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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: p(X1, X2)\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: 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];
5 [label="5: !_2 | p(X1, X2), SCOPE: 2, CLAUSE: 2\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 8 [arrowhead = none ];
8 -> 2;

9 [label="ONLY EVAL with clause\nq :- p(X1, X2).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 9 [arrowhead = none ];
9 -> 3;

10 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 10 [arrowhead = none ];
10 -> 4;

11 [label="ONLY EVAL with clause\np(a, b) :- !_2.\nand substitutionX1 -> a,\nX2 -> b", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 11 [arrowhead = none ];
11 -> 5;

12 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 12 [arrowhead = none ];
12 -> 6;

13 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 13 [arrowhead = none ];
13 -> 7;

}

(2) Obligation:

Clauses:

qA.

Query: qA()

(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:

qA_in_ → qA_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:

qA_in_ → qA_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:

qA_in_ → qA_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:

qA_in_ → qA_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