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

(0) Obligation:

Clauses:

p(X) :- ','(no(letter(X)), q(X)).
q(x) :- p(X1).
letter(x).
no(X) :- ','(X, ','(!, failure(a))).
no(X2).
failure(b).

Query: p(a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(no(letter(T4)), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(','(call(letter(T9)), ','(!_2, failure(a))), q(T9)) | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(letter(T9), ','(','(!_2, failure(a)), q(T9))) | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(letter(T9), ','(','(!_2, failure(a)), q(T9))), SCOPE: 4, CLAUSE: 2 | ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(','(!_2, failure(a)), q(x)) | ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(failure(a), q(x)), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: q(T14), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(X15), SCOPE: 7, CLAUSE: 0\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(no(letter(X24)), q(X24))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 3 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(call(letter(X31)), ','(!_8, failure(a))), q(X31)) | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(letter(X31), ','(','(!_8, failure(a)), q(X31))) | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(letter(X31), ','(','(!_8, failure(a)), q(X31))), SCOPE: 10, CLAUSE: 2 | ?_10 | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(','(!_8, failure(a)), q(x)) | ?_10 | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), q(x)), SCOPE: 11, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 29 [arrowhead = none ];
29 -> 2;

30 [label="ONLY EVAL with clause\np(X4) :- ','(no(letter(X4)), q(X4)).\nand substitutionT1 -> T4,\nX4 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 30 [arrowhead = none ];
30 -> 3;

31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 31 [arrowhead = none ];
31 -> 4;

32 [label="ONLY EVAL with clause\nno(X7) :- ','(call(X7), ','(!_2, failure(a))).\nand substitutionT4 -> T9,\nX7 -> letter(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 33 [arrowhead = none ];
33 -> 6;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 34 [arrowhead = none ];
34 -> 7;

35 [label="EVAL with clause\nletter(x).\nand substitutionT9 -> x", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 35 [arrowhead = none ];
35 -> 8;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 36 [arrowhead = none ];
36 -> 9;

37 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 37 [arrowhead = none ];
37 -> 10;

38 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 39 [arrowhead = none ];
39 -> 11;

40 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 40 [arrowhead = none ];
40 -> 12;

41 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="ONLY EVAL with clause\nno(X12).\nand substitutionT4 -> T14,\nX12 -> letter(T14),\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="EVAL with clause\nq(x) :- p(X15).\nand substitutionT14 -> x", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 44 [arrowhead = none ];
44 -> 17;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="ONLY EVAL with clause\np(X23) :- ','(no(letter(X23)), q(X23)).\nand substitutionX15 -> X24,\nX23 -> X24", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_8, failure(a))).\nand substitutionX24 -> X31,\nX30 -> letter(X31)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="ONLY EVAL with clause\nletter(x).\nand substitutionX31 -> x", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 53 [arrowhead = none ];
53 -> 26;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 54 [arrowhead = none ];
54 -> 27;

55 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 55 [arrowhead = none ];
55 -> 28;

}

(2) Obligation:

Triples:


Clauses:


Afs:

(3) TPisEmptyProof (EQUIVALENT transformation)

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

(4) YES