Left Termination of the query pattern p_in_1(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).

Queries:

p(a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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(T3)), q(T3))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 3 | ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 4 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(letter(T5), ','(!_2, ','(failure(a), q(T5)))) | ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 4 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(letter(T5), ','(!_2, ','(failure(a), q(T5)))), SCOPE: 3, CLAUSE: 2 | ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 4 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, ','(failure(a), q(x))) | ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 4 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(letter(T3)), q(T3)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(failure(a), q(x)), SCOPE: 4, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: q(T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: q(T7), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(X9), SCOPE: 6, CLAUSE: 0\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(letter(X12)), q(X12))\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(letter(X12)), q(X12)), SCOPE: 7, CLAUSE: 3 | ','(no(letter(X12)), q(X12)), SCOPE: 7, CLAUSE: 4 | ?_7\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(letter(X15), ','(!_7, ','(failure(a), q(X15)))) | ','(no(letter(X12)), q(X12)), SCOPE: 7, CLAUSE: 4 | ?_7\n\nX12 is free\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(letter(X15), ','(!_7, ','(failure(a), q(X15)))), SCOPE: 8, CLAUSE: 2 | ','(no(letter(X12)), q(X12)), SCOPE: 7, CLAUSE: 4 | ?_7\n\nX12 is free\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(!_7, ','(failure(a), q(x))) | ','(no(letter(X12)), q(X12)), SCOPE: 7, CLAUSE: 4 | ?_7\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(failure(a), q(x)), SCOPE: 9, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 25 [arrowhead = none ];
25 -> 2;

26 [label="EVAL with clause\np(X4) :- ','(no(letter(X4)), q(X4)).\nand substitution\nT1 -> T3\nX4 -> T3\nT2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 26 [arrowhead = none ];
26 -> 3;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 27 [arrowhead = none ];
27 -> 4;

28 [label="EVAL with clause\nno(X6) :- ','(X6, ','(!, failure(a))).\nand substitution\nT3 -> T5\nX6 -> letter(T5)\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 28 [arrowhead = none ];
28 -> 5;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 29 [arrowhead = none ];
29 -> 6;

30 [label="EVAL with clause\nletter(x).\nand substitution\nT5 -> x", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 30 [arrowhead = none ];
30 -> 7;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 32 [arrowhead = none ];
32 -> 9;

33 [label="EVAL with clause\nno(X8).\nand substitution\nT3 -> T7\nX8 -> letter(T7)\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 33 [arrowhead = none ];
33 -> 12;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 36 [arrowhead = none ];
36 -> 13;

37 [label="EVAL with clause\nq(x) :- p(X9).\nand substitution\nT7 -> x", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 38 [arrowhead = none ];
38 -> 15;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 39 [arrowhead = none ];
39 -> 16;

40 [label="EVAL with clause\np(X11) :- ','(no(letter(X11)), q(X11)).\nand substitution\nX9 -> X12\nX11 -> X12", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 40 [arrowhead = none ];
40 -> 17;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 41 [arrowhead = none ];
41 -> 18;

42 [label="EVAL with clause\nno(X14) :- ','(X14, ','(!, failure(a))).\nand substitution\nX12 -> X15\nX14 -> letter(X15)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 42 [arrowhead = none ];
42 -> 19;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 43 [arrowhead = none ];
43 -> 20;

44 [label="EVAL with clause\nletter(x).\nand substitution\nX15 -> x", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 44 [arrowhead = none ];
44 -> 21;

45 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 45 [arrowhead = none ];
45 -> 22;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 46 [arrowhead = none ];
46 -> 23;

47 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 47 [arrowhead = none ];
47 -> 24;

}

(2) Obligation:

Clauses:

(3) PPisEmptyProof (EQUIVALENT transformation)

The Prolog program is empty. Hence, it trivially terminates.

(4) TRUE