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Specification set-basic
generic specification
   parameter elem
   using nat
target
   sorts set;
   constants 
       ∅ : set;
   
   functions 
       . ++ . : set × elem → set prio 9 left;
       # . : set → nat;
   
   predicates 
       . ∈ . : elem × set;
       . ⊆ . : set × set;
   
   variables 
       s : set;
       s0 : set;
       s1 : set;
       s2 : set;
   
   
   induction 
      set generated by ∅, ++;
   
   axioms 
      Extension: s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2);
      used for: ls;
      
      In-empty: ¬ a ∈ ∅;
      used for: s, ls;
      
      In-insert: a ∈ s ++ b ↔ a = b ∨ a ∈ s;
      used for: s, ls;
      
      Size-empty: # ∅ = 0;
      used for: s, ls;
      
      Size-insert: ¬ a ∈ s → #(s ++ a) = # s + 1;
      used for: s, ls;
      
      Subset: s1 ⊆ s2 ↔ (∀ a. a ∈ s1 → a ∈ s2);
      used for: ls;
   
   
end generic specification