(0) Obligation:
Clauses:f(X) :- g(s(s(s(X)))).
f(s(X)) :- f(X).
g(s(s(s(s(X))))) :- f(X).
Queries:
f(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:f1(s(T8)) :- f1(T8).
f1(s(T12)) :- f1(T12).
Clauses:
fc1(s(T8)) :- fc1(T8).
fc1(s(T12)) :- fc1(T12).
Afs:
f1(x1) = f1(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:f1_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
F1_IN_G(s(T8)) → U1_G(T8, f1_in_g(T8))
F1_IN_G(s(T8)) → F1_IN_G(T8)
F1_IN_G(s(T12)) → U2_G(T12, f1_in_g(T12))
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:The TRS P consists of the following rules:
F1_IN_G(s(T8)) → U1_G(T8, f1_in_g(T8))
F1_IN_G(s(T8)) → F1_IN_G(T8)
F1_IN_G(s(T12)) → U2_G(T12, f1_in_g(T12))
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
F1_IN_G(s(T8)) → F1_IN_G(T8)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(7) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(8) Obligation:
Q DP problem:The TRS P consists of the following rules:
F1_IN_G(s(T8)) → F1_IN_G(T8)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.From the DPs we obtained the following set of size-change graphs:
- F1_IN_G(s(T8)) → F1_IN_G(T8)
The graph contains the following edges 1 > 1