{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations


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

open import Fields.FieldOfFractions.Setoid I

fieldOfFractionsPlus : fieldOfFractionsSet → fieldOfFractionsSet → fieldOfFractionsSet
fieldOfFractionsSet.num (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = (a * d) + (b * c)
fieldOfFractionsSet.denom (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = b * d
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))

--record { num = ((a * d) + (b * c)) ; denom = b * d ; denomNonzero = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0)) }

plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
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
  where
    open Setoid S
    open Ring R
    open Equivalence eq
    have1 : (c * h) ∼ (d * g)
    have1 = ch=dg
    have2 : (a * f) ∼ (b * e)
    have2 = af=be
    need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
    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))