(0) Obligation:
Clauses:
list([]).
list(.(X1, Ts)) :- list(Ts).
Query: list(g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
listA(.(X1, .(X2, X3))) :- listA(X3).
Clauses:
listcA([]).
listcA(.(X1, [])).
listcA(.(X1, .(X2, X3))) :- listcA(X3).
Afs:
listA(x1) = listA(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
listA_in: (b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
LISTA_IN_G(.(X1, .(X2, X3))) → U1_G(X1, X2, X3, listA_in_g(X3))
LISTA_IN_G(.(X1, .(X2, X3))) → LISTA_IN_G(X3)
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:
LISTA_IN_G(.(X1, .(X2, X3))) → U1_G(X1, X2, X3, listA_in_g(X3))
LISTA_IN_G(.(X1, .(X2, X3))) → LISTA_IN_G(X3)
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:
LISTA_IN_G(.(X1, .(X2, X3))) → LISTA_IN_G(X3)
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:
LISTA_IN_G(.(X1, .(X2, X3))) → LISTA_IN_G(X3)
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:
- LISTA_IN_G(.(X1, .(X2, X3))) → LISTA_IN_G(X3)
The graph contains the following edges 1 > 1
(10) YES