(0) Obligation:

Clauses:

a :- b.
b :- c.
c :- d.
d :- ','(e, !).
e.
e :- a.

Query: a()

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

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

aA_in_aA_out_

Pi is empty.

(3) 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:

aA_in_aA_out_

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

(4) Obligation:

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

aA_in_aA_out_

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

(5) PisEmptyProof (EQUIVALENT transformation)

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

(6) YES