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Specification list-set
enrich
   list-del
with
   functions 
       . ∪ . : list('a) × list('a) → list('a);
       . \ . : list('a) × list('a) → list('a);
      comment: x \ y : remove from x all elements that are in y;
       filter : list('a) × list('a) → list('a);
      comment: filter(x,y) : remove from x all elements that are not in y.;
   
   predicates 
       . ⊆ . : list('a) × list('a);
   
   variables 
       x : list('a);
       y : list('a);
       z : list('a);
       a : 'a;
       b : 'a;
       c : 'a;
       m : nat;
       n : nat;
   
   
   axioms 
      subset: x ⊆ y ↔ (∀ a. a ∈ x → a ∈ y);
      used for: ls;
      
      union: x ∪ y = rmdup(x + y);
      
      diff-e: [] \ y = [];
      used for: s, ls;
      
      diff-y: a ∈ y → (a ' + x) \ y = x \ y;
      used for: s, ls, lc;
      
      diff-n: ¬ a ∈ y → (a ' + x) \ y = a + (x \ y);
      used for: s, ls;
      
      filt-e: filter([], y) = [];
      used for: s, ls;
      
      filt-y: a ∈ y → filter(a ' + x, y) = a ' + filter(x, y);
      used for: s, ls, lc;
      
      filt-n: ¬ a ∈ y → filter(a ' + x, y) = filter(x, y);
      used for: s, ls;
   
   
end enrich