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agdaproofs/Fields/FieldOfFractions/Order.agda
Patrick Stevens 269f2aa14f N-ary expansions (#113)
2020-04-12 12:16:20 +01:00

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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Rings.Orders.Total.Lemmas
open import Rings.IntegralDomains.Definition
open import Functions
open import Setoids.Setoids
open import Setoids.Orders
open import Sets.EquivalenceRelations
module Fields.FieldOfFractions.Order {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (I : IntegralDomain R) (order : TotallyOrderedRing pRing) where
open import Fields.FieldOfFractions.Setoid I
open import Fields.FieldOfFractions.Ring I
open import Fields.FieldOfFractions.Addition I
open import Fields.FieldOfFractions.Lemmas I
open Ring R
open Setoid S
open Equivalence eq
open SetoidTotalOrder (TotallyOrderedRing.total order)
open import Rings.Orders.Partial.Lemmas
open PartiallyOrderedRing pRing
fieldOfFractionsComparison : Rel fieldOfFractionsSet
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
private
abstract
fieldOfFractionsOrderWellDefinedLeft : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ fieldOfFractionsSetoid x z → fieldOfFractionsComparison z y
fieldOfFractionsOrderWellDefinedLeft {(numX ,, (denomX , denomX!=0))} {(numY ,, (denomY , denomY!=0))} {(numZ ,, (denomZ , denomZ!=0))} x<y x=z with totality (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = s
where
have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
have = ringCanMultiplyByPositive pRing 0<denomZ x<y
p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
p = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
q : ((denomX * numZ) * denomY) < ((numY * denomX) * denomZ)
q = SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=z reflexive) reflexive p
r : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) *Commutative) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) q
s : (numZ * denomY) < (numY * denomZ)
s = ringCanCancelPositive order 0<denomX r
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (symmetric x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 r
where
p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
p = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y)
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
q = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined x=z reflexive) p
r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX r
where
p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
p = ringCanMultiplyByPositive pRing 0<denomZ x<y
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
q = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined x=z reflexive)))) p
r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 q
where
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByPositive pRing 0<denomZ x<y
q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
q = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Commutative (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
where
p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
p = ringCanMultiplyByNegative pRing denomZ<0 (SetoidPartialOrder.<WellDefined pOrder *Commutative reflexive x<y)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
where
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByNegative pRing denomZ<0 x<y
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder denomY<0 0<denomY))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inr x = exFalso (denomX!=0 (symmetric x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inr 0=denomZ = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomZ))
fieldOfFractionsOrderWellDefinedRight : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ (fieldOfFractionsSetoid) y z → fieldOfFractionsComparison x z
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with totality (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with totality (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
swapLemma : {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
swapLemma = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
private
abstract
irreflexive : (a : fieldOfFractionsSet) (pr : fieldOfFractionsComparison a a) → False
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr with totality (Ring.0R R) aDenom
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) with totality (Ring.0R R) aDenom
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inl (inl _) = SetoidPartialOrder.irreflexive pOrder pr
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inl (inr aDenom<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<aDenom aDenom<0))
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) with totality (Ring.0R R) aDenom
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inl (inl 0<aDenom) = SetoidPartialOrder.irreflexive pOrder pr
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inl (inr _) = SetoidPartialOrder.irreflexive pOrder pr
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive : (a b c : fieldOfFractionsSet) (a<b : fieldOfFractionsComparison a b) (b<c : fieldOfFractionsComparison b c) → fieldOfFractionsComparison a c
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c with totality (Ring.0R R) denomA
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB p
where
inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
inter = ringCanMultiplyByPositive pRing 0<denomC a<b
p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
p = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomC a<b)))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder swapLemma swapLemma (ringCanMultiplyByNegative pRing denomC<0 a<b)))
where
have : ((numC * denomA) * denomB) < ((numB * denomC) * denomA)
have = SetoidPartialOrder.<WellDefined pOrder swapLemma reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
... | (inl (inl 0<denomC)) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
... | (inl (inr _)) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
where
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
where
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByPositive pRing 0<denomC a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
where
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
where
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
where
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrder : SetoidPartialOrder fieldOfFractionsSetoid fieldOfFractionsComparison
SetoidPartialOrder.<WellDefined (fieldOfFractionsOrder) {a} {b} {c} {d} a=b c=d a<c = fieldOfFractionsOrderWellDefinedRight {b} {c} {d} (fieldOfFractionsOrderWellDefinedLeft {a} {c} {b} a<c a=b) c=d
SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {a} pr = irreflexive a pr
SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {a} {b} {c} a<b b<c = <transitive a b c a<b b<c
private
<totality : (a b : fieldOfFractionsSet) → ((fieldOfFractionsComparison a b) || (fieldOfFractionsComparison b a)) || (Setoid._∼_ fieldOfFractionsSetoid a b)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) with totality (numA * denomB) (numB * denomA)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inl x) = inl (inl x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inr x) = inl (inr x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) with totality (Ring.0R R) denomA
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) with totality (numB * denomA) (numA * denomB)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inl x) = inl (inl x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inr x) = inl (inr x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inr x = inr (Equivalence.symmetric (Setoid.eq S) (Equivalence.transitive (Setoid.eq S) (Ring.*Commutative R) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) with totality (numB * denomA) (numA * denomB)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inl x) = inl (inl x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inr x) = inl (inr x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) x) (Ring.*Commutative R))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) with totality (numA * denomB) (numB * denomA)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inl x) = inl (inl x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inr x) = inl (inr x)
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) x y = <totality x y
private
abstract
ineqLemma : {x y : A} → (Ring.0R R) < (x * y) → (Ring.0R R) < x → (Ring.0R R) < y
ineqLemma {x} {y} 0<xy 0<x with totality (Ring.0R R) y
ineqLemma {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
ineqLemma {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing y<0 0<x))))
ineqLemma {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
ineqLemma' : {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
ineqLemma' {x} {y} 0<xy x<0 with totality (Ring.0R R) y
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing x<0 0<y))))
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
ineqLemma'' : {x y : A} → (x * y) < (Ring.0R R) → (Ring.0R R) < x → y < (Ring.0R R)
ineqLemma'' {x} {y} xy<0 0<x with totality (Ring.0R R) y
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (orderRespectsMultiplication 0<x 0<y)))
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
ineqLemma''' : {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
ineqLemma''' {x} {y} xy<0 x<0 with totality (Ring.0R R) y
... | inl (inl 0<y) = 0<y
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing y<0 x<0))))
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
private
<orderRespectsAddition : (a b : fieldOfFractionsSet) (a<b : fieldOfFractionsComparison a b) (c : fieldOfFractionsSet) → fieldOfFractionsComparison (fieldOfFractionsPlus a c) (fieldOfFractionsPlus b c)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) with totality (Ring.0R R) (denomA * denomC)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) with totality (Ring.0R R) (denomB * denomC)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
where
0<dC : 0R < denomC
0<dC with totality 0R denomC
0<dC | inl (inl x) = x
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
p = SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<dC a<b)
ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
ans = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (PartiallyOrderedRing.orderRespectsAddition pRing p ((denomA * numC) * denomB))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
where
dC<0 : denomC < 0R
dC<0 with totality 0R denomC
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive pRing x dB<0))))
... | inl (inr x) = x
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dA)))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
where
0<dC : 0R < denomC
0<dC with totality 0R denomC
0<dC | inl (inl x) = x
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
dC<0 : denomC < 0R
dC<0 with totality 0R denomC
dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dC))))
dC<0 | inl (inr x) = x
dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dAdC dA<0
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByNegative pRing dC<0 a<b
have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
have' = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
have'' : ((denomA * (numB * denomC)) + (denomA * (denomB * numC))) < ((denomB * (numA * denomC)) + (denomB * (denomA * numC)))
have'' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (PartiallyOrderedRing.orderRespectsAddition pRing have' (denomA * (denomB * numC)))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso bad
where
0<dC : 0R < denomC
0<dC = ineqLemma 0<dAdC 0<dA
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dBdC<0 0<dB
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC ans)
where
0<dC : 0R < denomC
0<dC = ineqLemma 0<dAdC 0<dA
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByPositive pRing 0<dC a<b
have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
have' = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) have) _
ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
ans = SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dBdC<0 0<dB
have : (((numA * denomC) * denomB) + ((denomB * numC) * denomA)) < (((numB * denomC) * denomA) + ((denomB * numC) * denomA))
have = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing dC<0 a<b)) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = exFalso bad
where
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dAdC dA<0
0<dC : 0R < denomC
0<dC = ineqLemma''' dBdC<0 dB<0
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC | f = exFalso (denomC!=0 (f denomB!=0))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) with totality (Ring.0R R) (denomB * denomC)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
0<dC : 0R < denomC
0<dC = ineqLemma 0<dBdC 0<dB
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
where
0<dC : 0R < denomC
0<dC = ineqLemma 0<dBdC 0<dB
have : (((numB * denomA) * denomC) + ((denomA * numC) * denomB)) < (((numA * denomB) * denomC) + ((denomA * numC) * denomB))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dBdC dB<0
0<dC : 0R < denomC
0<dC = ineqLemma''' dAdC<0 dA<0
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inr 0=dA = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) 0=dA))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
have : (((numB * denomA) * denomC) + ((denomB * numC) * denomA)) < (((numA * denomB) * denomC) + ((denomB * numC) * denomA))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dC<0 0<dC))
where
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
0<dC : 0R < denomC
0<dC = ineqLemma''' dBdC<0 dB<0
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
0<dC : 0R < denomC
0<dC = ineqLemma''' dAdC<0 dA<0
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dBdC<0 0<dB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
where
0<dC : 0R < denomC
0<dC = ineqLemma''' dAdC<0 dA<0
have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
... | f = exFalso (denomC!=0 (f denomB!=0))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inr (0=dAdC) with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdC)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inr 0=dAdC | f = exFalso (denomC!=0 (f denomA!=0))
fieldOfFractionsPOrderedRing : PartiallyOrderedRing fieldOfFractionsRing (SetoidTotalOrder.partial fieldOfFractionsTotalOrder)
PartiallyOrderedRing.orderRespectsAddition fieldOfFractionsPOrderedRing {a} {b} a<b c = <orderRespectsAddition a b a<b c
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} t u with totality (Ring.0R R) (Ring.1R R)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) with totality (Ring.0R R) (denomA * denomB)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with totality (Ring.0R R) denomB
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
0<nA : 0R < numA
0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
0<nB : 0R < numB
0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
0<nAnB : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing 0<nB 0<nA)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dB))))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dB<0 0<dA))))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
nA<0 : numA < 0R
nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
0<nAnB : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative pRing nA<0 nB<0)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso f
where
f : False
f with PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder bl dAdB<0)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
where
0<nB : 0R < numB
0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
nA<0 : numA < 0R
nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
ans : (numA * numB) < 0R
ans = SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nA<0 0<nB)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
where
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
0<nA : 0R < numA
0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
nAnB<0 : (numA * numB) < 0R
nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nB<0 0<nA)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
where
h : 0R < (denomA * denomB)
h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0)
f : False
f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dAdB<0 h)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
... | f = exFalso (denomB!=0 (f denomA!=0))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 1<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative pRing 1<0 1<0))))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
fieldOfFractionsOrderInherited : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
fieldOfFractionsOrderInherited {x} {y} x<y with totality 0R 1R
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
fieldOfFractionsOrderInherited {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)
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