Specification string

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specification
   using char, nat
   data types 
      String = []
             | . + .(. head : Char;. tail : String)
             ;
   
   variables 
       s : String;
       head : Char;
       tail : String;
   
   size functions 
       # . : String → Nat;
   
   functions 
       . + . : String × String → String;
       . + . : String × Char → String;
       str : Char → String;
   
   predicates 
       . ∈ . : Char × String;
   
   variables 
       c : Char;
       s : String;
   
   axioms 
      Nil: [] + s = s;
      used for: s, ls;
      
      Cons: (c + s1) + s2 = c + (s1 + s2);
      used for: s, ls;
      
      One: str(c) = c + [];
      used for: s, ls;
      
      Last: s + c = s + str(c);
      used for: s, ls;
      
      in-empty: ¬ c ∈ [];
      used for: s, ls;
      
      in-rec: c ∈ c0 + s ↔ c = c0 ∨ c ∈ s;
      used for: s, ls;
   
   
end specification