Left Termination of the query pattern p_in_2(g, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

p(X, Y) :- ','(less(X, Y), ','(!, p(s(X), Y))).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

p(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(less(T5, T6), ','(!_1, p(s(T5), T6)))\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(less(T5, T6), ','(!_1, p(s(T5), T6))), SCOPE: 2, CLAUSE: 1 | ','(less(T5, T6), ','(!_1, p(s(T5), T6))), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, p(s(0), s(T9))) | ','(less(0, s(T9)), ','(!_1, p(s(0), s(T9)))), SCOPE: 2, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(less(T5, T6), ','(!_1, p(s(T5), T6))), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nless(T5, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(s(0), s(T9))\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(s(0), s(T9)), SCOPE: 3, CLAUSE: 0\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(less(s(0), s(T14)), ','(!_3, p(s(s(0)), s(T14))))\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: less(s(0), s(T14))\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(!_3, p(s(s(0)), s(T14)))\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: less(s(0), s(T14)), SCOPE: 4, CLAUSE: 1 | less(s(0), s(T14)), SCOPE: 4, CLAUSE: 2\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(s(0), s(T14)), SCOPE: 4, CLAUSE: 2\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(0, T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(0, T21), SCOPE: 5, CLAUSE: 1 | less(0, T21), SCOPE: 5, CLAUSE: 2\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(0, T21), SCOPE: 5, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(0, T21), SCOPE: 5, CLAUSE: 2\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(s(s(0)), s(T14))\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(s(s(0)), s(T14)), SCOPE: 6, CLAUSE: 0\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(less(s(s(0)), s(T33)), ','(!_6, p(s(s(s(0))), s(T33))))\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(s(s(0)), s(T33))\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_6, p(s(s(s(0))), s(T33)))\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(s(s(0)), s(T33)), SCOPE: 7, CLAUSE: 1 | less(s(s(0)), s(T33)), SCOPE: 7, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(s(s(0)), s(T33)), SCOPE: 7, CLAUSE: 2\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(s(0), T39)\n\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(s(0), T39), SCOPE: 8, CLAUSE: 1 | less(s(0), T39), SCOPE: 8, CLAUSE: 2\n\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: less(s(0), T39), SCOPE: 8, CLAUSE: 2\n\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(0, T43)\n\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(s(s(s(0))), s(T33))\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: p(s(s(s(0))), s(T33)), SCOPE: 9, CLAUSE: 0\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(less(s(s(s(0))), s(T50)), ','(!_9, p(s(s(s(s(0)))), s(T50))))\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: less(s(s(s(0))), s(T50))\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(!_9, p(s(s(s(s(0)))), s(T50)))\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: less(s(s(s(0))), s(T50)), SCOPE: 10, CLAUSE: 1 | less(s(s(s(0))), s(T50)), SCOPE: 10, CLAUSE: 2\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(s(s(s(0))), s(T50)), SCOPE: 10, CLAUSE: 2\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: less(s(s(0)), T56)\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(s(s(0)), T56), SCOPE: 11, CLAUSE: 1 | less(s(s(0)), T56), SCOPE: 11, CLAUSE: 2\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(s(s(0)), T56), SCOPE: 11, CLAUSE: 2\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(s(0), T60)\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(s(s(s(s(0)))), s(T50))\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: p(s(s(s(s(0)))), s(T50)), SCOPE: 12, CLAUSE: 0\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(less(s(s(s(s(0)))), s(T66)), ','(!_12, p(s(s(s(s(s(0))))), s(T66))))\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: less(s(s(s(s(0)))), s(T66))\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(!_12, p(s(s(s(s(s(0))))), s(T66)))\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(s(s(s(s(0)))), s(T66)), SCOPE: 13, CLAUSE: 1 | less(s(s(s(s(0)))), s(T66)), SCOPE: 13, CLAUSE: 2\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(s(s(s(s(0)))), s(T66)), SCOPE: 13, CLAUSE: 2\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(s(s(s(0))), T72)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(s(s(s(0))), T72), SCOPE: 14, CLAUSE: 1 | less(s(s(s(0))), T72), SCOPE: 14, CLAUSE: 2\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(s(s(s(0))), T72), SCOPE: 14, CLAUSE: 2\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(s(s(0)), T76)\n\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: p(s(s(s(s(s(0))))), s(T66))\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: p(s(s(s(s(s(0))))), s(T66)), SCOPE: 15, CLAUSE: 0\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(less(s(s(s(s(s(0))))), s(T82)), ','(!_15, p(s(s(s(s(s(s(0)))))), s(T82))))\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(s(s(s(s(s(0))))), s(T82))\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(!_15, p(s(s(s(s(s(s(0)))))), s(T82)))\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(s(s(s(s(s(0))))), s(T82)), SCOPE: 16, CLAUSE: 1 | less(s(s(s(s(s(0))))), s(T82)), SCOPE: 16, CLAUSE: 2\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(s(s(s(s(s(0))))), s(T82)), SCOPE: 16, CLAUSE: 2\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(s(s(s(s(0)))), T88)\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: less(s(s(s(s(0)))), T88), SCOPE: 17, CLAUSE: 1 | less(s(s(s(s(0)))), T88), SCOPE: 17, CLAUSE: 2\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(s(s(s(s(0)))), T88), SCOPE: 17, CLAUSE: 2\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: less(s(s(s(0))), T92)\n\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: p(s(s(s(s(s(s(0)))))), s(T82))\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(s(s(s(s(s(s(0)))))), s(T82)), SCOPE: 18, CLAUSE: 0\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(less(s(s(s(s(s(s(0)))))), s(T98)), ','(!_18, p(s(s(s(s(s(s(s(0))))))), s(T98))))\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(s(s(s(s(s(s(0)))))), s(T98))\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(!_18, p(s(s(s(s(s(s(s(0))))))), s(T98)))\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(s(s(s(s(s(s(0)))))), s(T98)), SCOPE: 19, CLAUSE: 1 | less(s(s(s(s(s(s(0)))))), s(T98)), SCOPE: 19, CLAUSE: 2\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(s(s(s(s(s(s(0)))))), s(T98)), SCOPE: 19, CLAUSE: 2\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(s(s(s(s(s(0))))), T104)\n\nT104 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: less(s(s(s(s(s(0))))), T104), SCOPE: 20, CLAUSE: 1 | less(s(s(s(s(s(0))))), T104), SCOPE: 20, CLAUSE: 2\n\nT104 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: less(s(s(s(s(s(0))))), T104), SCOPE: 20, CLAUSE: 2\n\nT104 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(s(s(s(s(0)))), T108)\n\nT108 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: p(s(s(s(s(s(s(s(0))))))), s(T98))\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: p(s(s(s(s(s(s(s(0))))))), s(T98)), SCOPE: 21, CLAUSE: 0\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(less(s(s(s(s(s(s(s(0))))))), s(T114)), ','(!_21, p(s(s(s(s(s(s(s(s(0)))))))), s(T114))))\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(less(s(s(s(s(s(s(s(0))))))), s(T114)), ','(!_21, p(s(T116), s(T114))))\n\nT114 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: less(s(s(s(s(s(s(s(0))))))), s(T114))\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(!_21, p(s(T116), s(T114)))\n\nT114 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: less(s(s(s(s(s(s(s(0))))))), s(T114)), SCOPE: 22, CLAUSE: 1 | less(s(s(s(s(s(s(s(0))))))), s(T114)), SCOPE: 22, CLAUSE: 2\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: less(s(s(s(s(s(s(s(0))))))), s(T114)), SCOPE: 22, CLAUSE: 2\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: less(s(s(s(s(s(s(0)))))), T122)\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(s(s(s(s(s(s(0)))))), T122), SCOPE: 23, CLAUSE: 1 | less(s(s(s(s(s(s(0)))))), T122), SCOPE: 23, CLAUSE: 2\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: less(s(s(s(s(s(s(0)))))), T122), SCOPE: 23, CLAUSE: 2\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: less(s(s(s(s(s(0))))), T126)\n\nT126 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: p(s(T116), s(T114))\n\nT114 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(less(T134, T135), ','(!_1, p(s(s(T134)), s(T135))))\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: less(T134, T135)\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(!_1, p(s(s(T134)), s(T135)))\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(T134, T135), SCOPE: 24, CLAUSE: 1 | less(T134, T135), SCOPE: 24, CLAUSE: 2\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: less(T134, T135), SCOPE: 24, CLAUSE: 1\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: less(T134, T135), SCOPE: 24, CLAUSE: 2\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: less(T149, T150)\n\nT149 is ground\nT150 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: p(s(s(T134)), s(T135))\n\nT134 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 109 [arrowhead = none ];
109 -> 2;

110 [label="ONLY EVAL with clause\np(X4, X5) :- ','(less(X4, X5), ','(!_1, p(s(X4), X5))).\nand substitution\nT1 -> T5\nX4 -> T5\nT2 -> T6\nX5 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 110 [arrowhead = none ];
110 -> 3;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 111 [arrowhead = none ];
111 -> 4;

112 [label="EVAL with clause\nless(0, s(X8)).\nand substitution\nT5 -> 0\nX8 -> T9\nT6 -> s(T9)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 112 [arrowhead = none ];
112 -> 5;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 113 [arrowhead = none ];
113 -> 6;

114 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 114 [arrowhead = none ];
114 -> 7;

115 [label="EVAL with clause\nless(s(X345), s(X346)) :- less(X345, X346).\nand substitution\nX345 -> T134\nT5 -> s(T134)\nX346 -> T135\nT6 -> s(T135)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 115 [arrowhead = none ];
115 -> 96;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 116 [arrowhead = none ];
116 -> 97;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 117 [arrowhead = none ];
117 -> 8;

118 [label="ONLY EVAL with clause\np(X20, X21) :- ','(less(X20, X21), ','(!_3, p(s(X20), X21))).\nand substitution\nX20 -> s(0)\nT9 -> T14\nX21 -> s(T14)", fontsize=14, style = filled, fillcolor = white];
8 -> 118 [arrowhead = none ];
118 -> 9;

119 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
9 -> 119 [arrowhead = none ];
119 -> 10;

120 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
9 -> 120 [arrowhead = none ];
120 -> 11;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 121 [arrowhead = none ];
121 -> 12;

122 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 122 [arrowhead = none ];
122 -> 22;

123 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 123 [arrowhead = none ];
123 -> 13;

124 [label="ONLY EVAL with clause\nless(s(X37), s(X38)) :- less(X37, X38).\nand substitution\nX37 -> 0\nT14 -> T21\nX38 -> T21", fontsize=14, style = filled, fillcolor = white];
13 -> 124 [arrowhead = none ];
124 -> 14;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 125 [arrowhead = none ];
125 -> 15;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 126 [arrowhead = none ];
126 -> 16;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 127 [arrowhead = none ];
127 -> 17;

128 [label="EVAL with clause\nless(0, s(X45)).\nand substitution\nX45 -> T28\nT21 -> s(T28)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 128 [arrowhead = none ];
128 -> 18;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 129 [arrowhead = none ];
129 -> 19;

130 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
17 -> 130 [arrowhead = none ];
130 -> 21;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 131 [arrowhead = none ];
131 -> 20;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 132 [arrowhead = none ];
132 -> 23;

133 [label="ONLY EVAL with clause\np(X59, X60) :- ','(less(X59, X60), ','(!_6, p(s(X59), X60))).\nand substitution\nX59 -> s(s(0))\nT14 -> T33\nX60 -> s(T33)", fontsize=14, style = filled, fillcolor = white];
23 -> 133 [arrowhead = none ];
133 -> 24;

134 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 134 [arrowhead = none ];
134 -> 25;

135 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
24 -> 135 [arrowhead = none ];
135 -> 26;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 136 [arrowhead = none ];
136 -> 27;

137 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 137 [arrowhead = none ];
137 -> 34;

138 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 138 [arrowhead = none ];
138 -> 28;

139 [label="ONLY EVAL with clause\nless(s(X78), s(X79)) :- less(X78, X79).\nand substitution\nX78 -> s(0)\nT33 -> T39\nX79 -> T39", fontsize=14, style = filled, fillcolor = white];
28 -> 139 [arrowhead = none ];
139 -> 29;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 140 [arrowhead = none ];
140 -> 30;

141 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 141 [arrowhead = none ];
141 -> 31;

142 [label="EVAL with clause\nless(s(X89), s(X90)) :- less(X89, X90).\nand substitution\nX89 -> 0\nX90 -> T43\nT39 -> s(T43)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 142 [arrowhead = none ];
142 -> 32;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 143 [arrowhead = none ];
143 -> 33;

144 [label="INSTANCE with matching:\nT21 -> T43", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 144 [arrowhead = none , style=dashed];
144 -> 14[style=dashed];

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 145 [arrowhead = none ];
145 -> 35;

146 [label="ONLY EVAL with clause\np(X105, X106) :- ','(less(X105, X106), ','(!_9, p(s(X105), X106))).\nand substitution\nX105 -> s(s(s(0)))\nT33 -> T50\nX106 -> s(T50)", fontsize=14, style = filled, fillcolor = white];
35 -> 146 [arrowhead = none ];
146 -> 36;

147 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 147 [arrowhead = none ];
147 -> 37;

148 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
36 -> 148 [arrowhead = none ];
148 -> 38;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 149 [arrowhead = none ];
149 -> 39;

150 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 150 [arrowhead = none ];
150 -> 46;

151 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
39 -> 151 [arrowhead = none ];
151 -> 40;

152 [label="ONLY EVAL with clause\nless(s(X124), s(X125)) :- less(X124, X125).\nand substitution\nX124 -> s(s(0))\nT50 -> T56\nX125 -> T56", fontsize=14, style = filled, fillcolor = white];
40 -> 152 [arrowhead = none ];
152 -> 41;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 153 [arrowhead = none ];
153 -> 42;

154 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
42 -> 154 [arrowhead = none ];
154 -> 43;

155 [label="EVAL with clause\nless(s(X135), s(X136)) :- less(X135, X136).\nand substitution\nX135 -> s(0)\nX136 -> T60\nT56 -> s(T60)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 155 [arrowhead = none ];
155 -> 44;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 156 [arrowhead = none ];
156 -> 45;

157 [label="INSTANCE with matching:\nT39 -> T60", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 157 [arrowhead = none , style=dashed];
157 -> 29[style=dashed];

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 158 [arrowhead = none ];
158 -> 47;

159 [label="ONLY EVAL with clause\np(X153, X154) :- ','(less(X153, X154), ','(!_12, p(s(X153), X154))).\nand substitution\nX153 -> s(s(s(s(0))))\nT50 -> T66\nX154 -> s(T66)", fontsize=14, style = filled, fillcolor = white];
47 -> 159 [arrowhead = none ];
159 -> 48;

160 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 160 [arrowhead = none ];
160 -> 49;

161 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
48 -> 161 [arrowhead = none ];
161 -> 50;

162 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 162 [arrowhead = none ];
162 -> 51;

163 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 163 [arrowhead = none ];
163 -> 58;

164 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
51 -> 164 [arrowhead = none ];
164 -> 52;

165 [label="ONLY EVAL with clause\nless(s(X172), s(X173)) :- less(X172, X173).\nand substitution\nX172 -> s(s(s(0)))\nT66 -> T72\nX173 -> T72", fontsize=14, style = filled, fillcolor = white];
52 -> 165 [arrowhead = none ];
165 -> 53;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 166 [arrowhead = none ];
166 -> 54;

167 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
54 -> 167 [arrowhead = none ];
167 -> 55;

168 [label="EVAL with clause\nless(s(X183), s(X184)) :- less(X183, X184).\nand substitution\nX183 -> s(s(0))\nX184 -> T76\nT72 -> s(T76)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 168 [arrowhead = none ];
168 -> 56;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 169 [arrowhead = none ];
169 -> 57;

170 [label="INSTANCE with matching:\nT56 -> T76", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 170 [arrowhead = none , style=dashed];
170 -> 41[style=dashed];

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 171 [arrowhead = none ];
171 -> 59;

172 [label="ONLY EVAL with clause\np(X201, X202) :- ','(less(X201, X202), ','(!_15, p(s(X201), X202))).\nand substitution\nX201 -> s(s(s(s(s(0)))))\nT66 -> T82\nX202 -> s(T82)", fontsize=14, style = filled, fillcolor = white];
59 -> 172 [arrowhead = none ];
172 -> 60;

173 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 173 [arrowhead = none ];
173 -> 61;

174 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
60 -> 174 [arrowhead = none ];
174 -> 62;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 175 [arrowhead = none ];
175 -> 63;

176 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 176 [arrowhead = none ];
176 -> 70;

177 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
63 -> 177 [arrowhead = none ];
177 -> 64;

178 [label="ONLY EVAL with clause\nless(s(X220), s(X221)) :- less(X220, X221).\nand substitution\nX220 -> s(s(s(s(0))))\nT82 -> T88\nX221 -> T88", fontsize=14, style = filled, fillcolor = white];
64 -> 178 [arrowhead = none ];
178 -> 65;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 179 [arrowhead = none ];
179 -> 66;

180 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
66 -> 180 [arrowhead = none ];
180 -> 67;

181 [label="EVAL with clause\nless(s(X231), s(X232)) :- less(X231, X232).\nand substitution\nX231 -> s(s(s(0)))\nX232 -> T92\nT88 -> s(T92)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 181 [arrowhead = none ];
181 -> 68;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 182 [arrowhead = none ];
182 -> 69;

183 [label="INSTANCE with matching:\nT72 -> T92", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 183 [arrowhead = none , style=dashed];
183 -> 53[style=dashed];

184 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 184 [arrowhead = none ];
184 -> 71;

185 [label="ONLY EVAL with clause\np(X249, X250) :- ','(less(X249, X250), ','(!_18, p(s(X249), X250))).\nand substitution\nX249 -> s(s(s(s(s(s(0))))))\nT82 -> T98\nX250 -> s(T98)", fontsize=14, style = filled, fillcolor = white];
71 -> 185 [arrowhead = none ];
185 -> 72;

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 186 [arrowhead = none ];
186 -> 73;

187 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
72 -> 187 [arrowhead = none ];
187 -> 74;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 188 [arrowhead = none ];
188 -> 75;

189 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 189 [arrowhead = none ];
189 -> 82;

190 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
75 -> 190 [arrowhead = none ];
190 -> 76;

191 [label="ONLY EVAL with clause\nless(s(X268), s(X269)) :- less(X268, X269).\nand substitution\nX268 -> s(s(s(s(s(0)))))\nT98 -> T104\nX269 -> T104", fontsize=14, style = filled, fillcolor = white];
76 -> 191 [arrowhead = none ];
191 -> 77;

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 192 [arrowhead = none ];
192 -> 78;

193 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
78 -> 193 [arrowhead = none ];
193 -> 79;

194 [label="EVAL with clause\nless(s(X279), s(X280)) :- less(X279, X280).\nand substitution\nX279 -> s(s(s(s(0))))\nX280 -> T108\nT104 -> s(T108)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 194 [arrowhead = none ];
194 -> 80;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 195 [arrowhead = none ];
195 -> 81;

196 [label="INSTANCE with matching:\nT88 -> T108", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 196 [arrowhead = none , style=dashed];
196 -> 65[style=dashed];

197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 197 [arrowhead = none ];
197 -> 83;

198 [label="ONLY EVAL with clause\np(X297, X298) :- ','(less(X297, X298), ','(!_21, p(s(X297), X298))).\nand substitution\nX297 -> s(s(s(s(s(s(s(0)))))))\nT98 -> T114\nX298 -> s(T114)", fontsize=14, style = filled, fillcolor = white];
83 -> 198 [arrowhead = none ];
198 -> 84;

199 [label="GENERALIZATION\nT116 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT116 is ground", fontsize=14, style = filled, fillcolor = violet];
84 -> 199 [arrowhead = none ];
199 -> 85;

200 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 200 [arrowhead = none ];
200 -> 86;

201 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
85 -> 201 [arrowhead = none ];
201 -> 87;

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 202 [arrowhead = none ];
202 -> 88;

203 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 203 [arrowhead = none ];
203 -> 95;

204 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 204 [arrowhead = none ];
204 -> 89;

205 [label="ONLY EVAL with clause\nless(s(X320), s(X321)) :- less(X320, X321).\nand substitution\nX320 -> s(s(s(s(s(s(0))))))\nT114 -> T122\nX321 -> T122", fontsize=14, style = filled, fillcolor = white];
89 -> 205 [arrowhead = none ];
205 -> 90;

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 206 [arrowhead = none ];
206 -> 91;

207 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
91 -> 207 [arrowhead = none ];
207 -> 92;

208 [label="EVAL with clause\nless(s(X331), s(X332)) :- less(X331, X332).\nand substitution\nX331 -> s(s(s(s(s(0)))))\nX332 -> T126\nT122 -> s(T126)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 208 [arrowhead = none ];
208 -> 93;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 209 [arrowhead = none ];
209 -> 94;

210 [label="INSTANCE with matching:\nT104 -> T126", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 210 [arrowhead = none , style=dashed];
210 -> 77[style=dashed];

211 [label="INSTANCE with matching:\nT1 -> s(T116)\nT2 -> s(T114)", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 211 [arrowhead = none , style=dashed];
211 -> 1[style=dashed];

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
96 -> 212 [arrowhead = none ];
212 -> 98;

213 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
96 -> 213 [arrowhead = none ];
213 -> 99;

214 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 214 [arrowhead = none ];
214 -> 100;

215 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 215 [arrowhead = none ];
215 -> 108;

216 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
100 -> 216 [arrowhead = none ];
216 -> 101;

217 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
100 -> 217 [arrowhead = none ];
217 -> 102;

218 [label="EVAL with clause\nless(0, s(X357)).\nand substitution\nT134 -> 0\nX357 -> T144\nT135 -> s(T144)", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 218 [arrowhead = none ];
218 -> 103;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 219 [arrowhead = none ];
219 -> 104;

220 [label="EVAL with clause\nless(s(X363), s(X364)) :- less(X363, X364).\nand substitution\nX363 -> T149\nT134 -> s(T149)\nX364 -> T150\nT135 -> s(T150)", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 220 [arrowhead = none ];
220 -> 106;

221 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 221 [arrowhead = none ];
221 -> 107;

222 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 222 [arrowhead = none ];
222 -> 105;

223 [label="INSTANCE with matching:\nT134 -> T149\nT135 -> T150", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 223 [arrowhead = none , style=dashed];
223 -> 98[style=dashed];

224 [label="INSTANCE with matching:\nT1 -> s(s(T134))\nT2 -> s(T135)", fontsize=14, style = filled, fillcolor = lightgrey];
108 -> 224 [arrowhead = none , style=dashed];
224 -> 1[style=dashed];

}

(2) Obligation:

Triples:

less29(s(T43)) :- less14(T43).
less41(s(T60)) :- less29(T60).
less53(s(T76)) :- less41(T76).
less65(s(T92)) :- less53(T92).
less77(s(T108)) :- less65(T108).
less98(s(T149), s(T150)) :- less98(T149, T150).
p85(s(T126), T116) :- less77(T126).
p85(T114, T116) :- ','(lessc86(T114), p1(s(T116), s(T114))).
p1(0, s(T21)) :- less14(T21).
p1(0, s(T39)) :- ','(lessc10(T39), less29(T39)).
p1(0, s(T56)) :- ','(lessc10(T56), ','(lessc25(T56), less41(T56))).
p1(0, s(T72)) :- ','(lessc10(T72), ','(lessc25(T72), ','(lessc37(T72), less53(T72)))).
p1(0, s(T88)) :- ','(lessc10(T88), ','(lessc25(T88), ','(lessc37(T88), ','(lessc49(T88), less65(T88))))).
p1(0, s(T104)) :- ','(lessc10(T104), ','(lessc25(T104), ','(lessc37(T104), ','(lessc49(T104), ','(lessc61(T104), less77(T104)))))).
p1(0, s(T114)) :- ','(lessc10(T114), ','(lessc25(T114), ','(lessc37(T114), ','(lessc49(T114), ','(lessc61(T114), ','(lessc73(T114), p85(T114, s(s(s(s(s(s(s(0)))))))))))))).
p1(s(T134), s(T135)) :- less98(T134, T135).
p1(s(T134), s(T135)) :- ','(lessc98(T134, T135), p1(s(s(T134)), s(T135))).

Clauses:

lessc14(s(T28)).
lessc29(s(T43)) :- lessc14(T43).
lessc41(s(T60)) :- lessc29(T60).
lessc53(s(T76)) :- lessc41(T76).
lessc65(s(T92)) :- lessc53(T92).
lessc77(s(T108)) :- lessc65(T108).
pc1(0, s(T114)) :- ','(lessc10(T114), ','(lessc25(T114), ','(lessc37(T114), ','(lessc49(T114), ','(lessc61(T114), ','(lessc73(T114), qc85(T114, s(s(s(s(s(s(s(0)))))))))))))).
pc1(s(T134), s(T135)) :- ','(lessc98(T134, T135), pc1(s(s(T134)), s(T135))).
lessc98(0, s(T144)).
lessc98(s(T149), s(T150)) :- lessc98(T149, T150).
qc85(T114, T116) :- ','(lessc86(T114), pc1(s(T116), s(T114))).
lessc10(T21) :- lessc14(T21).
lessc25(T39) :- lessc29(T39).
lessc37(T56) :- lessc41(T56).
lessc49(T72) :- lessc53(T72).
lessc61(T88) :- lessc65(T88).
lessc73(T104) :- lessc77(T104).
lessc86(s(T126)) :- lessc77(T126).

Afs:

p1(x1, x2)  =  p1(x1, x2)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

less98(s(T149), s(T150)) :- less98(T149, T150).
p85(T114, T116) :- ','(lessc86(T114), p1(s(T116), s(T114))).
p1(0, s(T114)) :- ','(lessc10(T114), ','(lessc25(T114), ','(lessc37(T114), ','(lessc49(T114), ','(lessc61(T114), ','(lessc73(T114), p85(T114, s(s(s(s(s(s(s(0)))))))))))))).
p1(s(T134), s(T135)) :- less98(T134, T135).
p1(s(T134), s(T135)) :- ','(lessc98(T134, T135), p1(s(s(T134)), s(T135))).

Clauses:

lessc14(s(T28)).
lessc29(s(T43)) :- lessc14(T43).
lessc41(s(T60)) :- lessc29(T60).
lessc53(s(T76)) :- lessc41(T76).
lessc65(s(T92)) :- lessc53(T92).
lessc77(s(T108)) :- lessc65(T108).
pc1(0, s(T114)) :- ','(lessc10(T114), ','(lessc25(T114), ','(lessc37(T114), ','(lessc49(T114), ','(lessc61(T114), ','(lessc73(T114), qc85(T114, s(s(s(s(s(s(s(0)))))))))))))).
pc1(s(T134), s(T135)) :- ','(lessc98(T134, T135), pc1(s(s(T134)), s(T135))).
lessc98(0, s(T144)).
lessc98(s(T149), s(T150)) :- lessc98(T149, T150).
qc85(T114, T116) :- ','(lessc86(T114), pc1(s(T116), s(T114))).
lessc10(T21) :- lessc14(T21).
lessc25(T39) :- lessc29(T39).
lessc37(T56) :- lessc41(T56).
lessc49(T72) :- lessc53(T72).
lessc61(T88) :- lessc65(T88).
lessc73(T104) :- lessc77(T104).
lessc86(s(T126)) :- lessc77(T126).

Afs:

p1(x1, x2)  =  p1(x1, x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b,b)
lessc10_in: (b)
lessc25_in: (b)
lessc29_in: (b)
lessc37_in: (b)
lessc41_in: (b)
lessc49_in: (b)
lessc53_in: (b)
lessc61_in: (b)
lessc65_in: (b)
lessc73_in: (b)
lessc77_in: (b)
p85_in: (b,b)
lessc86_in: (b)
less98_in: (b,b)
lessc98_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_GG(0, s(T114)) → U4_GG(T114, lessc10_in_g(T114))
U4_GG(T114, lessc10_out_g(T114)) → U5_GG(T114, lessc25_in_g(T114))
U5_GG(T114, lessc25_out_g(T114)) → U6_GG(T114, lessc37_in_g(T114))
U6_GG(T114, lessc37_out_g(T114)) → U7_GG(T114, lessc49_in_g(T114))
U7_GG(T114, lessc49_out_g(T114)) → U8_GG(T114, lessc61_in_g(T114))
U8_GG(T114, lessc61_out_g(T114)) → U9_GG(T114, lessc73_in_g(T114))
U9_GG(T114, lessc73_out_g(T114)) → U10_GG(T114, p85_in_gg(T114, s(s(s(s(s(s(s(0)))))))))
U9_GG(T114, lessc73_out_g(T114)) → P85_IN_GG(T114, s(s(s(s(s(s(s(0))))))))
P85_IN_GG(T114, T116) → U2_GG(T114, T116, lessc86_in_g(T114))
U2_GG(T114, T116, lessc86_out_g(T114)) → U3_GG(T114, T116, p1_in_gg(s(T116), s(T114)))
U2_GG(T114, T116, lessc86_out_g(T114)) → P1_IN_GG(s(T116), s(T114))
P1_IN_GG(s(T134), s(T135)) → U11_GG(T134, T135, less98_in_gg(T134, T135))
P1_IN_GG(s(T134), s(T135)) → LESS98_IN_GG(T134, T135)
LESS98_IN_GG(s(T149), s(T150)) → U1_GG(T149, T150, less98_in_gg(T149, T150))
LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)
P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → U13_GG(T134, T135, p1_in_gg(s(s(T134)), s(T135)))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc10_in_g(T21) → U32_g(T21, lessc14_in_g(T21))
lessc14_in_g(s(T28)) → lessc14_out_g(s(T28))
U32_g(T21, lessc14_out_g(T21)) → lessc10_out_g(T21)
lessc25_in_g(T39) → U33_g(T39, lessc29_in_g(T39))
lessc29_in_g(s(T43)) → U15_g(T43, lessc14_in_g(T43))
U15_g(T43, lessc14_out_g(T43)) → lessc29_out_g(s(T43))
U33_g(T39, lessc29_out_g(T39)) → lessc25_out_g(T39)
lessc37_in_g(T56) → U34_g(T56, lessc41_in_g(T56))
lessc41_in_g(s(T60)) → U16_g(T60, lessc29_in_g(T60))
U16_g(T60, lessc29_out_g(T60)) → lessc41_out_g(s(T60))
U34_g(T56, lessc41_out_g(T56)) → lessc37_out_g(T56)
lessc49_in_g(T72) → U35_g(T72, lessc53_in_g(T72))
lessc53_in_g(s(T76)) → U17_g(T76, lessc41_in_g(T76))
U17_g(T76, lessc41_out_g(T76)) → lessc53_out_g(s(T76))
U35_g(T72, lessc53_out_g(T72)) → lessc49_out_g(T72)
lessc61_in_g(T88) → U36_g(T88, lessc65_in_g(T88))
lessc65_in_g(s(T92)) → U18_g(T92, lessc53_in_g(T92))
U18_g(T92, lessc53_out_g(T92)) → lessc65_out_g(s(T92))
U36_g(T88, lessc65_out_g(T88)) → lessc61_out_g(T88)
lessc73_in_g(T104) → U37_g(T104, lessc77_in_g(T104))
lessc77_in_g(s(T108)) → U19_g(T108, lessc65_in_g(T108))
U19_g(T108, lessc65_out_g(T108)) → lessc77_out_g(s(T108))
U37_g(T104, lessc77_out_g(T104)) → lessc73_out_g(T104)
lessc86_in_g(s(T126)) → U38_g(T126, lessc77_in_g(T126))
U38_g(T126, lessc77_out_g(T126)) → lessc86_out_g(s(T126))
lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

P1_IN_GG(0, s(T114)) → U4_GG(T114, lessc10_in_g(T114))
U4_GG(T114, lessc10_out_g(T114)) → U5_GG(T114, lessc25_in_g(T114))
U5_GG(T114, lessc25_out_g(T114)) → U6_GG(T114, lessc37_in_g(T114))
U6_GG(T114, lessc37_out_g(T114)) → U7_GG(T114, lessc49_in_g(T114))
U7_GG(T114, lessc49_out_g(T114)) → U8_GG(T114, lessc61_in_g(T114))
U8_GG(T114, lessc61_out_g(T114)) → U9_GG(T114, lessc73_in_g(T114))
U9_GG(T114, lessc73_out_g(T114)) → U10_GG(T114, p85_in_gg(T114, s(s(s(s(s(s(s(0)))))))))
U9_GG(T114, lessc73_out_g(T114)) → P85_IN_GG(T114, s(s(s(s(s(s(s(0))))))))
P85_IN_GG(T114, T116) → U2_GG(T114, T116, lessc86_in_g(T114))
U2_GG(T114, T116, lessc86_out_g(T114)) → U3_GG(T114, T116, p1_in_gg(s(T116), s(T114)))
U2_GG(T114, T116, lessc86_out_g(T114)) → P1_IN_GG(s(T116), s(T114))
P1_IN_GG(s(T134), s(T135)) → U11_GG(T134, T135, less98_in_gg(T134, T135))
P1_IN_GG(s(T134), s(T135)) → LESS98_IN_GG(T134, T135)
LESS98_IN_GG(s(T149), s(T150)) → U1_GG(T149, T150, less98_in_gg(T149, T150))
LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)
P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → U13_GG(T134, T135, p1_in_gg(s(s(T134)), s(T135)))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc10_in_g(T21) → U32_g(T21, lessc14_in_g(T21))
lessc14_in_g(s(T28)) → lessc14_out_g(s(T28))
U32_g(T21, lessc14_out_g(T21)) → lessc10_out_g(T21)
lessc25_in_g(T39) → U33_g(T39, lessc29_in_g(T39))
lessc29_in_g(s(T43)) → U15_g(T43, lessc14_in_g(T43))
U15_g(T43, lessc14_out_g(T43)) → lessc29_out_g(s(T43))
U33_g(T39, lessc29_out_g(T39)) → lessc25_out_g(T39)
lessc37_in_g(T56) → U34_g(T56, lessc41_in_g(T56))
lessc41_in_g(s(T60)) → U16_g(T60, lessc29_in_g(T60))
U16_g(T60, lessc29_out_g(T60)) → lessc41_out_g(s(T60))
U34_g(T56, lessc41_out_g(T56)) → lessc37_out_g(T56)
lessc49_in_g(T72) → U35_g(T72, lessc53_in_g(T72))
lessc53_in_g(s(T76)) → U17_g(T76, lessc41_in_g(T76))
U17_g(T76, lessc41_out_g(T76)) → lessc53_out_g(s(T76))
U35_g(T72, lessc53_out_g(T72)) → lessc49_out_g(T72)
lessc61_in_g(T88) → U36_g(T88, lessc65_in_g(T88))
lessc65_in_g(s(T92)) → U18_g(T92, lessc53_in_g(T92))
U18_g(T92, lessc53_out_g(T92)) → lessc65_out_g(s(T92))
U36_g(T88, lessc65_out_g(T88)) → lessc61_out_g(T88)
lessc73_in_g(T104) → U37_g(T104, lessc77_in_g(T104))
lessc77_in_g(s(T108)) → U19_g(T108, lessc65_in_g(T108))
U19_g(T108, lessc65_out_g(T108)) → lessc77_out_g(s(T108))
U37_g(T104, lessc77_out_g(T104)) → lessc73_out_g(T104)
lessc86_in_g(s(T126)) → U38_g(T126, lessc77_in_g(T126))
U38_g(T126, lessc77_out_g(T126)) → lessc86_out_g(s(T126))
lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)

The TRS R consists of the following rules:

lessc10_in_g(T21) → U32_g(T21, lessc14_in_g(T21))
lessc14_in_g(s(T28)) → lessc14_out_g(s(T28))
U32_g(T21, lessc14_out_g(T21)) → lessc10_out_g(T21)
lessc25_in_g(T39) → U33_g(T39, lessc29_in_g(T39))
lessc29_in_g(s(T43)) → U15_g(T43, lessc14_in_g(T43))
U15_g(T43, lessc14_out_g(T43)) → lessc29_out_g(s(T43))
U33_g(T39, lessc29_out_g(T39)) → lessc25_out_g(T39)
lessc37_in_g(T56) → U34_g(T56, lessc41_in_g(T56))
lessc41_in_g(s(T60)) → U16_g(T60, lessc29_in_g(T60))
U16_g(T60, lessc29_out_g(T60)) → lessc41_out_g(s(T60))
U34_g(T56, lessc41_out_g(T56)) → lessc37_out_g(T56)
lessc49_in_g(T72) → U35_g(T72, lessc53_in_g(T72))
lessc53_in_g(s(T76)) → U17_g(T76, lessc41_in_g(T76))
U17_g(T76, lessc41_out_g(T76)) → lessc53_out_g(s(T76))
U35_g(T72, lessc53_out_g(T72)) → lessc49_out_g(T72)
lessc61_in_g(T88) → U36_g(T88, lessc65_in_g(T88))
lessc65_in_g(s(T92)) → U18_g(T92, lessc53_in_g(T92))
U18_g(T92, lessc53_out_g(T92)) → lessc65_out_g(s(T92))
U36_g(T88, lessc65_out_g(T88)) → lessc61_out_g(T88)
lessc73_in_g(T104) → U37_g(T104, lessc77_in_g(T104))
lessc77_in_g(s(T108)) → U19_g(T108, lessc65_in_g(T108))
U19_g(T108, lessc65_out_g(T108)) → lessc77_out_g(s(T108))
U37_g(T104, lessc77_out_g(T104)) → lessc73_out_g(T104)
lessc86_in_g(s(T126)) → U38_g(T126, lessc77_in_g(T126))
U38_g(T126, lessc77_out_g(T126)) → lessc86_out_g(s(T126))
lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)

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

(14) 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:

• LESS98_IN_GG(s(T149), s(T150)) → LESS98_IN_GG(T149, T150)
  The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc10_in_g(T21) → U32_g(T21, lessc14_in_g(T21))
lessc14_in_g(s(T28)) → lessc14_out_g(s(T28))
U32_g(T21, lessc14_out_g(T21)) → lessc10_out_g(T21)
lessc25_in_g(T39) → U33_g(T39, lessc29_in_g(T39))
lessc29_in_g(s(T43)) → U15_g(T43, lessc14_in_g(T43))
U15_g(T43, lessc14_out_g(T43)) → lessc29_out_g(s(T43))
U33_g(T39, lessc29_out_g(T39)) → lessc25_out_g(T39)
lessc37_in_g(T56) → U34_g(T56, lessc41_in_g(T56))
lessc41_in_g(s(T60)) → U16_g(T60, lessc29_in_g(T60))
U16_g(T60, lessc29_out_g(T60)) → lessc41_out_g(s(T60))
U34_g(T56, lessc41_out_g(T56)) → lessc37_out_g(T56)
lessc49_in_g(T72) → U35_g(T72, lessc53_in_g(T72))
lessc53_in_g(s(T76)) → U17_g(T76, lessc41_in_g(T76))
U17_g(T76, lessc41_out_g(T76)) → lessc53_out_g(s(T76))
U35_g(T72, lessc53_out_g(T72)) → lessc49_out_g(T72)
lessc61_in_g(T88) → U36_g(T88, lessc65_in_g(T88))
lessc65_in_g(s(T92)) → U18_g(T92, lessc53_in_g(T92))
U18_g(T92, lessc53_out_g(T92)) → lessc65_out_g(s(T92))
U36_g(T88, lessc65_out_g(T88)) → lessc61_out_g(T88)
lessc73_in_g(T104) → U37_g(T104, lessc77_in_g(T104))
lessc77_in_g(s(T108)) → U19_g(T108, lessc65_in_g(T108))
U19_g(T108, lessc65_out_g(T108)) → lessc77_out_g(s(T108))
U37_g(T104, lessc77_out_g(T104)) → lessc73_out_g(T104)
lessc86_in_g(s(T126)) → U38_g(T126, lessc77_in_g(T126))
U38_g(T126, lessc77_out_g(T126)) → lessc86_out_g(s(T126))
lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_GG(s(T134), s(T135)) → U12_GG(T134, T135, lessc98_in_gg(T134, T135)) at position [2] we obtained the following new rules [LPAR04]:

P1_IN_GG(s(0), s(s(x0))) → U12_GG(0, s(x0), lessc98_out_gg(0, s(x0)))
P1_IN_GG(s(s(x0)), s(s(x1))) → U12_GG(s(x0), s(x1), U29_gg(x0, x1, lessc98_in_gg(x0, x1)))

(22) Obligation:

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

U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))
P1_IN_GG(s(0), s(s(x0))) → U12_GG(0, s(x0), lessc98_out_gg(0, s(x0)))
P1_IN_GG(s(s(x0)), s(s(x1))) → U12_GG(s(x0), s(x1), U29_gg(x0, x1, lessc98_in_gg(x0, x1)))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(24) Obligation:

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

P1_IN_GG(s(s(x0)), s(s(x1))) → U12_GG(s(x0), s(x1), U29_gg(x0, x1, lessc98_in_gg(x0, x1)))
U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(25) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GG(T134, T135, lessc98_out_gg(T134, T135)) → P1_IN_GG(s(s(T134)), s(T135)) we obtained the following new rules [LPAR04]:

U12_GG(s(z0), s(z1), lessc98_out_gg(s(z0), s(z1))) → P1_IN_GG(s(s(s(z0))), s(s(z1)))

(26) Obligation:

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

P1_IN_GG(s(s(x0)), s(s(x1))) → U12_GG(s(x0), s(x1), U29_gg(x0, x1, lessc98_in_gg(x0, x1)))
U12_GG(s(z0), s(z1), lessc98_out_gg(s(z0), s(z1))) → P1_IN_GG(s(s(s(z0))), s(s(z1)))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(27) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_GG(s(s(x0)), s(s(x1))) → U12_GG(s(x0), s(x1), U29_gg(x0, x1, lessc98_in_gg(x0, x1))) we obtained the following new rules [LPAR04]:

P1_IN_GG(s(s(s(z0))), s(s(z1))) → U12_GG(s(s(z0)), s(z1), U29_gg(s(z0), z1, lessc98_in_gg(s(z0), z1)))

(28) Obligation:

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

U12_GG(s(z0), s(z1), lessc98_out_gg(s(z0), s(z1))) → P1_IN_GG(s(s(s(z0))), s(s(z1)))
P1_IN_GG(s(s(s(z0))), s(s(z1))) → U12_GG(s(s(z0)), s(z1), U29_gg(s(z0), z1, lessc98_in_gg(s(z0), z1)))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GG(s(z0), s(z1), lessc98_out_gg(s(z0), s(z1))) → P1_IN_GG(s(s(s(z0))), s(s(z1))) we obtained the following new rules [LPAR04]:

U12_GG(s(s(z0)), s(z1), lessc98_out_gg(s(s(z0)), s(z1))) → P1_IN_GG(s(s(s(s(z0)))), s(s(z1)))

(30) Obligation:

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

P1_IN_GG(s(s(s(z0))), s(s(z1))) → U12_GG(s(s(z0)), s(z1), U29_gg(s(z0), z1, lessc98_in_gg(s(z0), z1)))
U12_GG(s(s(z0)), s(z1), lessc98_out_gg(s(s(z0)), s(z1))) → P1_IN_GG(s(s(s(s(z0)))), s(s(z1)))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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

(31) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P1_IN_GG(s(s(s(z0))), s(s(z1))) → U12_GG(s(s(z0)), s(z1), U29_gg(s(z0), z1, lessc98_in_gg(s(z0), z1))) we obtained the following new rules [LPAR04]:

P1_IN_GG(s(s(s(s(z0)))), s(s(z1))) → U12_GG(s(s(s(z0))), s(z1), U29_gg(s(s(z0)), z1, lessc98_in_gg(s(s(z0)), z1)))

(32) Obligation:

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

U12_GG(s(s(z0)), s(z1), lessc98_out_gg(s(s(z0)), s(z1))) → P1_IN_GG(s(s(s(s(z0)))), s(s(z1)))
P1_IN_GG(s(s(s(s(z0)))), s(s(z1))) → U12_GG(s(s(s(z0))), s(z1), U29_gg(s(s(z0)), z1, lessc98_in_gg(s(s(z0)), z1)))

The TRS R consists of the following rules:

lessc98_in_gg(0, s(T144)) → lessc98_out_gg(0, s(T144))
lessc98_in_gg(s(T149), s(T150)) → U29_gg(T149, T150, lessc98_in_gg(T149, T150))
U29_gg(T149, T150, lessc98_out_gg(T149, T150)) → lessc98_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

lessc98_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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