Left Termination of the query pattern even_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 PiDP
↳7 PiDPToQDPProof (⇔)
↳8 QDP
↳9 QDPSizeChangeProof (⇔)
↳10 YES

(0) Obligation:

Clauses:

even(0).
even(s(X)) :- odd(X).
odd(s(X)) :- even(X).

Queries:

even(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

even1(s(s(T6))) :- even1(T6).

Clauses:

evenc1(0).
evenc1(s(s(T6))) :- evenc1(T6).

Afs:

even1(x1)  =  even1(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:
even1_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

EVEN1_IN_G(s(s(T6))) → U1_G(T6, even1_in_g(T6))
EVEN1_IN_G(s(s(T6))) → EVEN1_IN_G(T6)

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:

EVEN1_IN_G(s(s(T6))) → U1_G(T6, even1_in_g(T6))
EVEN1_IN_G(s(s(T6))) → EVEN1_IN_G(T6)

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:

EVEN1_IN_G(s(s(T6))) → EVEN1_IN_G(T6)

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:

EVEN1_IN_G(s(s(T6))) → EVEN1_IN_G(T6)

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:

  • EVEN1_IN_G(s(s(T6))) → EVEN1_IN_G(T6)
    The graph contains the following edges 1 > 1

(10) YES