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

(0) Obligation:

Clauses:

p(0).
p(s(X)) :- ','(q(X), ','(!, r)).
p(X) :- r.
q(0).
q(s(X)) :- ','(p(X), ','(!, r)).
q(X) :- r.
r.

Query: p(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | p(0), SCOPE: 1, CLAUSE: 1 | p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\np(T1) !=? p(0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(0), SCOPE: 1, CLAUSE: 1 | p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: r, SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(q(T3), ','(!_1, r)) | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX5 is free\np(T1) !=? p(0)\np(T1) !=? p(s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 3 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(!_1, r) | ','(q(0), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(0), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(0)), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: r, SCOPE: 4, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(','(p(T10), ','(!_3, r)), ','(!_1, r))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T10)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(!_3, r), ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(r, ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(r, ','(!_1, r)), SCOPE: 5, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(!_1, r)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: r, SCOPE: 6, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(r, ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(r, ','(!_1, r)), SCOPE: 7, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(!_1, r)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: r, SCOPE: 8, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: r, SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: r, SCOPE: 10, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 2;

52 [label="EVAL with clause\np(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="EVAL with clause\np(s(X5)) :- ','(q(X5), ','(!_1, r)).\nand substitutionX5 -> T3,\nT1 -> s(T3)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="BACKTRACK\nfor clause: p(s(X)) :- ','(q(X), ','(!, r))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 57 [arrowhead = none ];
57 -> 6;

58 [label="ONLY EVAL with clause\np(X3) :- r.\nand substitutionX3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="ONLY EVAL with clause\np(X20) :- r.\nand substitutionT1 -> T18,\nX20 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 63 [arrowhead = none ];
63 -> 47;

64 [label="EVAL with clause\nq(0).\nand substitutionT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 64 [arrowhead = none ];
64 -> 14;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 65 [arrowhead = none ];
65 -> 15;

66 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 66 [arrowhead = none ];
66 -> 16;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 67 [arrowhead = none ];
67 -> 20;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 68 [arrowhead = none ];
68 -> 21;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 69 [arrowhead = none ];
69 -> 17;

70 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 70 [arrowhead = none ];
70 -> 18;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 71 [arrowhead = none ];
71 -> 19;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 72 [arrowhead = none ];
72 -> 22;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 73 [arrowhead = none ];
73 -> 23;

74 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 74 [arrowhead = none ];
74 -> 42;

75 [label="EVAL with clause\nq(s(X12)) :- ','(p(X12), ','(!_3, r)).\nand substitutionX12 -> T10,\nT3 -> s(T10)", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="ONLY EVAL with clause\nq(X15) :- r.\nand substitutionT3 -> T13,\nX15 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 77 [arrowhead = none ];
77 -> 35;

78 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 78 [arrowhead = none ];
78 -> 26;

79 [label="SPLIT 2\nnew knowledge:\nT10 is ground", fontsize=14, style = filled, fillcolor = violet];
24 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="INSTANCE with matching:\nT1 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 80 [arrowhead = none , style=dashed];
80 -> 1[style=dashed];

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 83 [arrowhead = none ];
83 -> 30;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 84 [arrowhead = none ];
84 -> 31;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 85 [arrowhead = none ];
85 -> 32;

86 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 88 [arrowhead = none ];
88 -> 36;

89 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 89 [arrowhead = none ];
89 -> 37;

90 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 91 [arrowhead = none ];
91 -> 39;

92 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 92 [arrowhead = none ];
92 -> 40;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 93 [arrowhead = none ];
93 -> 41;

94 [label="ONLY EVAL with clause\np(X18) :- r.\nand substitutionT3 -> T16,\nX18 -> s(T16)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 98 [arrowhead = none ];
98 -> 48;

99 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 99 [arrowhead = none ];
99 -> 49;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 100 [arrowhead = none ];
100 -> 50;

}

(2) Obligation:

Clauses:

pA(0).
pA(0).
pA(s(0)).
pA(s(s(T10))) :- pA(T10).
pA(s(s(T10))) :- pA(T10).
pA(s(T13)).
pA(s(T16)).
pA(T18).

Query: pA(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
pA_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U1_g(T10, pA_out_g(T10)) → pA_out_g(s(s(T10)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
0  =  0
pA_out_g(x1)  =  pA_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U1_g(T10, pA_out_g(T10)) → pA_out_g(s(s(T10)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
0  =  0
pA_out_g(x1)  =  pA_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(T10))) → U1_G(T10, pA_in_g(T10))
PA_IN_G(s(s(T10))) → PA_IN_G(T10)

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U1_g(T10, pA_out_g(T10)) → pA_out_g(s(s(T10)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
0  =  0
pA_out_g(x1)  =  pA_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
PA_IN_G(x1)  =  PA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

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

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(T10))) → U1_G(T10, pA_in_g(T10))
PA_IN_G(s(s(T10))) → PA_IN_G(T10)

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U1_g(T10, pA_out_g(T10)) → pA_out_g(s(s(T10)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
0  =  0
pA_out_g(x1)  =  pA_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
PA_IN_G(x1)  =  PA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(T10))) → PA_IN_G(T10)

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U1_g(T10, pA_out_g(T10)) → pA_out_g(s(s(T10)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
0  =  0
pA_out_g(x1)  =  pA_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
PA_IN_G(x1)  =  PA_IN_G(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(T10))) → PA_IN_G(T10)

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

(11) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PA_IN_G(s(s(T10))) → PA_IN_G(T10)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(13) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• PA_IN_G(s(s(T10))) → PA_IN_G(T10)
  The graph contains the following edges 1 > 1

(14) YES