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 ( . ) : Bag('a) × 'a → Nat;
   
   variables 
       bag : Bag('a);
       a : 'a;
       b : 'a;
   
   
   induction 
      Bag('a) generated by ∅, ++;
   
   axioms 
      extension: bag0 = bag1 ↔ (∀ a. bag0.count(a) = bag1.count(a));
      
      count-empty: ∅.count(a) = 0;
      used for: s, ls;
      
      count-insert: (bag ++ b).count(a) = (a = b ⊃ 1;0) + bag.count(a);
      used for: ls;
   
   
end enrich