Left Termination of the query pattern
insert(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).

Query: insert(a,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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\nX3 is free\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\nX3 is free\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", 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\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(T12, T13, T14), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", 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\nT113 is ground\nT114 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(T112, T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", 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\nT113 is ground\nT114 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(T112, T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 4\n\nT112 is ground\nT113 is ground\nT114 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(T112, T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(less(T112, T116), insert(T116, T114, T117)), SCOPE: 5, CLAUSE: 5\n\nT112 is ground\nT113 is ground\nT114 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(T112, T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: insert(s(T124), T114, T123)\n\nT113 is ground\nT114 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(0, T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", 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\nT113 is ground\nT114 is ground\nT131 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(s(T131), T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(T131, T133)\n\nT113 is ground\nT114 is ground\nT131 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T1, tree(s(T131), T113, T114), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: insert(s(T137), T114, T138)\n\nT113 is ground\nT114 is ground\nT131 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T139, tree(s(T131), T113, T114), T140) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(less(T155, T151), insert(T155, T152, T156)) | insert(T1, tree(T151, T152, T153), T3), SCOPE: 1, CLAUSE: 3\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nX3 is free\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(less(T155, T151), insert(T155, T152, T156)), SCOPE: 6, CLAUSE: 4 | ','(less(T155, T151), insert(T155, T152, T156)), SCOPE: 6, CLAUSE: 5 | ?_6 | insert(T1, tree(T151, T152, T153), T3), SCOPE: 1, CLAUSE: 3\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(less(T155, T151), insert(T155, T152, T156)), SCOPE: 6, CLAUSE: 4\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(less(T155, T151), insert(T155, T152, T156)), SCOPE: 6, CLAUSE: 5 | ?_6 | insert(T1, tree(T151, T152, T153), T3), SCOPE: 1, CLAUSE: 3\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: insert(0, T152, T162)\n\nT152 is ground\nT153 is ground\nT161 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(s(T161), T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(T155, T151), insert(T155, T152, T156)), SCOPE: 6, CLAUSE: 5\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ?_6 | insert(T1, tree(T151, T152, T153), T3), SCOPE: 1, CLAUSE: 3\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(less(T173, T172), insert(s(T173), T152, T174))\n\nT152 is ground\nT153 is ground\nT172 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(s(T172), T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: insert(T1, tree(T151, T152, T153), T3), SCOPE: 1, CLAUSE: 3\n\nT151 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T151, T152, T153), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(less(T188, T192), insert(T192, T190, T193))\n\nT188 is ground\nT189 is ground\nT190 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T188, T189, T190), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(less(T202, T206), insert(T206, T204, T207))\n\nT202 is ground\nT203 is ground\nT204 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T202, T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(less(T202, T206), insert(T206, T204, T207)), SCOPE: 7, CLAUSE: 4 | ','(less(T202, T206), insert(T206, T204, T207)), SCOPE: 7, CLAUSE: 5\n\nT202 is ground\nT203 is ground\nT204 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T202, T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(less(T202, T206), insert(T206, T204, T207)), SCOPE: 7, CLAUSE: 4\n\nT202 is ground\nT203 is ground\nT204 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T202, T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(less(T202, T206), insert(T206, T204, T207)), SCOPE: 7, CLAUSE: 5\n\nT202 is ground\nT203 is ground\nT204 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(T202, T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: insert(s(T214), T204, T213)\n\nT203 is ground\nT204 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(0, T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(less(T221, T223), insert(s(T223), T204, T224))\n\nT203 is ground\nT204 is ground\nT221 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(s(T221), T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(T221, T223)\n\nT203 is ground\nT204 is ground\nT221 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T1, tree(s(T221), T203, T204), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: insert(s(T227), T204, T228)\n\nT203 is ground\nT204 is ground\nT221 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T229, tree(s(T221), T203, T204), T230) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 82 [arrowhead = none ];
82 -> 2;

83 [label="EVAL with clause\ninsert(X3, void, tree(X3, void, void)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> void,\nT3 -> tree(T5, void, void)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 83 [arrowhead = none ];
83 -> 3;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 84 [arrowhead = none ];
84 -> 4;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 85 [arrowhead = none ];
85 -> 5;

86 [label="EVAL with clause\ninsert(X20, tree(X20, X21, X22), tree(X20, X21, X22)).\nand substitutionT1 -> T12,\nX20 -> T12,\nX21 -> T13,\nX22 -> T14,\nT2 -> tree(T12, T13, T14),\nT3 -> tree(T12, T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 87 [arrowhead = none ];
87 -> 10;

88 [label="BACKTRACK\nfor clause: insert(X, tree(X, Left, Right), tree(X, Left, Right))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 88 [arrowhead = none ];
88 -> 6;

89 [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 = lightgreen];
6 -> 89 [arrowhead = none ];
89 -> 7;

90 [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 = lightgreen];
7 -> 90 [arrowhead = none ];
90 -> 8;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 91 [arrowhead = none ];
91 -> 11;

92 [label="EVAL with clause\ninsert(X137, tree(X138, X139, X140), tree(X138, X141, X140)) :- ','(less(X137, X138), insert(X137, X139, X141)).\nand substitutionT1 -> T155,\nX137 -> T155,\nX138 -> T151,\nX139 -> T152,\nX140 -> T153,\nT2 -> tree(T151, T152, T153),\nX141 -> T156,\nT3 -> tree(T151, T156, T153),\nT150 -> T155,\nT154 -> T156", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 92 [arrowhead = none ];
92 -> 57;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 93 [arrowhead = none ];
93 -> 58;

94 [label="EVAL with clause\ninsert(X28, tree(X29, X30, X31), tree(X29, X32, X31)) :- ','(less(X28, X29), insert(X28, X30, X32)).\nand substitutionT1 -> T25,\nX28 -> T25,\nT12 -> T21,\nX29 -> T21,\nT13 -> T22,\nX30 -> T22,\nT14 -> T23,\nX31 -> T23,\nX32 -> T26,\nT3 -> tree(T21, T26, T23),\nT20 -> T25,\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 94 [arrowhead = none ];
94 -> 12;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 95 [arrowhead = none ];
95 -> 13;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 96 [arrowhead = none ];
96 -> 14;

97 [label="EVAL with clause\ninsert(X108, tree(X109, X110, X111), tree(X109, X110, X112)) :- ','(less(X109, X108), insert(X108, X111, X112)).\nand substitutionT1 -> T116,\nX108 -> T116,\nT12 -> T112,\nX109 -> T112,\nT13 -> T113,\nX110 -> T113,\nT14 -> T114,\nX111 -> T114,\nX112 -> T117,\nT3 -> tree(T112, T113, T117),\nT111 -> T116,\nT115 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 99 [arrowhead = none ];
99 -> 15;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 100 [arrowhead = none ];
100 -> 16;

101 [label="EVAL with clause\nless(0, s(X37)).\nand substitutionT25 -> 0,\nX37 -> T31,\nT21 -> s(T31),\nT26 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 101 [arrowhead = none ];
101 -> 17;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 102 [arrowhead = none ];
102 -> 18;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 103 [arrowhead = none ];
103 -> 19;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 104 [arrowhead = none ];
104 -> 20;

105 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T22\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 105 [arrowhead = none , style=dashed];
105 -> 1[style=dashed];

106 [label="EVAL with clause\nless(s(X48), s(X49)) :- less(X48, X49).\nand substitutionX48 -> T43,\nT25 -> s(T43),\nX49 -> T42,\nT21 -> s(T42),\nT41 -> T43,\nT26 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 106 [arrowhead = none ];
106 -> 21;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 107 [arrowhead = none ];
107 -> 22;

108 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 108 [arrowhead = none ];
108 -> 33;

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 109 [arrowhead = none ];
109 -> 23;

110 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nT42 is ground\nreplacements:T43 -> T47,\nT44 -> T48", fontsize=14, style = filled, fillcolor = violet];
21 -> 110 [arrowhead = none ];
110 -> 24;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 111 [arrowhead = none ];
111 -> 25;

112 [label="INSTANCE with matching:\nT1 -> s(T47)\nT2 -> T22\nT3 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 112 [arrowhead = none , style=dashed];
112 -> 1[style=dashed];

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 113 [arrowhead = none ];
113 -> 26;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 114 [arrowhead = none ];
114 -> 27;

115 [label="EVAL with clause\nless(0, s(X58)).\nand substitutionT43 -> 0,\nX58 -> T55,\nT42 -> s(T55)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 115 [arrowhead = none ];
115 -> 28;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 116 [arrowhead = none ];
116 -> 29;

117 [label="EVAL with clause\nless(s(X63), s(X64)) :- less(X63, X64).\nand substitutionX63 -> T62,\nT43 -> s(T62),\nX64 -> T61,\nT42 -> s(T61),\nT60 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 117 [arrowhead = none ];
117 -> 31;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 118 [arrowhead = none ];
118 -> 32;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="INSTANCE with matching:\nT43 -> T62\nT42 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 120 [arrowhead = none , style=dashed];
120 -> 23[style=dashed];

121 [label="EVAL with clause\ninsert(X79, tree(X80, X81, X82), tree(X80, X81, X83)) :- ','(less(X80, X79), insert(X79, X82, X83)).\nand substitutionT1 -> T82,\nX79 -> T82,\nT21 -> T78,\nX80 -> T78,\nT22 -> T79,\nX81 -> T79,\nT23 -> T80,\nX82 -> T80,\nX83 -> T83,\nT3 -> tree(T78, T79, T83),\nT77 -> T82,\nT81 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 122 [arrowhead = none ];
122 -> 35;

123 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 123 [arrowhead = none ];
123 -> 36;

124 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nreplacements:T82 -> T86,\nT83 -> T87", fontsize=14, style = filled, fillcolor = violet];
34 -> 124 [arrowhead = none ];
124 -> 37;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 125 [arrowhead = none ];
125 -> 38;

126 [label="INSTANCE with matching:\nT1 -> T86\nT2 -> T80\nT3 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 126 [arrowhead = none , style=dashed];
126 -> 1[style=dashed];

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 127 [arrowhead = none ];
127 -> 39;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 128 [arrowhead = none ];
128 -> 40;

129 [label="EVAL with clause\nless(0, s(X92)).\nand substitutionT78 -> 0,\nX92 -> T94,\nT82 -> s(T94)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 129 [arrowhead = none ];
129 -> 41;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 130 [arrowhead = none ];
130 -> 42;

131 [label="EVAL with clause\nless(s(X97), s(X98)) :- less(X97, X98).\nand substitutionX97 -> T99,\nT78 -> s(T99),\nX98 -> T101,\nT82 -> s(T101),\nT100 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 131 [arrowhead = none ];
131 -> 44;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 132 [arrowhead = none ];
132 -> 45;

133 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 133 [arrowhead = none ];
133 -> 43;

134 [label="INSTANCE with matching:\nT78 -> T99\nT82 -> T101", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 134 [arrowhead = none , style=dashed];
134 -> 36[style=dashed];

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 135 [arrowhead = none ];
135 -> 48;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="EVAL with clause\nless(0, s(X117)).\nand substitutionT112 -> 0,\nX117 -> T124,\nT116 -> s(T124),\nT117 -> T123,\nT122 -> T124", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 139 [arrowhead = none ];
139 -> 52;

140 [label="EVAL with clause\nless(s(X124), s(X125)) :- less(X124, X125).\nand substitutionX124 -> T131,\nT112 -> s(T131),\nX125 -> T133,\nT116 -> s(T133),\nT132 -> T133,\nT117 -> T134", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 140 [arrowhead = none ];
140 -> 53;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 141 [arrowhead = none ];
141 -> 54;

142 [label="INSTANCE with matching:\nT1 -> s(T124)\nT2 -> T114\nT3 -> T123", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 142 [arrowhead = none , style=dashed];
142 -> 1[style=dashed];

143 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 143 [arrowhead = none ];
143 -> 55;

144 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nreplacements:T133 -> T137,\nT134 -> T138,\nT1 -> T139,\nT3 -> T140", fontsize=14, style = filled, fillcolor = violet];
53 -> 144 [arrowhead = none ];
144 -> 56;

145 [label="INSTANCE with matching:\nT78 -> T131\nT82 -> T133", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 145 [arrowhead = none , style=dashed];
145 -> 36[style=dashed];

146 [label="INSTANCE with matching:\nT1 -> s(T137)\nT2 -> T114\nT3 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 146 [arrowhead = none , style=dashed];
146 -> 1[style=dashed];

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 147 [arrowhead = none ];
147 -> 59;

148 [label="EVAL with clause\ninsert(X183, tree(X184, X185, X186), tree(X184, X185, X187)) :- ','(less(X184, X183), insert(X183, X186, X187)).\nand substitutionT1 -> T206,\nX183 -> T206,\nX184 -> T202,\nX185 -> T203,\nX186 -> T204,\nT2 -> tree(T202, T203, T204),\nX187 -> T207,\nT3 -> tree(T202, T203, T207),\nT201 -> T206,\nT205 -> T207", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 148 [arrowhead = none ];
148 -> 71;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 149 [arrowhead = none ];
149 -> 72;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 150 [arrowhead = none ];
150 -> 60;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 151 [arrowhead = none ];
151 -> 61;

152 [label="EVAL with clause\nless(0, s(X146)).\nand substitutionT155 -> 0,\nX146 -> T161,\nT151 -> s(T161),\nT156 -> T162", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 152 [arrowhead = none ];
152 -> 62;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 153 [arrowhead = none ];
153 -> 63;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 154 [arrowhead = none ];
154 -> 64;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 155 [arrowhead = none ];
155 -> 65;

156 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T152\nT3 -> T162", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 156 [arrowhead = none , style=dashed];
156 -> 1[style=dashed];

157 [label="EVAL with clause\nless(s(X157), s(X158)) :- less(X157, X158).\nand substitutionX157 -> T173,\nT155 -> s(T173),\nX158 -> T172,\nT151 -> s(T172),\nT171 -> T173,\nT156 -> T174", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 157 [arrowhead = none ];
157 -> 66;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 158 [arrowhead = none ];
158 -> 67;

159 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 159 [arrowhead = none ];
159 -> 68;

160 [label="INSTANCE with matching:\nT43 -> T173\nT42 -> T172\nT22 -> T152\nT44 -> T174", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 160 [arrowhead = none , style=dashed];
160 -> 21[style=dashed];

161 [label="EVAL with clause\ninsert(X171, tree(X172, X173, X174), tree(X172, X173, X175)) :- ','(less(X172, X171), insert(X171, X174, X175)).\nand substitutionT1 -> T192,\nX171 -> T192,\nT151 -> T188,\nX172 -> T188,\nT152 -> T189,\nX173 -> T189,\nT153 -> T190,\nX174 -> T190,\nX175 -> T193,\nT3 -> tree(T188, T189, T193),\nT187 -> T192,\nT191 -> T193", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 161 [arrowhead = none ];
161 -> 69;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 162 [arrowhead = none ];
162 -> 70;

163 [label="INSTANCE with matching:\nT78 -> T188\nT82 -> T192\nT80 -> T190\nT83 -> T193", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 163 [arrowhead = none , style=dashed];
163 -> 34[style=dashed];

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 164 [arrowhead = none ];
164 -> 73;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 165 [arrowhead = none ];
165 -> 74;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 166 [arrowhead = none ];
166 -> 75;

167 [label="EVAL with clause\nless(0, s(X192)).\nand substitutionT202 -> 0,\nX192 -> T214,\nT206 -> s(T214),\nT207 -> T213,\nT212 -> T214", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 168 [arrowhead = none ];
168 -> 77;

169 [label="EVAL with clause\nless(s(X199), s(X200)) :- less(X199, X200).\nand substitutionX199 -> T221,\nT202 -> s(T221),\nX200 -> T223,\nT206 -> s(T223),\nT222 -> T223,\nT207 -> T224", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 169 [arrowhead = none ];
169 -> 78;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 170 [arrowhead = none ];
170 -> 79;

171 [label="INSTANCE with matching:\nT1 -> s(T214)\nT2 -> T204\nT3 -> T213", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 171 [arrowhead = none , style=dashed];
171 -> 1[style=dashed];

172 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
78 -> 172 [arrowhead = none ];
172 -> 80;

173 [label="SPLIT 2\nnew knowledge:\nT221 is ground\nreplacements:T223 -> T227,\nT224 -> T228,\nT1 -> T229,\nT3 -> T230", fontsize=14, style = filled, fillcolor = violet];
78 -> 173 [arrowhead = none ];
173 -> 81;

174 [label="INSTANCE with matching:\nT78 -> T221\nT82 -> T223", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 174 [arrowhead = none , style=dashed];
174 -> 36[style=dashed];

175 [label="INSTANCE with matching:\nT1 -> s(T227)\nT2 -> T204\nT3 -> T228", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 175 [arrowhead = none , style=dashed];
175 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

lessA(0, s(T55)).
lessA(s(T62), s(T61)) :- lessA(T62, T61).
lessB(0, s(T94)).
lessB(s(T99), s(T101)) :- lessB(T99, T101).
pC(T43, T42, T22, T44) :- lessA(T43, T42).
pC(T47, T42, T22, T48) :- ','(lessA(T47, T42), insertD(s(T47), T22, T48)).
pE(T78, T82, T80, T83) :- lessB(T78, T82).
pE(T78, T86, T80, T87) :- ','(lessB(T78, T86), insertD(T86, T80, T87)).
insertD(T5, void, tree(T5, void, void)).
insertD(T12, tree(T12, T13, T14), tree(T12, T13, T14)).
insertD(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) :- insertD(0, T22, T32).
insertD(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) :- pC(T43, T42, T22, T44).
insertD(T82, tree(T78, T79, T80), tree(T78, T79, T83)) :- pE(T78, T82, T80, T83).
insertD(s(T124), tree(0, T113, T114), tree(0, T113, T123)) :- insertD(s(T124), T114, T123).
insertD(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) :- lessB(T131, T133).
insertD(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) :- ','(lessB(T131, T137), insertD(s(T137), T114, T138)).
insertD(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) :- insertD(0, T152, T162).
insertD(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) :- pC(T173, T172, T152, T174).
insertD(T192, tree(T188, T189, T190), tree(T188, T189, T193)) :- pE(T188, T192, T190, T193).
insertD(s(T214), tree(0, T203, T204), tree(0, T203, T213)) :- insertD(s(T214), T204, T213).
insertD(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) :- lessB(T221, T223).
insertD(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) :- ','(lessB(T221, T227), insertD(s(T227), T204, T228)).

Query: insertD(a,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
insertD_in: (f,b,f) (b,b,f)
pC_in: (b,b,b,f) (f,b,b,f)
lessA_in: (b,b) (f,b)
pE_in: (b,b,b,f) (b,f,b,f)
lessB_in: (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

INSERTD_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_AGA(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
INSERTD_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_GGA(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_GGA(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_GGGA(T43, T42, T22, T44)
PC_IN_GGGA(T43, T42, T22, T44) → U3_GGGA(T43, T42, T22, T44, lessA_in_gg(T43, T42))
PC_IN_GGGA(T43, T42, T22, T44) → LESSA_IN_GG(T43, T42)
LESSA_IN_GG(s(T62), s(T61)) → U1_GG(T62, T61, lessA_in_gg(T62, T61))
LESSA_IN_GG(s(T62), s(T61)) → LESSA_IN_GG(T62, T61)
PC_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_GGGA(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_GGA(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GGGA(T78, T82, T80, T83)
PE_IN_GGGA(T78, T82, T80, T83) → U6_GGGA(T78, T82, T80, T83, lessB_in_gg(T78, T82))
PE_IN_GGGA(T78, T82, T80, T83) → LESSB_IN_GG(T78, T82)
LESSB_IN_GG(s(T99), s(T101)) → U2_GG(T99, T101, lessB_in_gg(T99, T101))
LESSB_IN_GG(s(T99), s(T101)) → LESSB_IN_GG(T99, T101)
PE_IN_GGGA(T78, T86, T80, T87) → U7_GGGA(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_GGGA(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80, T87)
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_GGA(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_GGA(s(T124), T114, T123)
INSERTD_IN_GGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_GGA(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
INSERTD_IN_GGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → LESSB_IN_GG(T131, T133)
INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_GGA(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_GGA(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114, T138)
INSERTD_IN_GGA(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_GGA(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
INSERTD_IN_GGA(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_GGA(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
INSERTD_IN_GGA(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_GGA(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
INSERTD_IN_GGA(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_GGA(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
INSERTD_IN_GGA(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_GGA(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_GGA(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_GGA(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204, T228)
INSERTD_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_AGA(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
INSERTD_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_AGGA(T43, T42, T22, T44)
PC_IN_AGGA(T43, T42, T22, T44) → U3_AGGA(T43, T42, T22, T44, lessA_in_ag(T43, T42))
PC_IN_AGGA(T43, T42, T22, T44) → LESSA_IN_AG(T43, T42)
LESSA_IN_AG(s(T62), s(T61)) → U1_AG(T62, T61, lessA_in_ag(T62, T61))
LESSA_IN_AG(s(T62), s(T61)) → LESSA_IN_AG(T62, T61)
PC_IN_AGGA(T47, T42, T22, T48) → U4_AGGA(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_AGGA(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_AGGA(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U4_AGGA(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_AGA(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GAGA(T78, T82, T80, T83)
PE_IN_GAGA(T78, T82, T80, T83) → U6_GAGA(T78, T82, T80, T83, lessB_in_ga(T78, T82))
PE_IN_GAGA(T78, T82, T80, T83) → LESSB_IN_GA(T78, T82)
LESSB_IN_GA(s(T99), s(T101)) → U2_GA(T99, T101, lessB_in_ga(T99, T101))
LESSB_IN_GA(s(T99), s(T101)) → LESSB_IN_GA(T99, T101)
PE_IN_GAGA(T78, T86, T80, T87) → U7_GAGA(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_GAGA(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → INSERTD_IN_AGA(T86, T80, T87)
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_AGA(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_AGA(s(T124), T114, T123)
INSERTD_IN_AGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_AGA(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
INSERTD_IN_AGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → LESSB_IN_GA(T131, T133)
INSERTD_IN_AGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_AGA(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_AGA(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → INSERTD_IN_AGA(s(T137), T114, T138)
INSERTD_IN_AGA(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_AGA(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
INSERTD_IN_AGA(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_AGA(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
INSERTD_IN_AGA(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_AGA(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
INSERTD_IN_AGA(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_AGA(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
INSERTD_IN_AGA(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_AGA(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
INSERTD_IN_AGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_AGA(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_AGA(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → INSERTD_IN_AGA(s(T227), T204, T228)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
INSERTD_IN_AGA(x1, x2, x3)  =  INSERTD_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x1, x2, x3, x5)
INSERTD_IN_GGA(x1, x2, x3)  =  INSERTD_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
PC_IN_GGGA(x1, x2, x3, x4)  =  PC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
LESSA_IN_GG(x1, x2)  =  LESSA_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)
U11_GGA(x1, x2, x3, x4, x5, x6)  =  U11_GGA(x1, x2, x3, x4, x6)
PE_IN_GGGA(x1, x2, x3, x4)  =  PE_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5)  =  U6_GGGA(x1, x2, x3, x5)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
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)  =  U16_GGA(x1, x2, x3, x5)
U17_GGA(x1, x2, x3, x4, x5, x6)  =  U17_GGA(x1, x2, x3, x4, x6)
U18_GGA(x1, x2, x3, x4, x5, x6)  =  U18_GGA(x1, x2, x3, x4, x6)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5, x6)  =  U20_GGA(x1, x2, x3, x4, x6)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)
U22_GGA(x1, x2, x3, x4, x5, x6)  =  U22_GGA(x1, x2, x3, x4, x6)
U10_AGA(x1, x2, x3, x4, x5, x6)  =  U10_AGA(x2, x3, x4, x6)
PC_IN_AGGA(x1, x2, x3, x4)  =  PC_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
LESSA_IN_AG(x1, x2)  =  LESSA_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)
U11_AGA(x1, x2, x3, x4, x5, x6)  =  U11_AGA(x2, x3, x4, x6)
PE_IN_GAGA(x1, x2, x3, x4)  =  PE_IN_GAGA(x1, x3)
U6_GAGA(x1, x2, x3, x4, x5)  =  U6_GAGA(x1, x3, x5)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
U8_GAGA(x1, x2, x3, x4, x5)  =  U8_GAGA(x1, x3, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x3, x5)
U13_AGA(x1, x2, x3, x4, x5, x6)  =  U13_AGA(x2, x3, x4, x6)
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)  =  U16_AGA(x1, x2, x3, x5)
U17_AGA(x1, x2, x3, x4, x5, x6)  =  U17_AGA(x2, x3, x4, x6)
U18_AGA(x1, x2, x3, x4, x5, x6)  =  U18_AGA(x2, x3, x4, x6)
U19_AGA(x1, x2, x3, x4, x5)  =  U19_AGA(x2, x3, x5)
U20_AGA(x1, x2, x3, x4, x5, x6)  =  U20_AGA(x2, x3, x4, x6)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)
U22_AGA(x1, x2, x3, x4, x5, x6)  =  U22_AGA(x2, x3, x4, x6)

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

(6) Obligation:

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

INSERTD_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_AGA(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
INSERTD_IN_AGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_GGA(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_GGA(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_GGGA(T43, T42, T22, T44)
PC_IN_GGGA(T43, T42, T22, T44) → U3_GGGA(T43, T42, T22, T44, lessA_in_gg(T43, T42))
PC_IN_GGGA(T43, T42, T22, T44) → LESSA_IN_GG(T43, T42)
LESSA_IN_GG(s(T62), s(T61)) → U1_GG(T62, T61, lessA_in_gg(T62, T61))
LESSA_IN_GG(s(T62), s(T61)) → LESSA_IN_GG(T62, T61)
PC_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_GGGA(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_GGA(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GGGA(T78, T82, T80, T83)
PE_IN_GGGA(T78, T82, T80, T83) → U6_GGGA(T78, T82, T80, T83, lessB_in_gg(T78, T82))
PE_IN_GGGA(T78, T82, T80, T83) → LESSB_IN_GG(T78, T82)
LESSB_IN_GG(s(T99), s(T101)) → U2_GG(T99, T101, lessB_in_gg(T99, T101))
LESSB_IN_GG(s(T99), s(T101)) → LESSB_IN_GG(T99, T101)
PE_IN_GGGA(T78, T86, T80, T87) → U7_GGGA(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_GGGA(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80, T87)
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_GGA(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_GGA(s(T124), T114, T123)
INSERTD_IN_GGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_GGA(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
INSERTD_IN_GGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → LESSB_IN_GG(T131, T133)
INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_GGA(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_GGA(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114, T138)
INSERTD_IN_GGA(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_GGA(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
INSERTD_IN_GGA(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_GGA(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
INSERTD_IN_GGA(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_GGA(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
INSERTD_IN_GGA(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_GGA(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
INSERTD_IN_GGA(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_GGA(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_GGA(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_GGA(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204, T228)
INSERTD_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_AGA(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
INSERTD_IN_AGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_AGGA(T43, T42, T22, T44)
PC_IN_AGGA(T43, T42, T22, T44) → U3_AGGA(T43, T42, T22, T44, lessA_in_ag(T43, T42))
PC_IN_AGGA(T43, T42, T22, T44) → LESSA_IN_AG(T43, T42)
LESSA_IN_AG(s(T62), s(T61)) → U1_AG(T62, T61, lessA_in_ag(T62, T61))
LESSA_IN_AG(s(T62), s(T61)) → LESSA_IN_AG(T62, T61)
PC_IN_AGGA(T47, T42, T22, T48) → U4_AGGA(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_AGGA(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_AGGA(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U4_AGGA(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_AGA(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GAGA(T78, T82, T80, T83)
PE_IN_GAGA(T78, T82, T80, T83) → U6_GAGA(T78, T82, T80, T83, lessB_in_ga(T78, T82))
PE_IN_GAGA(T78, T82, T80, T83) → LESSB_IN_GA(T78, T82)
LESSB_IN_GA(s(T99), s(T101)) → U2_GA(T99, T101, lessB_in_ga(T99, T101))
LESSB_IN_GA(s(T99), s(T101)) → LESSB_IN_GA(T99, T101)
PE_IN_GAGA(T78, T86, T80, T87) → U7_GAGA(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_GAGA(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → INSERTD_IN_AGA(T86, T80, T87)
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_AGA(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_AGA(s(T124), T114, T123)
INSERTD_IN_AGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_AGA(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
INSERTD_IN_AGA(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → LESSB_IN_GA(T131, T133)
INSERTD_IN_AGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_AGA(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_AGA(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → INSERTD_IN_AGA(s(T137), T114, T138)
INSERTD_IN_AGA(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_AGA(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
INSERTD_IN_AGA(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_AGA(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
INSERTD_IN_AGA(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_AGA(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
INSERTD_IN_AGA(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_AGA(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
INSERTD_IN_AGA(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_AGA(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
INSERTD_IN_AGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_AGA(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_AGA(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → INSERTD_IN_AGA(s(T227), T204, T228)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
INSERTD_IN_AGA(x1, x2, x3)  =  INSERTD_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x1, x2, x3, x5)
INSERTD_IN_GGA(x1, x2, x3)  =  INSERTD_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
PC_IN_GGGA(x1, x2, x3, x4)  =  PC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
LESSA_IN_GG(x1, x2)  =  LESSA_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)
U11_GGA(x1, x2, x3, x4, x5, x6)  =  U11_GGA(x1, x2, x3, x4, x6)
PE_IN_GGGA(x1, x2, x3, x4)  =  PE_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5)  =  U6_GGGA(x1, x2, x3, x5)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
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)  =  U16_GGA(x1, x2, x3, x5)
U17_GGA(x1, x2, x3, x4, x5, x6)  =  U17_GGA(x1, x2, x3, x4, x6)
U18_GGA(x1, x2, x3, x4, x5, x6)  =  U18_GGA(x1, x2, x3, x4, x6)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5, x6)  =  U20_GGA(x1, x2, x3, x4, x6)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)
U22_GGA(x1, x2, x3, x4, x5, x6)  =  U22_GGA(x1, x2, x3, x4, x6)
U10_AGA(x1, x2, x3, x4, x5, x6)  =  U10_AGA(x2, x3, x4, x6)
PC_IN_AGGA(x1, x2, x3, x4)  =  PC_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
LESSA_IN_AG(x1, x2)  =  LESSA_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)
U11_AGA(x1, x2, x3, x4, x5, x6)  =  U11_AGA(x2, x3, x4, x6)
PE_IN_GAGA(x1, x2, x3, x4)  =  PE_IN_GAGA(x1, x3)
U6_GAGA(x1, x2, x3, x4, x5)  =  U6_GAGA(x1, x3, x5)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
U8_GAGA(x1, x2, x3, x4, x5)  =  U8_GAGA(x1, x3, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x3, x5)
U13_AGA(x1, x2, x3, x4, x5, x6)  =  U13_AGA(x2, x3, x4, x6)
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)  =  U16_AGA(x1, x2, x3, x5)
U17_AGA(x1, x2, x3, x4, x5, x6)  =  U17_AGA(x2, x3, x4, x6)
U18_AGA(x1, x2, x3, x4, x5, x6)  =  U18_AGA(x2, x3, x4, x6)
U19_AGA(x1, x2, x3, x4, x5)  =  U19_AGA(x2, x3, x5)
U20_AGA(x1, x2, x3, x4, x5, x6)  =  U20_AGA(x2, x3, x4, x6)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)
U22_AGA(x1, x2, x3, x4, x5, x6)  =  U22_AGA(x2, x3, x4, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSB_IN_GA(s(T99), s(T101)) → LESSB_IN_GA(T99, T101)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)

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:

LESSB_IN_GA(s(T99), s(T101)) → LESSB_IN_GA(T99, T101)

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

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:

LESSB_IN_GA(s(T99)) → LESSB_IN_GA(T99)

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:

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

(15) YES

(16) Obligation:

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

LESSA_IN_AG(s(T62), s(T61)) → LESSA_IN_AG(T62, T61)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
LESSA_IN_AG(x1, x2)  =  LESSA_IN_AG(x2)

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:

LESSA_IN_AG(s(T62), s(T61)) → LESSA_IN_AG(T62, T61)

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

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:

LESSA_IN_AG(s(T61)) → LESSA_IN_AG(T61)

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:

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

(22) YES

(23) Obligation:

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

LESSB_IN_GG(s(T99), s(T101)) → LESSB_IN_GG(T99, T101)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(x1, x2)

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:

LESSB_IN_GG(s(T99), s(T101)) → LESSB_IN_GG(T99, T101)

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

(26) PiDPToQDPProof (EQUIVALENT 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:

LESSB_IN_GG(s(T99), s(T101)) → LESSB_IN_GG(T99, T101)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(29) YES

(30) Obligation:

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

LESSA_IN_GG(s(T62), s(T61)) → LESSA_IN_GG(T62, T61)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
LESSA_IN_GG(x1, x2)  =  LESSA_IN_GG(x1, x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

LESSA_IN_GG(s(T62), s(T61)) → LESSA_IN_GG(T62, T61)

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

(33) PiDPToQDPProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LESSA_IN_GG(s(T62), s(T61)) → LESSA_IN_GG(T62, T61)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(36) YES

(37) Obligation:

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

INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GGGA(T78, T82, T80, T83)
PE_IN_GGGA(T78, T86, T80, T87) → U7_GGGA(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80, T87)
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_GGGA(T43, T42, T22, T44)
PC_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_GGA(s(T124), T114, T123)
INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_GGA(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114, T138)
INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_GGA(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204, T228)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
INSERTD_IN_GGA(x1, x2, x3)  =  INSERTD_IN_GGA(x1, x2)
PC_IN_GGGA(x1, x2, x3, x4)  =  PC_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
PE_IN_GGGA(x1, x2, x3, x4)  =  PE_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

INSERTD_IN_GGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GGGA(T78, T82, T80, T83)
PE_IN_GGGA(T78, T86, T80, T87) → U7_GGGA(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_GGGA(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80, T87)
INSERTD_IN_GGA(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → INSERTD_IN_GGA(0, T22, T32)
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → PC_IN_GGGA(T43, T42, T22, T44)
PC_IN_GGGA(T47, T42, T22, T48) → U4_GGGA(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → INSERTD_IN_GGA(s(T47), T22, T48)
INSERTD_IN_GGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_GGA(s(T124), T114, T123)
INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_GGA(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_GGA(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114, T138)
INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_GGA(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_GGA(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204, T228)

The TRS R consists of the following rules:

lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_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
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
INSERTD_IN_GGA(x1, x2, x3)  =  INSERTD_IN_GGA(x1, x2)
PC_IN_GGGA(x1, x2, x3, x4)  =  PC_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)
PE_IN_GGGA(x1, x2, x3, x4)  =  PE_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5)  =  U7_GGGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)
U21_GGA(x1, x2, x3, x4, x5, x6)  =  U21_GGA(x1, x2, x3, x4, x6)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

INSERTD_IN_GGA(T82, tree(T78, T79, T80)) → PE_IN_GGGA(T78, T82, T80)
PE_IN_GGGA(T78, T86, T80) → U7_GGGA(T78, T86, T80, lessB_in_gg(T78, T86))
U7_GGGA(T78, T86, T80, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80)
INSERTD_IN_GGA(0, tree(s(T31), T22, T23)) → INSERTD_IN_GGA(0, T22)
INSERTD_IN_GGA(s(T43), tree(s(T42), T22, T23)) → PC_IN_GGGA(T43, T42, T22)
PC_IN_GGGA(T47, T42, T22) → U4_GGGA(T47, T42, T22, lessA_in_gg(T47, T42))
U4_GGGA(T47, T42, T22, lessA_out_gg(T47, T42)) → INSERTD_IN_GGA(s(T47), T22)
INSERTD_IN_GGA(s(T124), tree(0, T113, T114)) → INSERTD_IN_GGA(s(T124), T114)
INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114)) → U14_GGA(T137, T131, T113, T114, lessB_in_gg(T131, T137))
U14_GGA(T137, T131, T113, T114, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114)
INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204)) → U21_GGA(T227, T221, T203, T204, lessB_in_gg(T221, T227))
U21_GGA(T227, T221, T203, T204, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204)

The TRS R consists of the following rules:

lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))

The set Q consists of the following terms:

lessB_in_gg(x0, x1)
lessA_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U1_gg(x0, x1, x2)

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

(42) 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:

• PE_IN_GGGA(T78, T86, T80) → U7_GGGA(T78, T86, T80, lessB_in_gg(T78, T86))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

• U7_GGGA(T78, T86, T80, lessB_out_gg(T78, T86)) → INSERTD_IN_GGA(T86, T80)
  The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2

• INSERTD_IN_GGA(T82, tree(T78, T79, T80)) → PE_IN_GGGA(T78, T82, T80)
  The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3

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

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

• INSERTD_IN_GGA(s(T124), tree(0, T113, T114)) → INSERTD_IN_GGA(s(T124), T114)
  The graph contains the following edges 1 >= 1, 2 > 2

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

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

• U14_GGA(T137, T131, T113, T114, lessB_out_gg(T131, T137)) → INSERTD_IN_GGA(s(T137), T114)
  The graph contains the following edges 4 >= 2

• U21_GGA(T227, T221, T203, T204, lessB_out_gg(T221, T227)) → INSERTD_IN_GGA(s(T227), T204)
  The graph contains the following edges 4 >= 2

• INSERTD_IN_GGA(s(T137), tree(s(T131), T113, T114)) → U14_GGA(T137, T131, T113, T114, lessB_in_gg(T131, T137))
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4

• INSERTD_IN_GGA(s(T227), tree(s(T221), T203, T204)) → U21_GGA(T227, T221, T203, T204, lessB_in_gg(T221, T227))
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4

(43) YES

(44) Obligation:

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

INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GAGA(T78, T82, T80, T83)
PE_IN_GAGA(T78, T86, T80, T87) → U7_GAGA(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → INSERTD_IN_AGA(T86, T80, T87)
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_AGA(s(T124), T114, T123)
INSERTD_IN_AGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_AGA(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → INSERTD_IN_AGA(s(T137), T114, T138)
INSERTD_IN_AGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_AGA(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → INSERTD_IN_AGA(s(T227), T204, T228)

The TRS R consists of the following rules:

insertD_in_aga(T5, void, tree(T5, void, void)) → insertD_out_aga(T5, void, tree(T5, void, void))
insertD_in_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_aga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_aga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(T5, void, tree(T5, void, void)) → insertD_out_gga(T5, void, tree(T5, void, void))
insertD_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insertD_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insertD_in_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23)) → U9_gga(T31, T22, T23, T32, insertD_in_gga(0, T22, T32))
insertD_in_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_gga(T43, T42, T22, T23, T44, pC_in_ggga(T43, T42, T22, T44))
pC_in_ggga(T43, T42, T22, T44) → U3_ggga(T43, T42, T22, T44, lessA_in_gg(T43, T42))
lessA_in_gg(0, s(T55)) → lessA_out_gg(0, s(T55))
lessA_in_gg(s(T62), s(T61)) → U1_gg(T62, T61, lessA_in_gg(T62, T61))
U1_gg(T62, T61, lessA_out_gg(T62, T61)) → lessA_out_gg(s(T62), s(T61))
U3_ggga(T43, T42, T22, T44, lessA_out_gg(T43, T42)) → pC_out_ggga(T43, T42, T22, T44)
pC_in_ggga(T47, T42, T22, T48) → U4_ggga(T47, T42, T22, T48, lessA_in_gg(T47, T42))
U4_ggga(T47, T42, T22, T48, lessA_out_gg(T47, T42)) → U5_ggga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
insertD_in_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_gga(T82, T78, T79, T80, T83, pE_in_ggga(T78, T82, T80, T83))
pE_in_ggga(T78, T82, T80, T83) → U6_ggga(T78, T82, T80, T83, lessB_in_gg(T78, T82))
lessB_in_gg(0, s(T94)) → lessB_out_gg(0, s(T94))
lessB_in_gg(s(T99), s(T101)) → U2_gg(T99, T101, lessB_in_gg(T99, T101))
U2_gg(T99, T101, lessB_out_gg(T99, T101)) → lessB_out_gg(s(T99), s(T101))
U6_ggga(T78, T82, T80, T83, lessB_out_gg(T78, T82)) → pE_out_ggga(T78, T82, T80, T83)
pE_in_ggga(T78, T86, T80, T87) → U7_ggga(T78, T86, T80, T87, lessB_in_gg(T78, T86))
U7_ggga(T78, T86, T80, T87, lessB_out_gg(T78, T86)) → U8_ggga(T78, T86, T80, T87, insertD_in_gga(T86, T80, T87))
insertD_in_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_gga(T124, T113, T114, T123, insertD_in_gga(s(T124), T114, T123))
insertD_in_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_gga(T133, T131, T113, T114, T134, lessB_in_gg(T131, T133))
U13_gga(T133, T131, T113, T114, T134, lessB_out_gg(T131, T133)) → insertD_out_gga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_gga(T137, T131, T113, T114, T138, lessB_in_gg(T131, T137))
U14_gga(T137, T131, T113, T114, T138, lessB_out_gg(T131, T137)) → U15_gga(T137, T131, T113, T114, T138, insertD_in_gga(s(T137), T114, T138))
insertD_in_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_gga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
insertD_in_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_gga(T173, T172, T152, T153, T174, pC_in_ggga(T173, T172, T152, T174))
U17_gga(T173, T172, T152, T153, T174, pC_out_ggga(T173, T172, T152, T174)) → insertD_out_gga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_gga(T192, T188, T189, T190, T193, pE_in_ggga(T188, T192, T190, T193))
U18_gga(T192, T188, T189, T190, T193, pE_out_ggga(T188, T192, T190, T193)) → insertD_out_gga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_gga(T214, T203, T204, T213, insertD_in_gga(s(T214), T204, T213))
insertD_in_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_gga(T223, T221, T203, T204, T224, lessB_in_gg(T221, T223))
U20_gga(T223, T221, T203, T204, T224, lessB_out_gg(T221, T223)) → insertD_out_gga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_gga(T227, T221, T203, T204, T228, lessB_in_gg(T221, T227))
U21_gga(T227, T221, T203, T204, T228, lessB_out_gg(T221, T227)) → U22_gga(T227, T221, T203, T204, T228, insertD_in_gga(s(T227), T204, T228))
U22_gga(T227, T221, T203, T204, T228, insertD_out_gga(s(T227), T204, T228)) → insertD_out_gga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_gga(T214, T203, T204, T213, insertD_out_gga(s(T214), T204, T213)) → insertD_out_gga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U16_gga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_gga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
U15_gga(T137, T131, T113, T114, T138, insertD_out_gga(s(T137), T114, T138)) → insertD_out_gga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_gga(T124, T113, T114, T123, insertD_out_gga(s(T124), T114, T123)) → insertD_out_gga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_ggga(T78, T86, T80, T87, insertD_out_gga(T86, T80, T87)) → pE_out_ggga(T78, T86, T80, T87)
U11_gga(T82, T78, T79, T80, T83, pE_out_ggga(T78, T82, T80, T83)) → insertD_out_gga(T82, tree(T78, T79, T80), tree(T78, T79, T83))
U5_ggga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_ggga(T47, T42, T22, T48)
U10_gga(T43, T42, T22, T23, T44, pC_out_ggga(T43, T42, T22, T44)) → insertD_out_gga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
U9_gga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_gga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
U9_aga(T31, T22, T23, T32, insertD_out_gga(0, T22, T32)) → insertD_out_aga(0, tree(s(T31), T22, T23), tree(s(T31), T32, T23))
insertD_in_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23)) → U10_aga(T43, T42, T22, T23, T44, pC_in_agga(T43, T42, T22, T44))
pC_in_agga(T43, T42, T22, T44) → U3_agga(T43, T42, T22, T44, lessA_in_ag(T43, T42))
lessA_in_ag(0, s(T55)) → lessA_out_ag(0, s(T55))
lessA_in_ag(s(T62), s(T61)) → U1_ag(T62, T61, lessA_in_ag(T62, T61))
U1_ag(T62, T61, lessA_out_ag(T62, T61)) → lessA_out_ag(s(T62), s(T61))
U3_agga(T43, T42, T22, T44, lessA_out_ag(T43, T42)) → pC_out_agga(T43, T42, T22, T44)
pC_in_agga(T47, T42, T22, T48) → U4_agga(T47, T42, T22, T48, lessA_in_ag(T47, T42))
U4_agga(T47, T42, T22, T48, lessA_out_ag(T47, T42)) → U5_agga(T47, T42, T22, T48, insertD_in_gga(s(T47), T22, T48))
U5_agga(T47, T42, T22, T48, insertD_out_gga(s(T47), T22, T48)) → pC_out_agga(T47, T42, T22, T48)
U10_aga(T43, T42, T22, T23, T44, pC_out_agga(T43, T42, T22, T44)) → insertD_out_aga(s(T43), tree(s(T42), T22, T23), tree(s(T42), T44, T23))
insertD_in_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → U11_aga(T82, T78, T79, T80, T83, pE_in_gaga(T78, T82, T80, T83))
pE_in_gaga(T78, T82, T80, T83) → U6_gaga(T78, T82, T80, T83, lessB_in_ga(T78, T82))
lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_out_ga(s(T99), s(T101))
U6_gaga(T78, T82, T80, T83, lessB_out_ga(T78, T82)) → pE_out_gaga(T78, T82, T80, T83)
pE_in_gaga(T78, T86, T80, T87) → U7_gaga(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_gaga(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → U8_gaga(T78, T86, T80, T87, insertD_in_aga(T86, T80, T87))
insertD_in_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → U12_aga(T124, T113, T114, T123, insertD_in_aga(s(T124), T114, T123))
insertD_in_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134)) → U13_aga(T133, T131, T113, T114, T134, lessB_in_ga(T131, T133))
U13_aga(T133, T131, T113, T114, T134, lessB_out_ga(T131, T133)) → insertD_out_aga(s(T133), tree(s(T131), T113, T114), tree(s(T131), T113, T134))
insertD_in_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_aga(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_aga(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → U15_aga(T137, T131, T113, T114, T138, insertD_in_aga(s(T137), T114, T138))
insertD_in_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153)) → U16_aga(T161, T152, T153, T162, insertD_in_gga(0, T152, T162))
U16_aga(T161, T152, T153, T162, insertD_out_gga(0, T152, T162)) → insertD_out_aga(0, tree(s(T161), T152, T153), tree(s(T161), T162, T153))
insertD_in_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153)) → U17_aga(T173, T172, T152, T153, T174, pC_in_agga(T173, T172, T152, T174))
U17_aga(T173, T172, T152, T153, T174, pC_out_agga(T173, T172, T152, T174)) → insertD_out_aga(s(T173), tree(s(T172), T152, T153), tree(s(T172), T174, T153))
insertD_in_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193)) → U18_aga(T192, T188, T189, T190, T193, pE_in_gaga(T188, T192, T190, T193))
U18_aga(T192, T188, T189, T190, T193, pE_out_gaga(T188, T192, T190, T193)) → insertD_out_aga(T192, tree(T188, T189, T190), tree(T188, T189, T193))
insertD_in_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213)) → U19_aga(T214, T203, T204, T213, insertD_in_aga(s(T214), T204, T213))
insertD_in_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224)) → U20_aga(T223, T221, T203, T204, T224, lessB_in_ga(T221, T223))
U20_aga(T223, T221, T203, T204, T224, lessB_out_ga(T221, T223)) → insertD_out_aga(s(T223), tree(s(T221), T203, T204), tree(s(T221), T203, T224))
insertD_in_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_aga(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_aga(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → U22_aga(T227, T221, T203, T204, T228, insertD_in_aga(s(T227), T204, T228))
U22_aga(T227, T221, T203, T204, T228, insertD_out_aga(s(T227), T204, T228)) → insertD_out_aga(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228))
U19_aga(T214, T203, T204, T213, insertD_out_aga(s(T214), T204, T213)) → insertD_out_aga(s(T214), tree(0, T203, T204), tree(0, T203, T213))
U15_aga(T137, T131, T113, T114, T138, insertD_out_aga(s(T137), T114, T138)) → insertD_out_aga(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138))
U12_aga(T124, T113, T114, T123, insertD_out_aga(s(T124), T114, T123)) → insertD_out_aga(s(T124), tree(0, T113, T114), tree(0, T113, T123))
U8_gaga(T78, T86, T80, T87, insertD_out_aga(T86, T80, T87)) → pE_out_gaga(T78, T86, T80, T87)
U11_aga(T82, T78, T79, T80, T83, pE_out_gaga(T78, T82, T80, T83)) → insertD_out_aga(T82, tree(T78, T79, T80), tree(T78, T79, T83))

The argument filtering Pi contains the following mapping:
insertD_in_aga(x1, x2, x3)  =  insertD_in_aga(x2)
void  =  void
insertD_out_aga(x1, x2, x3)  =  insertD_out_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x1, x2, x3, x5)
insertD_in_gga(x1, x2, x3)  =  insertD_in_gga(x1, x2)
insertD_out_gga(x1, x2, x3)  =  insertD_out_gga(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x2, x3, x4, x6)
pC_in_ggga(x1, x2, x3, x4)  =  pC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
lessA_in_gg(x1, x2)  =  lessA_in_gg(x1, x2)
lessA_out_gg(x1, x2)  =  lessA_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
pC_out_ggga(x1, x2, x3, x4)  =  pC_out_ggga(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)
U11_gga(x1, x2, x3, x4, x5, x6)  =  U11_gga(x1, x2, x3, x4, x6)
pE_in_ggga(x1, x2, x3, x4)  =  pE_in_ggga(x1, x2, x3)
U6_ggga(x1, x2, x3, x4, x5)  =  U6_ggga(x1, x2, x3, x5)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
pE_out_ggga(x1, x2, x3, x4)  =  pE_out_ggga(x1, x2, x3)
U7_ggga(x1, x2, x3, x4, x5)  =  U7_ggga(x1, x2, x3, x5)
U8_ggga(x1, x2, x3, x4, x5)  =  U8_ggga(x1, x2, x3, x5)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5, x6)  =  U13_gga(x1, x2, x3, x4, x6)
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)  =  U16_gga(x1, x2, x3, x5)
U17_gga(x1, x2, x3, x4, x5, x6)  =  U17_gga(x1, x2, x3, x4, x6)
U18_gga(x1, x2, x3, x4, x5, x6)  =  U18_gga(x1, x2, x3, x4, x6)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5, x6)  =  U20_gga(x1, x2, x3, x4, x6)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U10_aga(x1, x2, x3, x4, x5, x6)  =  U10_aga(x2, x3, x4, x6)
pC_in_agga(x1, x2, x3, x4)  =  pC_in_agga(x2, x3)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
lessA_in_ag(x1, x2)  =  lessA_in_ag(x2)
lessA_out_ag(x1, x2)  =  lessA_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pC_out_agga(x1, x2, x3, x4)  =  pC_out_agga(x1, 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)
U11_aga(x1, x2, x3, x4, x5, x6)  =  U11_aga(x2, x3, x4, x6)
pE_in_gaga(x1, x2, x3, x4)  =  pE_in_gaga(x1, x3)
U6_gaga(x1, x2, x3, x4, x5)  =  U6_gaga(x1, x3, x5)
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
pE_out_gaga(x1, x2, x3, x4)  =  pE_out_gaga(x1, x3)
U7_gaga(x1, x2, x3, x4, x5)  =  U7_gaga(x1, x3, x5)
U8_gaga(x1, x2, x3, x4, x5)  =  U8_gaga(x1, x3, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x3, x5)
U13_aga(x1, x2, x3, x4, x5, x6)  =  U13_aga(x2, x3, x4, x6)
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)  =  U16_aga(x1, x2, x3, x5)
U17_aga(x1, x2, x3, x4, x5, x6)  =  U17_aga(x2, x3, x4, x6)
U18_aga(x1, x2, x3, x4, x5, x6)  =  U18_aga(x2, x3, x4, x6)
U19_aga(x1, x2, x3, x4, x5)  =  U19_aga(x2, x3, x5)
U20_aga(x1, x2, x3, x4, x5, x6)  =  U20_aga(x2, x3, x4, x6)
U21_aga(x1, x2, x3, x4, x5, x6)  =  U21_aga(x2, x3, x4, x6)
U22_aga(x1, x2, x3, x4, x5, x6)  =  U22_aga(x2, x3, x4, x6)
INSERTD_IN_AGA(x1, x2, x3)  =  INSERTD_IN_AGA(x2)
PE_IN_GAGA(x1, x2, x3, x4)  =  PE_IN_GAGA(x1, x3)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

INSERTD_IN_AGA(T82, tree(T78, T79, T80), tree(T78, T79, T83)) → PE_IN_GAGA(T78, T82, T80, T83)
PE_IN_GAGA(T78, T86, T80, T87) → U7_GAGA(T78, T86, T80, T87, lessB_in_ga(T78, T86))
U7_GAGA(T78, T86, T80, T87, lessB_out_ga(T78, T86)) → INSERTD_IN_AGA(T86, T80, T87)
INSERTD_IN_AGA(s(T124), tree(0, T113, T114), tree(0, T113, T123)) → INSERTD_IN_AGA(s(T124), T114, T123)
INSERTD_IN_AGA(s(T137), tree(s(T131), T113, T114), tree(s(T131), T113, T138)) → U14_AGA(T137, T131, T113, T114, T138, lessB_in_ga(T131, T137))
U14_AGA(T137, T131, T113, T114, T138, lessB_out_ga(T131, T137)) → INSERTD_IN_AGA(s(T137), T114, T138)
INSERTD_IN_AGA(s(T227), tree(s(T221), T203, T204), tree(s(T221), T203, T228)) → U21_AGA(T227, T221, T203, T204, T228, lessB_in_ga(T221, T227))
U21_AGA(T227, T221, T203, T204, T228, lessB_out_ga(T221, T227)) → INSERTD_IN_AGA(s(T227), T204, T228)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T94)) → lessB_out_ga(0, s(T94))
lessB_in_ga(s(T99), s(T101)) → U2_ga(T99, T101, lessB_in_ga(T99, T101))
U2_ga(T99, T101, lessB_out_ga(T99, T101)) → lessB_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
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
lessB_out_ga(x1, x2)  =  lessB_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
INSERTD_IN_AGA(x1, x2, x3)  =  INSERTD_IN_AGA(x2)
PE_IN_GAGA(x1, x2, x3, x4)  =  PE_IN_GAGA(x1, x3)
U7_GAGA(x1, x2, x3, x4, x5)  =  U7_GAGA(x1, x3, x5)
U14_AGA(x1, x2, x3, x4, x5, x6)  =  U14_AGA(x2, x3, x4, x6)
U21_AGA(x1, x2, x3, x4, x5, x6)  =  U21_AGA(x2, x3, x4, x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

INSERTD_IN_AGA(tree(T78, T79, T80)) → PE_IN_GAGA(T78, T80)
PE_IN_GAGA(T78, T80) → U7_GAGA(T78, T80, lessB_in_ga(T78))
U7_GAGA(T78, T80, lessB_out_ga(T78)) → INSERTD_IN_AGA(T80)
INSERTD_IN_AGA(tree(0, T113, T114)) → INSERTD_IN_AGA(T114)
INSERTD_IN_AGA(tree(s(T131), T113, T114)) → U14_AGA(T131, T113, T114, lessB_in_ga(T131))
U14_AGA(T131, T113, T114, lessB_out_ga(T131)) → INSERTD_IN_AGA(T114)
INSERTD_IN_AGA(tree(s(T221), T203, T204)) → U21_AGA(T221, T203, T204, lessB_in_ga(T221))
U21_AGA(T221, T203, T204, lessB_out_ga(T221)) → INSERTD_IN_AGA(T204)

The TRS R consists of the following rules:

lessB_in_ga(0) → lessB_out_ga(0)
lessB_in_ga(s(T99)) → U2_ga(T99, lessB_in_ga(T99))
U2_ga(T99, lessB_out_ga(T99)) → lessB_out_ga(s(T99))

The set Q consists of the following terms:

lessB_in_ga(x0)
U2_ga(x0, x1)

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

(49) 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:

• PE_IN_GAGA(T78, T80) → U7_GAGA(T78, T80, lessB_in_ga(T78))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U7_GAGA(T78, T80, lessB_out_ga(T78)) → INSERTD_IN_AGA(T80)
  The graph contains the following edges 2 >= 1

• INSERTD_IN_AGA(tree(T78, T79, T80)) → PE_IN_GAGA(T78, T80)
  The graph contains the following edges 1 > 1, 1 > 2

• INSERTD_IN_AGA(tree(0, T113, T114)) → INSERTD_IN_AGA(T114)
  The graph contains the following edges 1 > 1

• U14_AGA(T131, T113, T114, lessB_out_ga(T131)) → INSERTD_IN_AGA(T114)
  The graph contains the following edges 3 >= 1

• U21_AGA(T221, T203, T204, lessB_out_ga(T221)) → INSERTD_IN_AGA(T204)
  The graph contains the following edges 3 >= 1

• INSERTD_IN_AGA(tree(s(T131), T113, T114)) → U14_AGA(T131, T113, T114, lessB_in_ga(T131))
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

• INSERTD_IN_AGA(tree(s(T221), T203, T204)) → U21_AGA(T221, T203, T204, lessB_in_ga(T221))
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

(50) YES