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

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇔)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

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

Queries:

list(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

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

Queries:

list1(g).

(3) PrologToPiTRSProof (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 Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])
list1_in_g(.(T4, [])) → list1_out_g(.(T4, []))
list1_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, list1_in_g(T11))
U1_g(T4, T10, T11, list1_out_g(T11)) → list1_out_g(.(T4, .(T10, T11)))

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])
list1_in_g(.(T4, [])) → list1_out_g(.(T4, []))
list1_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, list1_in_g(T11))
U1_g(T4, T10, T11, list1_out_g(T11)) → list1_out_g(.(T4, .(T10, T11)))

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])
list1_in_g(.(T4, [])) → list1_out_g(.(T4, []))
list1_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, list1_in_g(T11))
U1_g(T4, T10, T11, list1_out_g(T11)) → list1_out_g(.(T4, .(T10, T11)))

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x4)
LIST1_IN_G(x1)  =  LIST1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x4)

We have to consider all (P,R,Pi)-chains

(6) 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)

The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])
list1_in_g(.(T4, [])) → list1_out_g(.(T4, []))
list1_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, list1_in_g(T11))
U1_g(T4, T10, T11, list1_out_g(T11)) → list1_out_g(.(T4, .(T10, T11)))

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x4)
LIST1_IN_G(x1)  =  LIST1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x4)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])
list1_in_g(.(T4, [])) → list1_out_g(.(T4, []))
list1_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, list1_in_g(T11))
U1_g(T4, T10, T11, list1_out_g(T11)) → list1_out_g(.(T4, .(T10, T11)))

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x4)
LIST1_IN_G(x1)  =  LIST1_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) 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

(11) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) 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.

(13) 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

(14) YES