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Specification oset
specification
   using actualize
            set-union
         with
            gelem
            
         by morphism
            
         end actualize
   predicates 
       . < . : elem × set;
       . < . : set × elem;
   
   partial functions 
       . .min : set → elem with λ s. s ≠ ∅;
       . .butmin : set → set with λ s. s ≠ ∅;
       . .max : set → elem with λ s. s ≠ ∅;
       . .butmax : set → set with λ s. s ≠ ∅;
   
   
   axioms 
      Min-in: s ≠ ∅ → s.min ∈ s;
      used for: lf;
      
      Min-min: a ∈ s → ¬ a < s.min;
      used for: lf;
      
      Butmin: s ≠ ∅ → s.butmin = s -- s.min;
      used for: ls;
      
      Max-in: s ≠ ∅ → s.max ∈ s;
      used for: lf;
      
      Max-max: a ∈ s → ¬ s.max < a;
      used for: lf;
      
      Butmax: s ≠ ∅ → s.butmax = s -- s.max;
      used for: ls;
      
      Less: a < s ↔ (∀ b. b ∈ s → a < b);
      used for: ls;
      
      Greater: s < a ↔ (∀ b. b ∈ s → b < a);
      used for: ls;
   
   
end specification