(0) Obligation:

Clauses:

even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).

Query: even(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

evenA(s(s(X1))) :- evenA(X1).

Clauses:

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

Afs:

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

EVENA_IN_G(s(s(X1))) → U1_G(X1, evenA_in_g(X1))
EVENA_IN_G(s(s(X1))) → EVENA_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:

EVENA_IN_G(s(s(X1))) → U1_G(X1, evenA_in_g(X1))
EVENA_IN_G(s(s(X1))) → EVENA_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:

EVENA_IN_G(s(s(X1))) → EVENA_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:

EVENA_IN_G(s(s(X1))) → EVENA_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:

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

(10) YES