(0) Obligation:

Clauses:

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

Queries:

even(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

even1(0).
even1(s(s(0))).
even1(s(s(s(s(0))))).
even1(s(s(s(s(T6))))) :- even1(s(s(T6))).

Queries:

even1(g).

(3) PrologToPiTRSProof (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 Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))

Pi is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))

The TRS R consists of the following rules:

even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(11) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

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

(14) YES