(0) Obligation:
Clauses:list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
tail([], []).
tail(.(X, Xs), Xs).
Queries:
list(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:list1(.(T8, T9)) :- list1(T9).
Clauses:
listc1([]).
listc1(.(T8, T9)) :- listc1(T9).
Afs:
list1(x1) = list1(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:list1_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
LIST1_IN_G(.(T8, T9)) → U1_G(T8, T9, list1_in_g(T9))
LIST1_IN_G(.(T8, T9)) → LIST1_IN_G(T9)
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:
LIST1_IN_G(.(T8, T9)) → U1_G(T8, T9, list1_in_g(T9))
LIST1_IN_G(.(T8, T9)) → LIST1_IN_G(T9)
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:
LIST1_IN_G(.(T8, T9)) → LIST1_IN_G(T9)
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:
LIST1_IN_G(.(T8, T9)) → LIST1_IN_G(T9)
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:
- LIST1_IN_G(.(T8, T9)) → LIST1_IN_G(T9)
The graph contains the following edges 1 > 1