(0) Obligation:
Clauses:
q(X) :- p(X, 0).
p(0, X1).
p(s(X), Y) :- p(X, s(Y)).
Query: q(g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
pA(s(X1), X2) :- pA(X1, s(X2)).
qB(s(s(s(s(s(s(s(s(X1))))))))) :- pA(X1, s(s(s(s(s(s(s(0)))))))).
Clauses:
pcA(0, X1).
pcA(s(X1), X2) :- pcA(X1, s(X2)).
Afs:
qB(x1) = qB(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:
qB_in: (b)
pA_in: (b,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
QB_IN_G(s(s(s(s(s(s(s(s(X1))))))))) → U2_G(X1, pA_in_gg(X1, s(s(s(s(s(s(s(0)))))))))
QB_IN_G(s(s(s(s(s(s(s(s(X1))))))))) → PA_IN_GG(X1, s(s(s(s(s(s(s(0))))))))
PA_IN_GG(s(X1), X2) → U1_GG(X1, X2, pA_in_gg(X1, s(X2)))
PA_IN_GG(s(X1), X2) → PA_IN_GG(X1, s(X2))
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:
QB_IN_G(s(s(s(s(s(s(s(s(X1))))))))) → U2_G(X1, pA_in_gg(X1, s(s(s(s(s(s(s(0)))))))))
QB_IN_G(s(s(s(s(s(s(s(s(X1))))))))) → PA_IN_GG(X1, s(s(s(s(s(s(s(0))))))))
PA_IN_GG(s(X1), X2) → U1_GG(X1, X2, pA_in_gg(X1, s(X2)))
PA_IN_GG(s(X1), X2) → PA_IN_GG(X1, s(X2))
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 3 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_GG(s(X1), X2) → PA_IN_GG(X1, s(X2))
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:
PA_IN_GG(s(X1), X2) → PA_IN_GG(X1, s(X2))
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:
- PA_IN_GG(s(X1), X2) → PA_IN_GG(X1, s(X2))
The graph contains the following edges 1 > 1
(10) YES