Tidy up groups (#64)

Numbers/Rationals/Definition.agda

@@ -0,0 +1,87 @@
+{-# OPTIONS --safe --warning=error #-}
+
+open import LogicalFormulae
+open import Numbers.Naturals.Naturals
+open import Numbers.Integers.Integers
+open import Groups.Groups
+open import Groups.Definition
+open import Groups.Lemmas
+open import Rings.Definition
+open import Rings.Orders.Total.Definition
+open import Rings.Orders.Partial.Definition
+open import Fields.Fields
+open import Numbers.Primes.PrimeNumbers
+open import Setoids.Setoids
+open import Setoids.Orders
+open import Functions
+open import Sets.EquivalenceRelations
+
+module Numbers.Rationals.Definition where
+
+open import Fields.FieldOfFractions.Setoid ℤIntDom
+open import Fields.FieldOfFractions.Addition ℤIntDom
+open import Fields.FieldOfFractions.Multiplication ℤIntDom
+open import Fields.FieldOfFractions.Ring ℤIntDom
+open import Fields.FieldOfFractions.Field ℤIntDom
+open import Fields.FieldOfFractions.Order ℤIntDom ℤOrderedRing
+
+ℚ : Set
+ℚ = fieldOfFractionsSet
+
+_+Q_ : ℚ → ℚ → ℚ
+a +Q b = fieldOfFractionsPlus a b
+
+_*Q_ : ℚ → ℚ → ℚ
+a *Q b = fieldOfFractionsTimes a b
+
+ℚRing : Ring fieldOfFractionsSetoid _+Q_ _*Q_
+ℚRing = fieldOfFractionsRing
+
+0Q : ℚ
+0Q = Ring.0R ℚRing
+
+injectionQ : ℤ → ℚ
+injectionQ z = z ,, (nonneg 1 , λ ())
+
+ℚField : Field ℚRing
+ℚField = fieldOfFractions
+
+_<Q_ : ℚ → ℚ → Set
+_<Q_ = fieldOfFractionsComparison
+
+_=Q_ : ℚ → ℚ → Set
+a =Q b = Setoid._∼_ fieldOfFractionsSetoid a b
+
+reflQ : {x : ℚ} → (x =Q x)
+reflQ {x} = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {x}
+
+_≤Q_ : ℚ → ℚ → Set
+a ≤Q b = (a <Q b) || (a =Q b)
+
+negateQ : ℚ → ℚ
+negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
+
+_-Q_ : ℚ → ℚ → ℚ
+a -Q b = a +Q negateQ b
+
+a-A : (a : ℚ) → (a -Q a) =Q 0Q
+a-A a = Group.invRight (Ring.additiveGroup ℚRing) {a}
+
+ℚPartialOrder : SetoidPartialOrder fieldOfFractionsSetoid fieldOfFractionsComparison
+ℚPartialOrder = fieldOfFractionsOrder
+
+ℚTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
+ℚTotalOrder = fieldOfFractionsTotalOrder
+
+open SetoidTotalOrder fieldOfFractionsTotalOrder
+open SetoidPartialOrder partial
+open Setoid fieldOfFractionsSetoid
+
+negateQWellDefined : (a b : ℚ) → (a =Q b) → (negateQ a) =Q (negateQ b)
+negateQWellDefined a b a=b = inverseWellDefined (Ring.additiveGroup ℚRing) {a} {b} a=b
+
+ℚPOrdered : PartiallyOrderedRing ℚRing partial
+ℚPOrdered = fieldOfFractionsPOrderedRing
+
+ℚOrdered : TotallyOrderedRing ℚPOrdered
+ℚOrdered = fieldOfFractionsOrderedRing