Left Termination of the query pattern insert_in_3(a, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

insert(X, void, tree(X, void, void)).
insert(X, tree(X, Left, Right), tree(X, Left, Right)).
insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)).
insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

insert(a,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: insert(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | insert(T1, void, T3), SCOPE: 1, CLAUSE: 1 | insert(T1, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, void, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: insert(T1, void, T3), SCOPE: 1, CLAUSE: 1 | insert(T1, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, void, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: insert(T1, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, void, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: insert(T1, void, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | insert(T1, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 2 | insert(T1, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: insert(T1, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 2 | insert(T1, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(less(T25, T21), insert(T25, T22, T26)) | insert(T1, tree(T21, T22, T23), T3), SCOPE: 1, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: insert(T1, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\ninsert(T1, tree(T12, T13, T14), T3) !=? insert(X32, tree(X33, X34, X35), tree(X33, X36, X35))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(less(T25, T21), insert(T25, T22, T26)), SCOPE: 2, CLAUSE: 4 | ','(less(T25, T21), insert(T25, T22, T26)), SCOPE: 2, CLAUSE: 5 | ?_2 | insert(T1, tree(T21, T22, T23), T3), SCOPE: 1, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(less(T25, T21), insert(T25, T22, T26)), SCOPE: 2, CLAUSE: 4\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(less(T25, T21), insert(T25, T22, T26)), SCOPE: 2, CLAUSE: 5 | ?_2 | insert(T1, tree(T21, T22, T23), T3), SCOPE: 1, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: insert(0, T22, T32)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(less(T25, T21), insert(T25, T22, T26)), SCOPE: 2, CLAUSE: 5\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ?_2 | insert(T1, tree(T21, T22, T23), T3), SCOPE: 1, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(less(T43, T42), insert(s(T43), T22, T44))\n\nT22 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T43, T42)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: insert(s(T47), T22, T48)\n\nT22 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(T43, T42), SCOPE: 3, CLAUSE: 4 | less(T43, T42), SCOPE: 3, CLAUSE: 5\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T43, T42), SCOPE: 3, CLAUSE: 4\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T43, T42), SCOPE: 3, CLAUSE: 5\n\nT42 is ground", 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: less(T62, T61)\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: insert(T1, tree(T21, T22, T23), T3), SCOPE: 1, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(less(T78, T82), insert(T82, T80, T83))\n\nT78 is ground\nT80 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T78, T82)\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: insert(T86, T80, T87)\n\nT80 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: less(T78, T82), SCOPE: 4, CLAUSE: 4 | less(T78, T82), SCOPE: 4, CLAUSE: 5\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: less(T78, T82), SCOPE: 4, CLAUSE: 4\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T78, T82), SCOPE: 4, CLAUSE: 5\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T99, T101)\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(less(T112, T116), insert(T116, T114, T117))\n\nT112 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 4 | ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 5\n\nT112 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 4\n\nT112 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 5\n\nT112 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: insert(s(T124), T114, T123)\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(less(T131, T133), insert(s(T133), T114, T134))\n\nT114 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(T131, T133)\n\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: insert(s(T137), T114, T138)\n\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(less(T153, T149), insert(T153, T150, T154)) | insert(T1, tree(T149, T150, T151), T3), SCOPE: 1, CLAUSE: 3\n\nT149 is ground\nT150 is ground\nT151 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))\ninsert(T1, T2, T3) !=? insert(X168, tree(X169, X170, X171), tree(X169, X172, X171))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(less(T153, T149), insert(T153, T150, T154)), SCOPE: 6, CLAUSE: 4 | ','(less(T153, T149), insert(T153, T150, T154)), SCOPE: 6, CLAUSE: 5 | ?_6 | insert(T1, tree(T149, T150, T151), T3), SCOPE: 1, CLAUSE: 3\n\nT149 is ground\nT150 is ground\nT151 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(less(T153, T149), insert(T153, T150, T154)), SCOPE: 6, CLAUSE: 4\n\nT149 is ground\nT150 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(less(T153, T149), insert(T153, T150, T154)), SCOPE: 6, CLAUSE: 5 | ?_6 | insert(T1, tree(T149, T150, T151), T3), SCOPE: 1, CLAUSE: 3\n\nT149 is ground\nT150 is ground\nT151 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: insert(0, T150, T160)\n\nT150 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(T153, T149), insert(T153, T150, T154)), SCOPE: 6, CLAUSE: 5\n\nT149 is ground\nT150 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ?_6 | insert(T1, tree(T149, T150, T151), T3), SCOPE: 1, CLAUSE: 3\n\nT149 is ground\nT150 is ground\nT151 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(less(T171, T170), insert(s(T171), T150, T172))\n\nT150 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: insert(T1, tree(T149, T150, T151), T3), SCOPE: 1, CLAUSE: 3\n\nT149 is ground\nT150 is ground\nT151 is ground\ninsert(T1, tree(T149, T150, T151), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(less(T186, T190), insert(T190, T188, T191))\n\nT186 is ground\nT188 is ground\ninsert(T1, tree(T186, T187, T188), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(less(T200, T204), insert(T204, T202, T205))\n\nT200 is ground\nT202 is ground\ninsert(T1, tree(T200, T201, T202), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(less(T200, T204), insert(T204, T202, T205)), SCOPE: 7, CLAUSE: 4 | ','(less(T200, T204), insert(T204, T202, T205)), SCOPE: 7, CLAUSE: 5\n\nT200 is ground\nT202 is ground\ninsert(T1, tree(T200, T201, T202), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(less(T200, T204), insert(T204, T202, T205)), SCOPE: 7, CLAUSE: 4\n\nT200 is ground\nT202 is ground\ninsert(T1, tree(T200, T201, T202), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(less(T200, T204), insert(T204, T202, T205)), SCOPE: 7, CLAUSE: 5\n\nT200 is ground\nT202 is ground\ninsert(T1, tree(T200, T201, T202), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: insert(s(T212), T202, T211)\n\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(less(T219, T221), insert(s(T221), T202, T222))\n\nT202 is ground\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 80 [arrowhead = none ];
80 -> 2;

81 [label="EVAL with clause\ninsert(X3, void, tree(X3, void, void)).\nand substitution\nT1 -> T5\nX3 -> T5\nT2 -> void\nT3 -> tree(T5, void, void)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 81 [arrowhead = none ];
81 -> 3;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 82 [arrowhead = none ];
82 -> 4;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 83 [arrowhead = none ];
83 -> 5;

84 [label="EVAL with clause\ninsert(X21, tree(X21, X22, X23), tree(X21, X22, X23)).\nand substitution\nT1 -> T12\nX21 -> T12\nX22 -> T13\nX23 -> T14\nT2 -> tree(T12, T13, T14)\nT3 -> tree(T12, T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 84 [arrowhead = none ];
84 -> 9;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 85 [arrowhead = none ];
85 -> 10;

86 [label="BACKTRACK\nfor clause: insert(X, tree(X, Left, Right), tree(X, Left, Right))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 86 [arrowhead = none ];
86 -> 6;

87 [label="BACKTRACK\nfor clause: insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 87 [arrowhead = none ];
87 -> 7;

88 [label="BACKTRACK\nfor clause: insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
7 -> 88 [arrowhead = none ];
88 -> 8;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 89 [arrowhead = none ];
89 -> 11;

90 [label="EVAL with clause\ninsert(X168, tree(X169, X170, X171), tree(X169, X172, X171)) :- ','(less(X168, X169), insert(X168, X170, X172)).\nand substitution\nT1 -> T153\nX168 -> T153\nX169 -> T149\nX170 -> T150\nX171 -> T151\nT2 -> tree(T149, T150, T151)\nX172 -> T154\nT3 -> tree(T149, T154, T151)\nT148 -> T153\nT152 -> T154", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 90 [arrowhead = none ];
90 -> 57;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 91 [arrowhead = none ];
91 -> 58;

92 [label="EVAL with clause\ninsert(X32, tree(X33, X34, X35), tree(X33, X36, X35)) :- ','(less(X32, X33), insert(X32, X34, X36)).\nand substitution\nT1 -> T25\nX32 -> T25\nT12 -> T21\nX33 -> T21\nT13 -> T22\nX34 -> T22\nT14 -> T23\nX35 -> T23\nX36 -> T26\nT3 -> tree(T21, T26, T23)\nT20 -> T25\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 92 [arrowhead = none ];
92 -> 12;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 93 [arrowhead = none ];
93 -> 13;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 94 [arrowhead = none ];
94 -> 14;

95 [label="EVAL with clause\ninsert(X131, tree(X132, X133, X134), tree(X132, X133, X135)) :- ','(less(X132, X131), insert(X131, X134, X135)).\nand substitution\nT1 -> T116\nX131 -> T116\nT12 -> T112\nX132 -> T112\nT13 -> T113\nX133 -> T113\nT14 -> T114\nX134 -> T114\nX135 -> T117\nT3 -> tree(T112, T113, T117)\nT111 -> T116\nT115 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 95 [arrowhead = none ];
95 -> 46;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 96 [arrowhead = none ];
96 -> 47;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 97 [arrowhead = none ];
97 -> 15;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 98 [arrowhead = none ];
98 -> 16;

99 [label="EVAL with clause\nless(0, s(X46)).\nand substitution\nT25 -> 0\nX46 -> T31\nT21 -> s(T31)\nT26 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 99 [arrowhead = none ];
99 -> 17;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 100 [arrowhead = none ];
100 -> 18;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 101 [arrowhead = none ];
101 -> 19;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 102 [arrowhead = none ];
102 -> 20;

103 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T22\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 103 [arrowhead = none , style=dashed];
103 -> 1[style=dashed];

104 [label="EVAL with clause\nless(s(X58), s(X59)) :- less(X58, X59).\nand substitution\nX58 -> T43\nT25 -> s(T43)\nX59 -> T42\nT21 -> s(T42)\nT41 -> T43\nT26 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 104 [arrowhead = none ];
104 -> 21;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 105 [arrowhead = none ];
105 -> 22;

106 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 106 [arrowhead = none ];
106 -> 33;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 107 [arrowhead = none ];
107 -> 23;

108 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nreplacements:\nT43 -> T47\nT44 -> T48", fontsize=14, style = filled, fillcolor = violet];
21 -> 108 [arrowhead = none ];
108 -> 24;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 109 [arrowhead = none ];
109 -> 25;

110 [label="INSTANCE with matching:\nT1 -> s(T47)\nT2 -> T22\nT3 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 110 [arrowhead = none , style=dashed];
110 -> 1[style=dashed];

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="EVAL with clause\nless(0, s(X70)).\nand substitution\nT43 -> 0\nX70 -> T55\nT42 -> s(T55)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 113 [arrowhead = none ];
113 -> 28;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 114 [arrowhead = none ];
114 -> 29;

115 [label="EVAL with clause\nless(s(X76), s(X77)) :- less(X76, X77).\nand substitution\nX76 -> T62\nT43 -> s(T62)\nX77 -> T61\nT42 -> s(T61)\nT60 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 115 [arrowhead = none ];
115 -> 31;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 116 [arrowhead = none ];
116 -> 32;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 117 [arrowhead = none ];
117 -> 30;

118 [label="INSTANCE with matching:\nT43 -> T62\nT42 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 118 [arrowhead = none , style=dashed];
118 -> 23[style=dashed];

119 [label="EVAL with clause\ninsert(X94, tree(X95, X96, X97), tree(X95, X96, X98)) :- ','(less(X95, X94), insert(X94, X97, X98)).\nand substitution\nT1 -> T82\nX94 -> T82\nT21 -> T78\nX95 -> T78\nT22 -> T79\nX96 -> T79\nT23 -> T80\nX97 -> T80\nX98 -> T83\nT3 -> tree(T78, T79, T83)\nT77 -> T82\nT81 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 119 [arrowhead = none ];
119 -> 34;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 120 [arrowhead = none ];
120 -> 35;

121 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 121 [arrowhead = none ];
121 -> 36;

122 [label="SPLIT 2\nreplacements:\nT82 -> T86\nT83 -> T87", fontsize=14, style = filled, fillcolor = violet];
34 -> 122 [arrowhead = none ];
122 -> 37;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 123 [arrowhead = none ];
123 -> 38;

124 [label="INSTANCE with matching:\nT1 -> T86\nT2 -> T80\nT3 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 124 [arrowhead = none , style=dashed];
124 -> 1[style=dashed];

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 125 [arrowhead = none ];
125 -> 39;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 126 [arrowhead = none ];
126 -> 40;

127 [label="EVAL with clause\nless(0, s(X112)).\nand substitution\nT78 -> 0\nX112 -> T94\nT82 -> s(T94)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 127 [arrowhead = none ];
127 -> 41;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 128 [arrowhead = none ];
128 -> 42;

129 [label="EVAL with clause\nless(s(X118), s(X119)) :- less(X118, X119).\nand substitution\nX118 -> T99\nT78 -> s(T99)\nX119 -> T101\nT82 -> s(T101)\nT100 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 129 [arrowhead = none ];
129 -> 44;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 130 [arrowhead = none ];
130 -> 45;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 131 [arrowhead = none ];
131 -> 43;

132 [label="INSTANCE with matching:\nT78 -> T99\nT82 -> T101", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 132 [arrowhead = none , style=dashed];
132 -> 36[style=dashed];

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 133 [arrowhead = none ];
133 -> 48;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 134 [arrowhead = none ];
134 -> 49;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 135 [arrowhead = none ];
135 -> 50;

136 [label="EVAL with clause\nless(0, s(X145)).\nand substitution\nT112 -> 0\nX145 -> T124\nT116 -> s(T124)\nT117 -> T123\nT122 -> T124", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 136 [arrowhead = none ];
136 -> 51;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 137 [arrowhead = none ];
137 -> 52;

138 [label="EVAL with clause\nless(s(X153), s(X154)) :- less(X153, X154).\nand substitution\nX153 -> T131\nT112 -> s(T131)\nX154 -> T133\nT116 -> s(T133)\nT132 -> T133\nT117 -> T134", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 138 [arrowhead = none ];
138 -> 53;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 139 [arrowhead = none ];
139 -> 54;

140 [label="INSTANCE with matching:\nT1 -> s(T124)\nT2 -> T114\nT3 -> T123", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 140 [arrowhead = none , style=dashed];
140 -> 1[style=dashed];

141 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 141 [arrowhead = none ];
141 -> 55;

142 [label="SPLIT 2\nreplacements:\nT133 -> T137\nT134 -> T138", fontsize=14, style = filled, fillcolor = violet];
53 -> 142 [arrowhead = none ];
142 -> 56;

143 [label="INSTANCE with matching:\nT78 -> T131\nT82 -> T133", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 143 [arrowhead = none , style=dashed];
143 -> 36[style=dashed];

144 [label="INSTANCE with matching:\nT1 -> s(T137)\nT2 -> T114\nT3 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 145 [arrowhead = none ];
145 -> 59;

146 [label="EVAL with clause\ninsert(X227, tree(X228, X229, X230), tree(X228, X229, X231)) :- ','(less(X228, X227), insert(X227, X230, X231)).\nand substitution\nT1 -> T204\nX227 -> T204\nX228 -> T200\nX229 -> T201\nX230 -> T202\nT2 -> tree(T200, T201, T202)\nX231 -> T205\nT3 -> tree(T200, T201, T205)\nT199 -> T204\nT203 -> T205", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 146 [arrowhead = none ];
146 -> 71;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 147 [arrowhead = none ];
147 -> 72;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 149 [arrowhead = none ];
149 -> 61;

150 [label="EVAL with clause\nless(0, s(X182)).\nand substitution\nT153 -> 0\nX182 -> T159\nT149 -> s(T159)\nT154 -> T160", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 150 [arrowhead = none ];
150 -> 62;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 151 [arrowhead = none ];
151 -> 63;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 152 [arrowhead = none ];
152 -> 64;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T150\nT3 -> T160", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 154 [arrowhead = none , style=dashed];
154 -> 1[style=dashed];

155 [label="EVAL with clause\nless(s(X194), s(X195)) :- less(X194, X195).\nand substitution\nX194 -> T171\nT153 -> s(T171)\nX195 -> T170\nT149 -> s(T170)\nT169 -> T171\nT154 -> T172", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 155 [arrowhead = none ];
155 -> 66;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 156 [arrowhead = none ];
156 -> 67;

157 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 157 [arrowhead = none ];
157 -> 68;

158 [label="INSTANCE with matching:\nT43 -> T171\nT42 -> T170\nT22 -> T150\nT44 -> T172", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 158 [arrowhead = none , style=dashed];
158 -> 21[style=dashed];

159 [label="EVAL with clause\ninsert(X210, tree(X211, X212, X213), tree(X211, X212, X214)) :- ','(less(X211, X210), insert(X210, X213, X214)).\nand substitution\nT1 -> T190\nX210 -> T190\nT149 -> T186\nX211 -> T186\nT150 -> T187\nX212 -> T187\nT151 -> T188\nX213 -> T188\nX214 -> T191\nT3 -> tree(T186, T187, T191)\nT185 -> T190\nT189 -> T191", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 159 [arrowhead = none ];
159 -> 69;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 160 [arrowhead = none ];
160 -> 70;

161 [label="INSTANCE with matching:\nT112 -> T186\nT116 -> T190\nT114 -> T188\nT117 -> T191", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 161 [arrowhead = none , style=dashed];
161 -> 46[style=dashed];

162 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 162 [arrowhead = none ];
162 -> 73;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 163 [arrowhead = none ];
163 -> 74;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 164 [arrowhead = none ];
164 -> 75;

165 [label="EVAL with clause\nless(0, s(X241)).\nand substitution\nT200 -> 0\nX241 -> T212\nT204 -> s(T212)\nT205 -> T211\nT210 -> T212", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 165 [arrowhead = none ];
165 -> 76;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 166 [arrowhead = none ];
166 -> 77;

167 [label="EVAL with clause\nless(s(X249), s(X250)) :- less(X249, X250).\nand substitution\nX249 -> T219\nT200 -> s(T219)\nX250 -> T221\nT204 -> s(T221)\nT220 -> T221\nT205 -> T222", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 167 [arrowhead = none ];
167 -> 78;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 168 [arrowhead = none ];
168 -> 79;

169 [label="INSTANCE with matching:\nT1 -> s(T212)\nT2 -> T202\nT3 -> T211", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 169 [arrowhead = none , style=dashed];
169 -> 1[style=dashed];

170 [label="INSTANCE with matching:\nT131 -> T219\nT133 -> T221\nT114 -> T202\nT134 -> T222", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 170 [arrowhead = none , style=dashed];
170 -> 53[style=dashed];

}

(2) Obligation:

Triples:

less23(s(T62), s(T61)) :- less23(T62, T61).
less36(s(T99), s(T101)) :- less36(T99, T101).
p21(T43, T42, T22, T44) :- less23(T43, T42).
p21(T47, T42, T22, T48) :- ','(lessc23(T47, T42), insert1(s(T47), T22, T48)).
p46(0, s(T124), T114, T123) :- insert1(s(T124), T114, T123).
p46(s(T131), s(T133), T114, T134) :- p53(T131, T133, T114, T134).
p53(T131, T133, T114, T134) :- less36(T131, T133).
p53(T131, T137, T114, T138) :- ','(lessc36(T131, T137), insert1(s(T137), T114, T138)).
insert1(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) :- insert1(0, T22, T32).
insert1(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) :- p21(T43, T42, T22, T44).
insert1(T82, tree(T78, T79, T80), tree(T78, T79, T83)) :- less36(T78, T82).
insert1(T86, tree(T78, T79, T80), tree(T78, T79, T87)) :- ','(lessc36(T78, T86), insert1(T86, T80, T87)).
insert1(T116, tree(T112, T113, T114), tree(T112, T113, T117)) :- p46(T112, T116, T114, T117).
insert1(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) :- insert1(0, T150, T160).
insert1(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) :- p21(T171, T170, T150, T172).
insert1(T190, tree(T186, T187, T188), tree(T186, T187, T191)) :- p46(T186, T190, T188, T191).
insert1(s(T212), tree(0, T201, T202), tree(0, T201, T211)) :- insert1(s(T212), T202, T211).
insert1(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) :- p53(T219, T221, T202, T222).

Clauses:

insertc1(T5, void, tree(T5, void, void)).
insertc1(T12, tree(T12, T13, T14), tree(T12, T13, T14)).
insertc1(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) :- insertc1(0, T22, T32).
insertc1(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) :- qc21(T43, T42, T22, T44).
insertc1(T86, tree(T78, T79, T80), tree(T78, T79, T87)) :- ','(lessc36(T78, T86), insertc1(T86, T80, T87)).
insertc1(T116, tree(T112, T113, T114), tree(T112, T113, T117)) :- qc46(T112, T116, T114, T117).
insertc1(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) :- insertc1(0, T150, T160).
insertc1(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) :- qc21(T171, T170, T150, T172).
insertc1(T190, tree(T186, T187, T188), tree(T186, T187, T191)) :- qc46(T186, T190, T188, T191).
insertc1(s(T212), tree(0, T201, T202), tree(0, T201, T211)) :- insertc1(s(T212), T202, T211).
insertc1(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) :- qc53(T219, T221, T202, T222).
lessc23(0, s(T55)).
lessc23(s(T62), s(T61)) :- lessc23(T62, T61).
lessc36(0, s(T94)).
lessc36(s(T99), s(T101)) :- lessc36(T99, T101).
qc21(T47, T42, T22, T48) :- ','(lessc23(T47, T42), insertc1(s(T47), T22, T48)).
qc46(0, s(T124), T114, T123) :- insertc1(s(T124), T114, T123).
qc46(s(T131), s(T133), T114, T134) :- qc53(T131, T133, T114, T134).
qc53(T131, T137, T114, T138) :- ','(lessc36(T131, T137), insertc1(s(T137), T114, T138)).

Afs:

insert1(x1, x2, x3)  =  insert1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
insert1_in: (f,b,f) (b,b,f)
p21_in: (b,b,b,f) (f,b,b,f)
less23_in: (b,b) (f,b)
lessc23_in: (b,b) (f,b)
less36_in: (b,b) (b,f)
lessc36_in: (b,b) (b,f)
p46_in: (b,b,b,f) (b,f,b,f)
p53_in: (b,b,b,f) (b,f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

INSERT1_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U11_AGA(T31, T22, T23, T32, insert1_in_gga(0, T22, T32))
INSERT1_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U11_GGA(T31, T22, T23, T32, insert1_in_gga(0, T22, T32))
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U12_GGA(T43, T42, T22, T23, T44, p21_in_ggga(T43, T42, T22, T44))
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_GGGA(T43, T42, T22, T44)
P21_IN_GGGA(T43, T42, T22, T44) → U3_GGGA(T43, T42, T22, T44, less23_in_gg(T43, T42))
P21_IN_GGGA(T43, T42, T22, T44) → LESS23_IN_GG(T43, T42)
LESS23_IN_GG(s(T62), s(T61)) → U1_GG(T62, T61, less23_in_gg(T62, T61))
LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)
P21_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessc23_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → U5_GGGA(T47, T42, T22, T48, insert1_in_gga(s(T47), T22, T48))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U13_GGA(T82, T78, T79, T80, T83, less36_in_gg(T78, T82))
INSERT1_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → LESS36_IN_GG(T78, T82)
LESS36_IN_GG(s(T99), s(T101)) → U2_GG(T99, T101, less36_in_gg(T99, T101))
LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)
INSERT1_IN_GGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_GGA(T86, T78, T79, T80, T87, lessc36_in_gg(T78, T86))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → U15_GGA(T86, T78, T79, T80, T87, insert1_in_gga(T86, T80, T87))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80, T87)
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → U16_GGA(T116, T112, T113, T114, T117, p46_in_ggga(T112, T116, T114, T117))
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GGGA(T112, T116, T114, T117)
P46_IN_GGGA(0, s(T124), T114, T123) → U6_GGGA(T124, T114, T123, insert1_in_gga(s(T124), T114, T123))
P46_IN_GGGA(0, s(T124), T114, T123) → INSERT1_IN_GGA(s(T124), T114, T123)
INSERT1_IN_GGA(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) → U17_GGA(T159, T150, T151, T160, insert1_in_gga(0, T150, T160))
INSERT1_IN_GGA(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) → U18_GGA(T171, T170, T150, T151, T172, p21_in_ggga(T171, T170, T150, T172))
INSERT1_IN_GGA(T190, tree(T186, T187, T188), tree(T186, T187, T191)) → U19_GGA(T190, T186, T187, T188, T191, p46_in_ggga(T186, T190, T188, T191))
P46_IN_GGGA(s(T131), s(T133), T114, T134) → U7_GGGA(T131, T133, T114, T134, p53_in_ggga(T131, T133, T114, T134))
P46_IN_GGGA(s(T131), s(T133), T114, T134) → P53_IN_GGGA(T131, T133, T114, T134)
P53_IN_GGGA(T131, T133, T114, T134) → U8_GGGA(T131, T133, T114, T134, less36_in_gg(T131, T133))
P53_IN_GGGA(T131, T133, T114, T134) → LESS36_IN_GG(T131, T133)
P53_IN_GGGA(T131, T137, T114, T138) → U9_GGGA(T131, T137, T114, T138, lessc36_in_gg(T131, T137))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → U10_GGGA(T131, T137, T114, T138, insert1_in_gga(s(T137), T114, T138))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114, T138)
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → U20_GGA(T212, T201, T202, T211, insert1_in_gga(s(T212), T202, T211))
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_GGA(s(T212), T202, T211)
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → U21_GGA(T221, T219, T201, T202, T222, p53_in_ggga(T219, T221, T202, T222))
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GGGA(T219, T221, T202, T222)
INSERT1_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U12_AGA(T43, T42, T22, T23, T44, p21_in_agga(T43, T42, T22, T44))
INSERT1_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_AGGA(T43, T42, T22, T44)
P21_IN_AGGA(T43, T42, T22, T44) → U3_AGGA(T43, T42, T22, T44, less23_in_ag(T43, T42))
P21_IN_AGGA(T43, T42, T22, T44) → LESS23_IN_AG(T43, T42)
LESS23_IN_AG(s(T62), s(T61)) → U1_AG(T62, T61, less23_in_ag(T62, T61))
LESS23_IN_AG(s(T62), s(T61)) → LESS23_IN_AG(T62, T61)
P21_IN_AGGA(T47, T42, T22, T48) → U4_AGGA(T47, T42, T22, T48, lessc23_in_ag(T47, T42))
U4_AGGA(T47, T42, T22, T48, lessc23_out_ag(T47, T42)) → U5_AGGA(T47, T42, T22, T48, insert1_in_gga(s(T47), T22, T48))
U4_AGGA(T47, T42, T22, T48, lessc23_out_ag(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U13_AGA(T82, T78, T79, T80, T83, less36_in_ga(T78, T82))
INSERT1_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → LESS36_IN_GA(T78, T82)
LESS36_IN_GA(s(T99), s(T101)) → U2_GA(T99, T101, less36_in_ga(T99, T101))
LESS36_IN_GA(s(T99), s(T101)) → LESS36_IN_GA(T99, T101)
INSERT1_IN_AGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_AGA(T86, T78, T79, T80, T87, lessc36_in_ga(T78, T86))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → U15_AGA(T86, T78, T79, T80, T87, insert1_in_aga(T86, T80, T87))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → INSERT1_IN_AGA(T86, T80, T87)
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → U16_AGA(T116, T112, T113, T114, T117, p46_in_gaga(T112, T116, T114, T117))
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GAGA(T112, T116, T114, T117)
P46_IN_GAGA(0, s(T124), T114, T123) → U6_GAGA(T124, T114, T123, insert1_in_aga(s(T124), T114, T123))
P46_IN_GAGA(0, s(T124), T114, T123) → INSERT1_IN_AGA(s(T124), T114, T123)
INSERT1_IN_AGA(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) → U17_AGA(T159, T150, T151, T160, insert1_in_gga(0, T150, T160))
INSERT1_IN_AGA(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) → U18_AGA(T171, T170, T150, T151, T172, p21_in_agga(T171, T170, T150, T172))
INSERT1_IN_AGA(T190, tree(T186, T187, T188), tree(T186, T187, T191)) → U19_AGA(T190, T186, T187, T188, T191, p46_in_gaga(T186, T190, T188, T191))
P46_IN_GAGA(s(T131), s(T133), T114, T134) → U7_GAGA(T131, T133, T114, T134, p53_in_gaga(T131, T133, T114, T134))
P46_IN_GAGA(s(T131), s(T133), T114, T134) → P53_IN_GAGA(T131, T133, T114, T134)
P53_IN_GAGA(T131, T133, T114, T134) → U8_GAGA(T131, T133, T114, T134, less36_in_ga(T131, T133))
P53_IN_GAGA(T131, T133, T114, T134) → LESS36_IN_GA(T131, T133)
P53_IN_GAGA(T131, T137, T114, T138) → U9_GAGA(T131, T137, T114, T138, lessc36_in_ga(T131, T137))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → U10_GAGA(T131, T137, T114, T138, insert1_in_aga(s(T137), T114, T138))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → INSERT1_IN_AGA(s(T137), T114, T138)
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → U20_AGA(T212, T201, T202, T211, insert1_in_aga(s(T212), T202, T211))
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_AGA(s(T212), T202, T211)
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → U21_AGA(T221, T219, T201, T202, T222, p53_in_gaga(T219, T221, T202, T222))
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GAGA(T219, T221, T202, T222)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
insert1_in_aga(x1, x2, x3)  =  insert1_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
insert1_in_gga(x1, x2, x3)  =  insert1_in_gga(x1, x2)
0  =  0
p21_in_ggga(x1, x2, x3, x4)  =  p21_in_ggga(x1, x2, x3)
less23_in_gg(x1, x2)  =  less23_in_gg(x1, x2)
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
p46_in_ggga(x1, x2, x3, x4)  =  p46_in_ggga(x1, x2, x3)
p53_in_ggga(x1, x2, x3, x4)  =  p53_in_ggga(x1, x2, x3)
p21_in_agga(x1, x2, x3, x4)  =  p21_in_agga(x2, x3)
less23_in_ag(x1, x2)  =  less23_in_ag(x2)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
less36_in_ga(x1, x2)  =  less36_in_ga(x1)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
p46_in_gaga(x1, x2, x3, x4)  =  p46_in_gaga(x1, x3)
p53_in_gaga(x1, x2, x3, x4)  =  p53_in_gaga(x1, x3)
INSERT1_IN_AGA(x1, x2, x3)  =  INSERT1_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x1, x2, x3, x5)
INSERT1_IN_GGA(x1, x2, x3)  =  INSERT1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
P21_IN_GGGA(x1, x2, x3, x4)  =  P21_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
LESS23_IN_GG(x1, x2)  =  LESS23_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
U5_GGGA(x1, x2, x3, x4, x5)  =  U5_GGGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
U15_GGA(x1, x2, x3, x4, x5, x6)  =  U15_GGA(x1, x2, x3, x4, x6)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x1, x2, x3, x4, x6)
P46_IN_GGGA(x1, x2, x3, x4)  =  P46_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x1, x2, x3, x5)
U18_GGA(x1, x2, x3, x4, x5, x6)  =  U18_GGA(x1, x2, x3, x4, x6)
U19_GGA(x1, x2, x3, x4, x5, x6)  =  U19_GGA(x1, x2, x3, x4, x6)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
P53_IN_GGGA(x1, x2, x3, x4)  =  P53_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
U9_GGGA(x1, x2, x3, x4, x5)  =  U9_GGGA(x1, x2, x3, x5)
U10_GGGA(x1, x2, x3, x4, x5)  =  U10_GGGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)
U12_AGA(x1, x2, x3, x4, x5, x6)  =  U12_AGA(x2, x3, x4, x6)
P21_IN_AGGA(x1, x2, x3, x4)  =  P21_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
LESS23_IN_AG(x1, x2)  =  LESS23_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_AGGA(x1, x2, x3, x4, x5)  =  U4_AGGA(x2, x3, x5)
U5_AGGA(x1, x2, x3, x4, x5)  =  U5_AGGA(x1, x2, x3, x5)
U13_AGA(x1, x2, x3, x4, x5, x6)  =  U13_AGA(x2, x3, x4, x6)
LESS36_IN_GA(x1, x2)  =  LESS36_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
U15_AGA(x1, x2, x3, x4, x5, x6)  =  U15_AGA(x2, x3, x4, x6)
U16_AGA(x1, x2, x3, x4, x5, x6)  =  U16_AGA(x2, x3, x4, x6)
P46_IN_GAGA(x1, x2, x3, x4)  =  P46_IN_GAGA(x1, x3)
U6_GAGA(x1, x2, x3, x4)  =  U6_GAGA(x2, x4)
U17_AGA(x1, x2, x3, x4, x5)  =  U17_AGA(x1, x2, x3, x5)
U18_AGA(x1, x2, x3, x4, x5, x6)  =  U18_AGA(x2, x3, x4, x6)
U19_AGA(x1, x2, x3, x4, x5, x6)  =  U19_AGA(x2, x3, x4, x6)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
P53_IN_GAGA(x1, x2, x3, x4)  =  P53_IN_GAGA(x1, x3)
U8_GAGA(x1, x2, x3, x4, x5)  =  U8_GAGA(x1, x3, x5)
U9_GAGA(x1, x2, x3, x4, x5)  =  U9_GAGA(x1, x3, x5)
U10_GAGA(x1, x2, x3, x4, x5)  =  U10_GAGA(x1, x3, x5)
U20_AGA(x1, x2, x3, x4, x5)  =  U20_AGA(x2, x3, x5)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

INSERT1_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U11_AGA(T31, T22, T23, T32, insert1_in_gga(0, T22, T32))
INSERT1_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U11_GGA(T31, T22, T23, T32, insert1_in_gga(0, T22, T32))
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U12_GGA(T43, T42, T22, T23, T44, p21_in_ggga(T43, T42, T22, T44))
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_GGGA(T43, T42, T22, T44)
P21_IN_GGGA(T43, T42, T22, T44) → U3_GGGA(T43, T42, T22, T44, less23_in_gg(T43, T42))
P21_IN_GGGA(T43, T42, T22, T44) → LESS23_IN_GG(T43, T42)
LESS23_IN_GG(s(T62), s(T61)) → U1_GG(T62, T61, less23_in_gg(T62, T61))
LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)
P21_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessc23_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → U5_GGGA(T47, T42, T22, T48, insert1_in_gga(s(T47), T22, T48))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U13_GGA(T82, T78, T79, T80, T83, less36_in_gg(T78, T82))
INSERT1_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → LESS36_IN_GG(T78, T82)
LESS36_IN_GG(s(T99), s(T101)) → U2_GG(T99, T101, less36_in_gg(T99, T101))
LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)
INSERT1_IN_GGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_GGA(T86, T78, T79, T80, T87, lessc36_in_gg(T78, T86))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → U15_GGA(T86, T78, T79, T80, T87, insert1_in_gga(T86, T80, T87))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80, T87)
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → U16_GGA(T116, T112, T113, T114, T117, p46_in_ggga(T112, T116, T114, T117))
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GGGA(T112, T116, T114, T117)
P46_IN_GGGA(0, s(T124), T114, T123) → U6_GGGA(T124, T114, T123, insert1_in_gga(s(T124), T114, T123))
P46_IN_GGGA(0, s(T124), T114, T123) → INSERT1_IN_GGA(s(T124), T114, T123)
INSERT1_IN_GGA(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) → U17_GGA(T159, T150, T151, T160, insert1_in_gga(0, T150, T160))
INSERT1_IN_GGA(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) → U18_GGA(T171, T170, T150, T151, T172, p21_in_ggga(T171, T170, T150, T172))
INSERT1_IN_GGA(T190, tree(T186, T187, T188), tree(T186, T187, T191)) → U19_GGA(T190, T186, T187, T188, T191, p46_in_ggga(T186, T190, T188, T191))
P46_IN_GGGA(s(T131), s(T133), T114, T134) → U7_GGGA(T131, T133, T114, T134, p53_in_ggga(T131, T133, T114, T134))
P46_IN_GGGA(s(T131), s(T133), T114, T134) → P53_IN_GGGA(T131, T133, T114, T134)
P53_IN_GGGA(T131, T133, T114, T134) → U8_GGGA(T131, T133, T114, T134, less36_in_gg(T131, T133))
P53_IN_GGGA(T131, T133, T114, T134) → LESS36_IN_GG(T131, T133)
P53_IN_GGGA(T131, T137, T114, T138) → U9_GGGA(T131, T137, T114, T138, lessc36_in_gg(T131, T137))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → U10_GGGA(T131, T137, T114, T138, insert1_in_gga(s(T137), T114, T138))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114, T138)
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → U20_GGA(T212, T201, T202, T211, insert1_in_gga(s(T212), T202, T211))
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_GGA(s(T212), T202, T211)
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → U21_GGA(T221, T219, T201, T202, T222, p53_in_ggga(T219, T221, T202, T222))
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GGGA(T219, T221, T202, T222)
INSERT1_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U12_AGA(T43, T42, T22, T23, T44, p21_in_agga(T43, T42, T22, T44))
INSERT1_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_AGGA(T43, T42, T22, T44)
P21_IN_AGGA(T43, T42, T22, T44) → U3_AGGA(T43, T42, T22, T44, less23_in_ag(T43, T42))
P21_IN_AGGA(T43, T42, T22, T44) → LESS23_IN_AG(T43, T42)
LESS23_IN_AG(s(T62), s(T61)) → U1_AG(T62, T61, less23_in_ag(T62, T61))
LESS23_IN_AG(s(T62), s(T61)) → LESS23_IN_AG(T62, T61)
P21_IN_AGGA(T47, T42, T22, T48) → U4_AGGA(T47, T42, T22, T48, lessc23_in_ag(T47, T42))
U4_AGGA(T47, T42, T22, T48, lessc23_out_ag(T47, T42)) → U5_AGGA(T47, T42, T22, T48, insert1_in_gga(s(T47), T22, T48))
U4_AGGA(T47, T42, T22, T48, lessc23_out_ag(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U13_AGA(T82, T78, T79, T80, T83, less36_in_ga(T78, T82))
INSERT1_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → LESS36_IN_GA(T78, T82)
LESS36_IN_GA(s(T99), s(T101)) → U2_GA(T99, T101, less36_in_ga(T99, T101))
LESS36_IN_GA(s(T99), s(T101)) → LESS36_IN_GA(T99, T101)
INSERT1_IN_AGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_AGA(T86, T78, T79, T80, T87, lessc36_in_ga(T78, T86))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → U15_AGA(T86, T78, T79, T80, T87, insert1_in_aga(T86, T80, T87))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → INSERT1_IN_AGA(T86, T80, T87)
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → U16_AGA(T116, T112, T113, T114, T117, p46_in_gaga(T112, T116, T114, T117))
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GAGA(T112, T116, T114, T117)
P46_IN_GAGA(0, s(T124), T114, T123) → U6_GAGA(T124, T114, T123, insert1_in_aga(s(T124), T114, T123))
P46_IN_GAGA(0, s(T124), T114, T123) → INSERT1_IN_AGA(s(T124), T114, T123)
INSERT1_IN_AGA(0, tree(s(T159), T150, T151), tree(s(T159), T160, T151)) → U17_AGA(T159, T150, T151, T160, insert1_in_gga(0, T150, T160))
INSERT1_IN_AGA(s(T171), tree(s(T170), T150, T151), tree(s(T170), T172, T151)) → U18_AGA(T171, T170, T150, T151, T172, p21_in_agga(T171, T170, T150, T172))
INSERT1_IN_AGA(T190, tree(T186, T187, T188), tree(T186, T187, T191)) → U19_AGA(T190, T186, T187, T188, T191, p46_in_gaga(T186, T190, T188, T191))
P46_IN_GAGA(s(T131), s(T133), T114, T134) → U7_GAGA(T131, T133, T114, T134, p53_in_gaga(T131, T133, T114, T134))
P46_IN_GAGA(s(T131), s(T133), T114, T134) → P53_IN_GAGA(T131, T133, T114, T134)
P53_IN_GAGA(T131, T133, T114, T134) → U8_GAGA(T131, T133, T114, T134, less36_in_ga(T131, T133))
P53_IN_GAGA(T131, T133, T114, T134) → LESS36_IN_GA(T131, T133)
P53_IN_GAGA(T131, T137, T114, T138) → U9_GAGA(T131, T137, T114, T138, lessc36_in_ga(T131, T137))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → U10_GAGA(T131, T137, T114, T138, insert1_in_aga(s(T137), T114, T138))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → INSERT1_IN_AGA(s(T137), T114, T138)
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → U20_AGA(T212, T201, T202, T211, insert1_in_aga(s(T212), T202, T211))
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_AGA(s(T212), T202, T211)
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → U21_AGA(T221, T219, T201, T202, T222, p53_in_gaga(T219, T221, T202, T222))
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GAGA(T219, T221, T202, T222)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
insert1_in_aga(x1, x2, x3)  =  insert1_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
insert1_in_gga(x1, x2, x3)  =  insert1_in_gga(x1, x2)
0  =  0
p21_in_ggga(x1, x2, x3, x4)  =  p21_in_ggga(x1, x2, x3)
less23_in_gg(x1, x2)  =  less23_in_gg(x1, x2)
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
p46_in_ggga(x1, x2, x3, x4)  =  p46_in_ggga(x1, x2, x3)
p53_in_ggga(x1, x2, x3, x4)  =  p53_in_ggga(x1, x2, x3)
p21_in_agga(x1, x2, x3, x4)  =  p21_in_agga(x2, x3)
less23_in_ag(x1, x2)  =  less23_in_ag(x2)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
less36_in_ga(x1, x2)  =  less36_in_ga(x1)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
p46_in_gaga(x1, x2, x3, x4)  =  p46_in_gaga(x1, x3)
p53_in_gaga(x1, x2, x3, x4)  =  p53_in_gaga(x1, x3)
INSERT1_IN_AGA(x1, x2, x3)  =  INSERT1_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x1, x2, x3, x5)
INSERT1_IN_GGA(x1, x2, x3)  =  INSERT1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
P21_IN_GGGA(x1, x2, x3, x4)  =  P21_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
LESS23_IN_GG(x1, x2)  =  LESS23_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
U5_GGGA(x1, x2, x3, x4, x5)  =  U5_GGGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
U15_GGA(x1, x2, x3, x4, x5, x6)  =  U15_GGA(x1, x2, x3, x4, x6)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x1, x2, x3, x4, x6)
P46_IN_GGGA(x1, x2, x3, x4)  =  P46_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x1, x2, x3, x5)
U18_GGA(x1, x2, x3, x4, x5, x6)  =  U18_GGA(x1, x2, x3, x4, x6)
U19_GGA(x1, x2, x3, x4, x5, x6)  =  U19_GGA(x1, x2, x3, x4, x6)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
P53_IN_GGGA(x1, x2, x3, x4)  =  P53_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
U9_GGGA(x1, x2, x3, x4, x5)  =  U9_GGGA(x1, x2, x3, x5)
U10_GGGA(x1, x2, x3, x4, x5)  =  U10_GGGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)
U12_AGA(x1, x2, x3, x4, x5, x6)  =  U12_AGA(x2, x3, x4, x6)
P21_IN_AGGA(x1, x2, x3, x4)  =  P21_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
LESS23_IN_AG(x1, x2)  =  LESS23_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_AGGA(x1, x2, x3, x4, x5)  =  U4_AGGA(x2, x3, x5)
U5_AGGA(x1, x2, x3, x4, x5)  =  U5_AGGA(x1, x2, x3, x5)
U13_AGA(x1, x2, x3, x4, x5, x6)  =  U13_AGA(x2, x3, x4, x6)
LESS36_IN_GA(x1, x2)  =  LESS36_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
U15_AGA(x1, x2, x3, x4, x5, x6)  =  U15_AGA(x2, x3, x4, x6)
U16_AGA(x1, x2, x3, x4, x5, x6)  =  U16_AGA(x2, x3, x4, x6)
P46_IN_GAGA(x1, x2, x3, x4)  =  P46_IN_GAGA(x1, x3)
U6_GAGA(x1, x2, x3, x4)  =  U6_GAGA(x2, x4)
U17_AGA(x1, x2, x3, x4, x5)  =  U17_AGA(x1, x2, x3, x5)
U18_AGA(x1, x2, x3, x4, x5, x6)  =  U18_AGA(x2, x3, x4, x6)
U19_AGA(x1, x2, x3, x4, x5, x6)  =  U19_AGA(x2, x3, x4, x6)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
P53_IN_GAGA(x1, x2, x3, x4)  =  P53_IN_GAGA(x1, x3)
U8_GAGA(x1, x2, x3, x4, x5)  =  U8_GAGA(x1, x3, x5)
U9_GAGA(x1, x2, x3, x4, x5)  =  U9_GAGA(x1, x3, x5)
U10_GAGA(x1, x2, x3, x4, x5)  =  U10_GAGA(x1, x3, x5)
U20_AGA(x1, x2, x3, x4, x5)  =  U20_AGA(x2, x3, x5)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 46 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS36_IN_GA(s(T99), s(T101)) → LESS36_IN_GA(T99, T101)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
LESS36_IN_GA(x1, x2)  =  LESS36_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESS36_IN_GA(s(T99), s(T101)) → LESS36_IN_GA(T99, T101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS36_IN_GA(x1, x2)  =  LESS36_IN_GA(x1)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

LESS36_IN_GA(s(T99)) → LESS36_IN_GA(T99)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS36_IN_GA(s(T99)) → LESS36_IN_GA(T99)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

LESS23_IN_AG(s(T62), s(T61)) → LESS23_IN_AG(T62, T61)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
LESS23_IN_AG(x1, x2)  =  LESS23_IN_AG(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESS23_IN_AG(s(T62), s(T61)) → LESS23_IN_AG(T62, T61)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS23_IN_AG(x1, x2)  =  LESS23_IN_AG(x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

LESS23_IN_AG(s(T61)) → LESS23_IN_AG(T61)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS23_IN_AG(s(T61)) → LESS23_IN_AG(T61)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)

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

(24) PiDPToQDPProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)

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

(26) 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:

• LESS36_IN_GG(s(T99), s(T101)) → LESS36_IN_GG(T99, T101)
  The graph contains the following edges 1 > 1, 2 > 2

(27) YES

(28) Obligation:

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

LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
LESS23_IN_GG(x1, x2)  =  LESS23_IN_GG(x1, x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)

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

(31) PiDPToQDPProof (EQUIVALENT transformation)

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

(32) Obligation:

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

LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)

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

(33) 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:

• LESS23_IN_GG(s(T62), s(T61)) → LESS23_IN_GG(T62, T61)
  The graph contains the following edges 1 > 1, 2 > 2

(34) YES

(35) Obligation:

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

INSERT1_IN_GGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_GGA(T86, T78, T79, T80, T87, lessc36_in_gg(T78, T86))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80, T87)
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GGGA(T112, T116, T114, T117)
P46_IN_GGGA(0, s(T124), T114, T123) → INSERT1_IN_GGA(s(T124), T114, T123)
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_GGGA(T43, T42, T22, T44)
P21_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessc23_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_GGA(s(T212), T202, T211)
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GGGA(T219, T221, T202, T222)
P53_IN_GGGA(T131, T137, T114, T138) → U9_GGGA(T131, T137, T114, T138, lessc36_in_gg(T131, T137))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114, T138)
P46_IN_GGGA(s(T131), s(T133), T114, T134) → P53_IN_GGGA(T131, T133, T114, T134)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
INSERT1_IN_GGA(x1, x2, x3)  =  INSERT1_IN_GGA(x1, x2)
P21_IN_GGGA(x1, x2, x3, x4)  =  P21_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
P46_IN_GGGA(x1, x2, x3, x4)  =  P46_IN_GGGA(x1, x2, x3)
P53_IN_GGGA(x1, x2, x3, x4)  =  P53_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5)  =  U9_GGGA(x1, x2, x3, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

INSERT1_IN_GGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_GGA(T86, T78, T79, T80, T87, lessc36_in_gg(T78, T86))
U14_GGA(T86, T78, T79, T80, T87, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80, T87)
INSERT1_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERT1_IN_GGA(0, T22, T32)
INSERT1_IN_GGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GGGA(T112, T116, T114, T117)
P46_IN_GGGA(0, s(T124), T114, T123) → INSERT1_IN_GGA(s(T124), T114, T123)
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → P21_IN_GGGA(T43, T42, T22, T44)
P21_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessc23_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22, T48)
INSERT1_IN_GGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_GGA(s(T212), T202, T211)
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GGGA(T219, T221, T202, T222)
P53_IN_GGGA(T131, T137, T114, T138) → U9_GGGA(T131, T137, T114, T138, lessc36_in_gg(T131, T137))
U9_GGGA(T131, T137, T114, T138, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114, T138)
P46_IN_GGGA(s(T131), s(T133), T114, T134) → P53_IN_GGGA(T131, T133, T114, T134)

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
INSERT1_IN_GGA(x1, x2, x3)  =  INSERT1_IN_GGA(x1, x2)
P21_IN_GGGA(x1, x2, x3, x4)  =  P21_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
P46_IN_GGGA(x1, x2, x3, x4)  =  P46_IN_GGGA(x1, x2, x3)
P53_IN_GGGA(x1, x2, x3, x4)  =  P53_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5)  =  U9_GGGA(x1, x2, x3, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

INSERT1_IN_GGA(T86, tree(T78, T79, T80)) → U14_GGA(T86, T78, T79, T80, lessc36_in_gg(T78, T86))
U14_GGA(T86, T78, T79, T80, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80)
INSERT1_IN_GGA(0, tree(s(T31), T22, T23)) → INSERT1_IN_GGA(0, T22)
INSERT1_IN_GGA(T116, tree(T112, T113, T114)) → P46_IN_GGGA(T112, T116, T114)
P46_IN_GGGA(0, s(T124), T114) → INSERT1_IN_GGA(s(T124), T114)
INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23)) → P21_IN_GGGA(T43, T42, T22)
P21_IN_GGGA(T47, T42, T22) → U4_GGGA(T47, T42, T22, lessc23_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22)
INSERT1_IN_GGA(s(T212), tree(0, T201, T202)) → INSERT1_IN_GGA(s(T212), T202)
INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202)) → P53_IN_GGGA(T219, T221, T202)
P53_IN_GGGA(T131, T137, T114) → U9_GGGA(T131, T137, T114, lessc36_in_gg(T131, T137))
U9_GGGA(T131, T137, T114, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114)
P46_IN_GGGA(s(T131), s(T133), T114) → P53_IN_GGGA(T131, T133, T114)

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))

The set Q consists of the following terms:

lessc36_in_gg(x0, x1)
lessc23_in_gg(x0, x1)
U34_gg(x0, x1, x2)
U33_gg(x0, x1, x2)

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

(40) 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:

• U14_GGA(T86, T78, T79, T80, lessc36_out_gg(T78, T86)) → INSERT1_IN_GGA(T86, T80)
  The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

• INSERT1_IN_GGA(T86, tree(T78, T79, T80)) → U14_GGA(T86, T78, T79, T80, lessc36_in_gg(T78, T86))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

• INSERT1_IN_GGA(0, tree(s(T31), T22, T23)) → INSERT1_IN_GGA(0, T22)
  The graph contains the following edges 1 >= 1, 2 > 2

• INSERT1_IN_GGA(s(T212), tree(0, T201, T202)) → INSERT1_IN_GGA(s(T212), T202)
  The graph contains the following edges 1 >= 1, 2 > 2

• INSERT1_IN_GGA(T116, tree(T112, T113, T114)) → P46_IN_GGGA(T112, T116, T114)
  The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3

• P46_IN_GGGA(0, s(T124), T114) → INSERT1_IN_GGA(s(T124), T114)
  The graph contains the following edges 2 >= 1, 3 >= 2

• P46_IN_GGGA(s(T131), s(T133), T114) → P53_IN_GGGA(T131, T133, T114)
  The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

• P21_IN_GGGA(T47, T42, T22) → U4_GGGA(T47, T42, T22, lessc23_in_gg(T47, T42))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

• P53_IN_GGGA(T131, T137, T114) → U9_GGGA(T131, T137, T114, lessc36_in_gg(T131, T137))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

• U4_GGGA(T47, T42, T22, lessc23_out_gg(T47, T42)) → INSERT1_IN_GGA(s(T47), T22)
  The graph contains the following edges 3 >= 2

• U9_GGGA(T131, T137, T114, lessc36_out_gg(T131, T137)) → INSERT1_IN_GGA(s(T137), T114)
  The graph contains the following edges 3 >= 2

• INSERT1_IN_GGA(s(T43), tree(s(T42), T22, T23)) → P21_IN_GGGA(T43, T42, T22)
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3

• INSERT1_IN_GGA(s(T221), tree(s(T219), T201, T202)) → P53_IN_GGGA(T219, T221, T202)
  The graph contains the following edges 2 > 1, 1 > 2, 2 > 3

(41) YES

(42) Obligation:

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

INSERT1_IN_AGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_AGA(T86, T78, T79, T80, T87, lessc36_in_ga(T78, T86))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → INSERT1_IN_AGA(T86, T80, T87)
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GAGA(T112, T116, T114, T117)
P46_IN_GAGA(0, s(T124), T114, T123) → INSERT1_IN_AGA(s(T124), T114, T123)
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_AGA(s(T212), T202, T211)
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GAGA(T219, T221, T202, T222)
P53_IN_GAGA(T131, T137, T114, T138) → U9_GAGA(T131, T137, T114, T138, lessc36_in_ga(T131, T137))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → INSERT1_IN_AGA(s(T137), T114, T138)
P46_IN_GAGA(s(T131), s(T133), T114, T134) → P53_IN_GAGA(T131, T133, T114, T134)

The TRS R consists of the following rules:

lessc23_in_gg(0, s(T55)) → lessc23_out_gg(0, s(T55))
lessc23_in_gg(s(T62), s(T61)) → U33_gg(T62, T61, lessc23_in_gg(T62, T61))
U33_gg(T62, T61, lessc23_out_gg(T62, T61)) → lessc23_out_gg(s(T62), s(T61))
lessc36_in_gg(0, s(T94)) → lessc36_out_gg(0, s(T94))
lessc36_in_gg(s(T99), s(T101)) → U34_gg(T99, T101, lessc36_in_gg(T99, T101))
U34_gg(T99, T101, lessc36_out_gg(T99, T101)) → lessc36_out_gg(s(T99), s(T101))
lessc23_in_ag(0, s(T55)) → lessc23_out_ag(0, s(T55))
lessc23_in_ag(s(T62), s(T61)) → U33_ag(T62, T61, lessc23_in_ag(T62, T61))
U33_ag(T62, T61, lessc23_out_ag(T62, T61)) → lessc23_out_ag(s(T62), s(T61))
lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessc23_in_gg(x1, x2)  =  lessc23_in_gg(x1, x2)
lessc23_out_gg(x1, x2)  =  lessc23_out_gg(x1, x2)
U33_gg(x1, x2, x3)  =  U33_gg(x1, x2, x3)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U34_gg(x1, x2, x3)  =  U34_gg(x1, x2, x3)
lessc23_in_ag(x1, x2)  =  lessc23_in_ag(x2)
lessc23_out_ag(x1, x2)  =  lessc23_out_ag(x1, x2)
U33_ag(x1, x2, x3)  =  U33_ag(x2, x3)
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
INSERT1_IN_AGA(x1, x2, x3)  =  INSERT1_IN_AGA(x2)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
P46_IN_GAGA(x1, x2, x3, x4)  =  P46_IN_GAGA(x1, x3)
P53_IN_GAGA(x1, x2, x3, x4)  =  P53_IN_GAGA(x1, x3)
U9_GAGA(x1, x2, x3, x4, x5)  =  U9_GAGA(x1, x3, x5)

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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) Obligation:

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

INSERT1_IN_AGA(T86, tree(T78, T79, T80), tree(T78, T79, T87)) → U14_AGA(T86, T78, T79, T80, T87, lessc36_in_ga(T78, T86))
U14_AGA(T86, T78, T79, T80, T87, lessc36_out_ga(T78, T86)) → INSERT1_IN_AGA(T86, T80, T87)
INSERT1_IN_AGA(T116, tree(T112, T113, T114), tree(T112, T113, T117)) → P46_IN_GAGA(T112, T116, T114, T117)
P46_IN_GAGA(0, s(T124), T114, T123) → INSERT1_IN_AGA(s(T124), T114, T123)
INSERT1_IN_AGA(s(T212), tree(0, T201, T202), tree(0, T201, T211)) → INSERT1_IN_AGA(s(T212), T202, T211)
INSERT1_IN_AGA(s(T221), tree(s(T219), T201, T202), tree(s(T219), T201, T222)) → P53_IN_GAGA(T219, T221, T202, T222)
P53_IN_GAGA(T131, T137, T114, T138) → U9_GAGA(T131, T137, T114, T138, lessc36_in_ga(T131, T137))
U9_GAGA(T131, T137, T114, T138, lessc36_out_ga(T131, T137)) → INSERT1_IN_AGA(s(T137), T114, T138)
P46_IN_GAGA(s(T131), s(T133), T114, T134) → P53_IN_GAGA(T131, T133, T114, T134)

The TRS R consists of the following rules:

lessc36_in_ga(0, s(T94)) → lessc36_out_ga(0, s(T94))
lessc36_in_ga(s(T99), s(T101)) → U34_ga(T99, T101, lessc36_in_ga(T99, T101))
U34_ga(T99, T101, lessc36_out_ga(T99, T101)) → lessc36_out_ga(s(T99), s(T101))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessc36_in_ga(x1, x2)  =  lessc36_in_ga(x1)
lessc36_out_ga(x1, x2)  =  lessc36_out_ga(x1)
U34_ga(x1, x2, x3)  =  U34_ga(x1, x3)
INSERT1_IN_AGA(x1, x2, x3)  =  INSERT1_IN_AGA(x2)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
P46_IN_GAGA(x1, x2, x3, x4)  =  P46_IN_GAGA(x1, x3)
P53_IN_GAGA(x1, x2, x3, x4)  =  P53_IN_GAGA(x1, x3)
U9_GAGA(x1, x2, x3, x4, x5)  =  U9_GAGA(x1, x3, x5)

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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) Obligation:

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

INSERT1_IN_AGA(tree(T78, T79, T80)) → U14_AGA(T78, T79, T80, lessc36_in_ga(T78))
U14_AGA(T78, T79, T80, lessc36_out_ga(T78)) → INSERT1_IN_AGA(T80)
INSERT1_IN_AGA(tree(T112, T113, T114)) → P46_IN_GAGA(T112, T114)
P46_IN_GAGA(0, T114) → INSERT1_IN_AGA(T114)
INSERT1_IN_AGA(tree(0, T201, T202)) → INSERT1_IN_AGA(T202)
INSERT1_IN_AGA(tree(s(T219), T201, T202)) → P53_IN_GAGA(T219, T202)
P53_IN_GAGA(T131, T114) → U9_GAGA(T131, T114, lessc36_in_ga(T131))
U9_GAGA(T131, T114, lessc36_out_ga(T131)) → INSERT1_IN_AGA(T114)
P46_IN_GAGA(s(T131), T114) → P53_IN_GAGA(T131, T114)

The TRS R consists of the following rules:

lessc36_in_ga(0) → lessc36_out_ga(0)
lessc36_in_ga(s(T99)) → U34_ga(T99, lessc36_in_ga(T99))
U34_ga(T99, lessc36_out_ga(T99)) → lessc36_out_ga(s(T99))

The set Q consists of the following terms:

lessc36_in_ga(x0)
U34_ga(x0, x1)

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

(47) 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:

• U14_AGA(T78, T79, T80, lessc36_out_ga(T78)) → INSERT1_IN_AGA(T80)
  The graph contains the following edges 3 >= 1

• INSERT1_IN_AGA(tree(0, T201, T202)) → INSERT1_IN_AGA(T202)
  The graph contains the following edges 1 > 1

• INSERT1_IN_AGA(tree(T78, T79, T80)) → U14_AGA(T78, T79, T80, lessc36_in_ga(T78))
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

• P46_IN_GAGA(0, T114) → INSERT1_IN_AGA(T114)
  The graph contains the following edges 2 >= 1

• U9_GAGA(T131, T114, lessc36_out_ga(T131)) → INSERT1_IN_AGA(T114)
  The graph contains the following edges 2 >= 1

• P46_IN_GAGA(s(T131), T114) → P53_IN_GAGA(T131, T114)
  The graph contains the following edges 1 > 1, 2 >= 2

• P53_IN_GAGA(T131, T114) → U9_GAGA(T131, T114, lessc36_in_ga(T131))
  The graph contains the following edges 1 >= 1, 2 >= 2

• INSERT1_IN_AGA(tree(T112, T113, T114)) → P46_IN_GAGA(T112, T114)
  The graph contains the following edges 1 > 1, 1 > 2

• INSERT1_IN_AGA(tree(s(T219), T201, T202)) → P53_IN_GAGA(T219, T202)
  The graph contains the following edges 1 > 1, 1 > 2

(48) YES