Specification fsarray

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enrich
   nat
with
   sorts FSArray[A];
   constants 
       size : Nat;
       mkarray : FSArray[A];
   
   functions 
       mkfsarray : A → FSArray[A];
       # . : FSArray[A] → Nat;
   
   variables 
       ar : FSArray[A];
       m : Nat;
       n : Nat;
       a : A;
       b : A;
       c : A;
   
   partial functions 
       . ( . ) : FSArray[A] × Nat → A with λ ar, n. n < size;
       . ( . := . ) : FSArray[A] × Nat × A → FSArray[A] with λ ar, n, a. n < size;
   
   induction 
      FSArray[A] generated by mkarray : FSArray[A],  . (. := .) : (FSArray[A] × Nat × A → FSArray[A]) ;
   
   axioms 
      Extension: ar1 = ar2 ↔ (∀ n. n < size → ar1(n) = ar2(n));
      used for: ls;
      
      Size-fsarray: # ar = size;
      used for: s, ls;
      
      Put-same: m < size → ar(m := a)(m) = a;
      used for: s, ls, lc;
      
      Put-other: m ≠ n → ar(n := a)(m) = ar(m);
      used for: s, ls, lc;
      
      get-mkar: m < size → mkfsarray(e)(m) = e;
      used for: s, ls;
   
   
end enrich