set n} (r : Ring A) (s : Ring B) → Ring (A && B)
--Ring.additiveGroup (directSumRing r s) = directSumGroup (Ring.additiveGroup r) (Ring.additiveGroup s)
@@ -66,10 +59,3 @@ module Rings.Definition where
-- field
-- ringHom : RingHom R S f
-- bijective : Bijection f
abs : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) (x : A) → A
abs R order x with SetoidTotalOrder.totality order (Ring.0R R) x
... | inl (inl 0<x) = x
... | inl (inr x<0) = Group.inverse (Ring.additiveGroup R) x
... | inr 0=x = Ring.0R R