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Specification option
specification
   data types 
      Option[A] = Some(. get : A)
                | None
                ;
   
   variables 
       a : A;
       opt : Option[A];
       get : A;
   
   functions 
       . map ( . ) : Option[A] × (A → B) → Option[B];
       . getOrElse ( . ) : Option[A] × A → A;
       . o . : (B → Option[C]) × (A → Option[B]) → A → Option[C];
   
   predicates 
       oinj : (A → Option[B]);
       osurj : (A → Option[B]);
   
   variables 
       opt : Option[A];
       f : A → B;
       of : A → Option[B];
       b : B;
       og : B → Option[C];
   
   axioms 
      map-Some: Some(a).map(f) = Some(f(a));
      used for: s, ls;
      
      map-None: None.map(f) = None;
      used for: s, ls;
      
      getOrElse-Some: Some(a).getOrElse(a0) = a;
      used for: s, ls;
      
      getOrElse-None: None.getOrElse(a0) = a0;
      used for: s, ls;
      
      compose-def: (og o of)(a) = (of(a) ≠ None ⊃ og(of(a).get);None);
      used for: s, ls;
      
      oinj-def: oinj(of) ↔ (∀ a1, a2. of(a1) ≠ None ∧ of(a1) = of(a2) → a1 = a2);
      used for: ls;
      
      osurj-def: osurj(of) ↔ (∀ b. ∃ a. of(a) = Some(b));
      used for: ls;
   
   
end specification