Runtime Complexity of the query pattern
q(g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

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

Query: q(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

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

Query: q(g)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

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: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(T4, 0)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T4, 0), SCOPE: 2, CLAUSE: 0 | p(T4, 0), SCOPE: 2, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | p(0, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T4, 0), SCOPE: 2, CLAUSE: 2\n\nT4 is ground\nX11 is free\np(T4, 0) !=? p(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(0, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: p(T7, s(0))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T7, s(0)), SCOPE: 3, CLAUSE: 0 | p(T7, s(0)), SCOPE: 3, CLAUSE: 2\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true | p(0, s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T7, s(0)), SCOPE: 3, CLAUSE: 2\n\nT7 is ground\nX24 is free\np(T7, s(0)) !=? p(0, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(0, s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(T10, s(s(0)))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: p(T10, s(s(0))), SCOPE: 4, CLAUSE: 0 | p(T10, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true | p(0, s(s(0))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(T10, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT10 is ground\nX37 is free\np(T10, s(s(0))) !=? p(0, X37)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(0, s(s(0))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: p(T13, s(s(s(0))))\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: p(T13, s(s(s(0)))), SCOPE: 5, CLAUSE: 0 | p(T13, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | p(0, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: p(T13, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT13 is ground\nX50 is free\np(T13, s(s(s(0)))) !=? p(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: p(0, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: p(T16, s(s(s(s(0)))))\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: p(T16, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 0 | p(T16, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true | p(0, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: p(T16, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT16 is ground\nX63 is free\np(T16, s(s(s(s(0))))) !=? p(0, X63)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: p(0, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: p(T19, s(s(s(s(s(0))))))\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: p(T19, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 0 | p(T19, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true | p(0, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: p(T19, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT19 is ground\nX76 is free\np(T19, s(s(s(s(s(0)))))) !=? p(0, X76)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: p(0, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: p(T22, s(s(s(s(s(s(0)))))))\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: p(T22, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 0 | p(T22, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true | p(0, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: p(T22, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT22 is ground\nX89 is free\np(T22, s(s(s(s(s(s(0))))))) !=? p(0, X89)", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: p(0, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: p(T25, s(s(s(s(s(s(s(0))))))))\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: p(T25, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 0 | p(T25, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: true | p(0, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: p(T25, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT25 is ground\nX102 is free\np(T25, s(s(s(s(s(s(s(0)))))))) !=? p(0, X102)", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: p(0, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: p(T28, s(s(s(s(s(s(s(s(0)))))))))\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: p(T28, s(T29))\n\nT28 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: p(T28, s(T29)), SCOPE: 10, CLAUSE: 0 | p(T28, s(T29)), SCOPE: 10, CLAUSE: 2\n\nT28 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: true | p(0, s(T34)), SCOPE: 10, CLAUSE: 2\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: p(T28, s(T29)), SCOPE: 10, CLAUSE: 2\n\nT28 is ground\nT29 is ground\nX117 is free\np(T28, s(T29)) !=? p(0, X117)", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: p(0, s(T34)), SCOPE: 10, CLAUSE: 2\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: p(T39, s(s(T40)))\n\nT39 is ground\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 128 [arrowhead = none ];
128 -> 4;

129 [label="ONLY EVAL with clause\nq(X6) :- p(X6, 0).\nand substitutionT1 -> T4,\nX6 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 129 [arrowhead = none ];
129 -> 6;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 130 [arrowhead = none ];
130 -> 14;

131 [label="EVAL with clause\np(0, X11).\nand substitutionT4 -> 0,\nX11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 131 [arrowhead = none ];
131 -> 15;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 132 [arrowhead = none ];
132 -> 18;

133 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 133 [arrowhead = none ];
133 -> 19;

134 [label="EVAL with clause\np(s(X18), X19) :- p(X18, s(X19)).\nand substitutionX18 -> T7,\nT4 -> s(T7),\nX19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 134 [arrowhead = none ];
134 -> 24;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 135 [arrowhead = none ];
135 -> 25;

136 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 136 [arrowhead = none ];
136 -> 22;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 137 [arrowhead = none ];
137 -> 28;

138 [label="EVAL with clause\np(0, X24).\nand substitutionT7 -> 0,\nX24 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 138 [arrowhead = none ];
138 -> 29;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 139 [arrowhead = none ];
139 -> 31;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 140 [arrowhead = none ];
140 -> 32;

141 [label="EVAL with clause\np(s(X31), X32) :- p(X31, s(X32)).\nand substitutionX31 -> T10,\nT7 -> s(T10),\nX32 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 141 [arrowhead = none ];
141 -> 38;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 142 [arrowhead = none ];
142 -> 39;

143 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 143 [arrowhead = none ];
143 -> 33;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 144 [arrowhead = none ];
144 -> 41;

145 [label="EVAL with clause\np(0, X37).\nand substitutionT10 -> 0,\nX37 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 145 [arrowhead = none ];
145 -> 43;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 146 [arrowhead = none ];
146 -> 44;

147 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 147 [arrowhead = none ];
147 -> 45;

148 [label="EVAL with clause\np(s(X44), X45) :- p(X44, s(X45)).\nand substitutionX44 -> T13,\nT10 -> s(T13),\nX45 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 148 [arrowhead = none ];
148 -> 49;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 149 [arrowhead = none ];
149 -> 50;

150 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 150 [arrowhead = none ];
150 -> 46;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 151 [arrowhead = none ];
151 -> 54;

152 [label="EVAL with clause\np(0, X50).\nand substitutionT13 -> 0,\nX50 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 152 [arrowhead = none ];
152 -> 55;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 153 [arrowhead = none ];
153 -> 56;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 154 [arrowhead = none ];
154 -> 57;

155 [label="EVAL with clause\np(s(X57), X58) :- p(X57, s(X58)).\nand substitutionX57 -> T16,\nT13 -> s(T16),\nX58 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 155 [arrowhead = none ];
155 -> 59;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 156 [arrowhead = none ];
156 -> 60;

157 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 157 [arrowhead = none ];
157 -> 58;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 158 [arrowhead = none ];
158 -> 61;

159 [label="EVAL with clause\np(0, X63).\nand substitutionT16 -> 0,\nX63 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 159 [arrowhead = none ];
159 -> 62;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 160 [arrowhead = none ];
160 -> 64;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 161 [arrowhead = none ];
161 -> 65;

162 [label="EVAL with clause\np(s(X70), X71) :- p(X70, s(X71)).\nand substitutionX70 -> T19,\nT16 -> s(T19),\nX71 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 162 [arrowhead = none ];
162 -> 69;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 163 [arrowhead = none ];
163 -> 70;

164 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 164 [arrowhead = none ];
164 -> 67;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 165 [arrowhead = none ];
165 -> 75;

166 [label="EVAL with clause\np(0, X76).\nand substitutionT19 -> 0,\nX76 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 166 [arrowhead = none ];
166 -> 77;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 167 [arrowhead = none ];
167 -> 79;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="EVAL with clause\np(s(X83), X84) :- p(X83, s(X84)).\nand substitutionX83 -> T22,\nT19 -> s(T22),\nX84 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 169 [arrowhead = none ];
169 -> 83;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 170 [arrowhead = none ];
170 -> 85;

171 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 171 [arrowhead = none ];
171 -> 81;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 172 [arrowhead = none ];
172 -> 89;

173 [label="EVAL with clause\np(0, X89).\nand substitutionT22 -> 0,\nX89 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 173 [arrowhead = none ];
173 -> 90;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 174 [arrowhead = none ];
174 -> 92;

175 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 175 [arrowhead = none ];
175 -> 93;

176 [label="EVAL with clause\np(s(X96), X97) :- p(X96, s(X97)).\nand substitutionX96 -> T25,\nT22 -> s(T25),\nX97 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 176 [arrowhead = none ];
176 -> 96;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 177 [arrowhead = none ];
177 -> 98;

178 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 178 [arrowhead = none ];
178 -> 95;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 179 [arrowhead = none ];
179 -> 103;

180 [label="EVAL with clause\np(0, X102).\nand substitutionT25 -> 0,\nX102 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 180 [arrowhead = none ];
180 -> 104;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 181 [arrowhead = none ];
181 -> 106;

182 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 182 [arrowhead = none ];
182 -> 108;

183 [label="EVAL with clause\np(s(X109), X110) :- p(X109, s(X110)).\nand substitutionX109 -> T28,\nT25 -> s(T28),\nX110 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 183 [arrowhead = none ];
183 -> 110;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 184 [arrowhead = none ];
184 -> 112;

185 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 185 [arrowhead = none ];
185 -> 109;

186 [label="GENERALIZATION\nT29 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT29 is ground", fontsize=14, style = filled, fillcolor = violet];
110 -> 186 [arrowhead = none ];
186 -> 117;

187 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
117 -> 187 [arrowhead = none ];
187 -> 120;

188 [label="EVAL with clause\np(0, X117).\nand substitutionT28 -> 0,\nT29 -> T34,\nX117 -> s(T34)", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 188 [arrowhead = none ];
188 -> 121;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 189 [arrowhead = none ];
189 -> 122;

190 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 190 [arrowhead = none ];
190 -> 123;

191 [label="EVAL with clause\np(s(X124), X125) :- p(X124, s(X125)).\nand substitutionX124 -> T39,\nT28 -> s(T39),\nT29 -> T40,\nX125 -> s(T40)", fontsize=14, style = filled, fillcolor = lightblue];
122 -> 191 [arrowhead = none ];
191 -> 126;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 192 [arrowhead = none ];
192 -> 127;

193 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 193 [arrowhead = none ];
193 -> 124;

194 [label="INSTANCE with matching:\nT28 -> T39\nT29 -> s(T40)", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 194 [arrowhead = none , style=dashed];
194 -> 117[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(0)) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(s(0)))) → f2_out1
f2_in(s(s(s(s(0))))) → f2_out1
f2_in(s(s(s(s(s(0)))))) → f2_out1
f2_in(s(s(s(s(s(s(0))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(0)))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f117_out1, s(s(s(s(s(s(s(s(z0))))))))) → f2_out1
f117_in(0, z0) → f117_out1
f117_in(s(z0), z1) → U2(f117_in(z0, s(z1)), s(z0), z1)
U2(f117_out1, s(z0), z1) → f117_out1

Tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c8(U1'(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))), F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(U2'(f117_in(z0, s(z1)), s(z0), z1), F117_IN(z0, s(z1)))

S tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c8(U1'(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))), F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(U2'(f117_in(z0, s(z1)), s(z0), z1), F117_IN(z0, s(z1)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f117_in, U2

Defined Pair Symbols:

F2_IN, F117_IN

Compound Symbols:

c8, c11

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(0)) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(s(0)))) → f2_out1
f2_in(s(s(s(s(0))))) → f2_out1
f2_in(s(s(s(s(s(0)))))) → f2_out1
f2_in(s(s(s(s(s(s(0))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(0)))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f117_out1, s(s(s(s(s(s(s(s(z0))))))))) → f2_out1
f117_in(0, z0) → f117_out1
f117_in(s(z0), z1) → U2(f117_in(z0, s(z1)), s(z0), z1)
U2(f117_out1, s(z0), z1) → f117_out1

Tuples:

F117_IN(s(z0), z1) → c11(U2'(f117_in(z0, s(z1)), s(z0), z1), F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(U1'(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))

S tuples:

F117_IN(s(z0), z1) → c11(U2'(f117_in(z0, s(z1)), s(z0), z1), F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(U1'(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f117_in, U2

Defined Pair Symbols:

F117_IN, F2_IN

Compound Symbols:

c11, c

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(0)) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(s(0)))) → f2_out1
f2_in(s(s(s(s(0))))) → f2_out1
f2_in(s(s(s(s(s(0)))))) → f2_out1
f2_in(s(s(s(s(s(s(0))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(0)))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f117_out1, s(s(s(s(s(s(s(s(z0))))))))) → f2_out1
f117_in(0, z0) → f117_out1
f117_in(s(z0), z1) → U2(f117_in(z0, s(z1)), s(z0), z1)
U2(f117_out1, s(z0), z1) → f117_out1

Tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f117_in, U2

Defined Pair Symbols:

F2_IN, F117_IN

Compound Symbols:

c, c11, c

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(0)) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(s(0)))) → f2_out1
f2_in(s(s(s(s(0))))) → f2_out1
f2_in(s(s(s(s(s(0)))))) → f2_out1
f2_in(s(s(s(s(s(s(0))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(0)))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f117_out1, s(s(s(s(s(s(s(s(z0))))))))) → f2_out1
f117_in(0, z0) → f117_out1
f117_in(s(z0), z1) → U2(f117_in(z0, s(z1)), s(z0), z1)
U2(f117_out1, s(z0), z1) → f117_out1

Tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:

F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))

K tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

Defined Rule Symbols:

f2_in, U1, f117_in, U2

Defined Pair Symbols:

F2_IN, F117_IN

Compound Symbols:

c, c11, c

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(F117_IN(x₁, x₂)) = [3]x₁ + [2]x₂   
POL(F2_IN(x₁)) = [3] + [3]x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(s(x₁)) = [3] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(0)) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(s(0)))) → f2_out1
f2_in(s(s(s(s(0))))) → f2_out1
f2_in(s(s(s(s(s(0)))))) → f2_out1
f2_in(s(s(s(s(s(s(0))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(0)))))))) → f2_out1
f2_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f117_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f117_out1, s(s(s(s(s(s(s(s(z0))))))))) → f2_out1
f117_in(0, z0) → f117_out1
f117_in(s(z0), z1) → U2(f117_in(z0, s(z1)), s(z0), z1)
U2(f117_out1, s(z0), z1) → f117_out1

Tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:none
K tuples:

F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F117_IN(z0, s(s(s(s(s(s(s(0)))))))))
F2_IN(s(s(s(s(s(s(s(s(z0))))))))) → c
F117_IN(s(z0), z1) → c11(F117_IN(z0, s(z1)))

Defined Rule Symbols:

f2_in, U1, f117_in, U2

Defined Pair Symbols:

F2_IN, F117_IN

Compound Symbols:

c, c11, c

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))

(15) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

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];
3 [label="3: q(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(T3, 0)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T3, 0), SCOPE: 2, CLAUSE: 0 | p(T3, 0), SCOPE: 2, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | p(0, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T3, 0), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX6 is free\np(T3, 0) !=? p(0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(0, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(T6, s(0))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(T6, s(0)), SCOPE: 3, CLAUSE: 0 | p(T6, s(0)), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true | p(0, s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T6, s(0)), SCOPE: 3, CLAUSE: 2\n\nT6 is ground\nX19 is free\np(T6, s(0)) !=? p(0, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(0, s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T9, s(s(0)))\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: p(T9, s(s(0))), SCOPE: 4, CLAUSE: 0 | p(T9, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true | p(0, s(s(0))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: p(T9, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nT9 is ground\nX32 is free\np(T9, s(s(0))) !=? p(0, X32)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(0, s(s(0))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: p(T12, s(s(s(0))))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 0 | p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true | p(0, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: p(T12, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\nT12 is ground\nX45 is free\np(T12, s(s(s(0)))) !=? p(0, X45)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: p(0, s(s(s(0)))), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: p(T15, s(s(s(s(0)))))\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 0 | p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true | p(0, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: p(T15, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\nT15 is ground\nX58 is free\np(T15, s(s(s(s(0))))) !=? p(0, X58)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: p(0, s(s(s(s(0))))), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: p(T18, s(s(s(s(s(0))))))\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 0 | p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true | p(0, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: p(T18, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\nT18 is ground\nX71 is free\np(T18, s(s(s(s(s(0)))))) !=? p(0, X71)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: p(0, s(s(s(s(s(0)))))), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: p(T21, s(s(s(s(s(s(0)))))))\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 0 | p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: true | p(0, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: p(T21, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\nT21 is ground\nX84 is free\np(T21, s(s(s(s(s(s(0))))))) !=? p(0, X84)", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: p(0, s(s(s(s(s(s(0))))))), SCOPE: 8, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: p(T24, s(s(s(s(s(s(s(0))))))))\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 0 | p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: true | p(0, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: p(T24, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\nT24 is ground\nX97 is free\np(T24, s(s(s(s(s(s(s(0)))))))) !=? p(0, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: p(0, s(s(s(s(s(s(s(0)))))))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: p(T27, s(s(s(s(s(s(s(s(0)))))))))\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: p(T27, s(T28))\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: p(T27, s(T28)), SCOPE: 10, CLAUSE: 0 | p(T27, s(T28)), SCOPE: 10, CLAUSE: 2\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: true | p(0, s(T33)), SCOPE: 10, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: p(T27, s(T28)), SCOPE: 10, CLAUSE: 2\n\nT27 is ground\nT28 is ground\nX112 is free\np(T27, s(T28)) !=? p(0, X112)", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: p(0, s(T33)), SCOPE: 10, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: p(T38, s(s(T39)))\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 135 [arrowhead = none ];
135 -> 3;

136 [label="ONLY EVAL with clause\nq(X3) :- p(X3, 0).\nand substitutionT1 -> T3,\nX3 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 136 [arrowhead = none ];
136 -> 5;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 137 [arrowhead = none ];
137 -> 7;

138 [label="EVAL with clause\np(0, X6).\nand substitutionT3 -> 0,\nX6 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 138 [arrowhead = none ];
138 -> 8;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 139 [arrowhead = none ];
139 -> 9;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 140 [arrowhead = none ];
140 -> 10;

141 [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];
9 -> 141 [arrowhead = none ];
141 -> 12;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 142 [arrowhead = none ];
142 -> 13;

143 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 143 [arrowhead = none ];
143 -> 11;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 144 [arrowhead = none ];
144 -> 16;

145 [label="EVAL with clause\np(0, X19).\nand substitutionT6 -> 0,\nX19 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 145 [arrowhead = none ];
145 -> 17;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 146 [arrowhead = none ];
146 -> 20;

147 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 147 [arrowhead = none ];
147 -> 21;

148 [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];
20 -> 148 [arrowhead = none ];
148 -> 26;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 149 [arrowhead = none ];
149 -> 27;

150 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 150 [arrowhead = none ];
150 -> 23;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 151 [arrowhead = none ];
151 -> 30;

152 [label="EVAL with clause\np(0, X32).\nand substitutionT9 -> 0,\nX32 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 152 [arrowhead = none ];
152 -> 34;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 153 [arrowhead = none ];
153 -> 35;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 154 [arrowhead = none ];
154 -> 36;

155 [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];
35 -> 155 [arrowhead = none ];
155 -> 40;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 156 [arrowhead = none ];
156 -> 42;

157 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 157 [arrowhead = none ];
157 -> 37;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 158 [arrowhead = none ];
158 -> 47;

159 [label="EVAL with clause\np(0, X45).\nand substitutionT12 -> 0,\nX45 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 159 [arrowhead = none ];
159 -> 48;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 160 [arrowhead = none ];
160 -> 51;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 161 [arrowhead = none ];
161 -> 52;

162 [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];
51 -> 162 [arrowhead = none ];
162 -> 63;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 163 [arrowhead = none ];
163 -> 66;

164 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 164 [arrowhead = none ];
164 -> 53;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 165 [arrowhead = none ];
165 -> 68;

166 [label="EVAL with clause\np(0, X58).\nand substitutionT15 -> 0,\nX58 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 166 [arrowhead = none ];
166 -> 71;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 167 [arrowhead = none ];
167 -> 72;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 168 [arrowhead = none ];
168 -> 73;

169 [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];
72 -> 169 [arrowhead = none ];
169 -> 76;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 170 [arrowhead = none ];
170 -> 78;

171 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 171 [arrowhead = none ];
171 -> 74;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 172 [arrowhead = none ];
172 -> 82;

173 [label="EVAL with clause\np(0, X71).\nand substitutionT18 -> 0,\nX71 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 173 [arrowhead = none ];
173 -> 84;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 174 [arrowhead = none ];
174 -> 86;

175 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 175 [arrowhead = none ];
175 -> 87;

176 [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];
86 -> 176 [arrowhead = none ];
176 -> 91;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 177 [arrowhead = none ];
177 -> 94;

178 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 178 [arrowhead = none ];
178 -> 88;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 179 [arrowhead = none ];
179 -> 97;

180 [label="EVAL with clause\np(0, X84).\nand substitutionT21 -> 0,\nX84 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 180 [arrowhead = none ];
180 -> 99;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 181 [arrowhead = none ];
181 -> 100;

182 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 182 [arrowhead = none ];
182 -> 101;

183 [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];
100 -> 183 [arrowhead = none ];
183 -> 105;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 184 [arrowhead = none ];
184 -> 107;

185 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 185 [arrowhead = none ];
185 -> 102;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 186 [arrowhead = none ];
186 -> 111;

187 [label="EVAL with clause\np(0, X97).\nand substitutionT24 -> 0,\nX97 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
111 -> 187 [arrowhead = none ];
187 -> 113;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 188 [arrowhead = none ];
188 -> 114;

189 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 189 [arrowhead = none ];
189 -> 115;

190 [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];
114 -> 190 [arrowhead = none ];
190 -> 118;

191 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 191 [arrowhead = none ];
191 -> 119;

192 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 192 [arrowhead = none ];
192 -> 116;

193 [label="GENERALIZATION\nT28 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT28 is ground", fontsize=14, style = filled, fillcolor = violet];
118 -> 193 [arrowhead = none ];
193 -> 125;

194 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 194 [arrowhead = none ];
194 -> 128;

195 [label="EVAL with clause\np(0, X112).\nand substitutionT27 -> 0,\nT28 -> T33,\nX112 -> s(T33)", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 195 [arrowhead = none ];
195 -> 129;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 196 [arrowhead = none ];
196 -> 130;

197 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 197 [arrowhead = none ];
197 -> 131;

198 [label="EVAL with clause\np(s(X119), X120) :- p(X119, s(X120)).\nand substitutionX119 -> T38,\nT27 -> s(T38),\nT28 -> T39,\nX120 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
130 -> 198 [arrowhead = none ];
198 -> 133;

199 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 199 [arrowhead = none ];
199 -> 134;

200 [label="BACKTRACK\nfor clause: p(s(X), Y) :- p(X, s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
131 -> 200 [arrowhead = none ];
200 -> 132;

201 [label="INSTANCE with matching:\nT27 -> T38\nT28 -> s(T39)", fontsize=14, style = filled, fillcolor = lightgrey];
133 -> 201 [arrowhead = none , style=dashed];
201 -> 125[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(s(0)))) → f1_out1
f1_in(s(s(s(s(0))))) → f1_out1
f1_in(s(s(s(s(s(0)))))) → f1_out1
f1_in(s(s(s(s(s(s(0))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(0)))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f125_out1, s(s(s(s(s(s(s(s(z0))))))))) → f1_out1
f125_in(0, z0) → f125_out1
f125_in(s(z0), z1) → U2(f125_in(z0, s(z1)), s(z0), z1)
U2(f125_out1, s(z0), z1) → f125_out1

Tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c8(U1'(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))), F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(U2'(f125_in(z0, s(z1)), s(z0), z1), F125_IN(z0, s(z1)))

S tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c8(U1'(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))), F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(U2'(f125_in(z0, s(z1)), s(z0), z1), F125_IN(z0, s(z1)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f125_in, U2

Defined Pair Symbols:

F1_IN, F125_IN

Compound Symbols:

c8, c11

(17) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(s(0)))) → f1_out1
f1_in(s(s(s(s(0))))) → f1_out1
f1_in(s(s(s(s(s(0)))))) → f1_out1
f1_in(s(s(s(s(s(s(0))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(0)))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f125_out1, s(s(s(s(s(s(s(s(z0))))))))) → f1_out1
f125_in(0, z0) → f125_out1
f125_in(s(z0), z1) → U2(f125_in(z0, s(z1)), s(z0), z1)
U2(f125_out1, s(z0), z1) → f125_out1

Tuples:

F125_IN(s(z0), z1) → c11(U2'(f125_in(z0, s(z1)), s(z0), z1), F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(U1'(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))

S tuples:

F125_IN(s(z0), z1) → c11(U2'(f125_in(z0, s(z1)), s(z0), z1), F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(U1'(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0))))))))))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f125_in, U2

Defined Pair Symbols:

F125_IN, F1_IN

Compound Symbols:

c11, c

(19) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(s(0)))) → f1_out1
f1_in(s(s(s(s(0))))) → f1_out1
f1_in(s(s(s(s(s(0)))))) → f1_out1
f1_in(s(s(s(s(s(s(0))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(0)))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f125_out1, s(s(s(s(s(s(s(s(z0))))))))) → f1_out1
f125_in(0, z0) → f125_out1
f125_in(s(z0), z1) → U2(f125_in(z0, s(z1)), s(z0), z1)
U2(f125_out1, s(z0), z1) → f125_out1

Tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f125_in, U2

Defined Pair Symbols:

F1_IN, F125_IN

Compound Symbols:

c, c11, c

(21) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(s(0)))) → f1_out1
f1_in(s(s(s(s(0))))) → f1_out1
f1_in(s(s(s(s(s(0)))))) → f1_out1
f1_in(s(s(s(s(s(s(0))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(0)))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f125_out1, s(s(s(s(s(s(s(s(z0))))))))) → f1_out1
f125_in(0, z0) → f125_out1
f125_in(s(z0), z1) → U2(f125_in(z0, s(z1)), s(z0), z1)
U2(f125_out1, s(z0), z1) → f125_out1

Tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:

F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))

K tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

Defined Rule Symbols:

f1_in, U1, f125_in, U2

Defined Pair Symbols:

F1_IN, F125_IN

Compound Symbols:

c, c11, c

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))

We considered the (Usable) Rules:none
And the Tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(F125_IN(x₁, x₂)) = [3]x₁ + [2]x₂   
POL(F1_IN(x₁)) = [3] + [3]x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(s(x₁)) = [3] + x₁   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(s(0)))) → f1_out1
f1_in(s(s(s(s(0))))) → f1_out1
f1_in(s(s(s(s(s(0)))))) → f1_out1
f1_in(s(s(s(s(s(s(0))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(0)))))))) → f1_out1
f1_in(s(s(s(s(s(s(s(s(z0))))))))) → U1(f125_in(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(s(s(s(s(s(z0)))))))))
U1(f125_out1, s(s(s(s(s(s(s(s(z0))))))))) → f1_out1
f125_in(0, z0) → f125_out1
f125_in(s(z0), z1) → U2(f125_in(z0, s(z1)), s(z0), z1)
U2(f125_out1, s(z0), z1) → f125_out1

Tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c

S tuples:none
K tuples:

F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c(F125_IN(z0, s(s(s(s(s(s(s(0)))))))))
F1_IN(s(s(s(s(s(s(s(s(z0))))))))) → c
F125_IN(s(z0), z1) → c11(F125_IN(z0, s(z1)))

Defined Rule Symbols:

f1_in, U1, f125_in, U2

Defined Pair Symbols:

F1_IN, F125_IN

Compound Symbols:

c, c11, c