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

(0) Obligation:

Clauses:

less(0, s(X2)).
less(X, Y) :- ','(no(zero(X)), ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1)))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X3).
failure(b).

Query: less(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: less(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX5 is free\nless(T1, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))) | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))) | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(failure(a), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(failure(a), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 5, CLAUSE: 7\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22))))\n\nT10 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T10, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT10 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T10, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(','(call(zero(T15)), ','(!_6, failure(a))), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))) | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(zero(T15), ','(','(!_6, failure(a)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22))))) | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(zero(T15), ','(','(!_6, failure(a)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22))))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(','(!_6, failure(a)), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ?_8 | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(failure(a), ','(p(0, X21), ','(p(T12, X22), less(X21, X22))))\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(failure(a), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 9, CLAUSE: 7\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22)))\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 2 | ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 3\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 3\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(p(T12, X22), less(T21, X22))\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 2 | ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 3\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 2\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 3\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="B: less(T21, 0)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: less(T21, 0), SCOPE: 12, CLAUSE: 0 | less(T21, 0), SCOPE: 12, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(T21, 0), SCOPE: 12, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56))))\n\nT24 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 5 | ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT24 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(','(call(zero(T27)), ','(!_13, failure(a))), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))) | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(zero(T27), ','(','(!_13, failure(a)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56))))) | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(zero(T27), ','(','(!_13, failure(a)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56))))), SCOPE: 15, CLAUSE: 4 | ?_15 | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(','(!_13, failure(a)), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))) | ?_15 | ?_14 | ','(no(zero(0)), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ?_15 | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(failure(a), ','(p(0, X55), ','(p(0, X56), less(X55, X56))))\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: ','(failure(a), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 16, CLAUSE: 7\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: ','(p(T30, X55), ','(p(0, X56), less(X55, X56)))\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 2 | ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 3\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 3\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: ','(p(0, X56), less(T33, X56))\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 2 | ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 3\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 2\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 3\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: less(T33, 0)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: less(T21, T37)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 169 [arrowhead = none ];
169 -> 4;

170 [label="EVAL with clause\nless(0, s(X5)).\nand substitutionT1 -> 0,\nX5 -> T4,\nT2 -> s(T4)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 170 [arrowhead = none ];
170 -> 9;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 171 [arrowhead = none ];
171 -> 13;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 172 [arrowhead = none ];
172 -> 16;

173 [label="ONLY EVAL with clause\nless(X19, X20) :- ','(no(zero(X19)), ','(p(X19, X21), ','(p(X20, X22), less(X21, X22)))).\nand substitutionT1 -> T10,\nX19 -> T10,\nT2 -> T12,\nX20 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 173 [arrowhead = none ];
173 -> 47;

174 [label="ONLY EVAL with clause\nless(X8, X9) :- ','(no(zero(X8)), ','(p(X8, X10), ','(p(X9, X11), less(X10, X11)))).\nand substitutionX8 -> 0,\nT2 -> T7,\nX9 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 174 [arrowhead = none ];
174 -> 18;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 175 [arrowhead = none ];
175 -> 20;

176 [label="ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 176 [arrowhead = none ];
176 -> 24;

177 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 177 [arrowhead = none ];
177 -> 26;

178 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 178 [arrowhead = none ];
178 -> 29;

179 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 179 [arrowhead = none ];
179 -> 31;

180 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 180 [arrowhead = none ];
180 -> 33;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 181 [arrowhead = none ];
181 -> 37;

182 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 182 [arrowhead = none ];
182 -> 38;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 183 [arrowhead = none ];
183 -> 49;

184 [label="ONLY EVAL with clause\nno(X25) :- ','(call(X25), ','(!_6, failure(a))).\nand substitutionT10 -> T15,\nX25 -> zero(T15)", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 184 [arrowhead = none ];
184 -> 52;

185 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 185 [arrowhead = none ];
185 -> 53;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 186 [arrowhead = none ];
186 -> 57;

187 [label="EVAL with clause\nzero(0).\nand substitutionT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 187 [arrowhead = none ];
187 -> 58;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 188 [arrowhead = none ];
188 -> 60;

189 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 189 [arrowhead = none ];
189 -> 62;

190 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 190 [arrowhead = none ];
190 -> 66;

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 191 [arrowhead = none ];
191 -> 63;

192 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 192 [arrowhead = none ];
192 -> 64;

193 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 193 [arrowhead = none ];
193 -> 69;

194 [label="ONLY EVAL with clause\nno(X32).\nand substitutionT15 -> T18,\nX32 -> zero(T18)", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 194 [arrowhead = none ];
194 -> 72;

195 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 195 [arrowhead = none ];
195 -> 73;

196 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T18), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 196 [arrowhead = none ];
196 -> 74;

197 [label="EVAL with clause\np(s(X37), X37).\nand substitutionX37 -> T21,\nT18 -> s(T21),\nX21 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 197 [arrowhead = none ];
197 -> 77;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 198 [arrowhead = none ];
198 -> 79;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 199 [arrowhead = none ];
199 -> 82;

200 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 200 [arrowhead = none ];
200 -> 85;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 201 [arrowhead = none ];
201 -> 87;

202 [label="EVAL with clause\np(0, 0).\nand substitutionT12 -> 0,\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 202 [arrowhead = none ];
202 -> 88;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 203 [arrowhead = none ];
203 -> 89;

204 [label="EVAL with clause\np(s(X87), X87).\nand substitutionX87 -> T37,\nT12 -> s(T37),\nX22 -> T37,\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 204 [arrowhead = none ];
204 -> 166;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 205 [arrowhead = none ];
205 -> 168;

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 206 [arrowhead = none ];
206 -> 90;

207 [label="BACKTRACK\nfor clause: less(0, s(X2))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 207 [arrowhead = none ];
207 -> 91;

208 [label="ONLY EVAL with clause\nless(X53, X54) :- ','(no(zero(X53)), ','(p(X53, X55), ','(p(X54, X56), less(X55, X56)))).\nand substitutionT21 -> T24,\nX53 -> T24,\nX54 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 208 [arrowhead = none ];
208 -> 97;

209 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 209 [arrowhead = none ];
209 -> 101;

210 [label="ONLY EVAL with clause\nno(X61) :- ','(call(X61), ','(!_13, failure(a))).\nand substitutionT24 -> T27,\nX61 -> zero(T27)", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 210 [arrowhead = none ];
210 -> 106;

211 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
106 -> 211 [arrowhead = none ];
211 -> 107;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
107 -> 212 [arrowhead = none ];
212 -> 116;

213 [label="EVAL with clause\nzero(0).\nand substitutionT27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 213 [arrowhead = none ];
213 -> 120;

214 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 214 [arrowhead = none ];
214 -> 124;

215 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 215 [arrowhead = none ];
215 -> 125;

216 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 216 [arrowhead = none ];
216 -> 129;

217 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 217 [arrowhead = none ];
217 -> 126;

218 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
126 -> 218 [arrowhead = none ];
218 -> 127;

219 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 219 [arrowhead = none ];
219 -> 131;

220 [label="ONLY EVAL with clause\nno(X68).\nand substitutionT27 -> T30,\nX68 -> zero(T30)", fontsize=14, style = filled, fillcolor = lightblue];
131 -> 220 [arrowhead = none ];
220 -> 135;

221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
135 -> 221 [arrowhead = none ];
221 -> 137;

222 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T30), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 222 [arrowhead = none ];
222 -> 138;

223 [label="EVAL with clause\np(s(X73), X73).\nand substitutionX73 -> T33,\nT30 -> s(T33),\nX55 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
138 -> 223 [arrowhead = none ];
223 -> 141;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 224 [arrowhead = none ];
224 -> 142;

225 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
141 -> 225 [arrowhead = none ];
225 -> 146;

226 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 226 [arrowhead = none ];
226 -> 151;

227 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 227 [arrowhead = none ];
227 -> 152;

228 [label="ONLY EVAL with clause\np(0, 0).\nand substitutionX56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 228 [arrowhead = none ];
228 -> 157;

229 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 229 [arrowhead = none ];
229 -> 162;

230 [label="INSTANCE with matching:\nT21 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
157 -> 230 [arrowhead = none , style=dashed];
230 -> 88[style=dashed];

231 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
166 -> 231 [arrowhead = none , style=dashed];
231 -> 3[style=dashed];

}

(2) Obligation:

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

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(T21), 0) → U1_ga(T21, lessB_in_g(T21))
lessB_in_g(s(T33)) → U3_g(T33, lessB_in_g(T33))
U3_g(T33, lessB_out_g(T33)) → lessB_out_g(s(T33))
U1_ga(T21, lessB_out_g(T21)) → lessA_out_ga(s(T21), 0)
lessA_in_ga(s(T21), s(T37)) → U2_ga(T21, T37, lessA_in_ga(T21, T37))
U2_ga(T21, T37, lessA_out_ga(T21, T37)) → lessA_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)

(3) 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:

LESSA_IN_GA(s(T21), 0) → U1_GA(T21, lessB_in_g(T21))
LESSA_IN_GA(s(T21), 0) → LESSB_IN_G(T21)
LESSB_IN_G(s(T33)) → U3_G(T33, lessB_in_g(T33))
LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)
LESSA_IN_GA(s(T21), s(T37)) → U2_GA(T21, T37, lessA_in_ga(T21, T37))
LESSA_IN_GA(s(T21), s(T37)) → LESSA_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(T21), 0) → U1_ga(T21, lessB_in_g(T21))
lessB_in_g(s(T33)) → U3_g(T33, lessB_in_g(T33))
U3_g(T33, lessB_out_g(T33)) → lessB_out_g(s(T33))
U1_ga(T21, lessB_out_g(T21)) → lessA_out_ga(s(T21), 0)
lessA_in_ga(s(T21), s(T37)) → U2_ga(T21, T37, lessA_in_ga(T21, T37))
U2_ga(T21, T37, lessA_out_ga(T21, T37)) → lessA_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x1, x2)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(4) Obligation:

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

LESSA_IN_GA(s(T21), 0) → U1_GA(T21, lessB_in_g(T21))
LESSA_IN_GA(s(T21), 0) → LESSB_IN_G(T21)
LESSB_IN_G(s(T33)) → U3_G(T33, lessB_in_g(T33))
LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)
LESSA_IN_GA(s(T21), s(T37)) → U2_GA(T21, T37, lessA_in_ga(T21, T37))
LESSA_IN_GA(s(T21), s(T37)) → LESSA_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(T21), 0) → U1_ga(T21, lessB_in_g(T21))
lessB_in_g(s(T33)) → U3_g(T33, lessB_in_g(T33))
U3_g(T33, lessB_out_g(T33)) → lessB_out_g(s(T33))
U1_ga(T21, lessB_out_g(T21)) → lessA_out_ga(s(T21), 0)
lessA_in_ga(s(T21), s(T37)) → U2_ga(T21, T37, lessA_in_ga(T21, T37))
U2_ga(T21, T37, lessA_out_ga(T21, T37)) → lessA_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x1, x2)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(T21), 0) → U1_ga(T21, lessB_in_g(T21))
lessB_in_g(s(T33)) → U3_g(T33, lessB_in_g(T33))
U3_g(T33, lessB_out_g(T33)) → lessB_out_g(s(T33))
U1_ga(T21, lessB_out_g(T21)) → lessA_out_ga(s(T21), 0)
lessA_in_ga(s(T21), s(T37)) → U2_ga(T21, T37, lessA_in_ga(T21, T37))
U2_ga(T21, T37, lessA_out_ga(T21, T37)) → lessA_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)

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

(12) 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:

• LESSB_IN_G(s(T33)) → LESSB_IN_G(T33)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

LESSA_IN_GA(s(T21), s(T37)) → LESSA_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T4)) → lessA_out_ga(0, s(T4))
lessA_in_ga(s(T21), 0) → U1_ga(T21, lessB_in_g(T21))
lessB_in_g(s(T33)) → U3_g(T33, lessB_in_g(T33))
U3_g(T33, lessB_out_g(T33)) → lessB_out_g(s(T33))
U1_ga(T21, lessB_out_g(T21)) → lessA_out_ga(s(T21), 0)
lessA_in_ga(s(T21), s(T37)) → U2_ga(T21, T37, lessA_in_ga(T21, T37))
U2_ga(T21, T37, lessA_out_ga(T21, T37)) → lessA_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESSA_IN_GA(s(T21), s(T37)) → LESSA_IN_GA(T21, T37)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

LESSA_IN_GA(s(T21)) → LESSA_IN_GA(T21)

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

(19) 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:

• LESSA_IN_GA(s(T21)) → LESSA_IN_GA(T21)
  The graph contains the following edges 1 > 1

(20) YES