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Split out parts of Field of Fractions (#63)

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Patrick Stevens
2019-11-02 21:31:46 +00:00
committed by GitHub
parent 1325236359
commit e4daab7153
12 changed files with 374 additions and 224 deletions
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43
Numbers/Rationals.agda
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@@ -13,23 +13,28 @@ open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Sets.EquivalenceRelations
module Numbers.Rationals 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 ℤIntDom
ℚ = fieldOfFractionsSet
_+Q_ : ℚ → ℚ → ℚ
a +Q b = fieldOfFractionsPlus ℤIntDom a b
a +Q b = fieldOfFractionsPlus a b
_*Q_ : ℚ → ℚ → ℚ
a *Q b = fieldOfFractionsTimes ℤIntDom a b
a *Q b = fieldOfFractionsTimes a b
ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
ℚRing = fieldOfFractionsRing ℤIntDom
ℚRing : Ring fieldOfFractionsSetoid _+Q_ _*Q_
ℚRing = fieldOfFractionsRing
0Q : ℚ
0Q = Ring.0R ℚRing
@@ -38,16 +43,16 @@ injectionQ : ℤ → ℚ
injectionQ z = z ,, (nonneg 1 , λ ())
ℚField : Field ℚRing
ℚField = fieldOfFractions ℤIntDom
ℚField = fieldOfFractions
_<Q_ : ℚ → ℚ → Set
_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
_<Q_ = fieldOfFractionsComparison
_=Q_ : ℚ → ℚ → Set
a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
a =Q b = Setoid._∼_ fieldOfFractionsSetoid a b
reflQ : {x : ℚ} → (x =Q x)
reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
reflQ {x} = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {x}
_≤Q_ : ℚ → ℚ → Set
a ≤Q b = (a <Q b) || (a =Q b)
@@ -61,21 +66,21 @@ 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 ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
ℚPartialOrder : SetoidPartialOrder fieldOfFractionsSetoid fieldOfFractionsComparison
ℚPartialOrder = fieldOfFractionsOrder
ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
ℚTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
ℚTotalOrder = fieldOfFractionsTotalOrder
open SetoidTotalOrder (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
open SetoidTotalOrder fieldOfFractionsTotalOrder
open SetoidPartialOrder partial
open Setoid (fieldOfFractionsSetoid ℤIntDom)
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 ℤIntDom ℤOrderedRing
ℚPOrdered = fieldOfFractionsPOrderedRing
ℚOrdered : TotallyOrderedRing ℚPOrdered
ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing
ℚOrdered = fieldOfFractionsOrderedRing

2
Numbers/RationalsLemmas.agda
View File

@@ -15,8 +15,6 @@ open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Sets.EquivalenceRelations
open import Numbers.Rationals
open import Semirings.Definition