(0) Obligation:
Clauses:p(X, Y) :- ','(q(X, Y), r(X)).
q(a, 0).
q(X, s(Y)) :- q(X, Y).
r(b) :- r(b).
Queries:
p(g,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:p3(T11, s(T12)) :- p3(T11, T12).
p1(T5, T6) :- p3(T5, T6).
Clauses:
qc3(T11, s(T12)) :- qc3(T11, T12).
Afs:
p1(x1, x2) = p1(x1, x2)
(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:p1_in: (b,b)
p3_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_GG(T5, T6) → U2_GG(T5, T6, p3_in_gg(T5, T6))
P1_IN_GG(T5, T6) → P3_IN_GG(T5, T6)
P3_IN_GG(T11, s(T12)) → U1_GG(T11, T12, p3_in_gg(T11, T12))
P3_IN_GG(T11, s(T12)) → P3_IN_GG(T11, 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:
P1_IN_GG(T5, T6) → U2_GG(T5, T6, p3_in_gg(T5, T6))
P1_IN_GG(T5, T6) → P3_IN_GG(T5, T6)
P3_IN_GG(T11, s(T12)) → U1_GG(T11, T12, p3_in_gg(T11, T12))
P3_IN_GG(T11, s(T12)) → P3_IN_GG(T11, 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 3 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
P3_IN_GG(T11, s(T12)) → P3_IN_GG(T11, 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:
P3_IN_GG(T11, s(T12)) → P3_IN_GG(T11, 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:
- P3_IN_GG(T11, s(T12)) → P3_IN_GG(T11, T12)
The graph contains the following edges 1 >= 1, 2 > 2