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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)
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||||
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
|
||||
|
||||
Reference in New Issue
Block a user