Left Termination of the query pattern list_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 PiDP
↳7 PiDPToQDPProof (⇔)
↳8 QDP
↳9 QDPSizeChangeProof (⇔)
↳10 YES

(0) Obligation:

Clauses:

list([]).
list(.(X1, Ts)) :- list(Ts).

Queries:

list(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

list1(.(T4, .(T10, T11))) :- list1(T11).

Clauses:

listc1([]).
listc1(.(T4, [])).
listc1(.(T4, .(T10, T11))) :- listc1(T11).

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(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, list1_in_g(T11))
LIST1_IN_G(.(T4, .(T10, T11))) → LIST1_IN_G(T11)

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(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, list1_in_g(T11))
LIST1_IN_G(.(T4, .(T10, T11))) → LIST1_IN_G(T11)

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(.(T4, .(T10, T11))) → LIST1_IN_G(T11)

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(.(T4, .(T10, T11))) → LIST1_IN_G(T11)

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(.(T4, .(T10, T11))) → LIST1_IN_G(T11)
    The graph contains the following edges 1 > 1

(10) YES