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

(0) Obligation:

Clauses:

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

Query: p(a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: q(T4) | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: q(T4), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: q(T4), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(q(T15)), p(T15))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 3 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(','(call(q(T21)), ','(!_3, failure(a))), p(T21)) | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(q(T21), ','(','(!_3, failure(a)), p(T21))) | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(q(T21), ','(','(!_3, failure(a)), p(T21))), SCOPE: 5, CLAUSE: 2 | ?_5 | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(','(!_3, failure(a)), p(T26)) | ?_5 | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(failure(a), p(T26))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(failure(a), p(T26)), SCOPE: 6, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="ONLY EVAL with clause\np(X3) :- q(X3).\nand substitutionT1 -> T4,\nX3 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 7;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 57 [arrowhead = none ];
57 -> 15;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 58 [arrowhead = none ];
58 -> 16;

59 [label="ONLY EVAL with clause\nq(X8).\nand substitutionT4 -> T9,\nX8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 59 [arrowhead = none ];
59 -> 21;

60 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 60 [arrowhead = none ];
60 -> 23;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 61 [arrowhead = none ];
61 -> 22;

62 [label="ONLY EVAL with clause\np(X12) :- ','(no(q(X12)), p(X12)).\nand substitutionT1 -> T15,\nX12 -> T15,\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 62 [arrowhead = none ];
62 -> 29;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 63 [arrowhead = none ];
63 -> 30;

64 [label="ONLY EVAL with clause\nno(X16) :- ','(call(X16), ','(!_3, failure(a))).\nand substitutionT15 -> T21,\nX16 -> q(T21),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 64 [arrowhead = none ];
64 -> 35;

65 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 65 [arrowhead = none ];
65 -> 36;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 66 [arrowhead = none ];
66 -> 38;

67 [label="ONLY EVAL with clause\nq(X19).\nand substitutionT21 -> T26,\nX19 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 67 [arrowhead = none ];
67 -> 42;

68 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 68 [arrowhead = none ];
68 -> 44;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 69 [arrowhead = none ];
69 -> 51;

70 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 70 [arrowhead = none ];
70 -> 53;

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

pA_in_a(T9) → pA_out_a(T9)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a

(3) 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:

pA_in_a(T9) → pA_out_a(T9)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
P is empty.
The TRS R consists of the following rules:

pA_in_a(T9) → pA_out_a(T9)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a

We have to consider all (P,R,Pi)-chains

(5) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,R,Pi) chain.

(6) YES