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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)
|
||||
|
||||
|
||||
@@ -12,11 +12,11 @@ module Groups.Definition where
|
||||
record Group {lvl1 lvl2} {A : Set lvl1} (S : Setoid {lvl1} {lvl2} A) (_·_ : A → A → A) : Set (lsuc lvl1 ⊔ lvl2) where
|
||||
open Setoid S
|
||||
field
|
||||
wellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
|
||||
identity : A
|
||||
+WellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
|
||||
0G : A
|
||||
inverse : A → A
|
||||
multAssoc : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
|
||||
multIdentRight : {a : A} → (a · identity) ∼ a
|
||||
multIdentLeft : {a : A} → (identity · a) ∼ a
|
||||
invLeft : {a : A} → (inverse a) · a ∼ identity
|
||||
invRight : {a : A} → a · (inverse a) ∼ identity
|
||||
+Associative : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
|
||||
identRight : {a : A} → (a · 0G) ∼ a
|
||||
identLeft : {a : A} → (0G · a) ∼ a
|
||||
invLeft : {a : A} → (inverse a) · a ∼ 0G
|
||||
invRight : {a : A} → a · (inverse a) ∼ 0G
|
||||
|
||||
@@ -19,15 +19,15 @@ module Groups.Examples.ExampleSheet1 where
|
||||
{-
|
||||
Question 1: e is the unique solution of x^2 = x
|
||||
-}
|
||||
question1 : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} → (G : Group S _+_) → (x : A) → Setoid._∼_ S (x + x) x → Setoid._∼_ S x (Group.identity G)
|
||||
question1 {S = S} {_+_ = _+_} G x x+x=x = transitive (symmetric multIdentRight) (transitive (wellDefined reflexive (symmetric invRight)) (transitive multAssoc (transitive (wellDefined x+x=x reflexive) invRight)))
|
||||
question1 : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} → (G : Group S _+_) → (x : A) → Setoid._∼_ S (x + x) x → Setoid._∼_ S x (Group.0G G)
|
||||
question1 {S = S} {_+_ = _+_} G x x+x=x = transitive (symmetric identRight) (transitive (+WellDefined reflexive (symmetric invRight)) (transitive +Associative (transitive (+WellDefined x+x=x reflexive) invRight)))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
question1' : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} → (G : Group S _+_) → Setoid._∼_ S ((Group.identity G) + (Group.identity G)) (Group.identity G)
|
||||
question1' G = Group.multIdentRight G
|
||||
question1' : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} → (G : Group S _+_) → Setoid._∼_ S ((Group.0G G) + (Group.0G G)) (Group.0G G)
|
||||
question1' G = Group.identRight G
|
||||
|
||||
{-
|
||||
Question 2: intersection of subgroups is a subgroup; union of subgroups is a subgroup iff one is contained in the other.
|
||||
@@ -38,7 +38,7 @@ module Groups.Examples.ExampleSheet1 where
|
||||
ofElt : {x : A} → Sg B (λ b → Setoid._∼_ S (h1Inj b) x) → Sg C (λ c → Setoid._∼_ S (h2Inj c) x) → SubgroupIntersectionElement G H1 H2
|
||||
|
||||
subgroupIntersectionOp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {_+_ : A → A → A} {_+H1_ : B → B → B} {_+H2_ : C → C → C} (G : Group S _+_) {H1grp : Group T _+H1_} {H2grp : Group U _+H2_} {h1Inj : B → A} {h2Inj : C → A} {h1Hom : GroupHom H1grp G h1Inj} {h2Hom : GroupHom H2grp G h2Inj} (H1 : Subgroup G H1grp h1Hom) (H2 : Subgroup G H2grp h2Hom) → (r : SubgroupIntersectionElement G H1 H2) → (s : SubgroupIntersectionElement G H1 H2) → SubgroupIntersectionElement G H1 H2
|
||||
subgroupIntersectionOp {S = S} {_+_ = _+_} {_+H1_ = _+H1_} {_+H2_ = _+H2_} G {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2 (ofElt (b , prB) (c , prC)) (ofElt (b2 , prB2) (c2 , prC2)) = ofElt ((b +H1 b2) , GroupHom.groupHom h1Hom) ((c +H2 c2) , transitive (GroupHom.groupHom h2Hom) (transitive (Group.wellDefined G prC prC2) (Group.wellDefined G (symmetric prB) (symmetric prB2))))
|
||||
subgroupIntersectionOp {S = S} {_+_ = _+_} {_+H1_ = _+H1_} {_+H2_ = _+H2_} G {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2 (ofElt (b , prB) (c , prC)) (ofElt (b2 , prB2) (c2 , prC2)) = ofElt ((b +H1 b2) , GroupHom.groupHom h1Hom) ((c +H2 c2) , transitive (GroupHom.groupHom h2Hom) (transitive (Group.+WellDefined G prC prC2) (Group.+WellDefined G (symmetric prB) (symmetric prB2))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -50,20 +50,20 @@ module Groups.Examples.ExampleSheet1 where
|
||||
Equivalence.transitive (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2)) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst1 ,, snd1) (fst2 ,, snd2) = Equivalence.transitive (Setoid.eq T) fst1 fst2 ,, Equivalence.transitive (Setoid.eq U) snd1 snd2
|
||||
|
||||
subgroupIntersectionGroup : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {_+_ : A → A → A} {_+H1_ : B → B → B} {_+H2_ : C → C → C} (G : Group S _+_) {H1grp : Group T _+H1_} {H2grp : Group U _+H2_} {h1Inj : B → A} {h2Inj : C → A} {h1Hom : GroupHom H1grp G h1Inj} {h2Hom : GroupHom H2grp G h2Inj} (H1 : Subgroup G H1grp h1Hom) (H2 : Subgroup G H2grp h2Hom) → Group (subgroupIntersectionSetoid G H1 H2) (subgroupIntersectionOp G H1 H2)
|
||||
Group.wellDefined (subgroupIntersectionGroup {S = S} {T = T} {U = U} G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} {ofElt (_ , _ ) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (pr1 ,, pr2) (pr3 ,, pr4) = transitiveT (Group.wellDefined h1 pr1 reflexiveT) (Group.wellDefined h1 reflexiveT pr3) ,, transitiveU (Group.wellDefined h2 pr2 reflexiveU) ((Group.wellDefined h2 reflexiveU pr4))
|
||||
Group.+WellDefined (subgroupIntersectionGroup {S = S} {T = T} {U = U} G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} {ofElt (_ , _ ) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (pr1 ,, pr2) (pr3 ,, pr4) = transitiveT (Group.+WellDefined h1 pr1 reflexiveT) (Group.+WellDefined h1 reflexiveT pr3) ,, transitiveU (Group.+WellDefined h2 pr2 reflexiveU) ((Group.+WellDefined h2 reflexiveU pr4))
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T) renaming (transitive to transitiveT ; symmetric to symmetricT ; reflexive to reflexiveT)
|
||||
open Equivalence (Setoid.eq U) renaming (transitive to transitiveU ; symmetric to symmetricU ; reflexive to reflexiveU)
|
||||
Group.identity (subgroupIntersectionGroup G {H1grp = H1grp} {H2grp = H2grp} {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2) = ofElt {x = Group.identity G} (Group.identity H1grp , imageOfIdentityIsIdentity h1Hom) (Group.identity H2grp , imageOfIdentityIsIdentity h2Hom)
|
||||
Group.0G (subgroupIntersectionGroup G {H1grp = H1grp} {H2grp = H2grp} {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2) = ofElt {x = Group.0G G} (Group.0G H1grp , imageOfIdentityIsIdentity h1Hom) (Group.0G H2grp , imageOfIdentityIsIdentity h2Hom)
|
||||
Group.inverse (subgroupIntersectionGroup {S = S} G {H1grp = h1} {H2grp = h2} {h1Hom = h1hom} {h2Hom = h2hom} H1 H2) (ofElt (a , prA) (b , prB)) = ofElt (Group.inverse h1 a , homRespectsInverse h1hom) (Group.inverse h2 b , transitive (homRespectsInverse h2hom) (inverseWellDefined G (transitive prB (symmetric prA))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
Group.multAssoc (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} {ofElt (e , prE) (f , prF)} = Group.multAssoc h1 ,, Group.multAssoc h2
|
||||
Group.multIdentRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentRight h1 ,, Group.multIdentRight h2
|
||||
Group.multIdentLeft (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentLeft h1 ,, Group.multIdentLeft h2
|
||||
Group.+Associative (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} {ofElt (e , prE) (f , prF)} = Group.+Associative h1 ,, Group.+Associative h2
|
||||
Group.identRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.identRight h1 ,, Group.identRight h2
|
||||
Group.identLeft (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.identLeft h1 ,, Group.identLeft h2
|
||||
Group.invLeft (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.invLeft h1 ,, Group.invLeft h2
|
||||
Group.invRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.invRight h1 ,, Group.invRight h2
|
||||
|
||||
@@ -131,7 +131,7 @@ module Groups.Examples.ExampleSheet1 where
|
||||
open Equivalence eq
|
||||
bad : Setoid._∼_ S xCoeff2 0R
|
||||
bad with Field.allInvertible F xCoeff1 xCoeffNonzero1
|
||||
... | xinv , pr' = transitive (symmetric multIdentIsIdent) (transitive (multWellDefined (symmetric pr') reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive pr) (ringTimesZero R))))
|
||||
... | xinv , pr' = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric pr') reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive pr) (Ring.timesZero R))))
|
||||
LinearFunction.constant (composeLinearFunctions {_+_ = _+_} {_*_ = _*_} record { xCoeff = xCoeff1 ; xCoeffNonzero = xCoeffNonzero1 ; constant = constant1 } record { xCoeff = xCoeff2 ; xCoeffNonzero = xCoeffNonzero2 ; constant = constant2 }) = (xCoeff1 * constant2) + constant1
|
||||
|
||||
compositionIsCorrect : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {F : Field R} (f1 : LinearFunction F) (f2 : LinearFunction F) → {r : A} → Setoid._∼_ S (interpretLinearFunction (composeLinearFunctions f1 f2) r) (((interpretLinearFunction f1) ∘ (interpretLinearFunction f2)) r)
|
||||
@@ -141,7 +141,7 @@ module Groups.Examples.ExampleSheet1 where
|
||||
open Ring R
|
||||
open Equivalence eq
|
||||
ans : (((xCoeff * xCoeff') * r) + ((xCoeff * constant') + constant)) ∼ (xCoeff * ((xCoeff' * r) + constant')) + constant
|
||||
ans = transitive (Group.multAssoc additiveGroup) (Group.wellDefined additiveGroup (transitive (Group.wellDefined additiveGroup (symmetric (Ring.multAssoc R)) reflexive) (symmetric (Ring.multDistributes R))) (reflexive {constant}))
|
||||
ans = transitive (Group.+Associative additiveGroup) (Group.+WellDefined additiveGroup (transitive (Group.+WellDefined additiveGroup (symmetric (Ring.*Associative R)) reflexive) (symmetric (Ring.*DistributesOver+ R))) (reflexive {constant}))
|
||||
|
||||
linearFunctionsSetoid : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : Field R) → Setoid (LinearFunction I)
|
||||
Setoid._∼_ (linearFunctionsSetoid {S = S} I) f1 f2 = ((LinearFunction.xCoeff f1) ∼ (LinearFunction.xCoeff f2)) && ((LinearFunction.constant f1) ∼ (LinearFunction.constant f2))
|
||||
@@ -152,38 +152,38 @@ module Groups.Examples.ExampleSheet1 where
|
||||
Equivalence.transitive (Setoid.eq (linearFunctionsSetoid {S = S} I)) (fst1 ,, snd1) (fst2 ,, snd2) = Equivalence.transitive (Setoid.eq S) fst1 fst2 ,, Equivalence.transitive (Setoid.eq S) snd1 snd2
|
||||
|
||||
linearFunctionsGroup : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → Group (linearFunctionsSetoid F) (composeLinearFunctions)
|
||||
Group.wellDefined (linearFunctionsGroup {R = R} F) {record { xCoeff = xCoeffM ; xCoeffNonzero = xCoeffNonzeroM ; constant = constantM }} {record { xCoeff = xCoeffN ; xCoeffNonzero = xCoeffNonzeroN ; constant = constantN }} {record { xCoeff = xCoeffX ; xCoeffNonzero = xCoeffNonzeroX ; constant = constantX }} {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} (fst1 ,, snd1) (fst2 ,, snd2) = multWellDefined fst1 fst2 ,, Group.wellDefined additiveGroup (multWellDefined fst1 snd2) snd1
|
||||
Group.+WellDefined (linearFunctionsGroup {R = R} F) {record { xCoeff = xCoeffM ; xCoeffNonzero = xCoeffNonzeroM ; constant = constantM }} {record { xCoeff = xCoeffN ; xCoeffNonzero = xCoeffNonzeroN ; constant = constantN }} {record { xCoeff = xCoeffX ; xCoeffNonzero = xCoeffNonzeroX ; constant = constantX }} {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} (fst1 ,, snd1) (fst2 ,, snd2) = *WellDefined fst1 fst2 ,, Group.+WellDefined additiveGroup (*WellDefined fst1 snd2) snd1
|
||||
where
|
||||
open Ring R
|
||||
Group.identity (linearFunctionsGroup {S = S} {R = R} F) = record { xCoeff = Ring.1R R ; constant = Ring.0R R ; xCoeffNonzero = λ p → Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) p) }
|
||||
Group.0G (linearFunctionsGroup {S = S} {R = R} F) = record { xCoeff = Ring.1R R ; constant = Ring.0R R ; xCoeffNonzero = λ p → Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) p) }
|
||||
Group.inverse (linearFunctionsGroup {S = S} {_*_ = _*_} {R = R} F) record { xCoeff = xCoeff ; constant = c ; xCoeffNonzero = pr } with Field.allInvertible F xCoeff pr
|
||||
... | (inv , pr') = record { xCoeff = inv ; constant = inv * (Group.inverse (Ring.additiveGroup R) c) ; xCoeffNonzero = λ p → Field.nontrivial F (transitive (symmetric (transitive (Ring.multWellDefined R p reflexive) (transitive (Ring.multCommutative R) (ringTimesZero R)))) pr') }
|
||||
... | (inv , pr') = record { xCoeff = inv ; constant = inv * (Group.inverse (Ring.additiveGroup R) c) ; xCoeffNonzero = λ p → Field.nontrivial F (transitive (symmetric (transitive (Ring.*WellDefined R p reflexive) (transitive (Ring.*Commutative R) (Ring.timesZero R)))) pr') }
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
Group.multAssoc (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xA ; xCoeffNonzero = xANonzero ; constant = cA }} {record { xCoeff = xB ; xCoeffNonzero = xBNonzero ; constant = cB }} {record { xCoeff = xC ; xCoeffNonzero = xCNonzero ; constant = cC }} = Ring.multAssoc R ,, transitive (Group.wellDefined additiveGroup (transitive multDistributes (Group.wellDefined additiveGroup multAssoc reflexive)) reflexive) (symmetric (Group.multAssoc additiveGroup))
|
||||
Group.+Associative (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xA ; xCoeffNonzero = xANonzero ; constant = cA }} {record { xCoeff = xB ; xCoeffNonzero = xBNonzero ; constant = cB }} {record { xCoeff = xC ; xCoeffNonzero = xCNonzero ; constant = cC }} = Ring.*Associative R ,, transitive (Group.+WellDefined additiveGroup (transitive *DistributesOver+ (Group.+WellDefined additiveGroup *Associative reflexive)) reflexive) (symmetric (Group.+Associative additiveGroup))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
open Ring R
|
||||
Group.multIdentRight (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R) ,, transitive (Group.wellDefined additiveGroup (ringTimesZero R) reflexive) (Group.multIdentLeft additiveGroup)
|
||||
Group.identRight (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = transitive (Ring.*Commutative R) (Ring.identIsIdent R) ,, transitive (Group.+WellDefined additiveGroup (Ring.timesZero R) reflexive) (Group.identLeft additiveGroup)
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
Group.multIdentLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = multIdentIsIdent ,, transitive (Group.multIdentRight additiveGroup) multIdentIsIdent
|
||||
Group.identLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = identIsIdent ,, transitive (Group.identRight additiveGroup) identIsIdent
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Equivalence eq
|
||||
Group.invLeft (linearFunctionsGroup F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero
|
||||
Group.invLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} | inv , prInv = prInv ,, transitive (symmetric multDistributes) (transitive (multWellDefined reflexive (Group.invRight additiveGroup)) (ringTimesZero R))
|
||||
Group.invLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} | inv , prInv = prInv ,, transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive (Group.invRight additiveGroup)) (Ring.timesZero R))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Equivalence eq
|
||||
Group.invRight (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero
|
||||
... | inv , pr = transitive multCommutative pr ,, transitive (Group.wellDefined additiveGroup multAssoc reflexive) (transitive (Group.wellDefined additiveGroup (multWellDefined (transitive multCommutative pr) reflexive) reflexive) (transitive (Group.wellDefined additiveGroup multIdentIsIdent reflexive) (Group.invLeft additiveGroup)))
|
||||
... | inv , pr = transitive *Commutative pr ,, transitive (Group.+WellDefined additiveGroup *Associative reflexive) (transitive (Group.+WellDefined additiveGroup (*WellDefined (transitive *Commutative pr) reflexive) reflexive) (transitive (Group.+WellDefined additiveGroup identIsIdent reflexive) (Group.invLeft additiveGroup)))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
@@ -213,10 +213,10 @@ module Groups.Examples.ExampleSheet1 where
|
||||
fg = record { xCoeff = 1R + 1R ; xCoeffNonzero = pr ; constant = 1R }
|
||||
|
||||
oneWay : Setoid._∼_ (linearFunctionsSetoid F) gf (composeLinearFunctions g f)
|
||||
oneWay = symmetricS (transitiveS multCommutative multIdentIsIdent) ,, transitiveS (symmetricS (transitiveS multCommutative multIdentIsIdent)) (symmetricS (Group.multIdentRight additiveGroup))
|
||||
oneWay = symmetricS (transitiveS *Commutative identIsIdent) ,, transitiveS (symmetricS (transitiveS *Commutative identIsIdent)) (symmetricS (Group.identRight additiveGroup))
|
||||
|
||||
otherWay : Setoid._∼_ (linearFunctionsSetoid F) fg (composeLinearFunctions f g)
|
||||
otherWay = symmetricS multIdentIsIdent ,, transitiveS (symmetricS (Group.multIdentLeft additiveGroup)) (Group.wellDefined additiveGroup (symmetricS multIdentIsIdent) (reflexiveS {1R}))
|
||||
otherWay = symmetricS identIsIdent ,, transitiveS (symmetricS (Group.identLeft additiveGroup)) (Group.+WellDefined additiveGroup (symmetricS identIsIdent) (reflexiveS {1R}))
|
||||
|
||||
open Equivalence (Setoid.eq (linearFunctionsSetoid F))
|
||||
bad : Setoid._∼_ (linearFunctionsSetoid F) gf fg
|
||||
@@ -224,4 +224,4 @@ module Groups.Examples.ExampleSheet1 where
|
||||
|
||||
ans : False
|
||||
ans with bad
|
||||
ans | _ ,, contr = Field.nontrivial F (symmetricS (transitiveS {1R} {1R + (1R + Group.inverse additiveGroup 1R)} (transitiveS (symmetricS (Group.multIdentRight additiveGroup)) (Group.wellDefined additiveGroup reflexiveS (symmetricS (Group.invRight additiveGroup)))) (transitiveS (Group.multAssoc additiveGroup) (transitiveS (Group.wellDefined additiveGroup contr reflexiveS) (Group.invRight additiveGroup)))))
|
||||
ans | _ ,, contr = Field.nontrivial F (symmetricS (transitiveS {1R} {1R + (1R + Group.inverse additiveGroup 1R)} (transitiveS (symmetricS (Group.identRight additiveGroup)) (Group.+WellDefined additiveGroup reflexiveS (symmetricS (Group.invRight additiveGroup)))) (transitiveS (Group.+Associative additiveGroup) (transitiveS (Group.+WellDefined additiveGroup contr reflexiveS) (Group.invRight additiveGroup)))))
|
||||
|
||||
@@ -22,52 +22,52 @@ module Groups.Groups where
|
||||
groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
|
||||
groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
|
||||
where
|
||||
open Group g renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
|
||||
j = Equivalence.transitive eq (symmetric (multAssoc {x ^-1} {x} {y})) (wellDefined ~refl pr)
|
||||
j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
|
||||
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
|
||||
k = transitive j multAssoc
|
||||
l : e · y ∼ ((x ^-1) · x) · z
|
||||
l = transitive (wellDefined (symmetric invLeft) ~refl) k
|
||||
m : e · y ∼ e · z
|
||||
m = transitive l (wellDefined invLeft ~refl)
|
||||
n : y ∼ e · z
|
||||
n = transitive (symmetric multIdentLeft) m
|
||||
k = transitive j +Associative
|
||||
l : 0G · y ∼ ((x ^-1) · x) · z
|
||||
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
|
||||
m : 0G · y ∼ 0G · z
|
||||
m = transitive l (+WellDefined invLeft ~refl)
|
||||
n : y ∼ 0G · z
|
||||
n = transitive (symmetric identLeft) m
|
||||
o : y ∼ z
|
||||
o = transitive n multIdentLeft
|
||||
o = transitive n identLeft
|
||||
|
||||
groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
|
||||
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m multIdentRight
|
||||
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
|
||||
where
|
||||
open Group g renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
|
||||
i = wellDefined pr ~refl
|
||||
i = +WellDefined pr ~refl
|
||||
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
|
||||
j = transitive multAssoc i
|
||||
j' : y · e ∼ (z · x) · (x ^-1)
|
||||
j' = transitive (wellDefined ~refl (symmetric invRight)) j
|
||||
j = transitive +Associative i
|
||||
j' : y · 0G ∼ (z · x) · (x ^-1)
|
||||
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
|
||||
k : y ∼ (z · x) · (x ^-1)
|
||||
k = transitive (symmetric multIdentRight) j'
|
||||
k = transitive (symmetric identRight) j'
|
||||
l : y ∼ z · (x · (x ^-1))
|
||||
l = transitive k (symmetric multAssoc)
|
||||
m : y ∼ z · e
|
||||
m = transitive l (wellDefined ~refl invRight)
|
||||
l = transitive k (symmetric +Associative)
|
||||
m : y ∼ z · 0G
|
||||
m = transitive l (+WellDefined ~refl invRight)
|
||||
|
||||
replaceGroupOp : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
|
||||
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (wellDefined a~c b~d)) (transitive pr (wellDefined w~y x~z))
|
||||
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (+WellDefined a~c b~d)) (transitive pr (+WellDefined w~y x~z))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
|
||||
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (wellDefined reflexive c~x)
|
||||
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
@@ -84,69 +84,69 @@ module Groups.Groups where
|
||||
q : inverse x · x ∼ inverse y · x
|
||||
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
|
||||
|
||||
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.identity G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
|
||||
where
|
||||
open Group G renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y ∼ y · e
|
||||
j : y · e ∼ y · (x · (x ^-1))
|
||||
i : y ∼ y · 0G
|
||||
j : y · 0G ∼ y · (x · (x ^-1))
|
||||
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
|
||||
l : (y · x) · (x ^-1) ∼ e · (x ^-1)
|
||||
m : e · (x ^-1) ∼ x ^-1
|
||||
i = symmetric multIdentRight
|
||||
j = wellDefined ~refl (symmetric invRight)
|
||||
k = multAssoc
|
||||
l = wellDefined f ~refl
|
||||
m = multIdentLeft
|
||||
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
|
||||
m : 0G · (x ^-1) ∼ x ^-1
|
||||
i = symmetric identRight
|
||||
j = +WellDefined ~refl (symmetric invRight)
|
||||
k = +Associative
|
||||
l = +WellDefined f ~refl
|
||||
m = identLeft
|
||||
|
||||
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.identity G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
|
||||
where
|
||||
open Group G renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y · (x · y) ∼ y · e
|
||||
i : y · (x · y) ∼ y · 0G
|
||||
i' : y · (x · y) ∼ y
|
||||
j : (y · x) · y ∼ y
|
||||
k : (y · x) · y ∼ e · y
|
||||
l : y · x ∼ e
|
||||
i = wellDefined ~refl f
|
||||
i' = transitive i multIdentRight
|
||||
j = transitive (symmetric multAssoc) i'
|
||||
k = transitive j (symmetric multIdentLeft)
|
||||
l = groupsHaveRightCancellation G y (y · x) e k
|
||||
k : (y · x) · y ∼ 0G · y
|
||||
l : y · x ∼ 0G
|
||||
i = +WellDefined ~refl f
|
||||
i' = transitive i identRight
|
||||
j = transitive (symmetric +Associative) i'
|
||||
k = transitive j (symmetric identLeft)
|
||||
l = groupsHaveRightCancellation G y (y · x) 0G k
|
||||
|
||||
invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
|
||||
invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
|
||||
where
|
||||
open Group G renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
fourWayAssoc : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
|
||||
fourWayAssoc {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
|
||||
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
|
||||
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
|
||||
where
|
||||
open Group G renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
|
||||
p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
|
||||
p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
|
||||
p1 = Group.multAssoc G
|
||||
p2 = Group.wellDefined G (Group.multAssoc G) reflexive
|
||||
p3 = symmetric (Group.multAssoc G)
|
||||
p1 = Group.+Associative G
|
||||
p2 = Group.+WellDefined G (Group.+Associative G) reflexive
|
||||
p3 = symmetric (Group.+Associative G)
|
||||
|
||||
fourWayAssoc' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
|
||||
fourWayAssoc' {S = S} G = transitive (symmetric multAssoc) (symmetric (fourWayAssoc G))
|
||||
fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
|
||||
fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
fourWayAssoc'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
|
||||
fourWayAssoc'' {S = S} {_·_ = _·_} G = transitive multAssoc (symmetric (fourWayAssoc G))
|
||||
fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
|
||||
fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
@@ -155,27 +155,27 @@ module Groups.Groups where
|
||||
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
|
||||
invContravariant {S = S} {_·_} G {x} {y} = ans
|
||||
where
|
||||
open Group G renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
otherInv = (y ^-1) · (x ^-1)
|
||||
manyAssocs : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
|
||||
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
|
||||
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
|
||||
manyAssocs = fourWayAssoc G
|
||||
oneMult = symmetric (transitive (Group.wellDefined G reflexive (transitive (symmetric (Group.multIdentLeft G)) (Group.wellDefined G (symmetric (Group.invRight G)) reflexive))) manyAssocs)
|
||||
otherInvIsInverse : (x · y) · otherInv ∼ e
|
||||
many+Associatives = fourWay+Associative G
|
||||
oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
|
||||
otherInvIsInverse : (x · y) · otherInv ∼ 0G
|
||||
otherInvIsInverse = transitive oneMult (Group.invRight G)
|
||||
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
|
||||
ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
|
||||
|
||||
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.identity G)) (Group.identity G)
|
||||
invIdentity {S = S} G = transitive (symmetric multIdentLeft) invRight
|
||||
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
|
||||
invIdentity {S = S} G = transitive (symmetric identLeft) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.identity G) → Setoid._∼_ S a b
|
||||
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
|
||||
transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
|
||||
where
|
||||
open Setoid S
|
||||
@@ -186,29 +186,29 @@ module Groups.Groups where
|
||||
u : inverse (inverse a) ∼ inverse (inverse b)
|
||||
u = inverseWellDefined G t
|
||||
|
||||
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.identity G) → Setoid._∼_ S a (Group.inverse G b)
|
||||
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
|
||||
transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
lemma : a · (inverse (inverse b)) ∼ identity
|
||||
lemma = transitive (wellDefined reflexive (invTwice G b)) ab-1
|
||||
lemma : a · (inverse (inverse b)) ∼ 0G
|
||||
lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
|
||||
|
||||
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.identity G)
|
||||
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.wellDefined G a~b reflexive) invRight
|
||||
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
|
||||
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
directSumGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (h : Group T _·B_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
|
||||
Group.wellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.wellDefined g s u ,, Group.wellDefined h t v
|
||||
Group.identity (directSumGroup {A = A} {B} g h) = (Group.identity g ,, Group.identity h)
|
||||
Group.+WellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
|
||||
Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
|
||||
Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
|
||||
Group.multAssoc (directSumGroup {A = A} {B} g h) = Group.multAssoc g ,, Group.multAssoc h
|
||||
Group.multIdentRight (directSumGroup {A = A} {B} g h) = Group.multIdentRight g ,, Group.multIdentRight h
|
||||
Group.multIdentLeft (directSumGroup {A = A} {B} g h) = Group.multIdentLeft g ,, Group.multIdentLeft h
|
||||
Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
|
||||
Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
|
||||
Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
|
||||
Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
|
||||
Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
|
||||
|
||||
@@ -228,18 +228,18 @@ module Groups.Groups where
|
||||
field
|
||||
injective : SetoidInjection S T underf
|
||||
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.identity G)) (Group.identity H)
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
|
||||
where
|
||||
open Group H
|
||||
open Setoid T
|
||||
id2 : Setoid._∼_ S (Group.identity G) ((Group.identity G) ·A (Group.identity G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.multIdentRight G)
|
||||
r : f (Group.identity G) ∼ f (Group.identity G) ·B f (Group.identity G)
|
||||
s : identity ·B f (Group.identity G) ∼ f (Group.identity G) ·B f (Group.identity G)
|
||||
t : identity ∼ f (Group.identity G)
|
||||
t = groupsHaveRightCancellation H (f (Group.identity G)) identity (f (Group.identity G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) multIdentLeft r
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
t : 0G ∼ f (Group.0G G)
|
||||
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) identLeft r
|
||||
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
|
||||
|
||||
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
@@ -264,12 +264,12 @@ module Groups.Groups where
|
||||
|
||||
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
|
||||
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = wellDefined } ; surj = record { surjective = surj ; wellDefined = wellDefined } }
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
|
||||
where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid U renaming (_∼_ to _∼C_)
|
||||
wellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
wellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
|
||||
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
|
||||
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
|
||||
@@ -306,31 +306,31 @@ module Groups.Groups where
|
||||
open Group H
|
||||
open Equivalence eq
|
||||
ansSetoid : Setoid A
|
||||
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ identity
|
||||
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
|
||||
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
|
||||
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
|
||||
where
|
||||
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ identity
|
||||
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
|
||||
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
|
||||
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ identity
|
||||
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
|
||||
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
|
||||
i : f (n ·A Group.inverse G m) ∼ identity
|
||||
i = transitive (GroupHom.wellDefined fHom (Group.wellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
|
||||
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.wellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.multIdentLeft G)))) k
|
||||
i : f (n ·A Group.inverse G m) ∼ 0G
|
||||
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
|
||||
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
|
||||
where
|
||||
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ identity ·B identity
|
||||
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
|
||||
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
|
||||
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ identity
|
||||
h = transitive g multIdentLeft
|
||||
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ identity
|
||||
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
|
||||
h = transitive g identLeft
|
||||
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
|
||||
i = transitive (GroupHom.groupHom fHom) h
|
||||
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ identity
|
||||
j = transitive (GroupHom.wellDefined fHom (fourWayAssoc G)) i
|
||||
k : f (m ·A ((Group.identity G) ·A Group.inverse G o)) ∼ identity
|
||||
k = transitive (GroupHom.wellDefined fHom (Group.wellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.wellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
|
||||
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
|
||||
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
|
||||
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
|
||||
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
|
||||
|
||||
quotientGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
|
||||
Group.wellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
@@ -338,58 +338,58 @@ module Groups.Groups where
|
||||
p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
|
||||
p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
|
||||
p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
|
||||
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.identity H ·B f (Group.inverse G m))
|
||||
p6 : (f x) ·B (Group.identity H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
|
||||
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
|
||||
p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
|
||||
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
|
||||
p8 : f (x ·A (Group.inverse G m)) ∼ Group.identity H
|
||||
p1 = GroupHom.wellDefined fHom (Group.wellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
|
||||
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWayAssoc G))
|
||||
p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
|
||||
p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
|
||||
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
|
||||
p3 = GroupHom.groupHom fHom
|
||||
p4 = Group.wellDefined H reflexive (GroupHom.groupHom fHom)
|
||||
p5 = Group.wellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
|
||||
p6 = Group.wellDefined H reflexive (Group.multIdentLeft H)
|
||||
p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
|
||||
p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
|
||||
p6 = Group.+WellDefined H reflexive (Group.identLeft H)
|
||||
p7 = symmetric (GroupHom.groupHom fHom)
|
||||
p8 = x~m
|
||||
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.identity H
|
||||
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
|
||||
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
|
||||
Group.identity (quotientGroup G fHom) = Group.identity G
|
||||
Group.0G (quotientGroup G fHom) = Group.0G G
|
||||
Group.inverse (quotientGroup G fHom) = Group.inverse G
|
||||
Group.multAssoc (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.identity H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.multAssoc G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.multIdentRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((a ·A identity) ·A inverse a) ∼ Group.identity H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.wellDefined G (Group.multIdentRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.multIdentLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((identity ·A a) ·A (inverse a)) ∼ Group.identity H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.wellDefined G (Group.multIdentLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((inverse x ·A x) ·A (inverse identity)) ∼ (Group.identity H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (multIdentRight {identity}))) (imageOfIdentityIsIdentity fHom)
|
||||
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((x ·A inverse x) ·A (inverse identity)) ∼ (Group.identity H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (multIdentRight {identity}))) (imageOfIdentityIsIdentity fHom)
|
||||
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
|
||||
record Subgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid T renaming (_∼_ to _∼G_)
|
||||
@@ -413,7 +413,7 @@ module Groups.Groups where
|
||||
open Setoid T renaming (_∼_ to _∼T_)
|
||||
open Equivalence (Setoid.eq S)
|
||||
open Reflexive reflexiveEq
|
||||
have : f (x ·A inverse y) ∼T Group.identity H
|
||||
have : f (x ·A inverse y) ∼T Group.0G H
|
||||
have = x~y
|
||||
need : x ∼ y
|
||||
need = {!!}
|
||||
|
||||
@@ -28,17 +28,17 @@ module Groups.Groups2 where
|
||||
imageGroupOp {_+A_ = _+A_} fHom (ofElt a) (ofElt b) = ofElt (a +A b)
|
||||
|
||||
imageGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) → Group (imageGroupSetoid fHom) (imageGroupOp fHom)
|
||||
Group.wellDefined (imageGroup {T = T} {_+A_ = _+A_} {H = H} {f = f} fHom) {ofElt x} {ofElt y} {ofElt a} {ofElt b} x~a y~b = ans
|
||||
Group.+WellDefined (imageGroup {T = T} {_+A_ = _+A_} {H = H} {f = f} fHom) {ofElt x} {ofElt y} {ofElt a} {ofElt b} x~a y~b = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f (x +A y) ∼ f (a +A b)
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.wellDefined H x~a y~b) (symmetric (GroupHom.groupHom fHom)))
|
||||
Group.identity (imageGroup {G = G} fHom) = ofElt (Group.identity G)
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H x~a y~b) (symmetric (GroupHom.groupHom fHom)))
|
||||
Group.0G (imageGroup {G = G} fHom) = ofElt (Group.0G G)
|
||||
Group.inverse (imageGroup {G = G} fHom) (ofElt a) = ofElt (Group.inverse G a)
|
||||
Group.multAssoc (imageGroup {G = G} fHom) {ofElt a} {ofElt b} {ofElt c} = GroupHom.wellDefined fHom (Group.multAssoc G)
|
||||
Group.multIdentRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.multIdentRight G)
|
||||
Group.multIdentLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.multIdentLeft G)
|
||||
Group.+Associative (imageGroup {G = G} fHom) {ofElt a} {ofElt b} {ofElt c} = GroupHom.wellDefined fHom (Group.+Associative G)
|
||||
Group.identRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identRight G)
|
||||
Group.identLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identLeft G)
|
||||
Group.invLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invLeft G)
|
||||
Group.invRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invRight G)
|
||||
|
||||
@@ -64,8 +64,8 @@ module Groups.Groups2 where
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f (a +G Group.inverse G b) ∼ Group.identity H
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.wellDefined H pr reflexive) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom))))
|
||||
ans : f (a +G Group.inverse G b) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H pr reflexive) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom))))
|
||||
|
||||
groupFirstIsomorphismTheorem' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupIso (imageGroup fHom) (quotientGroup G fHom) (groupFirstIsomorphismIso fHom)
|
||||
GroupIso.groupHom (groupFirstIsomorphismTheorem' fHom) = groupFirstIsomorphismIsoHom fHom
|
||||
@@ -75,7 +75,7 @@ module Groups.Groups2 where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
need : f a ∼ f b
|
||||
need = transferToRight H (transitive (transitive (Group.wellDefined H reflexive (symmetric (homRespectsInverse fHom))) (symmetric (GroupHom.groupHom fHom))) pr)
|
||||
need = transferToRight H (transitive (transitive (Group.+WellDefined H reflexive (symmetric (homRespectsInverse fHom))) (symmetric (GroupHom.groupHom fHom))) pr)
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (groupFirstIsomorphismTheorem' fHom))) {x} {y} x~y = GroupHom.wellDefined (groupFirstIsomorphismIsoHom fHom) {x} {y} x~y
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (groupFirstIsomorphismTheorem' {G = G} fHom))) {a} = ofElt a , Equivalence.reflexive (Setoid.eq (quotientGroupSetoid G fHom))
|
||||
|
||||
@@ -89,7 +89,7 @@ module Groups.Groups2 where
|
||||
normal : {g : A} {h : B} → Sg B (λ fromH → (g ·A (f h)) ·A (Group.inverse G g) ∼ f fromH)
|
||||
|
||||
data GroupKernelElement {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
kerOfElt : (x : A) → (Setoid._∼_ T (f x) (Group.identity H)) → GroupKernelElement G hom
|
||||
kerOfElt : (x : A) → (Setoid._∼_ T (f x) (Group.0G H)) → GroupKernelElement G hom
|
||||
|
||||
groupKernel : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → Setoid (GroupKernelElement G hom)
|
||||
Setoid._∼_ (groupKernel {S = S} G {H} {f} fHom) (kerOfElt x fx=0) (kerOfElt y fy=0) = Setoid._∼_ S x y
|
||||
@@ -98,21 +98,21 @@ module Groups.Groups2 where
|
||||
Equivalence.transitive (Setoid.eq (groupKernel {S = S} G {H} {f} fHom)) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt z prZ} = Equivalence.transitive (Setoid.eq S)
|
||||
|
||||
groupKernelGroupOp : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → (GroupKernelElement G hom) → (GroupKernelElement G hom) → (GroupKernelElement G hom)
|
||||
groupKernelGroupOp {T = T} {_·A_ = _+A_} G {H = H} hom (kerOfElt x prX) (kerOfElt y prY) = kerOfElt (x +A y) (transitive (GroupHom.groupHom hom) (transitive (Group.wellDefined H prX prY) (Group.multIdentLeft H)))
|
||||
groupKernelGroupOp {T = T} {_·A_ = _+A_} G {H = H} hom (kerOfElt x prX) (kerOfElt y prY) = kerOfElt (x +A y) (transitive (GroupHom.groupHom hom) (transitive (Group.+WellDefined H prX prY) (Group.identLeft H)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
groupKernelGroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → Group (groupKernel G hom) (groupKernelGroupOp G hom)
|
||||
Group.wellDefined (groupKernelGroup G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt a prA} {kerOfElt b prB} = Group.wellDefined G
|
||||
Group.identity (groupKernelGroup G fHom) = kerOfElt (Group.identity G) (imageOfIdentityIsIdentity fHom)
|
||||
Group.+WellDefined (groupKernelGroup G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt a prA} {kerOfElt b prB} = Group.+WellDefined G
|
||||
Group.0G (groupKernelGroup G fHom) = kerOfElt (Group.0G G) (imageOfIdentityIsIdentity fHom)
|
||||
Group.inverse (groupKernelGroup {T = T} G {H = H} fHom) (kerOfElt x prX) = kerOfElt (Group.inverse G x) (transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H prX) (invIdentity H)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
Group.multAssoc (groupKernelGroup {S = S} {_·A_ = _·A_} G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt z prZ} = Group.multAssoc G
|
||||
Group.multIdentRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.multIdentRight G
|
||||
Group.multIdentLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.multIdentLeft G
|
||||
Group.+Associative (groupKernelGroup {S = S} {_·A_ = _·A_} G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt z prZ} = Group.+Associative G
|
||||
Group.identRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.identRight G
|
||||
Group.identLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.identLeft G
|
||||
Group.invLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invLeft G
|
||||
Group.invRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invRight G
|
||||
|
||||
@@ -135,26 +135,26 @@ module Groups.Groups2 where
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((g ·A h) ·A Group.inverse G g) ∼ Group.identity H
|
||||
ans = transitive (GroupHom.groupHom hom) (transitive (Group.wellDefined H (GroupHom.groupHom hom) reflexive) (transitive (Group.wellDefined H (Group.wellDefined H reflexive prH) reflexive) (transitive (Group.wellDefined H (Group.multIdentRight H) reflexive) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invRight G)) (imageOfIdentityIsIdentity hom))))))
|
||||
ans : f ((g ·A h) ·A Group.inverse G g) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.groupHom hom) (transitive (Group.+WellDefined H (GroupHom.groupHom hom) reflexive) (transitive (Group.+WellDefined H (Group.+WellDefined H reflexive prH) reflexive) (transitive (Group.+WellDefined H (Group.identRight H) reflexive) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invRight G)) (imageOfIdentityIsIdentity hom))))))
|
||||
|
||||
abelianGroupSubgroupIsNormal : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {underG : Group S _+A_} (G : AbelianGroup underG) {H : Group T _+B_} {f : B → A} {hom : GroupHom H underG f} (s : Subgroup underG H hom) → NormalSubgroup underG H hom
|
||||
NormalSubgroup.subgroup (abelianGroupSubgroupIsNormal G H) = H
|
||||
NormalSubgroup.normal (abelianGroupSubgroupIsNormal {S = S} {underG = G} record { commutative = commutative } H) {g} {h} = h , transitive (wellDefined commutative reflexive) (transitive (symmetric multAssoc) (transitive (wellDefined reflexive invRight) multIdentRight))
|
||||
NormalSubgroup.normal (abelianGroupSubgroupIsNormal {S = S} {underG = G} record { commutative = commutative } H) {g} {h} = h , transitive (+WellDefined commutative reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence (Setoid.eq S)
|
||||
|
||||
trivialGroup : Group (reflSetoid (FinSet 1)) λ _ _ → fzero
|
||||
Group.wellDefined trivialGroup _ _ = refl
|
||||
Group.identity trivialGroup = fzero
|
||||
Group.+WellDefined trivialGroup _ _ = refl
|
||||
Group.0G trivialGroup = fzero
|
||||
Group.inverse trivialGroup _ = fzero
|
||||
Group.multAssoc trivialGroup = refl
|
||||
Group.multIdentRight trivialGroup {fzero} = refl
|
||||
Group.multIdentRight trivialGroup {fsucc ()}
|
||||
Group.multIdentLeft trivialGroup {fzero} = refl
|
||||
Group.multIdentLeft trivialGroup {fsucc ()}
|
||||
Group.+Associative trivialGroup = refl
|
||||
Group.identRight trivialGroup {fzero} = refl
|
||||
Group.identRight trivialGroup {fsucc ()}
|
||||
Group.identLeft trivialGroup {fzero} = refl
|
||||
Group.identLeft trivialGroup {fsucc ()}
|
||||
Group.invLeft trivialGroup = refl
|
||||
Group.invRight trivialGroup = refl
|
||||
|
||||
|
||||
@@ -25,8 +25,8 @@ module Groups.LectureNotes.Lecture1 where
|
||||
-- TODO: R is a group with +
|
||||
|
||||
integersMinusNotGroup : Group (reflSetoid ℤ) (_-Z_) → False
|
||||
integersMinusNotGroup record { wellDefined = wellDefined ; identity = identity ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
|
||||
integersMinusNotGroup record { wellDefined = wellDefined ; identity = identity ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
|
||||
integersMinusNotGroup record { +WellDefined = wellDefined ; 0G = identity ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
|
||||
integersMinusNotGroup record { +WellDefined = wellDefined ; 0G = identity ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
|
||||
|
||||
negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b
|
||||
negSuccInjective {a} {.a} refl = refl
|
||||
@@ -35,17 +35,17 @@ module Groups.LectureNotes.Lecture1 where
|
||||
nonnegInjective {a} {.a} refl = refl
|
||||
|
||||
integersTimesNotGroup : Group (reflSetoid ℤ) (_*Z_) → False
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg zero) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg zero) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
|
||||
... | ()
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
|
||||
... | bl with inverse (nonneg zero)
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ (succ x))) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ (succ x))) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
|
||||
... | ()
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (negSucc x) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {nonneg 2}
|
||||
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (negSucc x) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {nonneg 2}
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
|
||||
|
||||
-- TODO: Q is not a group with *Q
|
||||
-- TODO: Q without 0 is a group with *Q
|
||||
@@ -119,18 +119,18 @@ module Groups.LectureNotes.Lecture1 where
|
||||
+WAssoc {c} {c} {c} = refl
|
||||
|
||||
weirdGroup : Group (reflSetoid Weird) _+W_
|
||||
Group.wellDefined weirdGroup = reflGroupWellDefined
|
||||
Group.identity weirdGroup = e
|
||||
Group.+WellDefined weirdGroup = reflGroupWellDefined
|
||||
Group.0G weirdGroup = e
|
||||
Group.inverse weirdGroup t = t
|
||||
Group.multAssoc weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t}
|
||||
Group.multIdentRight weirdGroup {e} = refl
|
||||
Group.multIdentRight weirdGroup {a} = refl
|
||||
Group.multIdentRight weirdGroup {b} = refl
|
||||
Group.multIdentRight weirdGroup {c} = refl
|
||||
Group.multIdentLeft weirdGroup {e} = refl
|
||||
Group.multIdentLeft weirdGroup {a} = refl
|
||||
Group.multIdentLeft weirdGroup {b} = refl
|
||||
Group.multIdentLeft weirdGroup {c} = refl
|
||||
Group.+Associative weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t}
|
||||
Group.identRight weirdGroup {e} = refl
|
||||
Group.identRight weirdGroup {a} = refl
|
||||
Group.identRight weirdGroup {b} = refl
|
||||
Group.identRight weirdGroup {c} = refl
|
||||
Group.identLeft weirdGroup {e} = refl
|
||||
Group.identLeft weirdGroup {a} = refl
|
||||
Group.identLeft weirdGroup {b} = refl
|
||||
Group.identLeft weirdGroup {c} = refl
|
||||
Group.invLeft weirdGroup {e} = refl
|
||||
Group.invLeft weirdGroup {a} = refl
|
||||
Group.invLeft weirdGroup {b} = refl
|
||||
|
||||
@@ -17,8 +17,8 @@ module Groups.Lemmas where
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
|
||||
invIdent : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → Setoid._∼_ S (Group.inverse G (Group.identity G)) (Group.identity G)
|
||||
invIdent {S = S} G = symmetric (transferToRight' G (Group.multIdentLeft G))
|
||||
invIdent : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → Setoid._∼_ S (Group.inverse G (Group.0G G)) (Group.0G G)
|
||||
invIdent {S = S} G = symmetric (transferToRight' G (Group.identLeft G))
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
@@ -31,9 +31,9 @@ module Groups.Lemmas where
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
|
||||
identityIsUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (e : A) → ((b : A) → (Setoid._∼_ S (b · e) b)) → (Setoid._∼_ S e (Group.identity G))
|
||||
identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric multIdentLeft) (fb e)
|
||||
identityIsUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (e : A) → ((b : A) → (Setoid._∼_ S (b · e) b)) → (Setoid._∼_ S e (Group.0G G))
|
||||
identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric identLeft) (fb 0G)
|
||||
where
|
||||
open Group g renaming (inverse to _^-1) renaming (identity to e)
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
@@ -57,13 +57,13 @@ module Groups.SymmetryGroups where
|
||||
ans {x} | a , b = b
|
||||
|
||||
symmetricGroup : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Group (symmetricSetoid S) (symmetricGroupOp {A = A})
|
||||
Group.wellDefined (symmetricGroup A) {sym {m} bijM} {sym {n} bijN} {sym {x} bijX} {sym {y} bijY} (ExtensionallyEqual.eq m~x) (ExtensionallyEqual.eq n~y) = ExtensionallyEqual.eq (transitive m~x (SetoidBijection.wellDefined bijX n~y))
|
||||
Group.+WellDefined (symmetricGroup A) {sym {m} bijM} {sym {n} bijN} {sym {x} bijX} {sym {y} bijY} (ExtensionallyEqual.eq m~x) (ExtensionallyEqual.eq n~y) = ExtensionallyEqual.eq (transitive m~x (SetoidBijection.wellDefined bijX n~y))
|
||||
where
|
||||
open Equivalence (Setoid.eq A)
|
||||
Group.identity (symmetricGroup A) = sym setoidIdIsBijective
|
||||
Group.0G (symmetricGroup A) = sym setoidIdIsBijective
|
||||
Group.inverse (symmetricGroup S) = symmetricGroupInv S
|
||||
Group.multAssoc (symmetricGroup A) {sym {f} bijF} {sym {g} bijG} {sym {h} bijH} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.multIdentRight (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.multIdentLeft (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.+Associative (symmetricGroup A) {sym {f} bijF} {sym {g} bijG} {sym {h} bijH} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.identRight (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.identLeft (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
|
||||
Group.invLeft (symmetricGroup S) {x} = symmetricGroupInvIsLeft S {x}
|
||||
Group.invRight (symmetricGroup S) {x} = symmetricGroupInvIsRight S {x}
|
||||
|
||||
@@ -9,9 +9,9 @@ open import Numbers.Naturals.Exponentiation
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Maybe
|
||||
open import Semirings.Definition
|
||||
open import Semirings.Solver
|
||||
import Semirings.Solver
|
||||
|
||||
module NatSolver = Semirings.Solver ℕSemiring multiplicationNIsCommutative
|
||||
open module NatSolver = Semirings.Solver ℕSemiring multiplicationNIsCommutative
|
||||
|
||||
module LectureNotes.NumbersAndSets.Lecture1 where
|
||||
|
||||
@@ -43,13 +43,13 @@ n3Bigger' zero = inr refl
|
||||
n3Bigger' (succ n) with n3Bigger (succ n)
|
||||
n3Bigger' (succ n) | inr f = f
|
||||
|
||||
_!!N_ = NatSolver._!!_
|
||||
|
||||
-- How to use the semiring solver
|
||||
-- The process is very mechanical; I haven't yet worked out how to do reflection,
|
||||
-- so there's quite a bit of transcribing expressions into the Expr form.
|
||||
-- The first two arguments to !!N are totally mindless in construction.
|
||||
-- The first two arguments to from-to-by are totally mindless in construction.
|
||||
proof : (n : ℕ) → ((n *N n) +N ((2 *N n) +N 1)) ≡ (n +N 1) *N (n +N 1)
|
||||
proof n =
|
||||
(plus (times (const n) (const n)) (plus (times (succ (succ zero)) (const n)) (succ zero)) !!N times (plus (const n) (succ zero)) (plus (const n) (succ zero)))
|
||||
(applyEquality (λ i → succ (n *N n) +N (n +N i)) ((const n !!N plus (const n) zero) refl))
|
||||
from plus (times (const n) (const n)) (plus (times (succ (succ zero)) (const n)) (succ zero))
|
||||
to times (plus (const n) (succ zero)) (plus (const n) (succ zero))
|
||||
by
|
||||
applyEquality (λ i → succ (n *N n) +N (n +N i)) ((from (const n) to (plus (const n) zero) by refl))
|
||||
|
||||
@@ -76,14 +76,14 @@ additiveInverse (nonneg (succ x)) = negSucc x
|
||||
additiveInverse (negSucc x) = nonneg (succ x)
|
||||
|
||||
ℤGroup : Group (reflSetoid ℤ) (_+Z_)
|
||||
Group.wellDefined ℤGroup refl refl = refl
|
||||
Group.identity ℤGroup = nonneg 0
|
||||
Group.+WellDefined ℤGroup refl refl = refl
|
||||
Group.0G ℤGroup = nonneg 0
|
||||
Group.inverse ℤGroup = additiveInverse
|
||||
Group.multAssoc ℤGroup {a} {b} {c} = +ZAssociative a b c
|
||||
Group.multIdentRight ℤGroup {nonneg zero} = refl
|
||||
Group.multIdentRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0)
|
||||
Group.multIdentRight ℤGroup {negSucc x} = refl
|
||||
Group.multIdentLeft ℤGroup = refl
|
||||
Group.+Associative ℤGroup {a} {b} {c} = +ZAssociative a b c
|
||||
Group.identRight ℤGroup {nonneg zero} = refl
|
||||
Group.identRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0)
|
||||
Group.identRight ℤGroup {negSucc x} = refl
|
||||
Group.identLeft ℤGroup = refl
|
||||
Group.invLeft ℤGroup {nonneg zero} = refl
|
||||
Group.invLeft ℤGroup {nonneg (succ x)} = additiveInverseExists x
|
||||
Group.invLeft ℤGroup {negSucc x} = transitivity (+ZCommutative (nonneg (succ x)) (negSucc x)) (additiveInverseExists x)
|
||||
|
||||
@@ -1,5 +1,6 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Definition
|
||||
open import Numbers.Integers.Addition
|
||||
open import Numbers.Integers.Multiplication
|
||||
@@ -15,3 +16,7 @@ open Numbers.Integers.Order using (_<Z_ ; ℤOrderedRing) public
|
||||
|
||||
_-Z_ : ℤ → ℤ → ℤ
|
||||
a -Z b = a +Z (Group.inverse ℤGroup b)
|
||||
|
||||
_^Z_ : ℤ → ℕ → ℤ
|
||||
a ^Z zero = nonneg 1
|
||||
a ^Z succ b = a *Z (a ^Z b)
|
||||
|
||||
@@ -180,13 +180,13 @@ distLemma a b rewrite Semiring.commutative ℕSemiring (a +N b *N a) b | Semirin
|
||||
|
||||
ℤRing : Ring (reflSetoid ℤ) _+Z_ _*Z_
|
||||
Ring.additiveGroup ℤRing = ℤGroup
|
||||
Ring.multWellDefined ℤRing refl refl = refl
|
||||
Ring.*WellDefined ℤRing refl refl = refl
|
||||
Ring.1R ℤRing = nonneg 1
|
||||
Ring.groupIsAbelian ℤRing {a} {b} = +ZCommutative a b
|
||||
Ring.multAssoc ℤRing {a} {b} {c} = *ZAssociative a b c
|
||||
Ring.multCommutative ℤRing {a} {b} = *ZCommutative a b
|
||||
Ring.multDistributes ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c
|
||||
Ring.multIdentIsIdent ℤRing {a} = *ZleftIdent a
|
||||
Ring.*Associative ℤRing {a} {b} {c} = *ZAssociative a b c
|
||||
Ring.*Commutative ℤRing {a} {b} = *ZCommutative a b
|
||||
Ring.*DistributesOver+ ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c
|
||||
Ring.identIsIdent ℤRing {a} = *ZleftIdent a
|
||||
|
||||
intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0)
|
||||
intDom (nonneg zero) (nonneg b) pr = inl refl
|
||||
|
||||
@@ -680,7 +680,7 @@ module Numbers.Modulo.IntegersModN where
|
||||
... | (inr n-a+sa=sn) = equalityZn _ _ refl
|
||||
|
||||
ℤnGroup : (n : ℕ) → (pr : 0 <N n) → Group (reflSetoid (ℤn n pr)) _+n_
|
||||
ℤnGroup n pr = record { identity = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; multAssoc = λ {a} {b} {c} → plusZnAssociative a b c ; multIdentRight = λ {a} → plusZnIdentityRight a ; multIdentLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; wellDefined = reflGroupWellDefined }
|
||||
ℤnGroup n pr = record { 0G = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; +Associative = λ {a} {b} {c} → plusZnAssociative a b c ; identRight = λ {a} → plusZnIdentityRight a ; identLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; +WellDefined = reflGroupWellDefined }
|
||||
where
|
||||
helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr }
|
||||
helpInvLeft a with inverseZn a
|
||||
|
||||
@@ -28,12 +28,12 @@ module Numbers.RationalsLemmas where
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<a+b) = inr (reflexive {a +Q b})
|
||||
where
|
||||
open Equivalence (Setoid.eq (fieldOfFractionsSetoid ℤIntDom))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inr a+b<0) = exFalso (exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {a +Q b} (symmetric {0Q} {0Q +Q 0Q} (Group.multIdentRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inr a+b<0) = exFalso (exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {a +Q b} (symmetric {0Q} {0Q +Q 0Q} (Group.identRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {0Q} (multIdentRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {0Q} (identRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
@@ -49,23 +49,23 @@ module Numbers.RationalsLemmas where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {a +Q (inverse b)} 0<a (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q a} {a} {inverse b +Q a} {a +Q inverse b} (multIdentLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {a +Q (inverse b)} 0<a (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q a} {a} {inverse b +Q a} {a +Q inverse b} (identLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inl 0<a+b) = inr (wellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inl 0<a+b) = inr (+WellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {a +Q b} (reflexive {0Q}) (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {a +Q b} (reflexive {0Q}) (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a +Q b} {a} {a +Q b} 0=a+b (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
|
||||
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a +Q b} {a} {a +Q b} 0=a+b (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
@@ -84,13 +84,13 @@ module Numbers.RationalsLemmas where
|
||||
open Equivalence eq
|
||||
blah : (inverse a +Q inverse b) <Q (inverse a +Q b)
|
||||
blah = SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b +Q inverse a} {inverse a +Q inverse b} {b +Q inverse a} {inverse a +Q b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (OrderedRing.orderRespectsAddition ℚOrdered {inverse b} {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b} {0Q} {b} (ringMinusFlipsOrder ℚOrdered {b} 0<b) 0<b) (inverse a))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {inverse a +Q b} 0<b (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q 0Q} {b} {b +Q inverse a} {inverse a +Q b} (multIdentRight {b}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q b} {b +Q 0Q} {inverse a +Q b} {b +Q inverse a} (Ring.groupIsAbelian ℚRing {0Q} {b}) (Ring.groupIsAbelian ℚRing {inverse a} {b}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse a} (ringMinusFlipsOrder'' ℚOrdered {a} a<0) b))))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {inverse a +Q b} 0<b (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q 0Q} {b} {b +Q inverse a} {inverse a +Q b} (identRight {b}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q b} {b +Q 0Q} {inverse a +Q b} {b +Q inverse a} (Ring.groupIsAbelian ℚRing {0Q} {b}) (Ring.groupIsAbelian ℚRing {inverse a} {b}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse a} (ringMinusFlipsOrder'' ℚOrdered {a} a<0) b))))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (multIdentRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+b))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (identRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
@@ -100,65 +100,65 @@ module Numbers.RationalsLemmas where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (symmetric {0Q} {a +Q b} 0=a+b) (reflexive {0Q}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (multIdentLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (symmetric {0Q} {a +Q b} 0=a+b) (reflexive {0Q}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (identLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} a<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {a} (reflexive {0Q}) (transitive {a +Q b} {a +Q 0Q} {a} (wellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b)) (multIdentRight {a})) 0<a+b)))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} a<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {a} (reflexive {0Q}) (transitive {a +Q b} {a +Q 0Q} {a} (+WellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b)) (identRight {a})) 0<a+b)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {(inverse b) +Q (inverse a)} {(inverse a) +Q 0Q} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {inverse a +Q 0Q} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (wellDefined {inverse a} {inverse b} {inverse a} {0Q} (reflexive {inverse a}) (transitive {inverse b} {inverse 0Q} {0Q} (inverseWellDefined (Ring.additiveGroup ℚRing) {b} {0Q} (symmetric {0Q} {b} 0=b)) (symmetric {0Q} {inverse 0Q} (invIdentity (Ring.additiveGroup ℚRing)))))))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {(inverse b) +Q (inverse a)} {(inverse a) +Q 0Q} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {inverse a +Q 0Q} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (+WellDefined {inverse a} {inverse b} {inverse a} {0Q} (reflexive {inverse a}) (transitive {inverse b} {inverse 0Q} {0Q} (inverseWellDefined (Ring.additiveGroup ℚRing) {b} {0Q} (symmetric {0Q} {b} 0=b)) (symmetric {0Q} {inverse 0Q} (invIdentity (Ring.additiveGroup ℚRing)))))))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {a +Q b} {0Q} (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) (symmetric {0Q} {a +Q b} 0=a+b)) (reflexive {0Q}) a<0))
|
||||
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {a +Q b} {0Q} (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) (symmetric {0Q} {a +Q b} 0=a+b)) (reflexive {0Q}) a<0))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q b
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inl 0<a+b) = inr (Group.wellDefined (Ring.additiveGroup ℚRing) {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b}))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inl 0<a+b) = inr (Group.+WellDefined (Ring.additiveGroup ℚRing) {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b}))
|
||||
where
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {b} {a +Q b} (reflexive {0Q}) (transitive {b} {0Q +Q b} {a +Q b} (symmetric {0Q +Q b} {b} (Group.multIdentLeft (Ring.additiveGroup ℚRing) {b})) (Group.wellDefined (Ring.additiveGroup ℚRing) {0Q} {b} {a} {b} 0=a (reflexive {b}))) 0<b)))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {b} {a +Q b} (reflexive {0Q}) (transitive {b} {0Q +Q b} {a +Q b} (symmetric {0Q +Q b} {b} (Group.identLeft (Ring.additiveGroup ℚRing) {b})) (Group.+WellDefined (Ring.additiveGroup ℚRing) {0Q} {b} {a} {b} 0=a (reflexive {b}))) 0<b)))
|
||||
where
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {b} {b} (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b}))) (reflexive {b}) 0<b))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {b} {b} (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b}))) (reflexive {b}) 0<b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {b} b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {b} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b})) 0<a+b)))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {b} b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {b} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b})) 0<a+b)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {inverse b +Q inverse a} {0Q +Q inverse b} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {0Q +Q inverse b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (wellDefined {inverse a} {inverse b} {0Q} {inverse b} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {inverse b}))))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {inverse b +Q inverse a} {0Q +Q inverse b} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {0Q +Q inverse b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (+WellDefined {inverse a} {inverse b} {0Q} {inverse b} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {inverse b}))))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {b} {0Q} {b} (reflexive {b}) (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b}))) b<0))
|
||||
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {b} {0Q} {b} (reflexive {b}) (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b}))) b<0))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
|
||||
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {0Q} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q 0Q} {0Q} (wellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (multIdentRight {0Q})) 0<a+b))
|
||||
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {0Q} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q 0Q} {0Q} (+WellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (identRight {0Q})) 0<a+b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
open Equivalence eq
|
||||
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (transitive {a +Q b} {0Q +Q 0Q} {0Q} (wellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (multIdentRight {0Q})) (reflexive {0Q}) a+b<0))
|
||||
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (transitive {a +Q b} {0Q +Q 0Q} {0Q} (+WellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (identRight {0Q})) (reflexive {0Q}) a+b<0))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
|
||||
@@ -7,6 +7,7 @@ open import Numbers.Naturals.Naturals
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
@@ -17,16 +18,21 @@ module Rings.Definition where
|
||||
additiveGroup : Group S _+_
|
||||
open Group additiveGroup
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
0R : A
|
||||
0R = identity
|
||||
0R = 0G
|
||||
_-R_ : A → A → A
|
||||
a -R b = a + (inverse b)
|
||||
field
|
||||
multWellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
|
||||
*WellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
|
||||
1R : A
|
||||
groupIsAbelian : {a b : A} → a + b ∼ b + a
|
||||
multAssoc : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
|
||||
multCommutative : {a b : A} → a * b ∼ b * a
|
||||
multDistributes : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
|
||||
multIdentIsIdent : {a : A} → 1R * a ∼ a
|
||||
*Associative : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
|
||||
*Commutative : {a b : A} → a * b ∼ b * a
|
||||
*DistributesOver+ : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
|
||||
identIsIdent : {a : A} → 1R * a ∼ a
|
||||
timesZero : {a : A} → a * 0R ∼ 0R
|
||||
timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
|
||||
|
||||
record OrderedRing {n m p} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) : Set (lsuc n ⊔ m ⊔ p) where
|
||||
open Ring R
|
||||
@@ -42,10 +48,10 @@ module Rings.Definition where
|
||||
--Ring.multWellDefined (directSumRing r s) (a ,, b) (c ,, d) = Ring.multWellDefined r a c ,, Ring.multWellDefined s b d
|
||||
--Ring.1R (directSumRing r s) = Ring.1R r ,, Ring.1R s
|
||||
--Ring.groupIsAbelian (directSumRing r s) = Ring.groupIsAbelian r ,, Ring.groupIsAbelian s
|
||||
--Ring.multAssoc (directSumRing r s) = Ring.multAssoc r ,, Ring.multAssoc s
|
||||
--Ring.assoc (directSumRing r s) = Ring.assoc r ,, Ring.assoc s
|
||||
--Ring.multCommutative (directSumRing r s) = Ring.multCommutative r ,, Ring.multCommutative s
|
||||
--Ring.multDistributes (directSumRing r s) = Ring.multDistributes r ,, Ring.multDistributes s
|
||||
--Ring.multIdentIsIdent (directSumRing r s) = Ring.multIdentIsIdent r ,, Ring.multIdentIsIdent s
|
||||
--Ring.identIsIdent (directSumRing r s) = Ring.identIsIdent r ,, Ring.identIsIdent s
|
||||
|
||||
record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Ring S
|
||||
|
||||
@@ -27,7 +27,7 @@ module Rings.IntegralDomains where
|
||||
t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
|
||||
t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
|
||||
t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
|
||||
t2 = transitive (transitive (Ring.multDistributes R) (Group.wellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (ringTimesZero R)))))) t1
|
||||
t2 = transitive (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R)))))) t1
|
||||
t3 : (a ∼ Ring.0R R) || ((b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R)
|
||||
t3 = IntegralDomain.intDom I t2
|
||||
t4 : (a ∼ Ring.0R R) || (b ∼ c)
|
||||
|
||||
@@ -11,22 +11,9 @@ open import Setoids.Orders
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Rings.Lemmas where
|
||||
ringTimesZero : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x : A} → Setoid._∼_ S (x * (Ring.0R R)) (Ring.0R R)
|
||||
ringTimesZero {S = S} {_+_ = _+_} {_*_ = _*_} R {x} = symmetric (transitive blah'' (transitive (Group.multAssoc additiveGroup) (transitive (wellDefined invLeft reflexive) multIdentLeft)))
|
||||
where
|
||||
open Ring R
|
||||
open Group additiveGroup
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
blah : (x * 0R) ∼ (x * 0R) + (x * 0R)
|
||||
blah = transitive (multWellDefined reflexive (symmetric multIdentRight)) multDistributes
|
||||
blah' : (inverse (x * 0R)) + (x * 0R) ∼ (inverse (x * 0R)) + ((x * 0R) + (x * 0R))
|
||||
blah' = wellDefined reflexive blah
|
||||
blah'' : 0R ∼ (inverse (x * 0R)) + ((x * 0R) + (x * 0R))
|
||||
blah'' = transitive (symmetric invLeft) blah'
|
||||
|
||||
ringMinusExtracts : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
|
||||
ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric multDistributes) (transitive (multWellDefined reflexive invLeft) (ringTimesZero R)))
|
||||
ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -40,17 +27,17 @@ module Rings.Lemmas where
|
||||
open OrderedRing oRing
|
||||
open Ring R
|
||||
|
||||
groupLemmaMoveIdentity : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.identity G) (Group.inverse G x)) → Setoid._∼_ S x (Group.identity G)
|
||||
groupLemmaMoveIdentity {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
|
||||
groupLemmaMove0G : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.0G G) (Group.inverse G x)) → Setoid._∼_ S x (Group.0G G)
|
||||
groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
p : inverse identity ∼ inverse (inverse x)
|
||||
p : inverse 0G ∼ inverse (inverse x)
|
||||
p = inverseWellDefined G pr
|
||||
|
||||
groupLemmaMoveIdentity' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.identity G) → (Setoid._∼_ S (Group.identity G) (Group.inverse G x))
|
||||
groupLemmaMoveIdentity' {S = S} G {x} pr = transferToRight' G (transitive multIdentLeft pr)
|
||||
groupLemmaMove0G' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.0G G) → (Setoid._∼_ S (Group.0G G) (Group.inverse G x))
|
||||
groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (transitive identLeft pr)
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
@@ -60,21 +47,21 @@ module Rings.Lemmas where
|
||||
ringMinusFlipsOrder {S = S} {_+_ = _+_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x = x} 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
|
||||
where
|
||||
bad : (Group.identity (Ring.additiveGroup R) + Group.identity (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
|
||||
bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
|
||||
bad = ringAddInequalities oRing 0<x 0<inv
|
||||
bad' : (Group.identity (Ring.additiveGroup R)) < (Group.identity (Ring.additiveGroup R))
|
||||
bad' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
|
||||
bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
|
||||
bad' = SetoidPartialOrder.wellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
|
||||
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inr inv<0) = inv<0
|
||||
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMoveIdentity (Ring.additiveGroup R) 0=inv) 0<x))
|
||||
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMove0G (Ring.additiveGroup R) 0=inv) 0<x))
|
||||
|
||||
ringMinusFlipsOrder' : {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} (oRing : OrderedRing R tOrder) → {x : A} → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R) → (Ring.0R R) < x
|
||||
ringMinusFlipsOrder' {R = R} {_<_ = _<_} {tOrder = tOrder} oRing {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
|
||||
ringMinusFlipsOrder' {R = R} {_<_} {tOrder = tOrder} oRing {x} -x<0 | inl (inl 0<x) = 0<x
|
||||
ringMinusFlipsOrder' {_+_ = _+_} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.multIdentRight (Ring.additiveGroup R)) bad))
|
||||
ringMinusFlipsOrder' {_+_ = _+_} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.identRight (Ring.additiveGroup R)) bad))
|
||||
where
|
||||
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.identity (Ring.additiveGroup R) + Group.identity (Ring.additiveGroup R))
|
||||
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
|
||||
bad = ringAddInequalities oRing -x<0 x<0
|
||||
ringMinusFlipsOrder' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric (groupLemmaMoveIdentity' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
|
||||
ringMinusFlipsOrder' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric (groupLemmaMove0G' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -86,23 +73,23 @@ module Rings.Lemmas where
|
||||
ringMinusFlipsOrder''' {S = S} {R = R} {pOrder = pOrder} oRing {x} 0<-x = SetoidPartialOrder.wellDefined pOrder (invInv (Ring.additiveGroup R)) (Equivalence.reflexive (Setoid.eq S)) (ringMinusFlipsOrder oRing 0<-x)
|
||||
|
||||
ringCanMultiplyByPositive : {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} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
|
||||
ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.multIdentRight additiveGroup) q'
|
||||
ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.identRight additiveGroup) q'
|
||||
where
|
||||
open Ring R
|
||||
open Equivalence (Setoid.eq S)
|
||||
have : 0R < (y + Group.inverse additiveGroup x)
|
||||
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing x<y (Group.inverse additiveGroup x))
|
||||
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (transitive multDistributes ((Group.wellDefined additiveGroup) multCommutative multCommutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (transitive *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
|
||||
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (transitive (transitive multCommutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup multCommutative))) p
|
||||
p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (transitive *Commutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup *Commutative))) p
|
||||
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
|
||||
q = OrderedRing.orderRespectsAddition oRing p' (x * c)
|
||||
q' : (x * c) < ((y * c) + 0R)
|
||||
q' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentLeft additiveGroup) (transitive (symmetric (Group.multAssoc additiveGroup)) (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
|
||||
q' = SetoidPartialOrder.wellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
|
||||
|
||||
ringCanCancelPositive : {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} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → (x * c) < (y * c) → x < y
|
||||
ringCanCancelPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.wellDefined pOrder (Group.multIdentLeft additiveGroup) (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.multIdentRight additiveGroup))) q''
|
||||
ringCanCancelPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.wellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) q''
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
@@ -110,41 +97,41 @@ module Rings.Lemmas where
|
||||
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (Group.inverse additiveGroup _))
|
||||
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (symmetric (transitive (multCommutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup multCommutative))))) have
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (transitive (*Commutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative))))) have
|
||||
q : 0R < ((y + Group.inverse additiveGroup x) * c)
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (symmetric multDistributes)) multCommutative) p
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p
|
||||
q' : 0R < (y + Group.inverse additiveGroup x)
|
||||
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
|
||||
q' | inl (inl pr) = pr
|
||||
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) k))
|
||||
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) k))
|
||||
where
|
||||
open Group additiveGroup
|
||||
f : ((y + inverse x) + (inverse (y + inverse x))) < (identity + inverse (y + inverse x))
|
||||
f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
|
||||
f = OrderedRing.orderRespectsAddition oRing y-x<0 _
|
||||
g : identity < inverse (y + inverse x)
|
||||
g = SetoidPartialOrder.wellDefined pOrder invRight multIdentLeft f
|
||||
h : (identity * c) < ((inverse (y + inverse x)) * c)
|
||||
g : 0G < inverse (y + inverse x)
|
||||
g = SetoidPartialOrder.wellDefined pOrder invRight identLeft f
|
||||
h : (0G * c) < ((inverse (y + inverse x)) * c)
|
||||
h = ringCanMultiplyByPositive oRing 0<c g
|
||||
i : (0R + (identity * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
|
||||
i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
|
||||
i = ringAddInequalities oRing q h
|
||||
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
|
||||
j = SetoidPartialOrder.wellDefined pOrder (transitive multIdentLeft (transitive multCommutative (ringTimesZero R))) (symmetric (transitive multCommutative (transitive multDistributes (Group.wellDefined additiveGroup multCommutative multCommutative)))) i
|
||||
j = SetoidPartialOrder.wellDefined pOrder (transitive identLeft (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative (transitive *DistributesOver+ (Group.+WellDefined additiveGroup *Commutative *Commutative)))) i
|
||||
k : 0R < (0R * c)
|
||||
k = SetoidPartialOrder.wellDefined pOrder reflexive (multWellDefined invRight reflexive) j
|
||||
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (multWellDefined x=y reflexive) reflexive xc<yc))
|
||||
k = SetoidPartialOrder.wellDefined pOrder reflexive (*WellDefined invRight reflexive) j
|
||||
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
open Group additiveGroup
|
||||
f : inverse identity ∼ inverse (y + inverse x)
|
||||
f : inverse 0G ∼ inverse (y + inverse x)
|
||||
f = inverseWellDefined additiveGroup 0=y-x
|
||||
g : identity ∼ (inverse y) + x
|
||||
g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (wellDefined reflexive (invInv additiveGroup))))
|
||||
g : 0G ∼ (inverse y) + x
|
||||
g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
|
||||
x=y : x ∼ y
|
||||
x=y = transferToRight additiveGroup (symmetric (transitive g groupIsAbelian))
|
||||
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
|
||||
q'' = OrderedRing.orderRespectsAddition oRing q' x
|
||||
|
||||
ringSwapNegatives : {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} (oRing : OrderedRing R tOrder) → {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
|
||||
ringSwapNegatives {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} oRing {x} {y} -x<-y = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.multIdentRight additiveGroup))) (Group.multIdentLeft additiveGroup) v
|
||||
ringSwapNegatives {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} oRing {x} {y} -x<-y = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
@@ -165,7 +152,7 @@ module Rings.Lemmas where
|
||||
p : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
|
||||
p = OrderedRing.orderRespectsAddition oRing c<0 _
|
||||
0<-c : 0R < (Group.inverse additiveGroup c)
|
||||
0<-c = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) (Group.multIdentLeft additiveGroup) p
|
||||
0<-c = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p
|
||||
t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
|
||||
t = ringCanMultiplyByPositive oRing 0<-c x<y
|
||||
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
|
||||
@@ -181,19 +168,19 @@ module Rings.Lemmas where
|
||||
p : 0R < ((y * c) + inverse (x * c))
|
||||
p = SetoidPartialOrder.wellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (inverse (x * c)))
|
||||
p1 : 0R < ((y * c) + ((inverse x) * c))
|
||||
p1 = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (transitive (inverseWellDefined additiveGroup multCommutative) (transitive (symmetric (ringMinusExtracts R)) multCommutative))) p
|
||||
p1 = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (inverseWellDefined additiveGroup *Commutative) (transitive (symmetric (ringMinusExtracts R)) *Commutative))) p
|
||||
p2 : 0R < ((y + inverse x) * c)
|
||||
p2 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) p1
|
||||
p2 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) p1
|
||||
q : (y + inverse x) < 0R
|
||||
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
|
||||
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
|
||||
where
|
||||
bad : 0R < c
|
||||
bad = ringCanCancelPositive oRing pr (SetoidPartialOrder.wellDefined pOrder (symmetric (transitive multCommutative (ringTimesZero R))) multCommutative p2)
|
||||
bad = ringCanCancelPositive oRing pr (SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) *Commutative p2)
|
||||
q | inl (inr pr) = pr
|
||||
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (multWellDefined x=y reflexive) reflexive xc<yc))
|
||||
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
x=y : x ∼ y
|
||||
x=y = transitive (symmetric multIdentLeft) (transitive (Group.wellDefined additiveGroup 0=y-x reflexive) (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive invLeft) multIdentRight)))
|
||||
x=y = transitive (symmetric identLeft) (transitive (Group.+WellDefined additiveGroup 0=y-x reflexive) (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive invLeft) identRight)))
|
||||
r : y < x
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (invLeft)) multIdentRight)) (Group.multIdentLeft additiveGroup) (OrderedRing.orderRespectsAddition oRing q x)
|
||||
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (invLeft)) identRight)) (Group.identLeft additiveGroup) (OrderedRing.orderRespectsAddition oRing q x)
|
||||
|
||||
@@ -21,3 +21,6 @@ record Semiring {a : _} {A : Set a} (Zero One : A) (_+_ : A → A → A) (_*_ :
|
||||
productOneRight = Monoid.idRight multMonoid
|
||||
sumZeroLeft = Monoid.idLeft monoid
|
||||
sumZeroRight = Monoid.idRight monoid
|
||||
|
||||
-- (b+c)(a+a) == b(a+a) + c(a+a) == ba+ba+ca+ca == (ba+ca) + (ba+ca)
|
||||
-- (b+c)(a+a) ==? (b+c)a+(b+c)a
|
||||
|
||||
@@ -3,7 +3,6 @@
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import LogicalFormulae
|
||||
open import Semirings.Definition
|
||||
open import Maybe
|
||||
|
||||
module Semirings.Solver {a : _} {A : Set a} {Zero One : A} {_+_ : A → A → A} {_*_ : A → A → A} (S : Semiring Zero One _+_ _*_) (comm : (a b : A) → a * b ≡ b * a) where
|
||||
|
||||
@@ -215,5 +214,5 @@ simplifyPreserves (times (times a b) (succ c)) = transitivity (simplePlusIsPlus
|
||||
simplifyPreserves (times (times a b) (plus c d)) = transitivity (simplePlusIsPlus (simplify (times (times a b) c)) (simplify (times (times a b) d))) (transitivity (equalPlus (simplifyPreserves (times (times a b) c)) (simplifyPreserves (times (times a b) d))) (equalityCommutative (+DistributesOver* _ _ _)))
|
||||
simplifyPreserves (times (times a b) (times c d)) = transitivity (equalTimes (simplifyPreserves a) (simplifyPreserves (times b (times c d)))) (*Associative (eval a) (eval b) _)
|
||||
|
||||
_!!_ : (x y : Expr) (pr : eval (simplify x) ≡ eval (simplify y)) → eval x ≡ eval y
|
||||
_!!_ = λ a b c → transitivity (equalityCommutative (simplifyPreserves a)) (transitivity c (simplifyPreserves b))
|
||||
from_to_by_ : (x y : Expr) (pr : eval (simplify x) ≡ eval (simplify y)) → eval x ≡ eval y
|
||||
from a to b by pr = transitivity (equalityCommutative (simplifyPreserves a)) (transitivity pr (simplifyPreserves b))
|
||||
|
||||
Reference in New Issue
Block a user