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Rename some confusing fields (#51)

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Patrick Stevens
2019-10-13 09:31:54 +01:00
committed by GitHub
parent 96d15c6017
commit 959071214e
21 changed files with 441 additions and 441 deletions
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Fields/FieldOfFractions.agda
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@@ -20,8 +20,8 @@ module Fields.FieldOfFractions where
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) 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)
Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c) Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c)
Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.multCommutative R Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.*Commutative R
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)) 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))
where where
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
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 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
@@ -30,13 +30,13 @@ module Fields.FieldOfFractions where
open Ring R open Ring R
open Equivalence eq open Equivalence eq
p : (a * d) * f ∼ (b * c) * f p : (a * d) * f ∼ (b * c) * f
p = Ring.multWellDefined R ad=bc reflexive p = Ring.*WellDefined R ad=bc reflexive
p2 : (a * f) * d ∼ b * (d * e) p2 : (a * f) * d ∼ b * (d * e)
p2 = transitive (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive multCommutative) multAssoc)) (transitive p (transitive (symmetric multAssoc) (multWellDefined reflexive cf=de))) p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
p3 : (a * f) * d ∼ (b * e) * d p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (multWellDefined reflexive multCommutative) multAssoc) p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
p4 : (d ∼ 0R) || ((a * f) ∼ (b * e)) p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
p4 = cancelIntDom I (transitive multCommutative (transitive p3 multCommutative)) p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
p5 : (a * f) ∼ (b * e) p5 : (a * f) ∼ (b * e)
p5 with p4 p5 with p4
p5 | inl d=0 = exFalso (d!=0 d=0) p5 | inl d=0 = exFalso (d!=0 d=0)
@@ -63,7 +63,7 @@ module Fields.FieldOfFractions where
ans pr | inr x = d!=0 x ans pr | inr x = d!=0 x
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) 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)
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 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
where where
open Setoid S open Setoid S
open Ring R open Ring R
@@ -73,79 +73,79 @@ module Fields.FieldOfFractions where
have2 : (a * f) ∼ (b * e) have2 : (a * f) ∼ (b * e)
have2 = af=be have2 = af=be
need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g)))) need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
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)) 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))
Group.identity (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I) Group.0G (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b
Group.multAssoc (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need Group.+Associative (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e))) need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e)))
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))))) 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)))))
Group.multIdentRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need Group.identRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((a * Ring.1R R) + (b * Group.identity (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a) need : (((a * Ring.1R R) + (b * Group.0G (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
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)) 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))
Group.multIdentLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need Group.identLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((Group.identity (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a) need : (((Group.0G (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
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) 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)
Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R)) need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
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)) 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))
Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.identity (Ring.additiveGroup R)) need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
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)) 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))
Ring.multWellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need Ring.*WellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g)) need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g))
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))))))))))) 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)))))))))))
Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I) Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a))) need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
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))) 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)))
Ring.multAssoc (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need Ring.*Associative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e)) need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
need = transitive (Ring.multWellDefined R (Ring.multAssoc R) (symmetric (Ring.multAssoc R))) (Ring.multCommutative R) need = transitive (Ring.*WellDefined R (Ring.*Associative R) (symmetric (Ring.*Associative R))) (Ring.*Commutative R)
Ring.multCommutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need Ring.*Commutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a)) need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
need = transitive (Ring.multCommutative R) (Ring.multWellDefined R (Ring.multCommutative R) (Ring.multCommutative R)) need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (Ring.*Commutative R))
Ring.multDistributes (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need Ring.*DistributesOver+ (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence eq open Equivalence eq
inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e))) inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e)))
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))))) 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)))))
need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e)))) need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e))))
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))))) 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)))))
Ring.multIdentIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need Ring.identIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a) need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a)
need = transitive (Ring.multWellDefined R (Ring.multIdentIsIdent R) reflexive) (transitive (Ring.multCommutative R) (Ring.multWellDefined R (symmetric (Ring.multIdentIsIdent R)) reflexive)) need = transitive (Ring.*WellDefined R (Ring.identIsIdent R) reflexive) (transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive))
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) 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 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 Setoid S
open Equivalence eq open Equivalence eq
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R) 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 : fst ∼ Ring.0R R → False
ans pr = prA need' ans pr = prA need'
where where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R) 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))) 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 (ringTimesZero R)) (transitive (Ring.multCommutative R) (transitive pr (Ring.multIdentIsIdent 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 where
open Setoid S open Setoid S
open Equivalence eq 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) 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.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 GroupHom.groupHom (RingHom.groupHom (homIntoFieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {x} {y} = need
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y))) 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)) 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.multCommutative R) (Ring.multWellDefined R reflexive x=y) GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
where where
open Equivalence (Setoid.eq S) 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) 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 where
open Equivalence (Setoid.eq S) 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 where
open Ring R open Ring R
open Setoid S open Setoid S

146
Fields/FieldOfFractionsOrder.agda
View File

@@ -48,11 +48,11 @@ module Fields.FieldOfFractionsOrder where
have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ) have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
have = ringCanMultiplyByPositive order 0<denomZ x<y have = ringCanMultiplyByPositive order 0<denomZ x<y
p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ) 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 : ((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 : ((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 : (numZ * denomY) < (numY * denomZ)
s = ringCanCancelPositive order 0<denomX r 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)) 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 Ring R
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY) 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 : ((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 : ((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} {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) | 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)) 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 : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
p = ringCanMultiplyByPositive order 0<denomZ x<y p = ringCanMultiplyByPositive order 0<denomZ x<y
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY) 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 : ((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 (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} {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)) 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 : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByPositive order 0<denomZ x<y p = ringCanMultiplyByPositive order 0<denomZ x<y
q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX) 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) | 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) | 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 (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) 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) 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} {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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ) 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) | 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 (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 = 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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) 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} {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 (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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 (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} {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 (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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 = 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) 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) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 (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) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 (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) 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) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 (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) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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)) 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 : {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 where
open Setoid S open Setoid S
open Ring R open Ring R
@@ -225,11 +225,11 @@ module Fields.FieldOfFractionsOrder where
inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC) inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
inter = ringCanMultiplyByPositive oRing 0<denomC a<b inter = ringCanMultiplyByPositive oRing 0<denomC a<b
p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB) 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) | 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 (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) 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 where
open Setoid S open Setoid S
open Ring R 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 _) 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 (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 _) | 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) | 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 (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) 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 _) 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 (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 _) | 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) | 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) | 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)) 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 _) 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 (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 _) | 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 (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) 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 (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 _) 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 (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 _) | 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) | 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)) | 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)) 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 : {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 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} {_<_} {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 where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Setoid S open Setoid S
open Ring R 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' : {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 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 where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0 ... | 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 where
open Setoid S open Setoid S
open Ring R open Ring R
@@ -380,7 +380,7 @@ module Fields.FieldOfFractionsOrder where
open Ring R open Ring R
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0 ... | 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 where
open Setoid S open Setoid S
open Ring R 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''' : {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 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 (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 where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence (Setoid.eq S) 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 where
open Setoid S open Setoid S
open Ring R 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) 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) 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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
@@ -413,12 +413,12 @@ module Fields.FieldOfFractionsOrder where
0<dC : 0R < denomC 0<dC : 0R < denomC
0<dC with SetoidTotalOrder.totality tOrder 0R denomC 0<dC with SetoidTotalOrder.totality tOrder 0R denomC
0<dC | inl (inl x) = x 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)) 0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA) p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
p = SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 0<dC a<b) 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 : ((((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 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 where
open Setoid S open Setoid S
@@ -426,11 +426,11 @@ module Fields.FieldOfFractionsOrder where
open Ring R open Ring R
dC<0 : denomC < 0R dC<0 : denomC < 0R
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC 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 ... | inl (inr x) = x
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x)) ... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False 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 (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) 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 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 : 0R < denomC
0<dC with SetoidTotalOrder.totality tOrder 0R denomC 0<dC with SetoidTotalOrder.totality tOrder 0R denomC
0<dC | inl (inl x) = x 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)) 0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
dC<0 : denomC < 0R dC<0 : denomC < 0R
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC 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 | inl (inr x) = x
dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x)) dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
@@ -460,9 +460,9 @@ module Fields.FieldOfFractionsOrder where
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC) have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByNegative oRing dC<0 a<b have = ringCanMultiplyByNegative oRing dC<0 a<b
have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC)) 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'' : ((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) | 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 (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 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 dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
bad : False bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq 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' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
have' = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) have) _ have' = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) have) _
ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB) 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 (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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
@@ -530,7 +530,7 @@ module Fields.FieldOfFractionsOrder where
0<dC = ineqLemma I oRing 0<dBdC 0<dB 0<dC = ineqLemma I oRing 0<dBdC 0<dB
dC<0 : denomC < 0R dC<0 : denomC < 0R
dC<0 = ineqLemma'' I oRing dAdC<0 0<dA 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
@@ -541,7 +541,7 @@ module Fields.FieldOfFractionsOrder where
have = OrderedRing.orderRespectsAddition oRing (ringCanMultiplyByNegative oRing dC<0 a<b) _ 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 (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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq 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 (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) 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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
@@ -592,7 +592,7 @@ module Fields.FieldOfFractionsOrder where
0<dC = ineqLemma''' I oRing dAdC<0 dA<0 0<dC = ineqLemma''' I oRing dAdC<0 dA<0
dC<0 : denomC < 0R dC<0 : denomC < 0R
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq 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) 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) 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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R open Ring R
0<nA : 0R < numA 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 : 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 : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.wellDefined pOrder (transitive multCommutative (ringTimesZero R)) reflexive (ringCanMultiplyByPositive oRing 0<nB 0<nA) 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 multCommutative (transitive multCommutative (ringTimesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dB)))) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R 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 (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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R open Ring R
nB<0 : numB < 0R 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 : 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 : 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) | 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 (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 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 : False
f with OrderedRing.orderRespectsMultiplication oRing 0<denomA 0<denomB f with OrderedRing.orderRespectsMultiplication oRing 0<denomA 0<denomB
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bl dAdB<0) ... | 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R open Ring R
0<nB : 0R < numB 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 : 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 : (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 (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) 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R open Ring R
nB<0 : numB < 0R 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 : 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 : (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 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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Ring R open Ring R
h : 0R < (denomA * denomB) 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 : False
f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder dAdB<0 h) 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)) 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) 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) ... | inl x = exFalso (denomA!=0 x)
... | inr x = exFalso (denomB!=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 where
open Setoid S open Setoid S
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)

26
Fields/Fields.agda
View File

@@ -20,43 +20,43 @@ module Fields.Fields where
open Group additiveGroup open Group additiveGroup
open Setoid S open Setoid S
field 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 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 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 {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 where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
open Ring R 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) a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric pr) reflexive x)
invA : A invA : A
invA = underlying (Field.allInvertible F a a!=0) invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a) q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0 q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr ... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.identity additiveGroup p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = multWellDefined reflexive ab=0 p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.identity additiveGroup p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R)) 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 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 (inr x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
open Ring R 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) a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (symmetric pr) x)
invA : A invA : A
invA = underlying (Field.allInvertible F a a!=0) invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a) q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0 q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr ... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.identity additiveGroup p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = multWellDefined reflexive ab=0 p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.identity additiveGroup p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R)) 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.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) IntegralDomain.nontrivial (orderedFieldIsIntDom {S = S} O F) pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)

14
Groups/Definition.agda
View File

@@ -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 record Group {lvl1 lvl2} {A : Set lvl1} (S : Setoid {lvl1} {lvl2} A) (_·_ : A → A → A) : Set (lsuc lvl1 ⊔ lvl2) where
open Setoid S open Setoid S
field field
wellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y) +WellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
identity : A 0G : A
inverse : A → A inverse : A → A
multAssoc : {a b c : A} → (a · (b · c)) ∼ (a · b) · c +Associative : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
multIdentRight : {a : A} → (a · identity) ∼ a identRight : {a : A} → (a · 0G) ∼ a
multIdentLeft : {a : A} → (identity · a) ∼ a identLeft : {a : A} → (0G · a) ∼ a
invLeft : {a : A} → (inverse a) · a ∼ identity invLeft : {a : A} → (inverse a) · a ∼ 0G
invRight : {a : A} → a · (inverse a) ∼ identity invRight : {a : A} → a · (inverse a) ∼ 0G

46
Groups/Examples/ExampleSheet1.agda
View File

@@ -19,15 +19,15 @@ module Groups.Examples.ExampleSheet1 where
{- {-
Question 1: e is the unique solution of x^2 = x 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 : {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 multIdentRight) (transitive (wellDefined reflexive (symmetric invRight)) (transitive multAssoc (transitive (wellDefined x+x=x reflexive) invRight))) 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 where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq 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' : {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.multIdentRight 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. 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 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 : {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 where
open Setoid S open Setoid S
open Equivalence eq 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 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) 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 where
open Group G open Group G
open Setoid T open Setoid T
open Equivalence (Setoid.eq T) renaming (transitive to transitiveT ; symmetric to symmetricT ; reflexive to reflexiveT) 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) 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)))) 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 where
open Setoid S open Setoid S
open Equivalence eq 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.+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.multIdentRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentRight h1 ,, Group.multIdentRight h2 Group.identRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.identRight h1 ,, Group.identRight h2
Group.multIdentLeft (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentLeft h1 ,, Group.multIdentLeft 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.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 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 open Equivalence eq
bad : Setoid._∼_ S xCoeff2 0R bad : Setoid._∼_ S xCoeff2 0R
bad with Field.allInvertible F xCoeff1 xCoeffNonzero1 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 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) 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 Ring R
open Equivalence eq open Equivalence eq
ans : (((xCoeff * xCoeff') * r) + ((xCoeff * constant') + constant)) ∼ (xCoeff * ((xCoeff' * r) + constant')) + constant 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) 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)) 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 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) 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 where
open Ring R 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 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 where
open Setoid S open Setoid S
open Equivalence eq 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 where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
open Ring R 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 where
open Ring R open Ring R
open Setoid S open Setoid S
open Equivalence eq 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 where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence eq open Equivalence eq
Group.invLeft (linearFunctionsGroup F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero 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 where
open Setoid S open Setoid S
open Ring R open Ring R
open Equivalence eq open Equivalence eq
Group.invRight (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero 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 where
open Setoid S open Setoid S
open Ring R open Ring R
@@ -213,10 +213,10 @@ module Groups.Examples.ExampleSheet1 where
fg = record { xCoeff = 1R + 1R ; xCoeffNonzero = pr ; constant = 1R } fg = record { xCoeff = 1R + 1R ; xCoeffNonzero = pr ; constant = 1R }
oneWay : Setoid._∼_ (linearFunctionsSetoid F) gf (composeLinearFunctions g f) 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 : 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)) open Equivalence (Setoid.eq (linearFunctionsSetoid F))
bad : Setoid._∼_ (linearFunctionsSetoid F) gf fg bad : Setoid._∼_ (linearFunctionsSetoid F) gf fg
@@ -224,4 +224,4 @@ module Groups.Examples.ExampleSheet1 where
ans : False ans : False
ans with bad 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)))))

244
Groups/Groups.agda
View File

@@ -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 : {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 groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
where where
open Group g renaming (inverse to _^-1) renaming (identity to e) open Group g renaming (inverse to _^-1)
open Setoid S open Setoid S
transitive = Equivalence.transitive (Setoid.eq S) transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S) symmetric = Equivalence.symmetric (Setoid.eq S)
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z) 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 : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
k = transitive j multAssoc k = transitive j +Associative
l : e · y ∼ ((x ^-1) · x) · z l : 0G · y ∼ ((x ^-1) · x) · z
l = transitive (wellDefined (symmetric invLeft) ~refl) k l = transitive (+WellDefined (symmetric invLeft) ~refl) k
m : e · y ∼ e · z m : 0G · y ∼ 0G · z
m = transitive l (wellDefined invLeft ~refl) m = transitive l (+WellDefined invLeft ~refl)
n : y ∼ e · z n : y ∼ 0G · z
n = transitive (symmetric multIdentLeft) m n = transitive (symmetric identLeft) m
o : y ∼ z 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 : {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 where
open Group g renaming (inverse to _^-1) renaming (identity to e) open Group g renaming (inverse to _^-1)
open Setoid S open Setoid S
transitive = Equivalence.transitive (Setoid.eq S) transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S) symmetric = Equivalence.symmetric (Setoid.eq S)
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1) 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 : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
j = transitive multAssoc i j = transitive +Associative i
j' : y · e ∼ (z · x) · (x ^-1) j' : y · 0G ∼ (z · x) · (x ^-1)
j' = transitive (wellDefined ~refl (symmetric invRight)) j j' = transitive (+WellDefined ~refl (symmetric invRight)) j
k : y ∼ (z · x) · (x ^-1) k : y ∼ (z · x) · (x ^-1)
k = transitive (symmetric multIdentRight) j' k = transitive (symmetric identRight) j'
l : y ∼ z · (x · (x ^-1)) l : y ∼ z · (x · (x ^-1))
l = transitive k (symmetric multAssoc) l = transitive k (symmetric +Associative)
m : y ∼ z · e m : y ∼ z · 0G
m = transitive l (wellDefined ~refl invRight) 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 : {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 where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq 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 : {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 where
open Group G open Group G
open Setoid S open Setoid S
@@ -84,69 +84,69 @@ module Groups.Groups where
q : inverse x · x ∼ inverse y · x q : inverse x · x ∼ inverse y · x
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y) 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))) rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
where where
open Group G renaming (inverse to _^-1) renaming (identity to e) open Group G renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
i : y ∼ y · e i : y ∼ y · 0G
j : y · e ∼ y · (x · (x ^-1)) j : y · 0G ∼ y · (x · (x ^-1))
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1) k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
l : (y · x) · (x ^-1) ∼ e · (x ^-1) l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
m : e · (x ^-1) ∼ x ^-1 m : 0G · (x ^-1) ∼ x ^-1
i = symmetric multIdentRight i = symmetric identRight
j = wellDefined ~refl (symmetric invRight) j = +WellDefined ~refl (symmetric invRight)
k = multAssoc k = +Associative
l = wellDefined f ~refl l = +WellDefined f ~refl
m = multIdentLeft 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 leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
where where
open Group G renaming (inverse to _^-1) renaming (identity to e) open Group G renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
i : y · (x · y) ∼ y · e i : y · (x · y) ∼ y · 0G
i' : y · (x · y) ∼ y i' : y · (x · y) ∼ y
j : (y · x) · y ∼ y j : (y · x) · y ∼ y
k : (y · x) · y ∼ e · y k : (y · x) · y ∼ 0G · y
l : y · x ∼ e l : y · x ∼ 0G
i = wellDefined ~refl f i = +WellDefined ~refl f
i' = transitive i multIdentRight i' = transitive i identRight
j = transitive (symmetric multAssoc) i' j = transitive (symmetric +Associative) i'
k = transitive j (symmetric multIdentLeft) k = transitive j (symmetric identLeft)
l = groupsHaveRightCancellation G y (y · x) e k 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 : {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) invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
where where
open Group G renaming (inverse to _^-1) renaming (identity to e) open Group G renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq 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)) 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))
fourWayAssoc {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3) fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
where where
open Group G renaming (inverse to _^-1) renaming (identity to e) open Group G renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
p2 : (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) p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
p1 = Group.multAssoc G p1 = Group.+Associative G
p2 = Group.wellDefined G (Group.multAssoc G) reflexive p2 = Group.+WellDefined G (Group.+Associative G) reflexive
p3 = symmetric (Group.multAssoc G) 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))) 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)))
fourWayAssoc' {S = S} G = transitive (symmetric multAssoc) (symmetric (fourWayAssoc G)) fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
where where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq 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))) 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)))
fourWayAssoc'' {S = S} {_·_ = _·_} G = transitive multAssoc (symmetric (fourWayAssoc G)) fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
where where
open Group G open Group G
open Setoid S 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 : {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 invContravariant {S = S} {_·_} G {x} {y} = ans
where where
open Group G renaming (inverse to _^-1) renaming (identity to e) open Group G renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
otherInv = (y ^-1) · (x ^-1) 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) oneMult : (x · y) · otherInv ∼ x · (x ^-1)
manyAssocs = fourWayAssoc G many+Associatives = fourWay+Associative G
oneMult = symmetric (transitive (Group.wellDefined G reflexive (transitive (symmetric (Group.multIdentLeft G)) (Group.wellDefined G (symmetric (Group.invRight G)) reflexive))) manyAssocs) 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 ∼ e otherInvIsInverse : (x · y) · otherInv ∼ 0G
otherInvIsInverse = transitive oneMult (Group.invRight G) otherInvIsInverse = transitive oneMult (Group.invRight G)
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1) ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
ans = symmetric (leftInversesAreUnique G otherInvIsInverse) 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 : {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 multIdentLeft) invRight invIdentity {S = S} G = transitive (symmetric identLeft) invRight
where where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq 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)) transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
where where
open Setoid S open Setoid S
@@ -186,29 +186,29 @@ module Groups.Groups where
u : inverse (inverse a) ∼ inverse (inverse b) u : inverse (inverse a) ∼ inverse (inverse b)
u = inverseWellDefined G t 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 transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
where where
open Setoid S open Setoid S
open Group G open Group G
open Equivalence eq open Equivalence eq
lemma : a · (inverse (inverse b)) ∼ identity lemma : a · (inverse (inverse b)) ∼ 0G
lemma = transitive (wellDefined reflexive (invTwice G b)) ab-1 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'' : {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 transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
where where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq 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)))) 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.+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.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.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.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
Group.multIdentRight (directSumGroup {A = A} {B} g h) = Group.multIdentRight g ,, Group.multIdentRight h Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
Group.multIdentLeft (directSumGroup {A = A} {B} g h) = Group.multIdentLeft g ,, Group.multIdentLeft 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.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 Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
@@ -228,18 +228,18 @@ module Groups.Groups where
field field
injective : SetoidInjection S T underf 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 imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
where where
open Group H open Group H
open Setoid T open Setoid T
id2 : Setoid._∼_ S (Group.identity G) ((Group.identity G) ·A (Group.identity G)) id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
id2 = Equivalence.symmetric (Setoid.eq S) (Group.multIdentRight G) id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
r : f (Group.identity G) ∼ f (Group.identity G) ·B f (Group.identity G) r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
s : identity ·B f (Group.identity G) ∼ f (Group.identity G) ·B f (Group.identity G) s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
t : identity ∼ f (Group.identity G) t : 0G ∼ f (Group.0G G)
t = groupsHaveRightCancellation H (f (Group.identity G)) identity (f (Group.identity G)) s t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
s = Equivalence.transitive (Setoid.eq T) multIdentLeft r s = Equivalence.transitive (Setoid.eq T) identLeft r
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom) 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) 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) 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.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 where
open Setoid S renaming (_∼_ to _∼A_) open Setoid S renaming (_∼_ to _∼A_)
open Setoid U renaming (_∼_ to _∼C_) open Setoid U renaming (_∼_ to _∼C_)
wellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y)) +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 = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x)) 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} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a} 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 Group H
open Equivalence eq open Equivalence eq
ansSetoid : Setoid A 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.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
where 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)) 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 h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
i : f (n ·A Group.inverse G m) ∼ identity 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 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 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 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 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 : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
h = transitive g multIdentLeft h = transitive g identLeft
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ identity i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
i = transitive (GroupHom.groupHom fHom) h i = transitive (GroupHom.groupHom fHom) h
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ identity j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
j = transitive (GroupHom.wellDefined fHom (fourWayAssoc G)) i j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
k : f (m ·A ((Group.identity G) ·A Group.inverse G o)) ∼ identity 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 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_ 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 where
open Setoid T open Setoid T
open Equivalence (Setoid.eq 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))) 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)) 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)) 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)) 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.identity H ·B f (Group.inverse G m)) ∼ (f x) ·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)) 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 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)) 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)) p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
p3 = GroupHom.groupHom fHom p3 = GroupHom.groupHom fHom
p4 = Group.wellDefined H reflexive (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) p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
p6 = Group.wellDefined H reflexive (Group.multIdentLeft H) p6 = Group.+WellDefined H reflexive (Group.identLeft H)
p7 = symmetric (GroupHom.groupHom fHom) p7 = symmetric (GroupHom.groupHom fHom)
p8 = x~m 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)))))) 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.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 where
open Setoid T open Setoid T
open Equivalence (Setoid.eq 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 : 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.multAssoc G))) (imageOfIdentityIsIdentity fHom) ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
Group.multIdentRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where where
open Group G open Group G
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S) transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((a ·A identity) ·A inverse a) ∼ Group.identity H ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.wellDefined G (Group.multIdentRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom) ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight 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 Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where where
open Group G open Group G
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S) transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((identity ·A a) ·A (inverse a)) ∼ Group.identity H ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.wellDefined G (Group.multIdentLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom) 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 Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where where
open Group G open Group G
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
ans : f ((inverse x ·A x) ·A (inverse identity)) ∼ (Group.identity H) 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)))) (multIdentRight {identity}))) (imageOfIdentityIsIdentity fHom) 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 Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where where
open Group G open Group G
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
ans : f ((x ·A inverse x) ·A (inverse identity)) ∼ (Group.identity H) 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))) (multIdentRight {identity}))) (imageOfIdentityIsIdentity fHom) 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 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_) open Setoid T renaming (_∼_ to _∼G_)
@@ -413,7 +413,7 @@ module Groups.Groups where
open Setoid T renaming (_∼_ to _∼T_) open Setoid T renaming (_∼_ to _∼T_)
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
open Reflexive reflexiveEq 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 have = x~y
need : x ∼ y need : x ∼ y
need = {!!} need = {!!}

52
Groups/Groups2.agda
View File

@@ -28,17 +28,17 @@ module Groups.Groups2 where
imageGroupOp {_+A_ = _+A_} fHom (ofElt a) (ofElt b) = ofElt (a +A b) 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) 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 where
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
ans : f (x +A y) ∼ f (a +A b) 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))) 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) Group.0G (imageGroup {G = G} fHom) = ofElt (Group.0G G)
Group.inverse (imageGroup {G = G} fHom) (ofElt a) = ofElt (Group.inverse G a) 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.+Associative (imageGroup {G = G} fHom) {ofElt a} {ofElt b} {ofElt c} = GroupHom.wellDefined fHom (Group.+Associative G)
Group.multIdentRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.multIdentRight G) Group.identRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identRight G)
Group.multIdentLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.multIdentLeft 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.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) Group.invRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invRight G)
@@ -64,8 +64,8 @@ module Groups.Groups2 where
where where
open Setoid T open Setoid T
open Equivalence (Setoid.eq T) open Equivalence (Setoid.eq T)
ans : f (a +G Group.inverse G b) ∼ Group.identity H 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)))) 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) 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 GroupIso.groupHom (groupFirstIsomorphismTheorem' fHom) = groupFirstIsomorphismIsoHom fHom
@@ -75,7 +75,7 @@ module Groups.Groups2 where
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
need : f a ∼ f b 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.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)) 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) 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 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) 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 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) 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 : {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 where
open Setoid T open Setoid T
open Equivalence eq 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) 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.+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.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))) 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 where
open Setoid T open Setoid T
open Equivalence eq 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.+Associative (groupKernelGroup {S = S} {_·A_ = _·A_} G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt z prZ} = Group.+Associative G
Group.multIdentRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.multIdentRight G Group.identRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.identRight G
Group.multIdentLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.multIdentLeft G Group.identLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.identLeft G
Group.invLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invLeft G Group.invLeft (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invLeft G
Group.invRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invRight G Group.invRight (groupKernelGroup G fHom) {kerOfElt x prX} = Group.invRight G
@@ -135,26 +135,26 @@ module Groups.Groups2 where
where where
open Setoid T open Setoid T
open Equivalence eq open Equivalence eq
ans : f ((g ·A h) ·A Group.inverse G g) ∼ Group.identity H 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.multIdentRight H) reflexive) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invRight G)) (imageOfIdentityIsIdentity hom)))))) 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 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.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 where
open Setoid S open Setoid S
open Group G open Group G
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
trivialGroup : Group (reflSetoid (FinSet 1)) λ _ _ → fzero trivialGroup : Group (reflSetoid (FinSet 1)) λ _ _ → fzero
Group.wellDefined trivialGroup _ _ = refl Group.+WellDefined trivialGroup _ _ = refl
Group.identity trivialGroup = fzero Group.0G trivialGroup = fzero
Group.inverse trivialGroup _ = fzero Group.inverse trivialGroup _ = fzero
Group.multAssoc trivialGroup = refl Group.+Associative trivialGroup = refl
Group.multIdentRight trivialGroup {fzero} = refl Group.identRight trivialGroup {fzero} = refl
Group.multIdentRight trivialGroup {fsucc ()} Group.identRight trivialGroup {fsucc ()}
Group.multIdentLeft trivialGroup {fzero} = refl Group.identLeft trivialGroup {fzero} = refl
Group.multIdentLeft trivialGroup {fsucc ()} Group.identLeft trivialGroup {fsucc ()}
Group.invLeft trivialGroup = refl Group.invLeft trivialGroup = refl
Group.invRight trivialGroup = refl Group.invRight trivialGroup = refl

42
Groups/LectureNotes/Lecture1.agda
View File

@@ -25,8 +25,8 @@ module Groups.LectureNotes.Lecture1 where
-- TODO: R is a group with + -- TODO: R is a group with +
integersMinusNotGroup : Group (reflSetoid ℤ) (_-Z_) → False 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 ; 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 ; 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 } | ()
negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b
negSuccInjective {a} {.a} refl = refl negSuccInjective {a} {.a} refl = refl
@@ -35,17 +35,17 @@ module Groups.LectureNotes.Lecture1 where
nonnegInjective {a} {.a} refl = refl nonnegInjective {a} {.a} refl = refl
integersTimesNotGroup : Group (reflSetoid ℤ) (_*Z_) → False 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) ... | 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 ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = 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 ; 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 ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x 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 ; 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 (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 ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = 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 } | ()
-- TODO: Q is not a group with *Q -- TODO: Q is not a group with *Q
-- TODO: Q without 0 is 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 +WAssoc {c} {c} {c} = refl
weirdGroup : Group (reflSetoid Weird) _+W_ weirdGroup : Group (reflSetoid Weird) _+W_
Group.wellDefined weirdGroup = reflGroupWellDefined Group.+WellDefined weirdGroup = reflGroupWellDefined
Group.identity weirdGroup = e Group.0G weirdGroup = e
Group.inverse weirdGroup t = t Group.inverse weirdGroup t = t
Group.multAssoc weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t} Group.+Associative weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t}
Group.multIdentRight weirdGroup {e} = refl Group.identRight weirdGroup {e} = refl
Group.multIdentRight weirdGroup {a} = refl Group.identRight weirdGroup {a} = refl
Group.multIdentRight weirdGroup {b} = refl Group.identRight weirdGroup {b} = refl
Group.multIdentRight weirdGroup {c} = refl Group.identRight weirdGroup {c} = refl
Group.multIdentLeft weirdGroup {e} = refl Group.identLeft weirdGroup {e} = refl
Group.multIdentLeft weirdGroup {a} = refl Group.identLeft weirdGroup {a} = refl
Group.multIdentLeft weirdGroup {b} = refl Group.identLeft weirdGroup {b} = refl
Group.multIdentLeft weirdGroup {c} = refl Group.identLeft weirdGroup {c} = refl
Group.invLeft weirdGroup {e} = refl Group.invLeft weirdGroup {e} = refl
Group.invLeft weirdGroup {a} = refl Group.invLeft weirdGroup {a} = refl
Group.invLeft weirdGroup {b} = refl Group.invLeft weirdGroup {b} = refl

10
Groups/Lemmas.agda
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@@ -17,8 +17,8 @@ module Groups.Lemmas where
open Group G open Group G
open Equivalence eq 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 : {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.multIdentLeft G)) invIdent {S = S} G = symmetric (transferToRight' G (Group.identLeft G))
where where
open Setoid S open Setoid S
open Group G open Group G
@@ -31,9 +31,9 @@ module Groups.Lemmas where
open Group G open Group G
open Equivalence eq 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 : {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 multIdentLeft) (fb e) identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric identLeft) (fb 0G)
where where
open Group g renaming (inverse to _^-1) renaming (identity to e) open Group g renaming (inverse to _^-1)
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq

10
Groups/SymmetryGroups.agda
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@@ -57,13 +57,13 @@ module Groups.SymmetryGroups where
ans {x} | a , b = b ans {x} | a , b = b
symmetricGroup : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Group (symmetricSetoid S) (symmetricGroupOp {A = A}) 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 where
open Equivalence (Setoid.eq A) open Equivalence (Setoid.eq A)
Group.identity (symmetricGroup A) = sym setoidIdIsBijective Group.0G (symmetricGroup A) = sym setoidIdIsBijective
Group.inverse (symmetricGroup S) = symmetricGroupInv S 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.+Associative (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.identRight (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.identLeft (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
Group.invLeft (symmetricGroup S) {x} = symmetricGroupInvIsLeft S {x} Group.invLeft (symmetricGroup S) {x} = symmetricGroupInvIsLeft S {x}
Group.invRight (symmetricGroup S) {x} = symmetricGroupInvIsRight S {x} Group.invRight (symmetricGroup S) {x} = symmetricGroupInvIsRight S {x}

14
LectureNotes/NumbersAndSets/Lecture1.agda
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@@ -9,9 +9,9 @@ open import Numbers.Naturals.Exponentiation
open import Numbers.Primes.PrimeNumbers open import Numbers.Primes.PrimeNumbers
open import Maybe open import Maybe
open import Semirings.Definition 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 module LectureNotes.NumbersAndSets.Lecture1 where
@@ -43,13 +43,13 @@ n3Bigger' zero = inr refl
n3Bigger' (succ n) with n3Bigger (succ n) n3Bigger' (succ n) with n3Bigger (succ n)
n3Bigger' (succ n) | inr f = f n3Bigger' (succ n) | inr f = f
_!!N_ = NatSolver._!!_
-- How to use the semiring solver -- How to use the semiring solver
-- The process is very mechanical; I haven't yet worked out how to do reflection, -- 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. -- 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 : ℕ) → ((n *N n) +N ((2 *N n) +N 1)) ≡ (n +N 1) *N (n +N 1)
proof n = 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))) from plus (times (const n) (const n)) (plus (times (succ (succ zero)) (const n)) (succ zero))
(applyEquality (λ i → succ (n *N n) +N (n +N i)) ((const n !!N plus (const n) zero) refl)) 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))

14
Numbers/Integers/Addition.agda
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@@ -76,14 +76,14 @@ additiveInverse (nonneg (succ x)) = negSucc x
additiveInverse (negSucc x) = nonneg (succ x) additiveInverse (negSucc x) = nonneg (succ x)
ℤGroup : Group (reflSetoid ℤ) (_+Z_) ℤGroup : Group (reflSetoid ℤ) (_+Z_)
Group.wellDefined ℤGroup refl refl = refl Group.+WellDefined ℤGroup refl refl = refl
Group.identity ℤGroup = nonneg 0 Group.0G ℤGroup = nonneg 0
Group.inverse ℤGroup = additiveInverse Group.inverse ℤGroup = additiveInverse
Group.multAssoc ℤGroup {a} {b} {c} = +ZAssociative a b c Group.+Associative ℤGroup {a} {b} {c} = +ZAssociative a b c
Group.multIdentRight ℤGroup {nonneg zero} = refl Group.identRight ℤGroup {nonneg zero} = refl
Group.multIdentRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0) Group.identRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0)
Group.multIdentRight ℤGroup {negSucc x} = refl Group.identRight ℤGroup {negSucc x} = refl
Group.multIdentLeft ℤGroup = refl Group.identLeft ℤGroup = refl
Group.invLeft ℤGroup {nonneg zero} = refl Group.invLeft ℤGroup {nonneg zero} = refl
Group.invLeft ℤGroup {nonneg (succ x)} = additiveInverseExists x Group.invLeft ℤGroup {nonneg (succ x)} = additiveInverseExists x
Group.invLeft ℤGroup {negSucc x} = transitivity (+ZCommutative (nonneg (succ x)) (negSucc x)) (additiveInverseExists x) Group.invLeft ℤGroup {negSucc x} = transitivity (+ZCommutative (nonneg (succ x)) (negSucc x)) (additiveInverseExists x)

5
Numbers/Integers/Integers.agda
View File

@@ -1,5 +1,6 @@
{-# OPTIONS --safe --warning=error --without-K #-} {-# OPTIONS --safe --warning=error --without-K #-}
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Definition open import Numbers.Integers.Definition
open import Numbers.Integers.Addition open import Numbers.Integers.Addition
open import Numbers.Integers.Multiplication open import Numbers.Integers.Multiplication
@@ -15,3 +16,7 @@ open Numbers.Integers.Order using (_<Z_ ; ℤOrderedRing) public
_-Z_ : ℤ → ℤ → ℤ _-Z_ : ℤ → ℤ → ℤ
a -Z b = a +Z (Group.inverse ℤGroup b) 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)

10
Numbers/Integers/Multiplication.agda
View File

@@ -180,13 +180,13 @@ distLemma a b rewrite Semiring.commutative ℕSemiring (a +N b *N a) b | Semirin
ℤRing : Ring (reflSetoid ℤ) _+Z_ _*Z_ ℤRing : Ring (reflSetoid ℤ) _+Z_ _*Z_
Ring.additiveGroup ℤRing = ℤGroup Ring.additiveGroup ℤRing = ℤGroup
Ring.multWellDefined ℤRing refl refl = refl Ring.*WellDefined ℤRing refl refl = refl
Ring.1R ℤRing = nonneg 1 Ring.1R ℤRing = nonneg 1
Ring.groupIsAbelian ℤRing {a} {b} = +ZCommutative a b Ring.groupIsAbelian ℤRing {a} {b} = +ZCommutative a b
Ring.multAssoc ℤRing {a} {b} {c} = *ZAssociative a b c Ring.*Associative ℤRing {a} {b} {c} = *ZAssociative a b c
Ring.multCommutative ℤRing {a} {b} = *ZCommutative a b Ring.*Commutative ℤRing {a} {b} = *ZCommutative a b
Ring.multDistributes ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c Ring.*DistributesOver+ ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c
Ring.multIdentIsIdent ℤRing {a} = *ZleftIdent a Ring.identIsIdent ℤRing {a} = *ZleftIdent a
intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0) intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0)
intDom (nonneg zero) (nonneg b) pr = inl refl intDom (nonneg zero) (nonneg b) pr = inl refl

2
Numbers/Modulo/IntegersModN.agda
View File

@@ -680,7 +680,7 @@ module Numbers.Modulo.IntegersModN where
... | (inr n-a+sa=sn) = equalityZn _ _ refl ... | (inr n-a+sa=sn) = equalityZn _ _ refl
ℤnGroup : (n : ℕ) → (pr : 0 <N n) → Group (reflSetoid (ℤn n pr)) _+n_ 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 where
helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr } helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr }
helpInvLeft a with inverseZn a helpInvLeft a with inverseZn a

40
Numbers/RationalsLemmas.agda
View File

@@ -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}) triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<a+b) = inr (reflexive {a +Q b})
where where
open Equivalence (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
@@ -49,23 +49,23 @@ module Numbers.RationalsLemmas where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
@@ -84,13 +84,13 @@ module Numbers.RationalsLemmas where
open Equivalence eq open Equivalence eq
blah : (inverse a +Q inverse b) <Q (inverse a +Q b) 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)) 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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) 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
@@ -100,65 +100,65 @@ module Numbers.RationalsLemmas where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq open Equivalence eq
triangleInequality {a} {b} | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q b 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) 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 where
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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) 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq 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 where
open Group (Ring.additiveGroup ℚRing) open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom) open Setoid (fieldOfFractionsSetoid ℤIntDom)

22
Rings/Definition.agda
View File

@@ -7,6 +7,7 @@ open import Numbers.Naturals.Naturals
open import Setoids.Orders open import Setoids.Orders
open import Setoids.Setoids open import Setoids.Setoids
open import Functions open import Functions
open import Sets.EquivalenceRelations
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
@@ -17,16 +18,21 @@ module Rings.Definition where
additiveGroup : Group S _+_ additiveGroup : Group S _+_
open Group additiveGroup open Group additiveGroup
open Setoid S open Setoid S
open Equivalence eq
0R : A 0R : A
0R = identity 0R = 0G
_-R_ : A → A → A
a -R b = a + (inverse b)
field 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 1R : A
groupIsAbelian : {a b : A} → a + b ∼ b + a groupIsAbelian : {a b : A} → a + b ∼ b + a
multAssoc : {a b c : A} → (a * (b * c)) ∼ (a * b) * c *Associative : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
multCommutative : {a b : A} → a * b ∼ b * a *Commutative : {a b : A} → a * b ∼ b * a
multDistributes : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c) *DistributesOver+ : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
multIdentIsIdent : {a : A} → 1R * a ∼ a 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 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 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.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.1R (directSumRing r s) = Ring.1R r ,, Ring.1R s
--Ring.groupIsAbelian (directSumRing r s) = Ring.groupIsAbelian r ,, Ring.groupIsAbelian 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.multCommutative (directSumRing r s) = Ring.multCommutative r ,, Ring.multCommutative s
--Ring.multDistributes (directSumRing r s) = Ring.multDistributes r ,, Ring.multDistributes 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 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 open Ring S

2
Rings/IntegralDomains.agda
View File

@@ -27,7 +27,7 @@ module Rings.IntegralDomains where
t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
t1 = transferToRight'' (Ring.additiveGroup R) ab=ac t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R 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 : (a ∼ Ring.0R R) || ((b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R)
t3 = IntegralDomain.intDom I t2 t3 = IntegralDomain.intDom I t2
t4 : (a ∼ Ring.0R R) || (b ∼ c) t4 : (a ∼ Ring.0R R) || (b ∼ c)

93
Rings/Lemmas.agda
View File

@@ -11,22 +11,9 @@ open import Setoids.Orders
open import Sets.EquivalenceRelations open import Sets.EquivalenceRelations
module Rings.Lemmas where 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 : {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 where
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
@@ -40,17 +27,17 @@ module Rings.Lemmas where
open OrderedRing oRing open OrderedRing oRing
open Ring R 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) 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)
groupLemmaMoveIdentity {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G)) groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
where where
open Group G open Group G
open Setoid S open Setoid S
open Equivalence eq open Equivalence eq
p : inverse identity ∼ inverse (inverse x) p : inverse 0G ∼ inverse (inverse x)
p = inverseWellDefined G pr 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)) 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))
groupLemmaMoveIdentity' {S = S} G {x} pr = transferToRight' G (transitive multIdentLeft pr) groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (transitive identLeft pr)
where where
open Group G open Group G
open Setoid S 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 = 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') ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
where 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 = ringAddInequalities oRing 0<x 0<inv
bad' : (Group.identity (Ring.additiveGroup R)) < (Group.identity (Ring.additiveGroup R)) bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
bad' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad 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 | 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' : {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 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} {_<_} {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 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 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 where
open Setoid S open Setoid S
open Equivalence eq 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) 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 : {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 where
open Ring R open Ring R
open Equivalence (Setoid.eq S) open Equivalence (Setoid.eq S)
have : 0R < (y + Group.inverse additiveGroup x) have : 0R < (y + Group.inverse additiveGroup x)
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing x<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 : 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' : 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 : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
q = OrderedRing.orderRespectsAddition oRing p' (x * c) q = OrderedRing.orderRespectsAddition oRing p' (x * c)
q' : (x * c) < ((y * c) + 0R) 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 : {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 where
open Ring R open Ring R
open Setoid S open Setoid S
@@ -110,41 +97,41 @@ module Rings.Lemmas where
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c))) 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 _)) 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 : 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 : 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' : 0R < (y + Group.inverse additiveGroup x)
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x) q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
q' | inl (inl pr) = pr 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 where
open Group additiveGroup 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 _ f = OrderedRing.orderRespectsAddition oRing y-x<0 _
g : identity < inverse (y + inverse x) g : 0G < inverse (y + inverse x)
g = SetoidPartialOrder.wellDefined pOrder invRight multIdentLeft f g = SetoidPartialOrder.wellDefined pOrder invRight identLeft f
h : (identity * c) < ((inverse (y + inverse x)) * c) h : (0G * c) < ((inverse (y + inverse x)) * c)
h = ringCanMultiplyByPositive oRing 0<c g 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 i = ringAddInequalities oRing q h
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c) 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 : 0R < (0R * c)
k = SetoidPartialOrder.wellDefined pOrder reflexive (multWellDefined invRight reflexive) j k = SetoidPartialOrder.wellDefined pOrder reflexive (*WellDefined invRight reflexive) j
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 where
open Group additiveGroup open Group additiveGroup
f : inverse identity ∼ inverse (y + inverse x) f : inverse 0G ∼ inverse (y + inverse x)
f = inverseWellDefined additiveGroup 0=y-x f = inverseWellDefined additiveGroup 0=y-x
g : identity ∼ (inverse y) + x g : 0G ∼ (inverse y) + x
g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (wellDefined reflexive (invInv additiveGroup)))) g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
x=y : x ∼ y x=y : x ∼ y
x=y = transferToRight additiveGroup (symmetric (transitive g groupIsAbelian)) x=y = transferToRight additiveGroup (symmetric (transitive g groupIsAbelian))
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x) q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
q'' = OrderedRing.orderRespectsAddition oRing q' 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 : {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 where
open Ring R open Ring R
open Setoid S open Setoid S
@@ -165,7 +152,7 @@ module Rings.Lemmas where
p : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c) p : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
p = OrderedRing.orderRespectsAddition oRing c<0 _ p = OrderedRing.orderRespectsAddition oRing c<0 _
0<-c : 0R < (Group.inverse additiveGroup c) 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 : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
t = ringCanMultiplyByPositive oRing 0<-c x<y t = ringCanMultiplyByPositive oRing 0<-c x<y
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c) 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 : 0R < ((y * c) + inverse (x * c))
p = SetoidPartialOrder.wellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (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 : 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 : 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 : (y + inverse x) < 0R
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x) q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0)) q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
where where
bad : 0R < c 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 | 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 where
x=y : x ∼ y 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 : 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)

3
Semirings/Definition.agda
View File

@@ -21,3 +21,6 @@ record Semiring {a : _} {A : Set a} (Zero One : A) (_+_ : A → A → A) (_*_ :
productOneRight = Monoid.idRight multMonoid productOneRight = Monoid.idRight multMonoid
sumZeroLeft = Monoid.idLeft monoid sumZeroLeft = Monoid.idLeft monoid
sumZeroRight = Monoid.idRight 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

5
Semirings/Solver.agda
View File

@@ -3,7 +3,6 @@
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae open import LogicalFormulae
open import Semirings.Definition 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 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) (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) _) 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 from_to_by_ : (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 a to b by pr = transitivity (equalityCommutative (simplifyPreserves a)) (transitivity pr (simplifyPreserves b))