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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: a\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: a, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: b\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: b, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: c\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: c, SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: d\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: d, SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(e, !_4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(e, !_4), SCOPE: 5, CLAUSE: 4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: !_4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="ONLY EVAL with clause\na :- b.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 39 [arrowhead = none ];
39 -> 7;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="ONLY EVAL with clause\nb :- c.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 41 [arrowhead = none ];
41 -> 13;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 42 [arrowhead = none ];
42 -> 16;

43 [label="ONLY EVAL with clause\nc :- d.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 43 [arrowhead = none ];
43 -> 19;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 44 [arrowhead = none ];
44 -> 22;

45 [label="ONLY EVAL with clause\nd :- ','(e, !_4).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 45 [arrowhead = none ];
45 -> 25;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 46 [arrowhead = none ];
46 -> 28;

47 [label="ONLY EVAL with clause\ne.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 47 [arrowhead = none ];
47 -> 30;

48 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 48 [arrowhead = none ];
48 -> 33;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 49 [arrowhead = none ];
49 -> 37;

}

(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