mirror of
https://github.com/Smaug123/agdaproofs
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34 lines
2.9 KiB
Agda
34 lines
2.9 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Definition
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open import Rings.Definition
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open import Rings.IntegralDomains.Definition
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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module Fields.FieldOfFractions.Addition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
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open import Fields.FieldOfFractions.Setoid I
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fieldOfFractionsPlus : fieldOfFractionsSet → fieldOfFractionsSet → fieldOfFractionsSet
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fieldOfFractionsSet.num (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = (a * d) + (b * c)
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fieldOfFractionsSet.denom (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = b * d
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fieldOfFractionsSet.denomNonzero (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0))
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--record { num = ((a * d) + (b * c)) ; denom = b * d ; denomNonzero = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0)) }
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plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
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plusWellDefined {record { num = a ; denom = b ; denomNonzero = b!=0 }} {record { num = c ; denom = d ; denomNonzero = d!=0 }} {record { num = e ; denom = f ; denomNonzero = f!=0 }} {record { num = g ; denom = h ; denomNonzero = h!=0 }} af=be ch=dg = need
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where
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open Setoid S
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open Ring R
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open Equivalence eq
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have1 : (c * h) ∼ (d * g)
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have1 = ch=dg
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have2 : (a * f) ∼ (b * e)
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have2 = af=be
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need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
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need = transitive (transitive (Ring.*Commutative R) (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) (transitive *Associative (transitive (*WellDefined (*Commutative) reflexive) (transitive (*WellDefined *Associative reflexive) (transitive (*WellDefined (*WellDefined have2 reflexive) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined (transitive (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)) *Associative) reflexive) (symmetric *Associative))))))))) (transitive *Commutative (transitive (transitive (symmetric *Associative) (*WellDefined reflexive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined have1 reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))))) *Associative))))) (symmetric (Ring.*DistributesOver+ R))
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