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Specification nat-prime
enrich
   nat-div
with
   functions 
       gcd : nat × nat → nat;
       dist : nat × nat → nat;
   
   predicates 
       prime : nat;
       . | . : nat × nat;
   
   
   axioms 
      Prime: prime(n) ↔ 1 < n ∧ (∀ m. m | n → m = 1 ∨ m = n);
      
      Divides: m | n ↔ (∃ n0. m * n0 = n);
      
      gcd-zero: gcd(0, n) = n;
      
      gcd-c: gcd(m, n) = gcd(n, m);
      
      gcd-rec1: n ≤ m → gcd(m, n) = gcd(m - n, n);
      
      dist-fst: m ≤ n → dist(m, n) = n - m;
      
      dist-snd: n < m → dist(m, n) = m - n;
   
   
end enrich