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

(0) Obligation:

Clauses:

p :- ','(q, ','(r, !)).
q.
q :- q.
r.

Query: p()

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: p, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(q, ','(r, !_1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 1 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(r, !_1) | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(r, !_1), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: !_1 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 24 [arrowhead = none ];
24 -> 6;

25 [label="ONLY EVAL with clause\np :- ','(q, ','(r, !_1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 25 [arrowhead = none ];
25 -> 8;

26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="ONLY EVAL with clause\nq.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 27 [arrowhead = none ];
27 -> 13;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 28 [arrowhead = none ];
28 -> 15;

29 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 29 [arrowhead = none ];
29 -> 17;

30 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 30 [arrowhead = none ];
30 -> 19;

31 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 31 [arrowhead = none ];
31 -> 23;

}

(2) Obligation:

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

pA_in_ → pA_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:

pA_in_ → pA_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:

pA_in_ → pA_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