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

(0) Obligation:

Clauses:

q :- p(X, Y).
p(a, b) :- !.
p(a, a) :- p(a, a).

Query: q()

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: q, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(X1, X2)\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(X1, X2), SCOPE: 2, CLAUSE: 1 | p(X1, X2), SCOPE: 2, CLAUSE: 2\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: !_2 | p(X1, X2), SCOPE: 2, CLAUSE: 2\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 18 [arrowhead = none ];
18 -> 4;

19 [label="ONLY EVAL with clause\nq :- p(X1, X2).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 19 [arrowhead = none ];
19 -> 7;

20 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 20 [arrowhead = none ];
20 -> 10;

21 [label="ONLY EVAL with clause\np(a, b) :- !_2.\nand substitutionX1 -> a,\nX2 -> b", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 21 [arrowhead = none ];
21 -> 12;

22 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 22 [arrowhead = none ];
22 -> 14;

23 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 23 [arrowhead = none ];
23 -> 17;

}

(2) Obligation:

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

qA_in_ → qA_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:

qA_in_ → qA_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:

qA_in_ → qA_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