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Irrelevant field of fractions (#115)

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Patrick Stevens
2020-04-13 14:09:23 +01:00
committed by GitHub
parent 46fcb023d4
commit 1bcb3f8537
10 changed files with 272 additions and 269 deletions
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13
Fields/FieldOfFractions/Ring.agda
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@@ -7,7 +7,6 @@ open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
module Fields.FieldOfFractions.Ring {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
open import Fields.FieldOfFractions.Setoid I
@@ -18,26 +17,26 @@ open import Fields.FieldOfFractions.Multiplication I
fieldOfFractionsRing : Ring fieldOfFractionsSetoid fieldOfFractionsPlus fieldOfFractionsTimes
Ring.additiveGroup fieldOfFractionsRing = fieldOfFractionsGroup
Ring.*WellDefined fieldOfFractionsRing {a} {b} {c} {d} = fieldOfFractionsTimesWellDefined {a} {b} {c} {d}
Ring.1R fieldOfFractionsRing = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Ring.groupIsAbelian fieldOfFractionsRing {a ,, (b , _)} {c ,, (d , _)} = need
Ring.1R fieldOfFractionsRing = record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }
Ring.groupIsAbelian fieldOfFractionsRing {record { num = a ; denom = b }} {record { num = c ; denom = d }} = need
where
open Setoid S
open Equivalence eq
need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
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.*Associative fieldOfFractionsRing {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
Ring.*Associative fieldOfFractionsRing {record { num = a ; denom = b }} {record { num = c ; denom = d }} {record { num = e ; denom = f }} = need
where
open Setoid S
open Equivalence eq
need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
need = transitive (Ring.*WellDefined R (Ring.*Associative R) (symmetric (Ring.*Associative R))) (Ring.*Commutative R)
Ring.*Commutative fieldOfFractionsRing {a ,, (b , _)} {c ,, (d , _)} = need
Ring.*Commutative fieldOfFractionsRing {record { num = a ; denom = b }} {record { num = c ; denom = d }} = need
where
open Setoid S
open Equivalence eq
need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (Ring.*Commutative R))
Ring.*DistributesOver+ fieldOfFractionsRing {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
Ring.*DistributesOver+ fieldOfFractionsRing {record { num = a ; denom = b }} {record { num = c ; denom = d }} {record { num = e ; denom = f }} = need
where
open Setoid S
open Ring R
@@ -46,7 +45,7 @@ Ring.*DistributesOver+ fieldOfFractionsRing {a ,, (b , _)} {c ,, (d , _)} {e ,,
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 = 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.identIsIdent fieldOfFractionsRing {a ,, (b , _)} = need
Ring.identIsIdent fieldOfFractionsRing {record { num = a ; denom = b }} = need
where
open Setoid S
open Equivalence eq