Specification bag-basic

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enrich
   nat
with
   sorts bag('a);
   constants 
       ∅ : bag('a);
   
   functions 
       . ++ . : bag('a) × 'a → bag('a) prio 9 left;
       count : 'a × bag('a) → nat;
   
   variables 
       bag : bag('a);
       a : 'a;
       b : 'a;
   
   
   induction 
      bag('a) generated by ∅, ++;
   
   axioms 
      extension: bag0 = bag1 ↔ (∀ a. count(a, bag0) = count(a, bag1));
      
      count-empty: count(a, ∅) = 0;
      used for: s, ls;
      
      count-insert: count(a, bag ++ b) = (a = b ⊃ 1;0) + count(a, bag);
      used for: ls;
   
   
end enrich