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

(0) Obligation:

Clauses:

num(0).
num(X) :- ','(no(zero(0)), ','(p(X, Y), num(Y))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: num(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: num(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: num(T1), SCOPE: 1, CLAUSE: 0 | num(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | num(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: num(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nnum(T1) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: num(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(zero(0)), ','(p(0, X4), num(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X4), num(X4))) | ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X4), num(X4)))) | ?_3 | ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X4), num(X4)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(','(!_2, failure(a)), ','(p(0, X4), num(X4))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X4), num(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(failure(a), ','(p(0, X4), num(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(failure(a), ','(p(0, X4), num(X4))), SCOPE: 5, CLAUSE: 7\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(no(zero(0)), ','(p(T3, X10), num(X10)))\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(','(call(zero(0)), ','(!_6, failure(a))), ','(p(T3, X10), num(X10))) | ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(zero(0), ','(','(!_6, failure(a)), ','(p(T3, X10), num(X10)))) | ?_7 | ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(zero(0), ','(','(!_6, failure(a)), ','(p(T3, X10), num(X10)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(','(!_6, failure(a)), ','(p(T3, X10), num(X10))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(T3, X10), num(X10))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(failure(a), ','(p(T3, X10), num(X10)))\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(failure(a), ','(p(T3, X10), num(X10))), SCOPE: 9, CLAUSE: 7\n\nT3 is ground\nX10 is free\nnum(T3) !=? num(0)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 60 [arrowhead = none ];
60 -> 6;

61 [label="EVAL with clause\nnum(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 62 [arrowhead = none ];
62 -> 12;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 63 [arrowhead = none ];
63 -> 17;

64 [label="ONLY EVAL with clause\nnum(X9) :- ','(no(zero(0)), ','(p(X9, X10), num(X10))).\nand substitutionT1 -> T3,\nX9 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 64 [arrowhead = none ];
64 -> 43;

65 [label="ONLY EVAL with clause\nnum(X3) :- ','(no(zero(0)), ','(p(X3, X4), num(X4))).\nand substitutionX3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 66 [arrowhead = none ];
66 -> 19;

67 [label="ONLY EVAL with clause\nno(X7) :- ','(call(X7), ','(!_2, failure(a))).\nand substitutionX7 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 68 [arrowhead = none ];
68 -> 23;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 69 [arrowhead = none ];
69 -> 27;

70 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 70 [arrowhead = none ];
70 -> 30;

71 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 71 [arrowhead = none ];
71 -> 31;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 72 [arrowhead = none ];
72 -> 35;

73 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 73 [arrowhead = none ];
73 -> 38;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 74 [arrowhead = none ];
74 -> 47;

75 [label="ONLY EVAL with clause\nno(X13) :- ','(call(X13), ','(!_6, failure(a))).\nand substitutionX13 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 75 [arrowhead = none ];
75 -> 49;

76 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 76 [arrowhead = none ];
76 -> 51;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 77 [arrowhead = none ];
77 -> 53;

78 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 78 [arrowhead = none ];
78 -> 55;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 79 [arrowhead = none ];
79 -> 56;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 80 [arrowhead = none ];
80 -> 57;

81 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 81 [arrowhead = none ];
81 -> 59;

}

(2) Obligation:

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

numA_in_g(0) → numA_out_g(0)

Pi is empty.

(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:

numA_in_g(0) → numA_out_g(0)

Pi is empty.
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:

numA_in_g(0) → numA_out_g(0)

Pi is empty.
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