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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: 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:

Triples:


Clauses:


Afs:

(3) TPisEmptyProof (EQUIVALENT transformation)

There are no more dependency triples. Hence, the dependency triple problem trivially terminates.

(4) YES