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

(0) Obligation:

Clauses:

q(X) :- p(X, 0).
p(0, X1).
p(s(X), Y) :- p(X, s(Y)).

Query: q(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: p(T3, 0)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: p(T3, 0), SCOPE: 2, CLAUSE: 1 | p(T3, 0), SCOPE: 2, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(T3, 0), SCOPE: 2, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(T3, 0), SCOPE: 2, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T6, s(0))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(T6, s(0)), SCOPE: 3, CLAUSE: 1 | p(T6, s(0)), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(T6, s(0)), SCOPE: 3, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T6, s(0)), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T9, s(s(0)))\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T9, s(s(0))), SCOPE: 4, CLAUSE: 1 | p(T9, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(T9, s(s(0))), SCOPE: 4, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(T9, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T12, s(s(s(0))))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 1 | p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(T15, s(s(s(s(0)))))\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 1 | p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 1\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(T18, s(s(s(s(s(0))))))\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 1 | p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 1\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: p(T21, s(s(s(s(s(s(0)))))))\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 1 | p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: p(T24, s(s(s(s(s(s(s(0))))))))\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 1 | p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 1\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: p(T27, s(s(s(s(s(s(s(s(0)))))))))\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: p(T27, s(T28))\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: p(T27, s(T28)), SCOPE: 10, CLAUSE: 1 | p(T27, s(T28)), SCOPE: 10, CLAUSE: 2\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: p(T27, s(T28)), SCOPE: 10, CLAUSE: 1\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(T27, s(T28)), SCOPE: 10, CLAUSE: 2\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: p(T40, s(s(T41)))\n\nT40 is ground\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="ONLY EVAL with clause\nq(X3) :- p(X3, 0).\nand substitutionT1 -> T3,\nX3 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 81 [arrowhead = none ];
81 -> 6;

82 [label="EVAL with clause\np(0, X8).\nand substitutionT3 -> 0,\nX8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 7;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 83 [arrowhead = none ];
83 -> 8;

84 [label="EVAL with clause\np(s(X13), X14) :- p(X13, s(X14)).\nand substitutionX13 -> T6,\nT3 -> s(T6),\nX14 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 85 [arrowhead = none ];
85 -> 11;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 88 [arrowhead = none ];
88 -> 13;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="EVAL with clause\np(0, X21).\nand substitutionT6 -> 0,\nX21 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="EVAL with clause\np(s(X26), X27) :- p(X26, s(X27)).\nand substitutionX26 -> T9,\nT6 -> s(T9),\nX27 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 18;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 93 [arrowhead = none ];
93 -> 19;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 95 [arrowhead = none ];
95 -> 20;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 97 [arrowhead = none ];
97 -> 22;

98 [label="EVAL with clause\np(0, X34).\nand substitutionT9 -> 0,\nX34 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 98 [arrowhead = none ];
98 -> 23;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 99 [arrowhead = none ];
99 -> 24;

100 [label="EVAL with clause\np(s(X39), X40) :- p(X39, s(X40)).\nand substitutionX39 -> T12,\nT9 -> s(T12),\nX40 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 104 [arrowhead = none ];
104 -> 29;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 105 [arrowhead = none ];
105 -> 30;

106 [label="EVAL with clause\np(0, X47).\nand substitutionT12 -> 0,\nX47 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 31;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="EVAL with clause\np(s(X52), X53) :- p(X52, s(X53)).\nand substitutionX52 -> T15,\nT12 -> s(T15),\nX53 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 34;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 109 [arrowhead = none ];
109 -> 35;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 111 [arrowhead = none ];
111 -> 36;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 112 [arrowhead = none ];
112 -> 37;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="EVAL with clause\np(0, X60).\nand substitutionT15 -> 0,\nX60 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 114 [arrowhead = none ];
114 -> 39;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\np(s(X65), X66) :- p(X65, s(X66)).\nand substitutionX65 -> T18,\nT15 -> s(T18),\nX66 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 118 [arrowhead = none ];
118 -> 41;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="EVAL with clause\np(0, X73).\nand substitutionT18 -> 0,\nX73 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 122 [arrowhead = none ];
122 -> 47;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\np(s(X78), X79) :- p(X78, s(X79)).\nand substitutionX78 -> T21,\nT18 -> s(T21),\nX79 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 125 [arrowhead = none ];
125 -> 51;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 126 [arrowhead = none ];
126 -> 49;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 127 [arrowhead = none ];
127 -> 52;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 128 [arrowhead = none ];
128 -> 53;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 129 [arrowhead = none ];
129 -> 54;

130 [label="EVAL with clause\np(0, X86).\nand substitutionT21 -> 0,\nX86 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL with clause\np(s(X91), X92) :- p(X91, s(X92)).\nand substitutionX91 -> T24,\nT21 -> s(T24),\nX92 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 132 [arrowhead = none ];
132 -> 58;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 133 [arrowhead = none ];
133 -> 59;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 134 [arrowhead = none ];
134 -> 57;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 135 [arrowhead = none ];
135 -> 60;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 136 [arrowhead = none ];
136 -> 61;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 137 [arrowhead = none ];
137 -> 62;

138 [label="EVAL with clause\np(0, X99).\nand substitutionT24 -> 0,\nX99 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="EVAL with clause\np(s(X104), X105) :- p(X104, s(X105)).\nand substitutionX104 -> T27,\nT24 -> s(T27),\nX105 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 141 [arrowhead = none ];
141 -> 67;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 142 [arrowhead = none ];
142 -> 65;

143 [label="GENERALIZATION\nT28 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT28 is ground", fontsize=14, style = filled, fillcolor = violet];
66 -> 143 [arrowhead = none ];
143 -> 68;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 144 [arrowhead = none ];
144 -> 69;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 145 [arrowhead = none ];
145 -> 70;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 146 [arrowhead = none ];
146 -> 71;

147 [label="EVAL with clause\np(0, X114).\nand substitutionT27 -> 0,\nT28 -> T35,\nX114 -> s(T35)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 147 [arrowhead = none ];
147 -> 72;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 148 [arrowhead = none ];
148 -> 73;

149 [label="EVAL with clause\np(s(X119), X120) :- p(X119, s(X120)).\nand substitutionX119 -> T40,\nT27 -> s(T40),\nT28 -> T41,\nX120 -> s(T41)", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 149 [arrowhead = none ];
149 -> 75;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 150 [arrowhead = none ];
150 -> 76;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 151 [arrowhead = none ];
151 -> 74;

152 [label="INSTANCE with matching:\nT27 -> T40\nT28 -> s(T41)", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 152 [arrowhead = none , style=dashed];
152 -> 68[style=dashed];

}

(2) Obligation:

Clauses:

pA(0, T35).
pA(s(T40), T41) :- pA(T40, s(T41)).
qB(0).
qB(s(0)).
qB(s(s(0))).
qB(s(s(s(0)))).
qB(s(s(s(s(0))))).
qB(s(s(s(s(s(0)))))).
qB(s(s(s(s(s(s(0))))))).
qB(s(s(s(s(s(s(s(0)))))))).
qB(s(s(s(s(s(s(s(s(T27))))))))) :- pA(T27, s(s(s(s(s(s(s(0)))))))).

Query: qB(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:
qB_in: (b)
pA_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qB_in_g(0) → qB_out_g(0)
qB_in_g(s(0)) → qB_out_g(s(0))
qB_in_g(s(s(0))) → qB_out_g(s(s(0)))
qB_in_g(s(s(s(0)))) → qB_out_g(s(s(s(0))))
qB_in_g(s(s(s(s(0))))) → qB_out_g(s(s(s(s(0)))))
qB_in_g(s(s(s(s(s(0)))))) → qB_out_g(s(s(s(s(s(0))))))
qB_in_g(s(s(s(s(s(s(0))))))) → qB_out_g(s(s(s(s(s(s(0)))))))
qB_in_g(s(s(s(s(s(s(s(0)))))))) → qB_out_g(s(s(s(s(s(s(s(0))))))))
qB_in_g(s(s(s(s(s(s(s(s(T27))))))))) → U2_g(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
pA_in_gg(0, T35) → pA_out_gg(0, T35)
pA_in_gg(s(T40), T41) → U1_gg(T40, T41, pA_in_gg(T40, s(T41)))
U1_gg(T40, T41, pA_out_gg(T40, s(T41))) → pA_out_gg(s(T40), T41)
U2_g(T27, pA_out_gg(T27, s(s(s(s(s(s(s(0))))))))) → qB_out_g(s(s(s(s(s(s(s(s(T27)))))))))

The argument filtering Pi contains the following mapping:
qB_in_g(x1)  =  qB_in_g(x1)
0  =  0
qB_out_g(x1)  =  qB_out_g
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x2)
pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
pA_out_gg(x1, x2)  =  pA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

qB_in_g(0) → qB_out_g(0)
qB_in_g(s(0)) → qB_out_g(s(0))
qB_in_g(s(s(0))) → qB_out_g(s(s(0)))
qB_in_g(s(s(s(0)))) → qB_out_g(s(s(s(0))))
qB_in_g(s(s(s(s(0))))) → qB_out_g(s(s(s(s(0)))))
qB_in_g(s(s(s(s(s(0)))))) → qB_out_g(s(s(s(s(s(0))))))
qB_in_g(s(s(s(s(s(s(0))))))) → qB_out_g(s(s(s(s(s(s(0)))))))
qB_in_g(s(s(s(s(s(s(s(0)))))))) → qB_out_g(s(s(s(s(s(s(s(0))))))))
qB_in_g(s(s(s(s(s(s(s(s(T27))))))))) → U2_g(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
pA_in_gg(0, T35) → pA_out_gg(0, T35)
pA_in_gg(s(T40), T41) → U1_gg(T40, T41, pA_in_gg(T40, s(T41)))
U1_gg(T40, T41, pA_out_gg(T40, s(T41))) → pA_out_gg(s(T40), T41)
U2_g(T27, pA_out_gg(T27, s(s(s(s(s(s(s(0))))))))) → qB_out_g(s(s(s(s(s(s(s(s(T27)))))))))

The argument filtering Pi contains the following mapping:
qB_in_g(x1)  =  qB_in_g(x1)
0  =  0
qB_out_g(x1)  =  qB_out_g
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x2)
pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
pA_out_gg(x1, x2)  =  pA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)

(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:

QB_IN_G(s(s(s(s(s(s(s(s(T27))))))))) → U2_G(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
QB_IN_G(s(s(s(s(s(s(s(s(T27))))))))) → PA_IN_GG(T27, s(s(s(s(s(s(s(0))))))))
PA_IN_GG(s(T40), T41) → U1_GG(T40, T41, pA_in_gg(T40, s(T41)))
PA_IN_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))

The TRS R consists of the following rules:

qB_in_g(0) → qB_out_g(0)
qB_in_g(s(0)) → qB_out_g(s(0))
qB_in_g(s(s(0))) → qB_out_g(s(s(0)))
qB_in_g(s(s(s(0)))) → qB_out_g(s(s(s(0))))
qB_in_g(s(s(s(s(0))))) → qB_out_g(s(s(s(s(0)))))
qB_in_g(s(s(s(s(s(0)))))) → qB_out_g(s(s(s(s(s(0))))))
qB_in_g(s(s(s(s(s(s(0))))))) → qB_out_g(s(s(s(s(s(s(0)))))))
qB_in_g(s(s(s(s(s(s(s(0)))))))) → qB_out_g(s(s(s(s(s(s(s(0))))))))
qB_in_g(s(s(s(s(s(s(s(s(T27))))))))) → U2_g(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
pA_in_gg(0, T35) → pA_out_gg(0, T35)
pA_in_gg(s(T40), T41) → U1_gg(T40, T41, pA_in_gg(T40, s(T41)))
U1_gg(T40, T41, pA_out_gg(T40, s(T41))) → pA_out_gg(s(T40), T41)
U2_g(T27, pA_out_gg(T27, s(s(s(s(s(s(s(0))))))))) → qB_out_g(s(s(s(s(s(s(s(s(T27)))))))))

The argument filtering Pi contains the following mapping:
qB_in_g(x1)  =  qB_in_g(x1)
0  =  0
qB_out_g(x1)  =  qB_out_g
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x2)
pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
pA_out_gg(x1, x2)  =  pA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
QB_IN_G(x1)  =  QB_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)

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

(6) Obligation:

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

QB_IN_G(s(s(s(s(s(s(s(s(T27))))))))) → U2_G(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
QB_IN_G(s(s(s(s(s(s(s(s(T27))))))))) → PA_IN_GG(T27, s(s(s(s(s(s(s(0))))))))
PA_IN_GG(s(T40), T41) → U1_GG(T40, T41, pA_in_gg(T40, s(T41)))
PA_IN_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))

The TRS R consists of the following rules:

qB_in_g(0) → qB_out_g(0)
qB_in_g(s(0)) → qB_out_g(s(0))
qB_in_g(s(s(0))) → qB_out_g(s(s(0)))
qB_in_g(s(s(s(0)))) → qB_out_g(s(s(s(0))))
qB_in_g(s(s(s(s(0))))) → qB_out_g(s(s(s(s(0)))))
qB_in_g(s(s(s(s(s(0)))))) → qB_out_g(s(s(s(s(s(0))))))
qB_in_g(s(s(s(s(s(s(0))))))) → qB_out_g(s(s(s(s(s(s(0)))))))
qB_in_g(s(s(s(s(s(s(s(0)))))))) → qB_out_g(s(s(s(s(s(s(s(0))))))))
qB_in_g(s(s(s(s(s(s(s(s(T27))))))))) → U2_g(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
pA_in_gg(0, T35) → pA_out_gg(0, T35)
pA_in_gg(s(T40), T41) → U1_gg(T40, T41, pA_in_gg(T40, s(T41)))
U1_gg(T40, T41, pA_out_gg(T40, s(T41))) → pA_out_gg(s(T40), T41)
U2_g(T27, pA_out_gg(T27, s(s(s(s(s(s(s(0))))))))) → qB_out_g(s(s(s(s(s(s(s(s(T27)))))))))

The argument filtering Pi contains the following mapping:
qB_in_g(x1)  =  qB_in_g(x1)
0  =  0
qB_out_g(x1)  =  qB_out_g
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x2)
pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
pA_out_gg(x1, x2)  =  pA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
QB_IN_G(x1)  =  QB_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)
PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

PA_IN_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))

The TRS R consists of the following rules:

qB_in_g(0) → qB_out_g(0)
qB_in_g(s(0)) → qB_out_g(s(0))
qB_in_g(s(s(0))) → qB_out_g(s(s(0)))
qB_in_g(s(s(s(0)))) → qB_out_g(s(s(s(0))))
qB_in_g(s(s(s(s(0))))) → qB_out_g(s(s(s(s(0)))))
qB_in_g(s(s(s(s(s(0)))))) → qB_out_g(s(s(s(s(s(0))))))
qB_in_g(s(s(s(s(s(s(0))))))) → qB_out_g(s(s(s(s(s(s(0)))))))
qB_in_g(s(s(s(s(s(s(s(0)))))))) → qB_out_g(s(s(s(s(s(s(s(0))))))))
qB_in_g(s(s(s(s(s(s(s(s(T27))))))))) → U2_g(T27, pA_in_gg(T27, s(s(s(s(s(s(s(0)))))))))
pA_in_gg(0, T35) → pA_out_gg(0, T35)
pA_in_gg(s(T40), T41) → U1_gg(T40, T41, pA_in_gg(T40, s(T41)))
U1_gg(T40, T41, pA_out_gg(T40, s(T41))) → pA_out_gg(s(T40), T41)
U2_g(T27, pA_out_gg(T27, s(s(s(s(s(s(s(0))))))))) → qB_out_g(s(s(s(s(s(s(s(s(T27)))))))))

The argument filtering Pi contains the following mapping:
qB_in_g(x1)  =  qB_in_g(x1)
0  =  0
qB_out_g(x1)  =  qB_out_g
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x2)
pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
pA_out_gg(x1, x2)  =  pA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)

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_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))

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_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))

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_GG(s(T40), T41) → PA_IN_GG(T40, s(T41))
  The graph contains the following edges 1 > 1

(14) YES