(0) Obligation:

Clauses:

p(0).
p(s(X)) :- ','(q(X), ','(!, r)).
p(X) :- r.
q(0).
q(s(X)) :- ','(p(X), ','(!, r)).
q(X) :- r.
r.

Query: p(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

pA(s(s(X1))) :- pA(X1).

Clauses:

pcA(0).
pcA(0).
pcA(s(0)).
pcA(s(s(X1))) :- pcA(X1).
pcA(s(X1)).
pcA(s(X1)).
pcA(X1).

Afs:

pA(x1)  =  pA(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:
pA_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(X1))) → U1_G(X1, pA_in_g(X1))
PA_IN_G(s(s(X1))) → PA_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:

PA_IN_G(s(s(X1))) → U1_G(X1, pA_in_g(X1))
PA_IN_G(s(s(X1))) → PA_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:

PA_IN_G(s(s(X1))) → PA_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:

PA_IN_G(s(s(X1))) → PA_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:

  • PA_IN_G(s(s(X1))) → PA_IN_G(X1)
    The graph contains the following edges 1 > 1

(10) YES