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Specification reduce
enrich subst with
   functions 
       . ^ . : (term × term → bool) × nat → term × term → bool;
      comment: iteration of any relation on terms.;
       
   predicates 
       . => . : term × term;
      comment: one step reduction.;
       
       . =>top . : term × term;
      comment: one step reduction with empty context.;
       
       . =>* . : term × term;
      comment: refl. trans clos. of one step red..;
       
      irred : term;
      comment: irreducible term.;
       
   variables 
      v, v1, v2, v3, t' : term; 
      R : term × term → bool; 
   
   axioms 
      one-step-red-def
      : ⊦ 
          t => t'
        ↔ (∃ C. 
              (∃ v1, v2. t = C[K _ v1 _ v2] ∧ t' = C[v1] ∧ value(v1) ∧ value(v2) ∧ context(C))
            ∨ (∃ v1, v2, v3. t = C[S _ v1 _ v2 _ v3] ∧ t' = C[v1 _ v3 _ (v2 _ v3)] ∧ value(v1) ∧ value(v2) ∧ value(v3) ∧ context(C)));
      topred-def
      : ⊦ 
          t =>top t'
        ↔   (∃ v1, v2. t = K _ v1 _ v2 ∧ t' = v1 ∧ value(v1) ∧ value(v2))
          ∨ (∃ v1, v2, v3. t = S _ v1 _ v2 _ v3 ∧ t' = v1 _ v3 _ (v2 _ v3) ∧ value(v1) ∧ value(v2) ∧ value(v3));
      it-base :  ⊦ (R ^ 0)(t, t') ↔ t = t';
      used for : s, ls;
       
      it-rec :  ⊦ (R ^ (n + 1))(t, t') ↔ (∃ t1. R(t, t1) ∧ (R ^ n)(t1, t'));
      used for : s, ls;
       
      reduce-def :  ⊦ t =>* t' ↔ (∃ n. (=> ^ n)(t, t'));
      irred-def :  ⊦ irred(t) ↔ ¬ (∃ t'. t => t');
end enrich