Left Termination of the query pattern in_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

in(X, tree(X, X1, X2)).
in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)).
in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)).
less(0, s(X5)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

in(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: in(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: in(T1, T2), SCOPE: 1, CLAUSE: 0 | in(T1, T2), SCOPE: 1, CLAUSE: 1 | in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | in(T6, T2), SCOPE: 1, CLAUSE: 1 | in(T6, T2), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: in(T1, T2), SCOPE: 1, CLAUSE: 1 | in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nin(T1, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: in(T6, T2), SCOPE: 1, CLAUSE: 1 | in(T6, T2), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(less(T13, T17), in(T13, T18)) | in(T13, T2), SCOPE: 1, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: in(T6, T2), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nin(T6, T2) !=? in(X19, tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(less(T13, T17), in(T13, T18)), SCOPE: 2, CLAUSE: 3 | ','(less(T13, T17), in(T13, T18)), SCOPE: 2, CLAUSE: 4 | ?_2 | in(T13, T2), SCOPE: 1, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(less(T13, T17), in(T13, T18)), SCOPE: 2, CLAUSE: 3\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(less(T13, T17), in(T13, T18)), SCOPE: 2, CLAUSE: 4 | ?_2 | in(T13, T2), SCOPE: 1, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: in(0, T24)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: in(0, T24), SCOPE: 3, CLAUSE: 0 | in(0, T24), SCOPE: 3, CLAUSE: 1 | in(0, T24), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: in(0, T24), SCOPE: 3, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: in(0, T24), SCOPE: 3, CLAUSE: 1 | in(0, T24), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: in(0, T24), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: in(0, T24), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(less(0, T54), in(0, T55))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(0, T54)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: in(0, T58)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(0, T54), SCOPE: 4, CLAUSE: 3 | less(0, T54), SCOPE: 4, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(0, T54), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(0, T54), SCOPE: 4, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(less(T79, 0), in(0, T80))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: less(T79, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: in(0, T82)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T79, 0), SCOPE: 5, CLAUSE: 3 | less(T79, 0), SCOPE: 5, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: less(T79, 0), SCOPE: 5, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(less(T13, T17), in(T13, T18)), SCOPE: 2, CLAUSE: 4\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ?_2 | in(T13, T2), SCOPE: 1, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(less(T97, T99), in(s(T97), T100))\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T97, T99)\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: in(s(T97), T103)\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T97, T99), SCOPE: 6, CLAUSE: 3 | less(T97, T99), SCOPE: 6, CLAUSE: 4\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: less(T97, T99), SCOPE: 6, CLAUSE: 3\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T97, T99), SCOPE: 6, CLAUSE: 4\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T115, T117)\n\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: in(T13, T2), SCOPE: 1, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(less(T138, T134), in(T134, T139))\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T138, T134)\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: in(T134, T142)\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(T138, T134), SCOPE: 7, CLAUSE: 3 | less(T138, T134), SCOPE: 7, CLAUSE: 4\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T138, T134), SCOPE: 7, CLAUSE: 3\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T138, T134), SCOPE: 7, CLAUSE: 4\n\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T156, T155)\n\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(less(T173, T169), in(T169, T174))\n\nT169 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(less(T173, T169), in(T169, T174)), SCOPE: 8, CLAUSE: 3 | ','(less(T173, T169), in(T169, T174)), SCOPE: 8, CLAUSE: 4\n\nT169 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(less(T173, T169), in(T169, T174)), SCOPE: 8, CLAUSE: 3\n\nT169 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(less(T173, T169), in(T169, T174)), SCOPE: 8, CLAUSE: 4\n\nT169 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: in(s(T179), T180)\n\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(less(T193, T192), in(s(T192), T194))\n\nT192 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T193, T192)\n\nT192 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: in(s(T192), T197)\n\nT192 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(less(T210, T214), in(T210, T215)) | in(T210, T2), SCOPE: 1, CLAUSE: 2\n\nT210 is ground\nin(T210, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nin(T1, T2) !=? in(X9, tree(X9, X10, X11))\nin(T1, T2) !=? in(X258, tree(X259, X260, X261))", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(less(T210, T214), in(T210, T215)), SCOPE: 9, CLAUSE: 3 | ','(less(T210, T214), in(T210, T215)), SCOPE: 9, CLAUSE: 4 | ?_9 | in(T210, T2), SCOPE: 1, CLAUSE: 2\n\nT210 is ground\nin(T210, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(less(T210, T214), in(T210, T215)), SCOPE: 9, CLAUSE: 3\n\nT210 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(less(T210, T214), in(T210, T215)), SCOPE: 9, CLAUSE: 4 | ?_9 | in(T210, T2), SCOPE: 1, CLAUSE: 2\n\nT210 is ground\nin(T210, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: in(0, T221)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(less(T210, T214), in(T210, T215)), SCOPE: 9, CLAUSE: 4\n\nT210 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ?_9 | in(T210, T2), SCOPE: 1, CLAUSE: 2\n\nT210 is ground\nin(T210, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(less(T234, T236), in(s(T234), T237))\n\nT234 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: in(T210, T2), SCOPE: 1, CLAUSE: 2\n\nT210 is ground\nin(T210, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(less(T252, T248), in(T248, T253))\n\nT248 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(less(T264, T260), in(T260, T265))\n\nT260 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(less(T264, T260), in(T260, T265)), SCOPE: 10, CLAUSE: 3 | ','(less(T264, T260), in(T260, T265)), SCOPE: 10, CLAUSE: 4\n\nT260 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(less(T264, T260), in(T260, T265)), SCOPE: 10, CLAUSE: 3\n\nT260 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(less(T264, T260), in(T260, T265)), SCOPE: 10, CLAUSE: 4\n\nT260 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: in(s(T270), T271)\n\nT270 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(less(T284, T283), in(s(T283), T285))\n\nT283 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 100 [arrowhead = none ];
100 -> 2;

101 [label="EVAL with clause\nin(X9, tree(X9, X10, X11)).\nand substitution\nT1 -> T6\nX9 -> T6\nX10 -> T7\nX11 -> T8\nT2 -> tree(T6, T7, T8)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 101 [arrowhead = none ];
101 -> 3;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 102 [arrowhead = none ];
102 -> 4;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 103 [arrowhead = none ];
103 -> 5;

104 [label="EVAL with clause\nin(X258, tree(X259, X260, X261)) :- ','(less(X258, X259), in(X258, X260)).\nand substitution\nT1 -> T210\nX258 -> T210\nX259 -> T214\nX260 -> T215\nX261 -> T213\nT2 -> tree(T214, T215, T213)\nT211 -> T214\nT212 -> T215", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 104 [arrowhead = none ];
104 -> 77;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 105 [arrowhead = none ];
105 -> 78;

106 [label="EVAL with clause\nin(X19, tree(X20, X21, X22)) :- ','(less(X19, X20), in(X19, X21)).\nand substitution\nT6 -> T13\nX19 -> T13\nX20 -> T17\nX21 -> T18\nX22 -> T16\nT2 -> tree(T17, T18, T16)\nT14 -> T17\nT15 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 106 [arrowhead = none ];
106 -> 6;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 107 [arrowhead = none ];
107 -> 7;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 108 [arrowhead = none ];
108 -> 8;

109 [label="EVAL with clause\nin(X216, tree(X217, X218, X219)) :- ','(less(X217, X216), in(X216, X219)).\nand substitution\nT6 -> T169\nX216 -> T169\nX217 -> T173\nX218 -> T171\nX219 -> T174\nT2 -> tree(T173, T171, T174)\nT170 -> T173\nT172 -> T174", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 109 [arrowhead = none ];
109 -> 66;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 110 [arrowhead = none ];
110 -> 67;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 111 [arrowhead = none ];
111 -> 9;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 112 [arrowhead = none ];
112 -> 10;

113 [label="EVAL with clause\nless(0, s(X31)).\nand substitution\nT13 -> 0\nX31 -> T23\nT17 -> s(T23)\nT18 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 113 [arrowhead = none ];
113 -> 11;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 114 [arrowhead = none ];
114 -> 12;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 115 [arrowhead = none ];
115 -> 39;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 116 [arrowhead = none ];
116 -> 40;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 117 [arrowhead = none ];
117 -> 13;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 118 [arrowhead = none ];
118 -> 14;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 119 [arrowhead = none ];
119 -> 15;

120 [label="EVAL with clause\nin(X51, tree(X51, X52, X53)).\nand substitution\nX51 -> 0\nX52 -> T37\nX53 -> T38\nT24 -> tree(0, T37, T38)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 120 [arrowhead = none ];
120 -> 16;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 121 [arrowhead = none ];
121 -> 17;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 122 [arrowhead = none ];
122 -> 19;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 123 [arrowhead = none ];
123 -> 20;

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 124 [arrowhead = none ];
124 -> 18;

125 [label="EVAL with clause\nin(X73, tree(X74, X75, X76)) :- ','(less(X73, X74), in(X73, X75)).\nand substitution\nX73 -> 0\nX74 -> T54\nX75 -> T55\nX76 -> T53\nT24 -> tree(T54, T55, T53)\nT51 -> T54\nT52 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 125 [arrowhead = none ];
125 -> 21;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 126 [arrowhead = none ];
126 -> 22;

127 [label="EVAL with clause\nin(X107, tree(X108, X109, X110)) :- ','(less(X108, X107), in(X107, X110)).\nand substitution\nX107 -> 0\nX108 -> T79\nX109 -> T77\nX110 -> T80\nT24 -> tree(T79, T77, T80)\nT76 -> T79\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 127 [arrowhead = none ];
127 -> 32;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 128 [arrowhead = none ];
128 -> 33;

129 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 129 [arrowhead = none ];
129 -> 23;

130 [label="SPLIT 2\nreplacements:\nT55 -> T58", fontsize=14, style = filled, fillcolor = violet];
21 -> 130 [arrowhead = none ];
130 -> 24;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 131 [arrowhead = none ];
131 -> 25;

132 [label="INSTANCE with matching:\nT24 -> T58", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 132 [arrowhead = none , style=dashed];
132 -> 11[style=dashed];

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 133 [arrowhead = none ];
133 -> 26;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 134 [arrowhead = none ];
134 -> 27;

135 [label="EVAL with clause\nless(0, s(X89)).\nand substitution\nX89 -> T65\nT54 -> s(T65)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 135 [arrowhead = none ];
135 -> 28;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 136 [arrowhead = none ];
136 -> 29;

137 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 137 [arrowhead = none ];
137 -> 31;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 138 [arrowhead = none ];
138 -> 30;

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 139 [arrowhead = none ];
139 -> 34;

140 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nreplacements:\nT80 -> T82", fontsize=14, style = filled, fillcolor = violet];
32 -> 140 [arrowhead = none ];
140 -> 35;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 141 [arrowhead = none ];
141 -> 36;

142 [label="INSTANCE with matching:\nT24 -> T82", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 142 [arrowhead = none , style=dashed];
142 -> 11[style=dashed];

143 [label="BACKTRACK\nfor clause: less(0, s(X5))because of non-unification", fontsize=14, style = filled, fillcolor = white];
36 -> 143 [arrowhead = none ];
143 -> 37;

144 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = white];
37 -> 144 [arrowhead = none ];
144 -> 38;

145 [label="EVAL with clause\nless(s(X140), s(X141)) :- less(X140, X141).\nand substitution\nX140 -> T97\nT13 -> s(T97)\nX141 -> T99\nT17 -> s(T99)\nT98 -> T99\nT18 -> T100", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 145 [arrowhead = none ];
145 -> 41;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 146 [arrowhead = none ];
146 -> 42;

147 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 147 [arrowhead = none ];
147 -> 53;

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 148 [arrowhead = none ];
148 -> 43;

149 [label="SPLIT 2\nreplacements:\nT100 -> T103", fontsize=14, style = filled, fillcolor = violet];
41 -> 149 [arrowhead = none ];
149 -> 44;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 150 [arrowhead = none ];
150 -> 45;

151 [label="INSTANCE with matching:\nT1 -> s(T97)\nT2 -> T103", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 151 [arrowhead = none , style=dashed];
151 -> 1[style=dashed];

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 152 [arrowhead = none ];
152 -> 46;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 153 [arrowhead = none ];
153 -> 47;

154 [label="EVAL with clause\nless(0, s(X152)).\nand substitution\nT97 -> 0\nX152 -> T110\nT99 -> s(T110)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 154 [arrowhead = none ];
154 -> 48;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 155 [arrowhead = none ];
155 -> 49;

156 [label="EVAL with clause\nless(s(X158), s(X159)) :- less(X158, X159).\nand substitution\nX158 -> T115\nT97 -> s(T115)\nX159 -> T117\nT99 -> s(T117)\nT116 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 156 [arrowhead = none ];
156 -> 51;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 157 [arrowhead = none ];
157 -> 52;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 158 [arrowhead = none ];
158 -> 50;

159 [label="INSTANCE with matching:\nT97 -> T115\nT99 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 159 [arrowhead = none , style=dashed];
159 -> 43[style=dashed];

160 [label="EVAL with clause\nin(X178, tree(X179, X180, X181)) :- ','(less(X179, X178), in(X178, X181)).\nand substitution\nT13 -> T134\nX178 -> T134\nX179 -> T138\nX180 -> T136\nX181 -> T139\nT2 -> tree(T138, T136, T139)\nT135 -> T138\nT137 -> T139", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 160 [arrowhead = none ];
160 -> 54;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 161 [arrowhead = none ];
161 -> 55;

162 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 162 [arrowhead = none ];
162 -> 56;

163 [label="SPLIT 2\nnew knowledge:\nT138 is ground\nreplacements:\nT139 -> T142", fontsize=14, style = filled, fillcolor = violet];
54 -> 163 [arrowhead = none ];
163 -> 57;

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 164 [arrowhead = none ];
164 -> 58;

165 [label="INSTANCE with matching:\nT1 -> T134\nT2 -> T142", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 165 [arrowhead = none , style=dashed];
165 -> 1[style=dashed];

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 166 [arrowhead = none ];
166 -> 59;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 167 [arrowhead = none ];
167 -> 60;

168 [label="EVAL with clause\nless(0, s(X194)).\nand substitution\nT138 -> 0\nX194 -> T149\nT134 -> s(T149)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 168 [arrowhead = none ];
168 -> 61;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 169 [arrowhead = none ];
169 -> 62;

170 [label="EVAL with clause\nless(s(X200), s(X201)) :- less(X200, X201).\nand substitution\nX200 -> T156\nT138 -> s(T156)\nX201 -> T155\nT134 -> s(T155)\nT154 -> T156", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 170 [arrowhead = none ];
170 -> 64;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 171 [arrowhead = none ];
171 -> 65;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 172 [arrowhead = none ];
172 -> 63;

173 [label="INSTANCE with matching:\nT138 -> T156\nT134 -> T155", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 173 [arrowhead = none , style=dashed];
173 -> 56[style=dashed];

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 174 [arrowhead = none ];
174 -> 68;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 175 [arrowhead = none ];
175 -> 69;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 176 [arrowhead = none ];
176 -> 70;

177 [label="EVAL with clause\nless(0, s(X228)).\nand substitution\nT173 -> 0\nX228 -> T179\nT169 -> s(T179)\nT174 -> T180", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 177 [arrowhead = none ];
177 -> 71;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 178 [arrowhead = none ];
178 -> 72;

179 [label="EVAL with clause\nless(s(X240), s(X241)) :- less(X240, X241).\nand substitution\nX240 -> T193\nT173 -> s(T193)\nX241 -> T192\nT169 -> s(T192)\nT191 -> T193\nT174 -> T194", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 179 [arrowhead = none ];
179 -> 73;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 180 [arrowhead = none ];
180 -> 74;

181 [label="INSTANCE with matching:\nT1 -> s(T179)\nT2 -> T180", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 181 [arrowhead = none , style=dashed];
181 -> 1[style=dashed];

182 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
73 -> 182 [arrowhead = none ];
182 -> 75;

183 [label="SPLIT 2\nnew knowledge:\nT193 is ground\nreplacements:\nT194 -> T197", fontsize=14, style = filled, fillcolor = violet];
73 -> 183 [arrowhead = none ];
183 -> 76;

184 [label="INSTANCE with matching:\nT138 -> T193\nT134 -> T192", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 184 [arrowhead = none , style=dashed];
184 -> 56[style=dashed];

185 [label="INSTANCE with matching:\nT1 -> s(T192)\nT2 -> T197", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 185 [arrowhead = none , style=dashed];
185 -> 1[style=dashed];

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 186 [arrowhead = none ];
186 -> 79;

187 [label="EVAL with clause\nin(X314, tree(X315, X316, X317)) :- ','(less(X315, X314), in(X314, X317)).\nand substitution\nT1 -> T260\nX314 -> T260\nX315 -> T264\nX316 -> T262\nX317 -> T265\nT2 -> tree(T264, T262, T265)\nT261 -> T264\nT263 -> T265", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 187 [arrowhead = none ];
187 -> 91;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 188 [arrowhead = none ];
188 -> 92;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 189 [arrowhead = none ];
189 -> 80;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 190 [arrowhead = none ];
190 -> 81;

191 [label="EVAL with clause\nless(0, s(X270)).\nand substitution\nT210 -> 0\nX270 -> T220\nT214 -> s(T220)\nT215 -> T221", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 191 [arrowhead = none ];
191 -> 82;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 192 [arrowhead = none ];
192 -> 83;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 193 [arrowhead = none ];
193 -> 84;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 194 [arrowhead = none ];
194 -> 85;

195 [label="INSTANCE with matching:\nT24 -> T221", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 195 [arrowhead = none , style=dashed];
195 -> 11[style=dashed];

196 [label="EVAL with clause\nless(s(X286), s(X287)) :- less(X286, X287).\nand substitution\nX286 -> T234\nT210 -> s(T234)\nX287 -> T236\nT214 -> s(T236)\nT235 -> T236\nT215 -> T237", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 196 [arrowhead = none ];
196 -> 86;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 197 [arrowhead = none ];
197 -> 87;

198 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 198 [arrowhead = none ];
198 -> 88;

199 [label="INSTANCE with matching:\nT97 -> T234\nT99 -> T236\nT100 -> T237", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 199 [arrowhead = none , style=dashed];
199 -> 41[style=dashed];

200 [label="EVAL with clause\nin(X300, tree(X301, X302, X303)) :- ','(less(X301, X300), in(X300, X303)).\nand substitution\nT210 -> T248\nX300 -> T248\nX301 -> T252\nX302 -> T250\nX303 -> T253\nT2 -> tree(T252, T250, T253)\nT249 -> T252\nT251 -> T253", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 200 [arrowhead = none ];
200 -> 89;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 201 [arrowhead = none ];
201 -> 90;

202 [label="INSTANCE with matching:\nT173 -> T252\nT169 -> T248\nT174 -> T253", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 202 [arrowhead = none , style=dashed];
202 -> 66[style=dashed];

203 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 203 [arrowhead = none ];
203 -> 93;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
93 -> 204 [arrowhead = none ];
204 -> 94;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
93 -> 205 [arrowhead = none ];
205 -> 95;

206 [label="EVAL with clause\nless(0, s(X326)).\nand substitution\nT264 -> 0\nX326 -> T270\nT260 -> s(T270)\nT265 -> T271", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 206 [arrowhead = none ];
206 -> 96;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 207 [arrowhead = none ];
207 -> 97;

208 [label="EVAL with clause\nless(s(X338), s(X339)) :- less(X338, X339).\nand substitution\nX338 -> T284\nT264 -> s(T284)\nX339 -> T283\nT260 -> s(T283)\nT282 -> T284\nT265 -> T285", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 208 [arrowhead = none ];
208 -> 98;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 209 [arrowhead = none ];
209 -> 99;

210 [label="INSTANCE with matching:\nT1 -> s(T270)\nT2 -> T271", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 210 [arrowhead = none , style=dashed];
210 -> 1[style=dashed];

211 [label="INSTANCE with matching:\nT193 -> T284\nT192 -> T283\nT194 -> T285", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 211 [arrowhead = none , style=dashed];
211 -> 73[style=dashed];

}

(2) Obligation:

Triples:

in11(tree(T54, T58, T53)) :- ','(lessc23(T54), in11(T58)).
in11(tree(T79, T77, T82)) :- ','(lessc34(T79), in11(T82)).
less43(s(T115), s(T117)) :- less43(T115, T117).
less56(s(T156), s(T155)) :- less56(T156, T155).
p41(T97, T99, T100) :- less43(T97, T99).
p41(T97, T99, T103) :- ','(lessc43(T97, T99), in1(s(T97), T103)).
p66(0, s(T179), T180) :- in1(s(T179), T180).
p66(s(T193), s(T192), T194) :- p73(T193, T192, T194).
p73(T193, T192, T194) :- less56(T193, T192).
p73(T193, T192, T197) :- ','(lessc56(T193, T192), in1(s(T192), T197)).
in1(0, tree(s(T23), T24, T16)) :- in11(T24).
in1(s(T97), tree(s(T99), T100, T16)) :- p41(T97, T99, T100).
in1(T134, tree(T138, T136, T139)) :- less56(T138, T134).
in1(T134, tree(T138, T136, T142)) :- ','(lessc56(T138, T134), in1(T134, T142)).
in1(T169, tree(T173, T171, T174)) :- p66(T173, T169, T174).
in1(0, tree(s(T220), T221, T213)) :- in11(T221).
in1(s(T234), tree(s(T236), T237, T213)) :- p41(T234, T236, T237).
in1(T248, tree(T252, T250, T253)) :- p66(T252, T248, T253).
in1(s(T270), tree(0, T262, T271)) :- in1(s(T270), T271).
in1(s(T283), tree(s(T284), T262, T285)) :- p73(T284, T283, T285).

Clauses:

inc11(tree(0, T37, T38)).
inc11(tree(T54, T58, T53)) :- ','(lessc23(T54), inc11(T58)).
inc11(tree(T79, T77, T82)) :- ','(lessc34(T79), inc11(T82)).
inc1(T6, tree(T6, T7, T8)).
inc1(0, tree(s(T23), T24, T16)) :- inc11(T24).
inc1(s(T97), tree(s(T99), T100, T16)) :- qc41(T97, T99, T100).
inc1(T134, tree(T138, T136, T142)) :- ','(lessc56(T138, T134), inc1(T134, T142)).
inc1(T169, tree(T173, T171, T174)) :- qc66(T173, T169, T174).
inc1(0, tree(s(T220), T221, T213)) :- inc11(T221).
inc1(s(T234), tree(s(T236), T237, T213)) :- qc41(T234, T236, T237).
inc1(T248, tree(T252, T250, T253)) :- qc66(T252, T248, T253).
inc1(s(T270), tree(0, T262, T271)) :- inc1(s(T270), T271).
inc1(s(T283), tree(s(T284), T262, T285)) :- qc73(T284, T283, T285).
lessc43(0, s(T110)).
lessc43(s(T115), s(T117)) :- lessc43(T115, T117).
lessc56(0, s(T149)).
lessc56(s(T156), s(T155)) :- lessc56(T156, T155).
qc41(T97, T99, T103) :- ','(lessc43(T97, T99), inc1(s(T97), T103)).
qc66(0, s(T179), T180) :- inc1(s(T179), T180).
qc66(s(T193), s(T192), T194) :- qc73(T193, T192, T194).
qc73(T193, T192, T197) :- ','(lessc56(T193, T192), inc1(s(T192), T197)).
lessc23(s(T65)).

Afs:

in1(x1, x2)  =  in1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

in11(tree(T54, T58, T53)) :- ','(lessc23(T54), in11(T58)).
less43(s(T115), s(T117)) :- less43(T115, T117).
less56(s(T156), s(T155)) :- less56(T156, T155).
p41(T97, T99, T100) :- less43(T97, T99).
p41(T97, T99, T103) :- ','(lessc43(T97, T99), in1(s(T97), T103)).
p66(0, s(T179), T180) :- in1(s(T179), T180).
p66(s(T193), s(T192), T194) :- p73(T193, T192, T194).
p73(T193, T192, T194) :- less56(T193, T192).
p73(T193, T192, T197) :- ','(lessc56(T193, T192), in1(s(T192), T197)).
in1(0, tree(s(T23), T24, T16)) :- in11(T24).
in1(s(T97), tree(s(T99), T100, T16)) :- p41(T97, T99, T100).
in1(T134, tree(T138, T136, T139)) :- less56(T138, T134).
in1(T134, tree(T138, T136, T142)) :- ','(lessc56(T138, T134), in1(T134, T142)).
in1(T169, tree(T173, T171, T174)) :- p66(T173, T169, T174).
in1(0, tree(s(T220), T221, T213)) :- in11(T221).
in1(s(T234), tree(s(T236), T237, T213)) :- p41(T234, T236, T237).
in1(T248, tree(T252, T250, T253)) :- p66(T252, T248, T253).
in1(s(T270), tree(0, T262, T271)) :- in1(s(T270), T271).
in1(s(T283), tree(s(T284), T262, T285)) :- p73(T284, T283, T285).

Clauses:

inc11(tree(0, T37, T38)).
inc11(tree(T54, T58, T53)) :- ','(lessc23(T54), inc11(T58)).
inc1(T6, tree(T6, T7, T8)).
inc1(0, tree(s(T23), T24, T16)) :- inc11(T24).
inc1(s(T97), tree(s(T99), T100, T16)) :- qc41(T97, T99, T100).
inc1(T134, tree(T138, T136, T142)) :- ','(lessc56(T138, T134), inc1(T134, T142)).
inc1(T169, tree(T173, T171, T174)) :- qc66(T173, T169, T174).
inc1(0, tree(s(T220), T221, T213)) :- inc11(T221).
inc1(s(T234), tree(s(T236), T237, T213)) :- qc41(T234, T236, T237).
inc1(T248, tree(T252, T250, T253)) :- qc66(T252, T248, T253).
inc1(s(T270), tree(0, T262, T271)) :- inc1(s(T270), T271).
inc1(s(T283), tree(s(T284), T262, T285)) :- qc73(T284, T283, T285).
lessc43(0, s(T110)).
lessc43(s(T115), s(T117)) :- lessc43(T115, T117).
lessc56(0, s(T149)).
lessc56(s(T156), s(T155)) :- lessc56(T156, T155).
qc41(T97, T99, T103) :- ','(lessc43(T97, T99), inc1(s(T97), T103)).
qc66(0, s(T179), T180) :- inc1(s(T179), T180).
qc66(s(T193), s(T192), T194) :- qc73(T193, T192, T194).
qc73(T193, T192, T197) :- ','(lessc56(T193, T192), inc1(s(T192), T197)).
lessc23(s(T65)).

Afs:

in1(x1, x2)  =  in1(x1)

(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:
in1_in: (b,f)
in11_in: (f)
p41_in: (b,f,f)
less43_in: (b,f)
lessc43_in: (b,f)
less56_in: (f,b)
lessc56_in: (f,b)
p66_in: (f,b,f)
p73_in: (f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

IN1_IN_GA(0, tree(s(T23), T24, T16)) → U13_GA(T23, T24, T16, in11_in_a(T24))
IN1_IN_GA(0, tree(s(T23), T24, T16)) → IN11_IN_A(T24)
IN11_IN_A(tree(T54, T58, T53)) → U1_A(T54, T58, T53, lessc23_in_a(T54))
U1_A(T54, T58, T53, lessc23_out_a(T54)) → U2_A(T54, T58, T53, in11_in_a(T58))
U1_A(T54, T58, T53, lessc23_out_a(T54)) → IN11_IN_A(T58)
IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → U14_GA(T97, T99, T100, T16, p41_in_gaa(T97, T99, T100))
IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → P41_IN_GAA(T97, T99, T100)
P41_IN_GAA(T97, T99, T100) → U5_GAA(T97, T99, T100, less43_in_ga(T97, T99))
P41_IN_GAA(T97, T99, T100) → LESS43_IN_GA(T97, T99)
LESS43_IN_GA(s(T115), s(T117)) → U3_GA(T115, T117, less43_in_ga(T115, T117))
LESS43_IN_GA(s(T115), s(T117)) → LESS43_IN_GA(T115, T117)
P41_IN_GAA(T97, T99, T103) → U6_GAA(T97, T99, T103, lessc43_in_ga(T97, T99))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → U7_GAA(T97, T99, T103, in1_in_ga(s(T97), T103))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → IN1_IN_GA(s(T97), T103)
IN1_IN_GA(T134, tree(T138, T136, T139)) → U15_GA(T134, T138, T136, T139, less56_in_ag(T138, T134))
IN1_IN_GA(T134, tree(T138, T136, T139)) → LESS56_IN_AG(T138, T134)
LESS56_IN_AG(s(T156), s(T155)) → U4_AG(T156, T155, less56_in_ag(T156, T155))
LESS56_IN_AG(s(T156), s(T155)) → LESS56_IN_AG(T156, T155)
IN1_IN_GA(T134, tree(T138, T136, T142)) → U16_GA(T134, T138, T136, T142, lessc56_in_ag(T138, T134))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → U17_GA(T134, T138, T136, T142, in1_in_ga(T134, T142))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134, T142)
IN1_IN_GA(T169, tree(T173, T171, T174)) → U18_GA(T169, T173, T171, T174, p66_in_aga(T173, T169, T174))
IN1_IN_GA(T169, tree(T173, T171, T174)) → P66_IN_AGA(T173, T169, T174)
P66_IN_AGA(0, s(T179), T180) → U8_AGA(T179, T180, in1_in_ga(s(T179), T180))
P66_IN_AGA(0, s(T179), T180) → IN1_IN_GA(s(T179), T180)
IN1_IN_GA(0, tree(s(T220), T221, T213)) → U19_GA(T220, T221, T213, in11_in_a(T221))
IN1_IN_GA(s(T234), tree(s(T236), T237, T213)) → U20_GA(T234, T236, T237, T213, p41_in_gaa(T234, T236, T237))
IN1_IN_GA(T248, tree(T252, T250, T253)) → U21_GA(T248, T252, T250, T253, p66_in_aga(T252, T248, T253))
P66_IN_AGA(s(T193), s(T192), T194) → U9_AGA(T193, T192, T194, p73_in_aga(T193, T192, T194))
P66_IN_AGA(s(T193), s(T192), T194) → P73_IN_AGA(T193, T192, T194)
P73_IN_AGA(T193, T192, T194) → U10_AGA(T193, T192, T194, less56_in_ag(T193, T192))
P73_IN_AGA(T193, T192, T194) → LESS56_IN_AG(T193, T192)
P73_IN_AGA(T193, T192, T197) → U11_AGA(T193, T192, T197, lessc56_in_ag(T193, T192))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → U12_AGA(T193, T192, T197, in1_in_ga(s(T192), T197))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192), T197)
IN1_IN_GA(s(T270), tree(0, T262, T271)) → U22_GA(T270, T262, T271, in1_in_ga(s(T270), T271))
IN1_IN_GA(s(T270), tree(0, T262, T271)) → IN1_IN_GA(s(T270), T271)
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → U23_GA(T283, T284, T262, T285, p73_in_aga(T284, T283, T285))
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → P73_IN_AGA(T284, T283, T285)

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
in1_in_ga(x1, x2)  =  in1_in_ga(x1)
0  =  0
in11_in_a(x1)  =  in11_in_a
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
p41_in_gaa(x1, x2, x3)  =  p41_in_gaa(x1)
less43_in_ga(x1, x2)  =  less43_in_ga(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
less56_in_ag(x1, x2)  =  less56_in_ag(x2)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
p66_in_aga(x1, x2, x3)  =  p66_in_aga(x2)
p73_in_aga(x1, x2, x3)  =  p73_in_aga(x2)
IN1_IN_GA(x1, x2)  =  IN1_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
IN11_IN_A(x1)  =  IN11_IN_A
U1_A(x1, x2, x3, x4)  =  U1_A(x4)
U2_A(x1, x2, x3, x4)  =  U2_A(x4)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x5)
P41_IN_GAA(x1, x2, x3)  =  P41_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
LESS43_IN_GA(x1, x2)  =  LESS43_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x5)
LESS56_IN_AG(x1, x2)  =  LESS56_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x5)
P66_IN_AGA(x1, x2, x3)  =  P66_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x1, x3)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x5)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
P73_IN_AGA(x1, x2, x3)  =  P73_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x2, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x5)

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:

IN1_IN_GA(0, tree(s(T23), T24, T16)) → U13_GA(T23, T24, T16, in11_in_a(T24))
IN1_IN_GA(0, tree(s(T23), T24, T16)) → IN11_IN_A(T24)
IN11_IN_A(tree(T54, T58, T53)) → U1_A(T54, T58, T53, lessc23_in_a(T54))
U1_A(T54, T58, T53, lessc23_out_a(T54)) → U2_A(T54, T58, T53, in11_in_a(T58))
U1_A(T54, T58, T53, lessc23_out_a(T54)) → IN11_IN_A(T58)
IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → U14_GA(T97, T99, T100, T16, p41_in_gaa(T97, T99, T100))
IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → P41_IN_GAA(T97, T99, T100)
P41_IN_GAA(T97, T99, T100) → U5_GAA(T97, T99, T100, less43_in_ga(T97, T99))
P41_IN_GAA(T97, T99, T100) → LESS43_IN_GA(T97, T99)
LESS43_IN_GA(s(T115), s(T117)) → U3_GA(T115, T117, less43_in_ga(T115, T117))
LESS43_IN_GA(s(T115), s(T117)) → LESS43_IN_GA(T115, T117)
P41_IN_GAA(T97, T99, T103) → U6_GAA(T97, T99, T103, lessc43_in_ga(T97, T99))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → U7_GAA(T97, T99, T103, in1_in_ga(s(T97), T103))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → IN1_IN_GA(s(T97), T103)
IN1_IN_GA(T134, tree(T138, T136, T139)) → U15_GA(T134, T138, T136, T139, less56_in_ag(T138, T134))
IN1_IN_GA(T134, tree(T138, T136, T139)) → LESS56_IN_AG(T138, T134)
LESS56_IN_AG(s(T156), s(T155)) → U4_AG(T156, T155, less56_in_ag(T156, T155))
LESS56_IN_AG(s(T156), s(T155)) → LESS56_IN_AG(T156, T155)
IN1_IN_GA(T134, tree(T138, T136, T142)) → U16_GA(T134, T138, T136, T142, lessc56_in_ag(T138, T134))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → U17_GA(T134, T138, T136, T142, in1_in_ga(T134, T142))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134, T142)
IN1_IN_GA(T169, tree(T173, T171, T174)) → U18_GA(T169, T173, T171, T174, p66_in_aga(T173, T169, T174))
IN1_IN_GA(T169, tree(T173, T171, T174)) → P66_IN_AGA(T173, T169, T174)
P66_IN_AGA(0, s(T179), T180) → U8_AGA(T179, T180, in1_in_ga(s(T179), T180))
P66_IN_AGA(0, s(T179), T180) → IN1_IN_GA(s(T179), T180)
IN1_IN_GA(0, tree(s(T220), T221, T213)) → U19_GA(T220, T221, T213, in11_in_a(T221))
IN1_IN_GA(s(T234), tree(s(T236), T237, T213)) → U20_GA(T234, T236, T237, T213, p41_in_gaa(T234, T236, T237))
IN1_IN_GA(T248, tree(T252, T250, T253)) → U21_GA(T248, T252, T250, T253, p66_in_aga(T252, T248, T253))
P66_IN_AGA(s(T193), s(T192), T194) → U9_AGA(T193, T192, T194, p73_in_aga(T193, T192, T194))
P66_IN_AGA(s(T193), s(T192), T194) → P73_IN_AGA(T193, T192, T194)
P73_IN_AGA(T193, T192, T194) → U10_AGA(T193, T192, T194, less56_in_ag(T193, T192))
P73_IN_AGA(T193, T192, T194) → LESS56_IN_AG(T193, T192)
P73_IN_AGA(T193, T192, T197) → U11_AGA(T193, T192, T197, lessc56_in_ag(T193, T192))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → U12_AGA(T193, T192, T197, in1_in_ga(s(T192), T197))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192), T197)
IN1_IN_GA(s(T270), tree(0, T262, T271)) → U22_GA(T270, T262, T271, in1_in_ga(s(T270), T271))
IN1_IN_GA(s(T270), tree(0, T262, T271)) → IN1_IN_GA(s(T270), T271)
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → U23_GA(T283, T284, T262, T285, p73_in_aga(T284, T283, T285))
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → P73_IN_AGA(T284, T283, T285)

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
in1_in_ga(x1, x2)  =  in1_in_ga(x1)
0  =  0
in11_in_a(x1)  =  in11_in_a
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
p41_in_gaa(x1, x2, x3)  =  p41_in_gaa(x1)
less43_in_ga(x1, x2)  =  less43_in_ga(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
less56_in_ag(x1, x2)  =  less56_in_ag(x2)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
p66_in_aga(x1, x2, x3)  =  p66_in_aga(x2)
p73_in_aga(x1, x2, x3)  =  p73_in_aga(x2)
IN1_IN_GA(x1, x2)  =  IN1_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
IN11_IN_A(x1)  =  IN11_IN_A
U1_A(x1, x2, x3, x4)  =  U1_A(x4)
U2_A(x1, x2, x3, x4)  =  U2_A(x4)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x5)
P41_IN_GAA(x1, x2, x3)  =  P41_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x4)
LESS43_IN_GA(x1, x2)  =  LESS43_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x5)
LESS56_IN_AG(x1, x2)  =  LESS56_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x5)
P66_IN_AGA(x1, x2, x3)  =  P66_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x1, x3)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x5)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
P73_IN_AGA(x1, x2, x3)  =  P73_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x2, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 23 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS56_IN_AG(s(T156), s(T155)) → LESS56_IN_AG(T156, T155)

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
0  =  0
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
LESS56_IN_AG(x1, x2)  =  LESS56_IN_AG(x2)

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:

LESS56_IN_AG(s(T156), s(T155)) → LESS56_IN_AG(T156, T155)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS56_IN_AG(x1, x2)  =  LESS56_IN_AG(x2)

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

(12) PiDPToQDPProof (SOUND 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:

LESS56_IN_AG(s(T155)) → LESS56_IN_AG(T155)

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:

• LESS56_IN_AG(s(T155)) → LESS56_IN_AG(T155)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESS43_IN_GA(s(T115), s(T117)) → LESS43_IN_GA(T115, T117)

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
0  =  0
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
LESS43_IN_GA(x1, x2)  =  LESS43_IN_GA(x1)

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:

LESS43_IN_GA(s(T115), s(T117)) → LESS43_IN_GA(T115, T117)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS43_IN_GA(x1, x2)  =  LESS43_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND 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:

LESS43_IN_GA(s(T115)) → LESS43_IN_GA(T115)

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

(21) 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:

• LESS43_IN_GA(s(T115)) → LESS43_IN_GA(T115)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

U1_A(T54, T58, T53, lessc23_out_a(T54)) → IN11_IN_A(T58)
IN11_IN_A(tree(T54, T58, T53)) → U1_A(T54, T58, T53, lessc23_in_a(T54))

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
0  =  0
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
IN11_IN_A(x1)  =  IN11_IN_A
U1_A(x1, x2, x3, x4)  =  U1_A(x4)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

U1_A(T54, T58, T53, lessc23_out_a(T54)) → IN11_IN_A(T58)
IN11_IN_A(tree(T54, T58, T53)) → U1_A(T54, T58, T53, lessc23_in_a(T54))

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))

The argument filtering Pi contains the following mapping:
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
IN11_IN_A(x1)  =  IN11_IN_A
U1_A(x1, x2, x3, x4)  =  U1_A(x4)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U1_A(lessc23_out_a) → IN11_IN_A
IN11_IN_A → U1_A(lessc23_in_a)

The TRS R consists of the following rules:

lessc23_in_a → lessc23_out_a

The set Q consists of the following terms:

lessc23_in_a

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

(28) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule IN11_IN_A → U1_A(lessc23_in_a) at position [0] we obtained the following new rules [LPAR04]:

IN11_IN_A → U1_A(lessc23_out_a)

(29) Obligation:

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

U1_A(lessc23_out_a) → IN11_IN_A
IN11_IN_A → U1_A(lessc23_out_a)

The TRS R consists of the following rules:

lessc23_in_a → lessc23_out_a

The set Q consists of the following terms:

lessc23_in_a

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

(30) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(31) Obligation:

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

U1_A(lessc23_out_a) → IN11_IN_A
IN11_IN_A → U1_A(lessc23_out_a)

R is empty.
The set Q consists of the following terms:

lessc23_in_a

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

(32) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

lessc23_in_a

(33) Obligation:

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

U1_A(lessc23_out_a) → IN11_IN_A
IN11_IN_A → U1_A(lessc23_out_a)

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

(34) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = IN11_IN_A evaluates to t =IN11_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

IN11_IN_A → U1_A(lessc23_out_a)
with rule IN11_IN_A → U1_A(lessc23_out_a) at position [] and matcher [ ]

U1_A(lessc23_out_a) → IN11_IN_A
with rule U1_A(lessc23_out_a) → IN11_IN_A

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(35) NO

(36) Obligation:

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

IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → P41_IN_GAA(T97, T99, T100)
P41_IN_GAA(T97, T99, T103) → U6_GAA(T97, T99, T103, lessc43_in_ga(T97, T99))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → IN1_IN_GA(s(T97), T103)
IN1_IN_GA(T134, tree(T138, T136, T142)) → U16_GA(T134, T138, T136, T142, lessc56_in_ag(T138, T134))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134, T142)
IN1_IN_GA(T169, tree(T173, T171, T174)) → P66_IN_AGA(T173, T169, T174)
P66_IN_AGA(0, s(T179), T180) → IN1_IN_GA(s(T179), T180)
IN1_IN_GA(s(T270), tree(0, T262, T271)) → IN1_IN_GA(s(T270), T271)
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → P73_IN_AGA(T284, T283, T285)
P73_IN_AGA(T193, T192, T197) → U11_AGA(T193, T192, T197, lessc56_in_ag(T193, T192))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192), T197)
P66_IN_AGA(s(T193), s(T192), T194) → P73_IN_AGA(T193, T192, T194)

The TRS R consists of the following rules:

lessc23_in_a(s(T65)) → lessc23_out_a(s(T65))
lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
0  =  0
lessc23_in_a(x1)  =  lessc23_in_a
lessc23_out_a(x1)  =  lessc23_out_a
s(x1)  =  s(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
IN1_IN_GA(x1, x2)  =  IN1_IN_GA(x1)
P41_IN_GAA(x1, x2, x3)  =  P41_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x5)
P66_IN_AGA(x1, x2, x3)  =  P66_IN_AGA(x2)
P73_IN_AGA(x1, x2, x3)  =  P73_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

IN1_IN_GA(s(T97), tree(s(T99), T100, T16)) → P41_IN_GAA(T97, T99, T100)
P41_IN_GAA(T97, T99, T103) → U6_GAA(T97, T99, T103, lessc43_in_ga(T97, T99))
U6_GAA(T97, T99, T103, lessc43_out_ga(T97, T99)) → IN1_IN_GA(s(T97), T103)
IN1_IN_GA(T134, tree(T138, T136, T142)) → U16_GA(T134, T138, T136, T142, lessc56_in_ag(T138, T134))
U16_GA(T134, T138, T136, T142, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134, T142)
IN1_IN_GA(T169, tree(T173, T171, T174)) → P66_IN_AGA(T173, T169, T174)
P66_IN_AGA(0, s(T179), T180) → IN1_IN_GA(s(T179), T180)
IN1_IN_GA(s(T270), tree(0, T262, T271)) → IN1_IN_GA(s(T270), T271)
IN1_IN_GA(s(T283), tree(s(T284), T262, T285)) → P73_IN_AGA(T284, T283, T285)
P73_IN_AGA(T193, T192, T197) → U11_AGA(T193, T192, T197, lessc56_in_ag(T193, T192))
U11_AGA(T193, T192, T197, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192), T197)
P66_IN_AGA(s(T193), s(T192), T194) → P73_IN_AGA(T193, T192, T194)

The TRS R consists of the following rules:

lessc43_in_ga(0, s(T110)) → lessc43_out_ga(0, s(T110))
lessc43_in_ga(s(T115), s(T117)) → U37_ga(T115, T117, lessc43_in_ga(T115, T117))
lessc56_in_ag(0, s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T156), s(T155)) → U38_ag(T156, T155, lessc56_in_ag(T156, T155))
U37_ga(T115, T117, lessc43_out_ga(T115, T117)) → lessc43_out_ga(s(T115), s(T117))
U38_ag(T156, T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
lessc43_in_ga(x1, x2)  =  lessc43_in_ga(x1)
lessc43_out_ga(x1, x2)  =  lessc43_out_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
lessc56_in_ag(x1, x2)  =  lessc56_in_ag(x2)
lessc56_out_ag(x1, x2)  =  lessc56_out_ag(x1, x2)
U38_ag(x1, x2, x3)  =  U38_ag(x2, x3)
IN1_IN_GA(x1, x2)  =  IN1_IN_GA(x1)
P41_IN_GAA(x1, x2, x3)  =  P41_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x5)
P66_IN_AGA(x1, x2, x3)  =  P66_IN_AGA(x2)
P73_IN_AGA(x1, x2, x3)  =  P73_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
P41_IN_GAA(T97) → U6_GAA(T97, lessc43_in_ga(T97))
U6_GAA(T97, lessc43_out_ga(T97)) → IN1_IN_GA(s(T97))
IN1_IN_GA(T134) → U16_GA(T134, lessc56_in_ag(T134))
U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
P73_IN_AGA(T192) → U11_AGA(T192, lessc56_in_ag(T192))
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(41) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P41_IN_GAA(T97) → U6_GAA(T97, lessc43_in_ga(T97)) at position [1] we obtained the following new rules [LPAR04]:

P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))

(42) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
U6_GAA(T97, lessc43_out_ga(T97)) → IN1_IN_GA(s(T97))
IN1_IN_GA(T134) → U16_GA(T134, lessc56_in_ag(T134))
U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
P73_IN_AGA(T192) → U11_AGA(T192, lessc56_in_ag(T192))
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(43) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule IN1_IN_GA(T134) → U16_GA(T134, lessc56_in_ag(T134)) at position [1] we obtained the following new rules [LPAR04]:

IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))

(44) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
U6_GAA(T97, lessc43_out_ga(T97)) → IN1_IN_GA(s(T97))
U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
P73_IN_AGA(T192) → U11_AGA(T192, lessc56_in_ag(T192))
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(45) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P73_IN_AGA(T192) → U11_AGA(T192, lessc56_in_ag(T192)) at position [1] we obtained the following new rules [LPAR04]:

P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))

(46) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
U6_GAA(T97, lessc43_out_ga(T97)) → IN1_IN_GA(s(T97))
U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(47) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GAA(T97, lessc43_out_ga(T97)) → IN1_IN_GA(s(T97)) we obtained the following new rules [LPAR04]:

U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))

(48) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(49) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_GA(T134, lessc56_out_ag(T138, T134)) → IN1_IN_GA(T134) we obtained the following new rules [LPAR04]:

U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))

(50) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
IN1_IN_GA(T169) → P66_IN_AGA(T169)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(51) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IN1_IN_GA(T169) → P66_IN_AGA(T169) we obtained the following new rules [LPAR04]:

IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))

(52) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))
IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(53) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U11_AGA(T192, lessc56_out_ag(T193, T192)) → IN1_IN_GA(s(T192)) we obtained the following new rules [LPAR04]:

U11_AGA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(s(z0)))

(54) Obligation:

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

IN1_IN_GA(s(T97)) → P41_IN_GAA(T97)
P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))
IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(s(z0)))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(55) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule IN1_IN_GA(s(T97)) → P41_IN_GAA(T97) we obtained the following new rules [LPAR04]:

IN1_IN_GA(s(0)) → P41_IN_GAA(0)
IN1_IN_GA(s(s(y_0))) → P41_IN_GAA(s(y_0))

(56) Obligation:

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

P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
IN1_IN_GA(s(T283)) → P73_IN_AGA(T283)
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))
IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(s(z0)))
IN1_IN_GA(s(0)) → P41_IN_GAA(0)
IN1_IN_GA(s(s(y_0))) → P41_IN_GAA(s(y_0))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(57) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule IN1_IN_GA(s(T283)) → P73_IN_AGA(T283) we obtained the following new rules [LPAR04]:

IN1_IN_GA(s(s(y_0))) → P73_IN_AGA(s(y_0))

(58) Obligation:

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

P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
P66_IN_AGA(s(T192)) → P73_IN_AGA(T192)
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))
IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(s(z0)))
IN1_IN_GA(s(0)) → P41_IN_GAA(0)
IN1_IN_GA(s(s(y_0))) → P41_IN_GAA(s(y_0))
IN1_IN_GA(s(s(y_0))) → P73_IN_AGA(s(y_0))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(59) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P66_IN_AGA(s(T192)) → P73_IN_AGA(T192) we obtained the following new rules [LPAR04]:

P66_IN_AGA(s(s(y_0))) → P73_IN_AGA(s(y_0))

(60) Obligation:

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

P66_IN_AGA(s(T179)) → IN1_IN_GA(s(T179))
IN1_IN_GA(s(T270)) → IN1_IN_GA(s(T270))
P41_IN_GAA(0) → U6_GAA(0, lessc43_out_ga(0))
P41_IN_GAA(s(x0)) → U6_GAA(s(x0), U37_ga(x0, lessc43_in_ga(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), lessc56_out_ag(0, s(x0)))
IN1_IN_GA(s(x0)) → U16_GA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), lessc56_out_ag(0, s(x0)))
P73_IN_AGA(s(x0)) → U11_AGA(s(x0), U38_ag(x0, lessc56_in_ag(x0)))
U6_GAA(0, lessc43_out_ga(0)) → IN1_IN_GA(s(0))
U6_GAA(s(z0), lessc43_out_ga(s(z0))) → IN1_IN_GA(s(s(z0)))
U16_GA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(z0))
U16_GA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(z0))
IN1_IN_GA(s(z0)) → P66_IN_AGA(s(z0))
IN1_IN_GA(s(0)) → P66_IN_AGA(s(0))
IN1_IN_GA(s(s(z0))) → P66_IN_AGA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(0, s(z0))) → IN1_IN_GA(s(s(z0)))
U11_AGA(s(z0), lessc56_out_ag(x1, s(z0))) → IN1_IN_GA(s(s(z0)))
IN1_IN_GA(s(0)) → P41_IN_GAA(0)
IN1_IN_GA(s(s(y_0))) → P41_IN_GAA(s(y_0))
IN1_IN_GA(s(s(y_0))) → P73_IN_AGA(s(y_0))
P66_IN_AGA(s(s(y_0))) → P73_IN_AGA(s(y_0))

The TRS R consists of the following rules:

lessc43_in_ga(0) → lessc43_out_ga(0)
lessc43_in_ga(s(T115)) → U37_ga(T115, lessc43_in_ga(T115))
lessc56_in_ag(s(T149)) → lessc56_out_ag(0, s(T149))
lessc56_in_ag(s(T155)) → U38_ag(T155, lessc56_in_ag(T155))
U37_ga(T115, lessc43_out_ga(T115)) → lessc43_out_ga(s(T115))
U38_ag(T155, lessc56_out_ag(T156, T155)) → lessc56_out_ag(s(T156), s(T155))

The set Q consists of the following terms:

lessc43_in_ga(x0)
lessc56_in_ag(x0)
U37_ga(x0, x1)
U38_ag(x0, x1)

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

(61) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = IN1_IN_GA(s(T270)) evaluates to t =IN1_IN_GA(s(T270))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from IN1_IN_GA(s(T270)) to IN1_IN_GA(s(T270)).

(62) NO