mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-13 15:48:39 +00:00
Irrelevant field of fractions (#115)
This commit is contained in:
Patrick Stevens
GitHub
@@ -27,26 +27,26 @@ 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))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) with totality (Ring.0R R) denomB
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
fieldOfFractionsOrderWellDefinedLeft {(record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 })} {(record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 })} {(record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 })} x<y x=z with totality (Ring.0R R) denomZ
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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
|
||||
@@ -58,10 +58,10 @@ private
|
||||
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
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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)
|
||||
@@ -69,13 +69,13 @@ private
|
||||
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
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) with totality (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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
|
||||
@@ -83,71 +83,71 @@ private
|
||||
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
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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)
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inl 0<denomZ) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inr denomZ<0) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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)
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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))
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inr x = exFalso (denomX!=0 (symmetric x))
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with totality (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y x=z | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
fieldOfFractionsOrderWellDefinedLeft {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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))
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z with totality (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inl 0<denomX) with totality (Ring.0R R) denomZ
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inr denomX<0) with totality (Ring.0R R) denomZ
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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 {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = denomZ!=0 }} x<y y=z | inl (inr denomX<0) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
fieldOfFractionsOrderWellDefinedRight {record { num = numX ; denom = denomX ; denomNonzero = denomX!=0 }} {record { num = numY ; denom = denomY ; denomNonzero = denomY!=0 }} {record { num = numZ ; denom = denomZ ; denomNonzero = 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)
|
||||
@@ -155,87 +155,87 @@ private
|
||||
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))
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr with totality (Ring.0R R) aDenom
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inl 0<aDenom) with totality (Ring.0R R) aDenom
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inl 0<aDenom) | inl (inl _) = SetoidPartialOrder.irreflexive pOrder pr
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inl 0<aDenom) | inl (inr aDenom<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<aDenom aDenom<0))
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inl 0<aDenom) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inr aDenom<0) with totality (Ring.0R R) aDenom
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inr aDenom<0) | inl (inl 0<aDenom) = SetoidPartialOrder.irreflexive pOrder pr
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inr aDenom<0) | inl (inr _) = SetoidPartialOrder.irreflexive pOrder pr
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = aDenom!=0 } pr | inl (inr aDenom<0) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
irreflexive record { num = aNum ; denom = aDenom ; denomNonzero = 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
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c with totality (Ring.0R R) denomA
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with totality (Ring.0R R) denomB
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) with totality (Ring.0R R) denomC
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -244,40 +244,40 @@ SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {a} {b} {c} a<b b<c = <tr
|
||||
|
||||
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))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) with totality (numA * denomB) (numB * denomA)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inl x) = inl (inl x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inr x) = inl (inr x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) with totality (Ring.0R R) denomA
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) with totality (numB * denomA) (numA * denomB)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inl x) = inl (inl x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inr x) = inl (inr x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) with totality (Ring.0R R) denomB
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) with totality (numB * denomA) (numA * denomB)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inl x) = inl (inl x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inr x) = inl (inr x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) with totality (numA * denomB) (numB * denomA)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inl x) = inl (inl x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inr x) = inl (inr x)
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<totality (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
|
||||
fieldOfFractionsTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) x y = <totality x y
|
||||
@@ -311,11 +311,11 @@ private
|
||||
|
||||
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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) with totality (Ring.0R R) (denomA * denomC)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) with totality (Ring.0R R) (denomB * denomC)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -326,7 +326,7 @@ private
|
||||
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
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -335,9 +335,9 @@ private
|
||||
... | 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
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -351,7 +351,7 @@ private
|
||||
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''))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -361,11 +361,11 @@ private
|
||||
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
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -373,7 +373,7 @@ private
|
||||
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)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -383,15 +383,15 @@ private
|
||||
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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
@@ -399,83 +399,83 @@ private
|
||||
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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inr 0=dBdC | f = exFalso (denomC!=0 (f denomB!=0))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) with totality (Ring.0R R) (denomB * denomC)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inr 0=dA = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) 0=dA))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inr dAdC<0) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inr (0=dAdC) with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdC)
|
||||
<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} t u with totality (Ring.0R R) (Ring.1R R)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) with totality (Ring.0R R) (denomA * denomB)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with totality (Ring.0R R) denomB
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
@@ -483,11 +483,11 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
|
||||
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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
@@ -495,16 +495,16 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
|
||||
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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
@@ -512,9 +512,9 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
|
||||
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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
@@ -522,18 +522,18 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
|
||||
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
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
|
||||
Reference in New Issue
Block a user