(0) Obligation:

Clauses:

list([]).
list(X) :- ','(no(empty(X)), ','(tail(X, T), list(T))).
tail([], []).
tail(.(X, Xs), Xs).
empty(X).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Queries:

list(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

list1([]).

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:
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([])

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g

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([])

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
P is empty.
The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g

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

(6) Obligation:

Pi DP problem:
P is empty.
The TRS R consists of the following rules:

list1_in_g([]) → list1_out_g([])

The argument filtering Pi contains the following mapping:
list1_in_g(x1)  =  list1_in_g(x1)
[]  =  []
list1_out_g(x1)  =  list1_out_g

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,R,Pi) chain.

(8) TRUE