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Rename some confusing fields (#51)
This commit is contained in:
Patrick Stevens
GitHub
@@ -20,8 +20,8 @@ module Fields.FieldOfFractions where
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fieldOfFractionsSetoid : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Setoid (fieldOfFractionsSet I)
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Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c)
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Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.multCommutative R
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Equivalence.symmetric (Setoid.eq (fieldOfFractionsSetoid {S = S} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} ad=bc = transitive (Ring.multCommutative R) (transitive (symmetric ad=bc) (Ring.multCommutative R))
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Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.*Commutative R
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Equivalence.symmetric (Setoid.eq (fieldOfFractionsSetoid {S = S} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} ad=bc = transitive (Ring.*Commutative R) (transitive (symmetric ad=bc) (Ring.*Commutative R))
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where
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open Equivalence (Setoid.eq S)
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Equivalence.transitive (Setoid.eq (fieldOfFractionsSetoid {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} ad=bc cf=de = p5
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@@ -30,13 +30,13 @@ module Fields.FieldOfFractions where
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open Ring R
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open Equivalence eq
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p : (a * d) * f ∼ (b * c) * f
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p = Ring.multWellDefined R ad=bc reflexive
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p = Ring.*WellDefined R ad=bc reflexive
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p2 : (a * f) * d ∼ b * (d * e)
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p2 = transitive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive p (transitive (symmetric multAssoc) (multWellDefined reflexive cf=de)))
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p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
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p3 : (a * f) * d ∼ (b * e) * d
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p3 = transitive p2 (transitive (multWellDefined reflexive multCommutative) multAssoc)
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p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
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p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
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p4 = cancelIntDom I (transitive multCommutative (transitive p3 multCommutative))
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p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
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p5 : (a * f) ∼ (b * e)
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p5 with p4
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p5 | inl d=0 = exFalso (d!=0 d=0)
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@@ -63,7 +63,7 @@ module Fields.FieldOfFractions where
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ans pr | inr x = d!=0 x
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fieldOfFractionsRing : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Ring (fieldOfFractionsSetoid I) (fieldOfFractionsPlus I) (fieldOfFractionsTimes I)
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Group.wellDefined (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
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Group.+WellDefined (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
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where
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open Setoid S
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open Ring R
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@@ -73,79 +73,79 @@ module Fields.FieldOfFractions where
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have2 : (a * f) ∼ (b * e)
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have2 = af=be
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need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
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need = transitive (transitive (Ring.multCommutative R) (transitive (Ring.multDistributes R) (Group.wellDefined (Ring.additiveGroup R) (transitive multAssoc (transitive (multWellDefined (multCommutative) reflexive) (transitive (multWellDefined multAssoc reflexive) (transitive (multWellDefined (multWellDefined have2 reflexive) reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined (transitive (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative)) multAssoc) reflexive) (symmetric multAssoc))))))))) (transitive multCommutative (transitive (transitive (symmetric multAssoc) (multWellDefined reflexive (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined have1 reflexive) (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative))))))) multAssoc))))) (symmetric (Ring.multDistributes R))
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Group.identity (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
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need = transitive (transitive (Ring.*Commutative R) (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) (transitive *Associative (transitive (*WellDefined (*Commutative) reflexive) (transitive (*WellDefined *Associative reflexive) (transitive (*WellDefined (*WellDefined have2 reflexive) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined (transitive (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)) *Associative) reflexive) (symmetric *Associative))))))))) (transitive *Commutative (transitive (transitive (symmetric *Associative) (*WellDefined reflexive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined have1 reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))))) *Associative))))) (symmetric (Ring.*DistributesOver+ R))
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Group.0G (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
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Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b
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Group.multAssoc (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
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Group.+Associative (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e)))
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need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Ring.multDistributes R)) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Group.wellDefined (Ring.additiveGroup R) (Ring.multAssoc R) (Ring.multAssoc R))) (transitive (Group.multAssoc (Ring.additiveGroup R)) (Group.wellDefined (Ring.additiveGroup R) (transitive (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multAssoc R) (Ring.multCommutative R)) (Ring.multCommutative R)) (symmetric (Ring.multDistributes R))) (Ring.multCommutative R)) reflexive)))))
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Group.multIdentRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
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need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.*Associative R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.*DistributesOver+ R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Associative R) (Ring.*Associative R))) (transitive (Group.+Associative (Ring.additiveGroup R)) (Group.+WellDefined (Ring.additiveGroup R) (transitive (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Associative R) (Ring.*Commutative R)) (Ring.*Commutative R)) (symmetric (Ring.*DistributesOver+ R))) (Ring.*Commutative R)) reflexive)))))
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Group.identRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((a * Ring.1R R) + (b * Group.identity (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
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need = transitive (transitive (Ring.multWellDefined R (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) reflexive) (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (ringTimesZero R)) (Group.multIdentRight (Ring.additiveGroup R)))) reflexive) (Ring.multCommutative R)) (symmetric (Ring.multWellDefined R (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) reflexive))
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Group.multIdentLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
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need : (((a * Ring.1R R) + (b * Group.0G (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
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need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.timesZero R)) (Group.identRight (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (symmetric (Ring.*WellDefined R (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive))
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Group.identLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((Group.identity (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
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need = transitive (transitive (Ring.multWellDefined R (transitive (Group.wellDefined (Ring.additiveGroup R) reflexive (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (transitive (Ring.multCommutative R) (ringTimesZero R)) reflexive) (Group.multIdentLeft (Ring.additiveGroup R)))) reflexive) (Ring.multCommutative R)) (Ring.multWellDefined R (symmetric (Ring.multIdentIsIdent R)) reflexive)
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need : (((Group.0G (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
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need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.timesZero R)) reflexive) (Group.identLeft (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive)
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Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R))
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need = transitive (transitive (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) reflexive) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (ringTimesZero R))))) (symmetric (ringTimesZero R))
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need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
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need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
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Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R))
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need = transitive (transitive (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R)) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) reflexive) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invRight (Ring.additiveGroup R))) (ringTimesZero R))))) (symmetric (ringTimesZero R))
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Ring.multWellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
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need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
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need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invRight (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
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Ring.*WellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
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where
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open Setoid S
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open Equivalence eq
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need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g))
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need = transitive (Ring.multWellDefined R reflexive (Ring.multCommutative R)) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) reflexive) (transitive (Ring.multWellDefined R (Ring.multWellDefined R reflexive ch=dg) reflexive) (transitive (Ring.multCommutative R) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (Ring.multCommutative R) reflexive) (transitive (Ring.multWellDefined R af=be reflexive) (transitive (Ring.multAssoc R) (transitive (Ring.multWellDefined R (transitive (symmetric (Ring.multAssoc R)) (transitive (Ring.multWellDefined R reflexive (Ring.multCommutative R)) (Ring.multAssoc R))) reflexive) (symmetric (Ring.multAssoc R)))))))))))
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need = transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) (transitive (Ring.*WellDefined R (Ring.*WellDefined R reflexive ch=dg) reflexive) (transitive (Ring.*Commutative R) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (Ring.*Commutative R) reflexive) (transitive (Ring.*WellDefined R af=be reflexive) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (transitive (symmetric (Ring.*Associative R)) (transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (Ring.*Associative R))) reflexive) (symmetric (Ring.*Associative R)))))))))))
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Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
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Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
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need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (Ring.multCommutative R) (transitive (Group.wellDefined (Ring.additiveGroup R) (Ring.multCommutative R) (Ring.multCommutative R)) (Ring.groupIsAbelian R)))
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Ring.multAssoc (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
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need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) (Ring.*Commutative R)) (Ring.groupIsAbelian R)))
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Ring.*Associative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
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need = transitive (Ring.multWellDefined R (Ring.multAssoc R) (symmetric (Ring.multAssoc R))) (Ring.multCommutative R)
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Ring.multCommutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
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need = transitive (Ring.*WellDefined R (Ring.*Associative R) (symmetric (Ring.*Associative R))) (Ring.*Commutative R)
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Ring.*Commutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
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need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (Ring.multCommutative R) (Ring.multCommutative R))
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Ring.multDistributes (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
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need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (Ring.*Commutative R))
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Ring.*DistributesOver+ (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
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where
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open Setoid S
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open Ring R
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open Equivalence eq
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inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e)))
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inter = transitive multAssoc (transitive multDistributes (Group.wellDefined additiveGroup (transitive multAssoc (transitive (multWellDefined (transitive (multWellDefined (multCommutative) reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc))) reflexive) (symmetric multAssoc))) (transitive multAssoc (transitive (multWellDefined (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) reflexive) (symmetric multAssoc)))))
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inter = transitive *Associative (transitive *DistributesOver+ (Group.+WellDefined additiveGroup (transitive *Associative (transitive (*WellDefined (transitive (*WellDefined (*Commutative) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative))) reflexive) (symmetric *Associative))) (transitive *Associative (transitive (*WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive) (symmetric *Associative)))))
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need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e))))
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need = transitive (Ring.multWellDefined R reflexive (Ring.multWellDefined R reflexive (Ring.multCommutative R))) (transitive (Ring.multWellDefined R reflexive (Ring.multAssoc R)) (transitive (Ring.multCommutative R) (transitive (Ring.multWellDefined R (Ring.multWellDefined R (symmetric (Ring.multAssoc R)) reflexive) reflexive) (transitive (symmetric (Ring.multAssoc R)) (Ring.multWellDefined R reflexive inter)))))
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Ring.multIdentIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
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need = transitive (Ring.*WellDefined R reflexive (Ring.*WellDefined R reflexive (Ring.*Commutative R))) (transitive (Ring.*WellDefined R reflexive (Ring.*Associative R)) (transitive (Ring.*Commutative R) (transitive (Ring.*WellDefined R (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) reflexive) (transitive (symmetric (Ring.*Associative R)) (Ring.*WellDefined R reflexive inter)))))
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Ring.identIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
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where
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open Setoid S
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open Equivalence eq
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need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a)
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need = transitive (Ring.multWellDefined R (Ring.multIdentIsIdent R) reflexive) (transitive (Ring.multCommutative R) (Ring.multWellDefined R (symmetric (Ring.multIdentIsIdent R)) reflexive))
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need = transitive (Ring.*WellDefined R (Ring.identIsIdent R) reflexive) (transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive))
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fieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Field (fieldOfFractionsRing I)
|
||||
Field.allInvertible (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
|
||||
@@ -153,13 +153,13 @@ module Fields.FieldOfFractions where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
|
||||
need = Ring.multWellDefined R (Ring.multCommutative R) reflexive
|
||||
need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
|
||||
ans : fst ∼ Ring.0R R → False
|
||||
ans pr = prA need'
|
||||
where
|
||||
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
|
||||
need' = transitive (Ring.multWellDefined R pr reflexive) (transitive (transitive (Ring.multCommutative R) (ringTimesZero R)) (symmetric (ringTimesZero R)))
|
||||
Field.nontrivial (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (ringTimesZero R)) (transitive (Ring.multCommutative R) (transitive pr (Ring.multIdentIsIdent R)))))
|
||||
need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
|
||||
Field.nontrivial (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (Ring.timesZero R)) (transitive (Ring.*Commutative R) (transitive pr (Ring.identIsIdent R)))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -171,22 +171,22 @@ module Fields.FieldOfFractions where
|
||||
|
||||
homIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → RingHom R (fieldOfFractionsRing I) (embedIntoFieldOfFractions I)
|
||||
RingHom.preserves1 (homIntoFieldOfFractions {S = S} I) = Equivalence.reflexive (Setoid.eq S)
|
||||
RingHom.ringHom (homIntoFieldOfFractions {S = S} {R = R} I) {a} {b} = Equivalence.transitive (Setoid.eq S) (Ring.multWellDefined R (Equivalence.reflexive (Setoid.eq S)) (Ring.multIdentIsIdent R)) (Ring.multCommutative R)
|
||||
RingHom.ringHom (homIntoFieldOfFractions {S = S} {R = R} I) {a} {b} = Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R (Equivalence.reflexive (Setoid.eq S)) (Ring.identIsIdent R)) (Ring.*Commutative R)
|
||||
GroupHom.groupHom (RingHom.groupHom (homIntoFieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {x} {y} = need
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y)))
|
||||
need = transitive (transitive (Ring.multWellDefined R reflexive (Ring.multIdentIsIdent R)) (transitive (Ring.multCommutative R) (transitive (Ring.multIdentIsIdent R) (Group.wellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R))) (symmetric (Ring.multIdentIsIdent R)))))) (symmetric (Ring.multIdentIsIdent R))
|
||||
GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.multCommutative R) (Ring.multWellDefined R reflexive x=y)
|
||||
need = transitive (transitive (Ring.*WellDefined R reflexive (Ring.identIsIdent R)) (transitive (Ring.*Commutative R) (transitive (Ring.identIsIdent R) (Group.+WellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.*Commutative R) (Ring.identIsIdent R))) (symmetric (Ring.identIsIdent R)))))) (symmetric (Ring.identIsIdent R))
|
||||
GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
|
||||
where
|
||||
open Equivalence (Setoid.eq S)
|
||||
|
||||
homIntoFieldOfFractionsIsInj : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → SetoidInjection S (fieldOfFractionsSetoid I) (embedIntoFieldOfFractions I)
|
||||
SetoidInjection.wellDefined (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x=y = transitive (Ring.multCommutative R) (Ring.multWellDefined R reflexive x=y)
|
||||
SetoidInjection.wellDefined (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
|
||||
where
|
||||
open Equivalence (Setoid.eq S)
|
||||
SetoidInjection.injective (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x~y = transitive (symmetric multIdentIsIdent) (transitive multCommutative (transitive x~y multIdentIsIdent))
|
||||
SetoidInjection.injective (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x~y = transitive (symmetric identIsIdent) (transitive *Commutative (transitive x~y identIsIdent))
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
|
||||
@@ -48,11 +48,11 @@ module Fields.FieldOfFractionsOrder where
|
||||
have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
|
||||
have = ringCanMultiplyByPositive order 0<denomZ x<y
|
||||
p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
|
||||
p = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) reflexive have
|
||||
p = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
|
||||
q : ((denomX * numZ) * denomY) < ((numY * denomX) * denomZ)
|
||||
q = SetoidPartialOrder.wellDefined pOrder (multWellDefined x=z reflexive) reflexive p
|
||||
q = SetoidPartialOrder.wellDefined pOrder (*WellDefined x=z reflexive) reflexive p
|
||||
r : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) multCommutative) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) q
|
||||
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 {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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))
|
||||
@@ -65,11 +65,11 @@ module Fields.FieldOfFractionsOrder where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByPositive order 0<denomZ x<y)
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y)
|
||||
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (multWellDefined x=z reflexive) p
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (*WellDefined x=z reflexive) p
|
||||
r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) multCommutative) q
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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))
|
||||
@@ -83,9 +83,9 @@ module Fields.FieldOfFractionsOrder where
|
||||
p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
|
||||
p = ringCanMultiplyByPositive order 0<denomZ x<y
|
||||
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (multWellDefined x=z reflexive)))) p
|
||||
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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) multCommutative) q
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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))
|
||||
@@ -96,23 +96,23 @@ module Fields.FieldOfFractionsOrder where
|
||||
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
|
||||
p = ringCanMultiplyByPositive order 0<denomZ x<y
|
||||
q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
|
||||
q = SetoidPartialOrder.wellDefined pOrder (transitive (multWellDefined multCommutative reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive x=z) (transitive multCommutative (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) p
|
||||
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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive x=z) (transitive (multWellDefined reflexive (multCommutative)) (transitive multAssoc (multWellDefined multCommutative reflexive))))) p)
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
|
||||
p = ringCanMultiplyByNegative order denomZ<0 (SetoidPartialOrder.wellDefined pOrder multCommutative reflexive x<y)
|
||||
p = ringCanMultiplyByNegative order denomZ<0 (SetoidPartialOrder.wellDefined pOrder *Commutative reflexive x<y)
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined x=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) p)
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -126,14 +126,14 @@ module Fields.FieldOfFractionsOrder where
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 (multWellDefined multCommutative reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive x=z) (transitive multAssoc (transitive multCommutative multAssoc))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {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 {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (multWellDefined multCommutative reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive x=z) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (multWellDefined multCommutative reflexive)))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -150,21 +150,21 @@ module Fields.FieldOfFractionsOrder where
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z with SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomZ
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order 0<denomZ x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByPositive order 0<denomZ x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order 0<denomZ x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive (multAssoc) (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -172,21 +172,21 @@ module Fields.FieldOfFractionsOrder where
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomZ
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByPositive order 0<denomZ x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order 0<denomZ x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order 0<denomZ x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 multAssoc) (transitive (multWellDefined reflexive multCommutative) (transitive multAssoc (transitive (multWellDefined y=z reflexive) (transitive (symmetric multAssoc) multCommutative))))) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByNegative order denomZ<0 x<y))
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {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 order denomZ<0 x<y))
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -195,7 +195,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
|
||||
swapLemma : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
|
||||
swapLemma {S = S} R = transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)
|
||||
swapLemma {S = S} R = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -225,11 +225,11 @@ module Fields.FieldOfFractionsOrder where
|
||||
inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
|
||||
inter = ringCanMultiplyByPositive oRing 0<denomC a<b
|
||||
p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
|
||||
p = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) reflexive inter) (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByPositive oRing 0<denomA b<c))
|
||||
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 oRing 0<denomA b<c))
|
||||
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {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))
|
||||
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {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))
|
||||
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
|
||||
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {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 oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)) (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (ringCanMultiplyByPositive oRing 0<denomC a<b)))
|
||||
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {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 oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)) (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive oRing 0<denomC a<b)))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -315,14 +315,14 @@ module Fields.FieldOfFractionsOrder where
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) with SetoidTotalOrder.totality tOrder (numA * denomB) (numB * denomA)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inl x) = inl (inl x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inr x) = inl (inr x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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.multCommutative R))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) with SetoidTotalOrder.totality tOrder (numB * denomA) (numA * denomB)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inl x) = inl (inl x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inr x) = inl (inr x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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.multCommutative R) x))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
@@ -332,14 +332,14 @@ module Fields.FieldOfFractionsOrder where
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) with SetoidTotalOrder.totality tOrder (numB * denomA) (numA * denomB)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inl x) = inl (inl x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inr x) = inl (inr x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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.multCommutative R))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) with SetoidTotalOrder.totality tOrder (numA * denomB) (numB * denomA)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inl x) = inl (inl x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inr x) = inl (inr x)
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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.multCommutative R))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (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))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
SetoidTotalOrder.totality (fieldOfFractionsTotalOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
@@ -347,12 +347,12 @@ module Fields.FieldOfFractionsOrder where
|
||||
ineqLemma : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (Ring.0R R) < (x * y) → (Ring.0R R) < x → (Ring.0R R) < y
|
||||
ineqLemma {S = S} {R = R} {_<_ = _<_} {tOrder = tOrder} I order {x} {y} 0<xy 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) y
|
||||
ineqLemma {S = S} {R = R} {_<_} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
|
||||
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative order y<0 0<x))))
|
||||
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order y<0 0<x))))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (multWellDefined reflexive (symmetric 0=y)) (ringTimesZero R)) 0<xy))
|
||||
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -360,13 +360,13 @@ module Fields.FieldOfFractionsOrder where
|
||||
|
||||
ineqLemma' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
|
||||
ineqLemma' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) y
|
||||
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder multCommutative (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative order x<0 0<y))))
|
||||
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order x<0 0<y))))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
... | inl (inr y<0) = y<0
|
||||
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (multWellDefined reflexive (symmetric 0=y)) (ringTimesZero R)) 0<xy))
|
||||
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -380,7 +380,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
... | inl (inr y<0) = y<0
|
||||
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (multWellDefined reflexive (symmetric 0=y)) (ringTimesZero R)) reflexive xy<0))
|
||||
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -389,12 +389,12 @@ module Fields.FieldOfFractionsOrder where
|
||||
ineqLemma''' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
|
||||
ineqLemma''' {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder} {tOrder} I order {x} {y} xy<0 x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) y
|
||||
... | inl (inl 0<y) = 0<y
|
||||
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder xy<0 (SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) reflexive (ringCanMultiplyByNegative order y<0 x<0))))
|
||||
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder xy<0 (SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative order y<0 x<0))))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (multWellDefined reflexive (symmetric 0=y)) (ringTimesZero R)) reflexive xy<0))
|
||||
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -405,7 +405,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) with SetoidTotalOrder.totality tOrder (Ring.0R R) (denomB * denomC)
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) ans))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -413,12 +413,12 @@ module Fields.FieldOfFractionsOrder where
|
||||
0<dC : 0R < denomC
|
||||
0<dC with SetoidTotalOrder.totality tOrder 0R denomC
|
||||
0<dC | inl (inl x) = x
|
||||
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
|
||||
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
|
||||
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
|
||||
p = SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 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 (multWellDefined multCommutative reflexive) (transitive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (multWellDefined multCommutative reflexive)))) (OrderedRing.orderRespectsAddition oRing p ((denomA * numC) * denomB))
|
||||
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)))) (OrderedRing.orderRespectsAddition oRing p ((denomA * numC) * denomB))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
|
||||
where
|
||||
open Setoid S
|
||||
@@ -426,11 +426,11 @@ module Fields.FieldOfFractionsOrder where
|
||||
open Ring R
|
||||
dC<0 : denomC < 0R
|
||||
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC
|
||||
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByPositive oRing x dB<0))))
|
||||
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive oRing x dB<0))))
|
||||
... | inl (inr x) = x
|
||||
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
bad : False
|
||||
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dA)))
|
||||
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dA)))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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
|
||||
@@ -441,16 +441,16 @@ module Fields.FieldOfFractionsOrder where
|
||||
0<dC : 0R < denomC
|
||||
0<dC with SetoidTotalOrder.totality tOrder 0R denomC
|
||||
0<dC | inl (inl x) = x
|
||||
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
|
||||
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
|
||||
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
dC<0 : denomC < 0R
|
||||
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC
|
||||
dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder multCommutative (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dC))))
|
||||
dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dC))))
|
||||
dC<0 | inl (inr x) = x
|
||||
dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
bad : False
|
||||
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0)
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric multDistributes) multCommutative) (transitive (symmetric multDistributes) multCommutative) have''))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -460,9 +460,9 @@ module Fields.FieldOfFractionsOrder where
|
||||
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
|
||||
have = ringCanMultiplyByNegative oRing dC<0 a<b
|
||||
have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
|
||||
have' = SetoidPartialOrder.wellDefined pOrder (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)) (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)) have
|
||||
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 multAssoc (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)))) (OrderedRing.orderRespectsAddition oRing have' (denomA * (denomB * numC)))
|
||||
have'' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (OrderedRing.orderRespectsAddition oRing have' (denomA * (denomB * numC)))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
@@ -478,7 +478,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
|
||||
bad : False
|
||||
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0)
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByPositive oRing 0<dC ans)
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 0<dC ans)
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -490,10 +490,10 @@ module Fields.FieldOfFractionsOrder where
|
||||
have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
|
||||
have' = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) have) _
|
||||
ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
|
||||
ans = SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) (transitive (Group.wellDefined additiveGroup multCommutative (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative))) (transitive (symmetric multDistributes) multCommutative)) have'
|
||||
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'
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (transitive (Group.wellDefined additiveGroup reflexive (transitive multAssoc (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)))) (symmetric multDistributes)) multCommutative)) (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) have))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -530,7 +530,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
0<dC = ineqLemma I oRing 0<dBdC 0<dB
|
||||
dC<0 : denomC < 0R
|
||||
dC<0 = ineqLemma'' I oRing dAdC<0 0<dA
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)) multCommutative) (transitive (symmetric multDistributes) multCommutative)) (transitive (Group.wellDefined additiveGroup (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)) (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative))) (transitive (symmetric multDistributes) multCommutative)) have))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -541,7 +541,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
have = OrderedRing.orderRespectsAddition oRing (ringCanMultiplyByNegative oRing dC<0 a<b) _
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (transitive (Group.wellDefined additiveGroup (transitive multCommutative (transitive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) multCommutative)) (transitive multAssoc (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc)))) (symmetric multDistributes)) multCommutative)) (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (transitive (Group.wellDefined additiveGroup (transitive multCommutative (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc))) reflexive) (symmetric multDistributes)) multCommutative)) have))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -563,7 +563,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (Group.wellDefined additiveGroup (transitive multCommutative (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc))) reflexive)) (transitive (symmetric multDistributes) multCommutative)) (transitive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (Group.wellDefined additiveGroup (transitive (transitive multAssoc (multWellDefined multCommutative reflexive)) multCommutative) (transitive multAssoc (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc))))) (transitive (symmetric multDistributes) multCommutative)) have))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -592,7 +592,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
0<dC = ineqLemma''' I oRing dAdC<0 dA<0
|
||||
dC<0 : denomC < 0R
|
||||
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multAssoc) (symmetric multAssoc) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (Group.wellDefined additiveGroup (transitive multCommutative (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc))) reflexive) (transitive (symmetric multDistributes) multCommutative))) (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (transitive (Group.wellDefined additiveGroup (transitive multCommutative (transitive (multWellDefined multCommutative reflexive) (symmetric multAssoc))) (transitive multCommutative (transitive (symmetric multAssoc) (multWellDefined reflexive multCommutative)))) (symmetric multDistributes)) multCommutative)) have))
|
||||
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 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
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
@@ -613,40 +613,40 @@ module Fields.FieldOfFractionsOrder where
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) with SetoidTotalOrder.totality tOrder (Ring.0R R) (denomA * denomB)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative (ringTimesZero R))) (symmetric (transitive multCommutative multIdentIsIdent)) 0<nAnB
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
0<nA : 0R < numA
|
||||
0<nA = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) (transitive multCommutative multIdentIsIdent) 0<a
|
||||
0<nA = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
|
||||
0<nB : 0R < numB
|
||||
0<nB = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) (transitive multCommutative multIdentIsIdent) 0<b
|
||||
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 multCommutative (ringTimesZero R)) reflexive (ringCanMultiplyByPositive oRing 0<nB 0<nA)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dB))))
|
||||
0<nAnB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive oRing 0<nB 0<nA)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing dA<0 0<dB))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dB<0 0<dA))))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing dB<0 0<dA))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative (ringTimesZero R))) (symmetric (transitive multCommutative multIdentIsIdent)) 0<nAnB
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
nB<0 : numB < 0R
|
||||
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative multIdentIsIdent) (transitive multCommutative (ringTimesZero R)) 0<b
|
||||
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
|
||||
nA<0 : numA < 0R
|
||||
nA<0 = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative multIdentIsIdent) (transitive multCommutative (ringTimesZero R)) 0<a
|
||||
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 multCommutative (ringTimesZero R)) multCommutative (ringCanMultiplyByNegative oRing nA<0 nB<0)
|
||||
0<nAnB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative oRing nA<0 nB<0)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
|
||||
@@ -659,37 +659,37 @@ module Fields.FieldOfFractionsOrder where
|
||||
f : False
|
||||
f with OrderedRing.orderRespectsMultiplication oRing 0<denomA 0<denomB
|
||||
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bl dAdB<0)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative multIdentIsIdent)) (symmetric (transitive multCommutative (ringTimesZero R))) ans
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
0<nB : 0R < numB
|
||||
0<nB = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) (transitive multCommutative multIdentIsIdent) 0<b
|
||||
0<nB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
|
||||
nA<0 : numA < 0R
|
||||
nA<0 = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative multIdentIsIdent) (transitive multCommutative (ringTimesZero R)) 0<a
|
||||
nA<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
|
||||
ans : (numA * numB) < 0R
|
||||
ans = SetoidPartialOrder.wellDefined pOrder multCommutative (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing nA<0 0<nB)
|
||||
ans = SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nA<0 0<nB)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative multIdentIsIdent)) (symmetric (transitive multCommutative (ringTimesZero R))) nAnB<0
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
nB<0 : numB < 0R
|
||||
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative multIdentIsIdent) (transitive multCommutative (ringTimesZero R)) 0<b
|
||||
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
|
||||
0<nA : 0R < numA
|
||||
0<nA = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) (transitive multCommutative multIdentIsIdent) 0<a
|
||||
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 multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing nB<0 0<nA)
|
||||
nAnB<0 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nB<0 0<nA)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Ring R
|
||||
h : 0R < (denomA * denomB)
|
||||
h = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) reflexive (ringCanMultiplyByNegative oRing denomB<0 denomA<0)
|
||||
h = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative oRing denomB<0 denomA<0)
|
||||
f : False
|
||||
f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder dAdB<0 h)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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))
|
||||
@@ -697,7 +697,7 @@ module Fields.FieldOfFractionsOrder where
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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)
|
||||
... | inl x = exFalso (denomA!=0 x)
|
||||
... | inr x = exFalso (denomB!=0 x)
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 multCommutative (ringTimesZero R)) multIdentIsIdent (ringCanMultiplyByNegative oRing 1<0 1<0))))
|
||||
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {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 oRing 1<0 1<0))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence (Setoid.eq S)
|
||||
|
||||
@@ -20,43 +20,43 @@ module Fields.Fields where
|
||||
open Group additiveGroup
|
||||
open Setoid S
|
||||
field
|
||||
allInvertible : (a : A) → ((a ∼ Group.identity (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
|
||||
allInvertible : (a : A) → ((a ∼ Group.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
|
||||
nontrivial : (0R ∼ 1R) → False
|
||||
|
||||
orderedFieldIsIntDom : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (O : OrderedRing R tOrder) (F : Field R) → IntegralDomain R
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 with SetoidTotalOrder.totality tOrder (Ring.0R R) a
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inl x) = inr (transitive (transitive (symmetric multIdentIsIdent) (multWellDefined q reflexive)) p')
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inl x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
open Ring R
|
||||
a!=0 : (a ∼ Group.identity additiveGroup) → False
|
||||
a!=0 : (a ∼ Group.0G additiveGroup) → False
|
||||
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric pr) reflexive x)
|
||||
invA : A
|
||||
invA = underlying (Field.allInvertible F a a!=0)
|
||||
q : 1R ∼ (invA * a)
|
||||
q with Field.allInvertible F a a!=0
|
||||
... | invA , pr = symmetric pr
|
||||
p : invA * (a * b) ∼ invA * Group.identity additiveGroup
|
||||
p = multWellDefined reflexive ab=0
|
||||
p' : (invA * a) * b ∼ Group.identity additiveGroup
|
||||
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R))
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inr x) = inr (transitive (transitive (symmetric multIdentIsIdent) (multWellDefined q reflexive)) p')
|
||||
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
|
||||
p = *WellDefined reflexive ab=0
|
||||
p' : (invA * a) * b ∼ Group.0G additiveGroup
|
||||
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inr x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
open Ring R
|
||||
a!=0 : (a ∼ Group.identity additiveGroup) → False
|
||||
a!=0 : (a ∼ Group.0G additiveGroup) → False
|
||||
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (symmetric pr) x)
|
||||
invA : A
|
||||
invA = underlying (Field.allInvertible F a a!=0)
|
||||
q : 1R ∼ (invA * a)
|
||||
q with Field.allInvertible F a a!=0
|
||||
... | invA , pr = symmetric pr
|
||||
p : invA * (a * b) ∼ invA * Group.identity additiveGroup
|
||||
p = multWellDefined reflexive ab=0
|
||||
p' : (invA * a) * b ∼ Group.identity additiveGroup
|
||||
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R))
|
||||
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
|
||||
p = *WellDefined reflexive ab=0
|
||||
p' : (invA * a) * b ∼ Group.0G additiveGroup
|
||||
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
|
||||
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
|
||||
IntegralDomain.nontrivial (orderedFieldIsIntDom {S = S} O F) pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
|
||||
|
||||
|
||||
Reference in New Issue
Block a user