(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).
Queries:
even(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:even1(s(s(T12))) :- even1(T12).
Clauses:
evenc1(0).
evenc1(s(s(0))).
evenc1(s(0)).
evenc1(s(s(T12))) :- evenc1(T12).
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(T12))) → U1_G(T12, even1_in_g(T12))
EVEN1_IN_G(s(s(T12))) → EVEN1_IN_G(T12)
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(T12))) → U1_G(T12, even1_in_g(T12))
EVEN1_IN_G(s(s(T12))) → EVEN1_IN_G(T12)
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(T12))) → EVEN1_IN_G(T12)
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(T12))) → EVEN1_IN_G(T12)
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(T12))) → EVEN1_IN_G(T12)
The graph contains the following edges 1 > 1