Specification either

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specification
   using option
   data types 
      Either[A, B] = Left(. getL : A)
                   | Right(. getR : B) with  . isRight
                   ;
   
   variables 
       either : Either[A, B];
       a : A;
       b : B;
       getL : A;
       getR : B;
   
   functions 
       . mapl ( . ) : Either[A, B] × (A → C) → Either[C, B];
       . mapr ( . ) : Either[A, B] × (B → C) → Either[A, C];
       . map ( . ) : Either[A, B] × (A → C) × (B → D) → Either[C, D];
       . fold ( . ) : Either[A, B] × (A → C) × (B → C) → C;
       . getLOrElse ( . ) : Either[A, B] × A → A;
       . getROrElse ( . ) : Either[A, B] × B → B;
       . optl : Either[A, B] → Option[A];
       . optr : Either[A, B] → Option[B];
   
   variables 
       a : A;
       b : B;
       c : C;
       d : D;
       oa : Option[A];
       ob : Option[B];
       fac : A → C;
       fbc : B → C;
       fbd : B → D;
   
   axioms 
      mapl-left: Left(a).mapl(fac) = Left(fac(a));
      used for: s, ls;
      
      mapl-right: Right(b).mapl(fac) = Right(b);
      used for: s, ls;
      
      mapr-left: Left(a).mapr(fbc) = Left(a);
      used for: s, ls;
      
      mapr-right: Right(b).mapr(fbc) = Right(fbc(b));
      used for: s, ls;
      
      map-left: Left(a).map(fac, fbd) = Left(fac(a));
      used for: s, ls;
      
      map-right: Right(b).map(fac, fbd) = Right(fbd(b));
      used for: s, ls;
      
      fold-left: Left(a).fold(fac, fbc) = fac(a);
      used for: s, ls;
      
      fold-right: Right(b).fold(fac, fbc) = fbc(b);
      used for: s, ls;
      
      optl-left: Left(a).optl = Some(a);
      used for: s, ls;
      
      optl-right: Right(b).optl = None;
      used for: s, ls;
      
      optr-left: Left(a).optr = None;
      used for: s, ls;
      
      optr-right: Right(b).optr = Some(b);
      used for: s, ls;
      
      getLOrElse-left: Left(a).getLOrElse(a0) = a;
      
      getLOrElse-right: Right(b).getLOrElse(a0) = a0;
      
      getROrElse-left: Left(a).getROrElse(b0) = b0;
      
      getROrElse-right: Right(b).getROrElse(b0) = b;
   
   
end specification