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) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="3: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(letter(T4)), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(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];
22 [label="22: ','(','(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];
24 [label="24: ','(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];
28 [label="28: ','(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];
32 [label="32: ','(','(!_2, failure(a)), q(x)) | ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(failure(a), q(x)), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: q(T14), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: p(X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: p(X15), SCOPE: 7, CLAUSE: 0\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(no(letter(X24)), q(X24))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(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];
71 [label="71: ','(','(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];
72 [label="72: ','(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];
73 [label="73: ','(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];
74 [label="74: ','(','(!_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];
75 [label="75: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(failure(a), q(x)), SCOPE: 11, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 78 [arrowhead = none ];
78 -> 6;

79 [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];
6 -> 79 [arrowhead = none ];
79 -> 7;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 80 [arrowhead = none ];
80 -> 11;

81 [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];
11 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 82 [arrowhead = none ];
82 -> 24;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="EVAL with clause\nletter(x).\nand substitutionT9 -> x", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 85 [arrowhead = none ];
85 -> 33;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 86 [arrowhead = none ];
86 -> 35;

87 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 87 [arrowhead = none ];
87 -> 41;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 88 [arrowhead = none ];
88 -> 37;

89 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 89 [arrowhead = none ];
89 -> 39;

90 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 90 [arrowhead = none ];
90 -> 43;

91 [label="ONLY EVAL with clause\nno(X12).\nand substitutionT4 -> T14,\nX12 -> letter(T14),\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 91 [arrowhead = none ];
91 -> 45;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 92 [arrowhead = none ];
92 -> 47;

93 [label="EVAL with clause\nq(x) :- p(X15).\nand substitutionT14 -> x", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 93 [arrowhead = none ];
93 -> 50;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 94 [arrowhead = none ];
94 -> 53;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 95 [arrowhead = none ];
95 -> 57;

96 [label="ONLY EVAL with clause\np(X23) :- ','(no(letter(X23)), q(X23)).\nand substitutionX15 -> X24,\nX23 -> X24", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 96 [arrowhead = none ];
96 -> 60;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 97 [arrowhead = none ];
97 -> 67;

98 [label="ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_8, failure(a))).\nand substitutionX24 -> X31,\nX30 -> letter(X31)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 98 [arrowhead = none ];
98 -> 71;

99 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 99 [arrowhead = none ];
99 -> 72;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 100 [arrowhead = none ];
100 -> 73;

101 [label="ONLY EVAL with clause\nletter(x).\nand substitutionX31 -> x", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 101 [arrowhead = none ];
101 -> 74;

102 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 102 [arrowhead = none ];
102 -> 75;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 103 [arrowhead = none ];
103 -> 76;

104 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 104 [arrowhead = none ];
104 -> 77;

}

(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) YES