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

(0) Obligation:

Clauses:

q(X) :- ','(not_zero(X), ','(p(X, Y), q(Y))).
p(0, 0).
p(s(X), X).
zero(0).
not_zero(X) :- ','(zero(X), ','(!, failure(a))).
not_zero(X1).
failure(b).

Query: q(g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: q(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(not_zero(T3), ','(p(T3, X4), q(X4)))\n\nT3 is ground\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(not_zero(T3), ','(p(T3, X4), q(X4))), SCOPE: 2, CLAUSE: 4 | ','(not_zero(T3), ','(p(T3, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(zero(T6), ','(!_2, failure(a))), ','(p(T6, X4), q(X4))) | ','(not_zero(T6), ','(p(T6, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(zero(T6), ','(!_2, failure(a))), ','(p(T6, X4), q(X4))), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(not_zero(T6), ','(p(T6, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(','(!_2, failure(a)), ','(p(0, X4), q(X4))) | ?_3 | ','(not_zero(0), ','(p(0, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_3 | ','(not_zero(T6), ','(p(T6, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX4 is free\nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(failure(a), ','(p(0, X4), q(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(failure(a), ','(p(0, X4), q(X4))), SCOPE: 4, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(not_zero(T6), ','(p(T6, X4), q(X4))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX4 is free\nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T9, X4), q(X4))\n\nT9 is ground\nX4 is free\nzero(T9) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(T9, X4), q(X4)), SCOPE: 5, CLAUSE: 1 | ','(p(T9, X4), q(X4)), SCOPE: 5, CLAUSE: 2\n\nT9 is ground\nX4 is free\nzero(T9) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(T9, X4), q(X4)), SCOPE: 5, CLAUSE: 2\n\nT9 is ground\nX4 is free\nzero(T9) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: q(T12)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 52 [arrowhead = none ];
52 -> 4;

53 [label="ONLY EVAL with clause\nq(X3) :- ','(not_zero(X3), ','(p(X3, X4), q(X4))).\nand substitutionT1 -> T3,\nX3 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 53 [arrowhead = none ];
53 -> 8;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="ONLY EVAL with clause\nnot_zero(X7) :- ','(zero(X7), ','(!_2, failure(a))).\nand substitutionT3 -> T6,\nX7 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 55 [arrowhead = none ];
55 -> 18;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="EVAL with clause\nzero(0).\nand substitutionT6 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 57 [arrowhead = none ];
57 -> 29;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 58 [arrowhead = none ];
58 -> 30;

59 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 59 [arrowhead = none ];
59 -> 31;

60 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 60 [arrowhead = none ];
60 -> 35;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 61 [arrowhead = none ];
61 -> 33;

62 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 62 [arrowhead = none ];
62 -> 34;

63 [label="ONLY EVAL with clause\nnot_zero(X16).\nand substitutionT6 -> T9,\nX16 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 63 [arrowhead = none ];
63 -> 38;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 64 [arrowhead = none ];
64 -> 44;

65 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T9), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 65 [arrowhead = none ];
65 -> 46;

66 [label="EVAL with clause\np(s(X22), X22).\nand substitutionX22 -> T12,\nT9 -> s(T12),\nX4 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 66 [arrowhead = none ];
66 -> 50;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 67 [arrowhead = none ];
67 -> 51;

68 [label="INSTANCE with matching:\nT1 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 68 [arrowhead = none , style=dashed];
68 -> 2[style=dashed];

}

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f2_in(s(T12)) → U1(f2_in(T12), s(T12))
U1(f2_out1, s(T12)) → f2_out1

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U1(x₁, x₂)) = x₁ + x₂   
POL(f2_in(x₁)) = 2 + 2·x₁   
POL(f2_out1) = 0   
POL(s(x₁)) = 1 + 2·x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f2_in(s(T12)) → U1(f2_in(T12), s(T12))
U1(f2_out1, s(T12)) → f2_out1

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) YES