(0) Obligation:
Clauses:
f(X) :- g(s(s(s(X)))).
f(s(X)) :- f(X).
g(s(s(s(s(X))))) :- f(X).
Query: f(g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
fA(s(X1)) :- fA(X1).
fA(s(X1)) :- fA(X1).
Clauses:
fcA(s(X1)) :- fcA(X1).
fcA(s(X1)) :- fcA(X1).
Afs:
fA(x1) = fA(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
fA_in: (b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(X1)) → U1_G(X1, fA_in_g(X1))
FA_IN_G(s(X1)) → FA_IN_G(X1)
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:
FA_IN_G(s(X1)) → U1_G(X1, fA_in_g(X1))
FA_IN_G(s(X1)) → FA_IN_G(X1)
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 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(X1)) → FA_IN_G(X1)
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:
FA_IN_G(s(X1)) → FA_IN_G(X1)
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:
- FA_IN_G(s(X1)) → FA_IN_G(X1)
The graph contains the following edges 1 > 1
(10) YES