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

(0) Obligation:

Clauses:

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

Query: a()

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: a\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: a, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: b\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: b, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: c\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: c, SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: d\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: d, SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(e, !_4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(e, !_4), SCOPE: 5, CLAUSE: 4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: !_4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 14 [arrowhead = none ];
14 -> 2;

15 [label="ONLY EVAL with clause\na :- b.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 15 [arrowhead = none ];
15 -> 3;

16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 16 [arrowhead = none ];
16 -> 4;

17 [label="ONLY EVAL with clause\nb :- c.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 17 [arrowhead = none ];
17 -> 5;

18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 18 [arrowhead = none ];
18 -> 6;

19 [label="ONLY EVAL with clause\nc :- d.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 19 [arrowhead = none ];
19 -> 7;

20 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 20 [arrowhead = none ];
20 -> 8;

21 [label="ONLY EVAL with clause\nd :- ','(e, !_4).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 21 [arrowhead = none ];
21 -> 9;

22 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 22 [arrowhead = none ];
22 -> 10;

23 [label="ONLY EVAL with clause\ne.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 23 [arrowhead = none ];
23 -> 11;

24 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 24 [arrowhead = none ];
24 -> 12;

25 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 25 [arrowhead = none ];
25 -> 13;

}

(2) Obligation:

Clauses:

aA.

Query: aA()

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. 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:

aA_in_ → aA_out_

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

aA_in_ → aA_out_

Pi is empty.

(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:

aA_in_ → aA_out_

Pi is empty.
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:

aA_in_ → aA_out_

Pi is empty.
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) YES