Left Termination of the query pattern
insert(g,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(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: insert(T1, T2, T3)\n\nT1 is ground\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\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | insert(T5, void, T3), SCOPE: 1, CLAUSE: 1 | insert(T5, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T5, void, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", 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\nT1 is ground\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(T5, void, T3), SCOPE: 1, CLAUSE: 1 | insert(T5, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T5, void, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: insert(T5, void, T3), SCOPE: 1, CLAUSE: 2 | insert(T5, void, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: insert(T5, void, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | insert(T12, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 2 | insert(T12, 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\nT1 is ground\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(T12, tree(T12, T13, T14), T3), SCOPE: 1, CLAUSE: 2 | insert(T12, 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(T19, T19), insert(T19, T20, T23)) | insert(T19, tree(T19, T20, T21), T3), SCOPE: 1, CLAUSE: 3\n\nT19 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: insert(T12, 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(T12, 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(T19, T19), insert(T19, T20, T23)), SCOPE: 2, CLAUSE: 4 | ','(less(T19, T19), insert(T19, T20, T23)), SCOPE: 2, CLAUSE: 5 | ?_2 | insert(T19, tree(T19, T20, T21), T3), SCOPE: 1, CLAUSE: 3\n\nT19 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(less(T19, T19), insert(T19, T20, T23)), SCOPE: 2, CLAUSE: 5 | ?_2 | insert(T19, tree(T19, T20, T21), T3), SCOPE: 1, CLAUSE: 3\n\nT19 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(less(T19, T19), insert(T19, T20, T23)), SCOPE: 2, CLAUSE: 5\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ?_2 | insert(T19, tree(T19, T20, T21), T3), SCOPE: 1, CLAUSE: 3\n\nT19 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="B: ','(less(T28, T28), insert(s(T28), T20, T23))\n\nT20 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="C: less(T28, T28)\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: insert(s(T28), T20, T23)\n\nT20 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: less(T28, T28), SCOPE: 3, CLAUSE: 4 | less(T28, T28), SCOPE: 3, CLAUSE: 5\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T28, T28), SCOPE: 3, CLAUSE: 5\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T31, T31)\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: insert(T19, tree(T19, T20, T21), T3), SCOPE: 1, CLAUSE: 3\n\nT19 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(less(T42, T42), insert(T42, T44, T46))\n\nT42 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T42, T42)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: insert(T42, T44, T46)\n\nT42 is ground\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(less(T53, T53), insert(T53, T55, T57))\n\nT53 is ground\nT54 is ground\nT55 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T53, tree(T53, T54, T55), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(less(T53, T53), insert(T53, T55, T57)), SCOPE: 4, CLAUSE: 4 | ','(less(T53, T53), insert(T53, T55, T57)), SCOPE: 4, CLAUSE: 5\n\nT53 is ground\nT54 is ground\nT55 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T53, tree(T53, T54, T55), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(less(T53, T53), insert(T53, T55, T57)), SCOPE: 4, CLAUSE: 5\n\nT53 is ground\nT54 is ground\nT55 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(T53, tree(T53, T54, T55), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(less(T60, T60), insert(s(T60), T55, T57))\n\nT54 is ground\nT55 is ground\nT60 is ground\nX28 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ninsert(s(T60), tree(s(T60), T54, T55), T3) !=? insert(X28, tree(X29, X30, X31), tree(X29, X32, X31))", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(less(T66, T67), insert(T66, T68, T71)) | insert(T66, tree(T67, T68, T69), T3), SCOPE: 1, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\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];
39 [label="39: ','(less(T66, T67), insert(T66, T68, T71)), SCOPE: 5, CLAUSE: 4 | ','(less(T66, T67), insert(T66, T68, T71)), SCOPE: 5, CLAUSE: 5 | ?_5 | insert(T66, tree(T67, T68, T69), T3), SCOPE: 1, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(less(T66, T67), insert(T66, T68, T71)), SCOPE: 5, CLAUSE: 4\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(less(T66, T67), insert(T66, T68, T71)), SCOPE: 5, CLAUSE: 5 | ?_5 | insert(T66, tree(T67, T68, T69), T3), SCOPE: 1, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: insert(0, T68, T71)\n\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(less(T66, T67), insert(T66, T68, T71)), SCOPE: 5, CLAUSE: 5\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_5 | insert(T66, tree(T67, T68, T69), T3), SCOPE: 1, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(less(T85, T86), insert(s(T85), T68, T71))\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="D: less(T85, T86)\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: insert(s(T85), T68, T71)\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T85, T86), SCOPE: 6, CLAUSE: 4 | less(T85, T86), SCOPE: 6, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T85, T86), SCOPE: 6, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(T85, T86), SCOPE: 6, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT85 is ground\nT86 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T85), tree(s(T86), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T100, T101)\n\nT68 is ground\nT69 is ground\nT100 is ground\nT101 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(T100)), tree(s(s(T101)), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(T100, T101), SCOPE: 7, CLAUSE: 4 | less(T100, T101), SCOPE: 7, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT100 is ground\nT101 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(T100)), tree(s(s(T101)), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T100, T101), SCOPE: 7, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT100 is ground\nT101 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(T100)), tree(s(s(T101)), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T100, T101), SCOPE: 7, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT100 is ground\nT101 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(T100)), tree(s(s(T101)), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T113, T114)\n\nT68 is ground\nT69 is ground\nT113 is ground\nT114 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(T113))), tree(s(s(s(T114))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: less(T113, T114), SCOPE: 8, CLAUSE: 4 | less(T113, T114), SCOPE: 8, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT113 is ground\nT114 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(T113))), tree(s(s(s(T114))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(T113, T114), SCOPE: 8, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT113 is ground\nT114 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(T113))), tree(s(s(s(T114))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: less(T113, T114), SCOPE: 8, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT113 is ground\nT114 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(T113))), tree(s(s(s(T114))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: less(T126, T127)\n\nT68 is ground\nT69 is ground\nT126 is ground\nT127 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(T126)))), tree(s(s(s(s(T127)))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: less(T126, T127), SCOPE: 9, CLAUSE: 4 | less(T126, T127), SCOPE: 9, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT126 is ground\nT127 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(T126)))), tree(s(s(s(s(T127)))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T126, T127), SCOPE: 9, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT126 is ground\nT127 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(T126)))), tree(s(s(s(s(T127)))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(T126, T127), SCOPE: 9, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT126 is ground\nT127 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(T126)))), tree(s(s(s(s(T127)))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(T139, T140)\n\nT68 is ground\nT69 is ground\nT139 is ground\nT140 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(T139))))), tree(s(s(s(s(s(T140))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: less(T139, T140), SCOPE: 10, CLAUSE: 4 | less(T139, T140), SCOPE: 10, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT139 is ground\nT140 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(T139))))), tree(s(s(s(s(s(T140))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: less(T139, T140), SCOPE: 10, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT139 is ground\nT140 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(T139))))), tree(s(s(s(s(s(T140))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: less(T139, T140), SCOPE: 10, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT139 is ground\nT140 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(T139))))), tree(s(s(s(s(s(T140))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: less(T152, T153)\n\nT68 is ground\nT69 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(T152)))))), tree(s(s(s(s(s(s(T153)))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: less(T152, T153), SCOPE: 11, CLAUSE: 4 | less(T152, T153), SCOPE: 11, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(T152)))))), tree(s(s(s(s(s(s(T153)))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(T152, T153), SCOPE: 11, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(T152)))))), tree(s(s(s(s(s(s(T153)))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: less(T152, T153), SCOPE: 11, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT152 is ground\nT153 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(T152)))))), tree(s(s(s(s(s(s(T153)))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: less(T165, T166)\n\nT68 is ground\nT69 is ground\nT165 is ground\nT166 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(s(T165))))))), tree(s(s(s(s(s(s(s(T166))))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: less(T165, T166), SCOPE: 12, CLAUSE: 4 | less(T165, T166), SCOPE: 12, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT165 is ground\nT166 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(s(T165))))))), tree(s(s(s(s(s(s(s(T166))))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: less(T165, T166), SCOPE: 12, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nT165 is ground\nT166 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(s(T165))))))), tree(s(s(s(s(s(s(s(T166))))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(T165, T166), SCOPE: 12, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nT165 is ground\nT166 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(s(T165))))))), tree(s(s(s(s(s(s(s(T166))))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: less(T178, T179)\n\nT68 is ground\nT69 is ground\nT178 is ground\nT179 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(s(s(s(s(s(s(s(T178)))))))), tree(s(s(s(s(s(s(s(s(T179)))))))), T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="E: less(T178, T179)\n\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: less(T178, T179), SCOPE: 13, CLAUSE: 4 | less(T178, T179), SCOPE: 13, CLAUSE: 5\n\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: less(T178, T179), SCOPE: 13, CLAUSE: 4\n\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: less(T178, T179), SCOPE: 13, CLAUSE: 5\n\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: less(T195, T196)\n\nT195 is ground\nT196 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: insert(T66, tree(T67, T68, T69), T3), SCOPE: 1, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nT68 is ground\nT69 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T66, tree(T67, T68, T69), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(less(T212, T211), insert(T211, T214, T216))\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T211, tree(T212, T213, T214), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: less(T212, T211)\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T211, tree(T212, T213, T214), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: insert(T211, T214, T216)\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T211, tree(T212, T213, T214), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(less(T229, T228), insert(T228, T231, T233))\n\nT228 is ground\nT229 is ground\nT230 is ground\nT231 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T228, tree(T229, T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(less(T229, T228), insert(T228, T231, T233)), SCOPE: 14, CLAUSE: 4 | ','(less(T229, T228), insert(T228, T231, T233)), SCOPE: 14, CLAUSE: 5\n\nT228 is ground\nT229 is ground\nT230 is ground\nT231 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T228, tree(T229, T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(less(T229, T228), insert(T228, T231, T233)), SCOPE: 14, CLAUSE: 4\n\nT228 is ground\nT229 is ground\nT230 is ground\nT231 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T228, tree(T229, T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(less(T229, T228), insert(T228, T231, T233)), SCOPE: 14, CLAUSE: 5\n\nT228 is ground\nT229 is ground\nT230 is ground\nT231 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(T228, tree(T229, T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: insert(s(T238), T231, T233)\n\nT231 is ground\nT238 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: ','(less(T245, T246), insert(s(T246), T231, T233))\n\nT230 is ground\nT231 is ground\nT245 is ground\nT246 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T246), tree(s(T245), T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: less(T245, T246)\n\nT230 is ground\nT231 is ground\nT245 is ground\nT246 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T246), tree(s(T245), T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: insert(s(T246), T231, T233)\n\nT230 is ground\nT231 is ground\nT245 is ground\nT246 is ground\nX20 is free\nX21 is free\nX22 is free\ninsert(s(T246), tree(s(T245), T230, T231), T3) !=? insert(X20, tree(X20, X21, X22), tree(X20, X21, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 131 [arrowhead = none ];
131 -> 2;

132 [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 -> 132 [arrowhead = none ];
132 -> 3;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 133 [arrowhead = none ];
133 -> 4;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 134 [arrowhead = none ];
134 -> 5;

135 [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 -> 135 [arrowhead = none ];
135 -> 9;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 136 [arrowhead = none ];
136 -> 10;

137 [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 -> 137 [arrowhead = none ];
137 -> 6;

138 [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 -> 138 [arrowhead = none ];
138 -> 7;

139 [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 -> 139 [arrowhead = none ];
139 -> 8;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 140 [arrowhead = none ];
140 -> 11;

141 [label="EVAL with clause\ninsert(X104, tree(X105, X106, X107), tree(X105, X108, X107)) :- ','(less(X104, X105), insert(X104, X106, X108)).\nand substitutionT1 -> T66,\nX104 -> T66,\nX105 -> T67,\nX106 -> T68,\nX107 -> T69,\nT2 -> tree(T67, T68, T69),\nX108 -> T71,\nT3 -> tree(T67, T71, T69),\nT70 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 141 [arrowhead = none ];
141 -> 37;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 142 [arrowhead = none ];
142 -> 38;

143 [label="EVAL with clause\ninsert(X28, tree(X29, X30, X31), tree(X29, X32, X31)) :- ','(less(X28, X29), insert(X28, X30, X32)).\nand substitutionT12 -> T19,\nX28 -> T19,\nX29 -> T19,\nT13 -> T20,\nX30 -> T20,\nT14 -> T21,\nX31 -> T21,\nX32 -> T23,\nT3 -> tree(T19, T23, T21),\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 143 [arrowhead = none ];
143 -> 12;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 144 [arrowhead = none ];
144 -> 13;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 145 [arrowhead = none ];
145 -> 14;

146 [label="EVAL with clause\ninsert(X85, tree(X86, X87, X88), tree(X86, X87, X89)) :- ','(less(X86, X85), insert(X85, X88, X89)).\nand substitutionT12 -> T53,\nX85 -> T53,\nX86 -> T53,\nT13 -> T54,\nX87 -> T54,\nT14 -> T55,\nX88 -> T55,\nX89 -> T57,\nT3 -> tree(T53, T54, T57),\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 146 [arrowhead = none ];
146 -> 31;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 147 [arrowhead = none ];
147 -> 32;

148 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 148 [arrowhead = none ];
148 -> 15;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 149 [arrowhead = none ];
149 -> 16;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 150 [arrowhead = none ];
150 -> 17;

151 [label="EVAL with clause\nless(s(X42), s(X43)) :- less(X42, X43).\nand substitutionX42 -> T28,\nT19 -> s(T28),\nX43 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 151 [arrowhead = none ];
151 -> 18;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 152 [arrowhead = none ];
152 -> 19;

153 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 153 [arrowhead = none ];
153 -> 26;

154 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 154 [arrowhead = none ];
154 -> 20;

155 [label="SPLIT 2\nnew knowledge:\nT28 is ground", fontsize=14, style = filled, fillcolor = violet];
18 -> 155 [arrowhead = none ];
155 -> 21;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 156 [arrowhead = none ];
156 -> 22;

157 [label="INSTANCE with matching:\nT1 -> s(T28)\nT2 -> T20\nT3 -> T23", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 157 [arrowhead = none , style=dashed];
157 -> 1[style=dashed];

158 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 158 [arrowhead = none ];
158 -> 23;

159 [label="EVAL with clause\nless(s(X53), s(X54)) :- less(X53, X54).\nand substitutionX53 -> T31,\nT28 -> s(T31),\nX54 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 159 [arrowhead = none ];
159 -> 24;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 160 [arrowhead = none ];
160 -> 25;

161 [label="INSTANCE with matching:\nT28 -> T31", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 161 [arrowhead = none , style=dashed];
161 -> 20[style=dashed];

162 [label="EVAL with clause\ninsert(X69, tree(X70, X71, X72), tree(X70, X71, X73)) :- ','(less(X70, X69), insert(X69, X72, X73)).\nand substitutionT19 -> T42,\nX69 -> T42,\nX70 -> T42,\nT20 -> T43,\nX71 -> T43,\nT21 -> T44,\nX72 -> T44,\nX73 -> T46,\nT3 -> tree(T42, T43, T46),\nT45 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 162 [arrowhead = none ];
162 -> 27;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 163 [arrowhead = none ];
163 -> 28;

164 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 164 [arrowhead = none ];
164 -> 29;

165 [label="SPLIT 2\nnew knowledge:\nT42 is ground", fontsize=14, style = filled, fillcolor = violet];
27 -> 165 [arrowhead = none ];
165 -> 30;

166 [label="INSTANCE with matching:\nT28 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 166 [arrowhead = none , style=dashed];
166 -> 20[style=dashed];

167 [label="INSTANCE with matching:\nT1 -> T42\nT2 -> T44\nT3 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 167 [arrowhead = none , style=dashed];
167 -> 1[style=dashed];

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 168 [arrowhead = none ];
168 -> 33;

169 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 169 [arrowhead = none ];
169 -> 34;

170 [label="EVAL with clause\nless(s(X95), s(X96)) :- less(X95, X96).\nand substitutionX95 -> T60,\nT53 -> s(T60),\nX96 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 170 [arrowhead = none ];
170 -> 35;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 171 [arrowhead = none ];
171 -> 36;

172 [label="INSTANCE with matching:\nT28 -> T60\nT20 -> T55\nT23 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 172 [arrowhead = none , style=dashed];
172 -> 18[style=dashed];

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 173 [arrowhead = none ];
173 -> 39;

174 [label="EVAL with clause\ninsert(X264, tree(X265, X266, X267), tree(X265, X266, X268)) :- ','(less(X265, X264), insert(X264, X267, X268)).\nand substitutionT1 -> T228,\nX264 -> T228,\nX265 -> T229,\nX266 -> T230,\nX267 -> T231,\nT2 -> tree(T229, T230, T231),\nX268 -> T233,\nT3 -> tree(T229, T230, T233),\nT232 -> T233", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 174 [arrowhead = none ];
174 -> 120;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 175 [arrowhead = none ];
175 -> 121;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 176 [arrowhead = none ];
176 -> 40;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 177 [arrowhead = none ];
177 -> 41;

178 [label="EVAL with clause\nless(0, s(X113)).\nand substitutionT66 -> 0,\nX113 -> T76,\nT67 -> s(T76)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 178 [arrowhead = none ];
178 -> 42;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 179 [arrowhead = none ];
179 -> 43;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 180 [arrowhead = none ];
180 -> 44;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 181 [arrowhead = none ];
181 -> 45;

182 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T68\nT3 -> T71", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 182 [arrowhead = none , style=dashed];
182 -> 1[style=dashed];

183 [label="EVAL with clause\nless(s(X124), s(X125)) :- less(X124, X125).\nand substitutionX124 -> T85,\nT66 -> s(T85),\nX125 -> T86,\nT67 -> s(T86)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 183 [arrowhead = none ];
183 -> 46;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 184 [arrowhead = none ];
184 -> 47;

185 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 185 [arrowhead = none ];
185 -> 115;

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 186 [arrowhead = none ];
186 -> 48;

187 [label="SPLIT 2\nnew knowledge:\nT85 is ground\nT86 is ground", fontsize=14, style = filled, fillcolor = violet];
46 -> 187 [arrowhead = none ];
187 -> 49;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 188 [arrowhead = none ];
188 -> 50;

189 [label="INSTANCE with matching:\nT1 -> s(T85)\nT2 -> T68\nT3 -> T71", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 189 [arrowhead = none , style=dashed];
189 -> 1[style=dashed];

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 190 [arrowhead = none ];
190 -> 51;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 191 [arrowhead = none ];
191 -> 52;

192 [label="EVAL with clause\nless(0, s(X134)).\nand substitutionT85 -> 0,\nX134 -> T95,\nT86 -> s(T95)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 192 [arrowhead = none ];
192 -> 53;

193 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 193 [arrowhead = none ];
193 -> 54;

194 [label="EVAL with clause\nless(s(X139), s(X140)) :- less(X139, X140).\nand substitutionX139 -> T100,\nT85 -> s(T100),\nX140 -> T101,\nT86 -> s(T101)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 194 [arrowhead = none ];
194 -> 56;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 195 [arrowhead = none ];
195 -> 57;

196 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 196 [arrowhead = none ];
196 -> 55;

197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 197 [arrowhead = none ];
197 -> 58;

198 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 198 [arrowhead = none ];
198 -> 59;

199 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 199 [arrowhead = none ];
199 -> 60;

200 [label="EVAL with clause\nless(0, s(X147)).\nand substitutionT100 -> 0,\nX147 -> T108,\nT101 -> s(T108)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 200 [arrowhead = none ];
200 -> 61;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 201 [arrowhead = none ];
201 -> 62;

202 [label="EVAL with clause\nless(s(X152), s(X153)) :- less(X152, X153).\nand substitutionX152 -> T113,\nT100 -> s(T113),\nX153 -> T114,\nT101 -> s(T114)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 202 [arrowhead = none ];
202 -> 64;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 203 [arrowhead = none ];
203 -> 65;

204 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 204 [arrowhead = none ];
204 -> 63;

205 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 205 [arrowhead = none ];
205 -> 66;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 206 [arrowhead = none ];
206 -> 67;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 207 [arrowhead = none ];
207 -> 68;

208 [label="EVAL with clause\nless(0, s(X160)).\nand substitutionT113 -> 0,\nX160 -> T121,\nT114 -> s(T121)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 208 [arrowhead = none ];
208 -> 69;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 209 [arrowhead = none ];
209 -> 70;

210 [label="EVAL with clause\nless(s(X165), s(X166)) :- less(X165, X166).\nand substitutionX165 -> T126,\nT113 -> s(T126),\nX166 -> T127,\nT114 -> s(T127)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 210 [arrowhead = none ];
210 -> 72;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 211 [arrowhead = none ];
211 -> 73;

212 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 212 [arrowhead = none ];
212 -> 71;

213 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 213 [arrowhead = none ];
213 -> 74;

214 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 214 [arrowhead = none ];
214 -> 75;

215 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 215 [arrowhead = none ];
215 -> 76;

216 [label="EVAL with clause\nless(0, s(X173)).\nand substitutionT126 -> 0,\nX173 -> T134,\nT127 -> s(T134)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 216 [arrowhead = none ];
216 -> 77;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 217 [arrowhead = none ];
217 -> 78;

218 [label="EVAL with clause\nless(s(X178), s(X179)) :- less(X178, X179).\nand substitutionX178 -> T139,\nT126 -> s(T139),\nX179 -> T140,\nT127 -> s(T140)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 218 [arrowhead = none ];
218 -> 80;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 219 [arrowhead = none ];
219 -> 81;

220 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 220 [arrowhead = none ];
220 -> 79;

221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 221 [arrowhead = none ];
221 -> 82;

222 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 222 [arrowhead = none ];
222 -> 83;

223 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 223 [arrowhead = none ];
223 -> 84;

224 [label="EVAL with clause\nless(0, s(X186)).\nand substitutionT139 -> 0,\nX186 -> T147,\nT140 -> s(T147)", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 224 [arrowhead = none ];
224 -> 85;

225 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 225 [arrowhead = none ];
225 -> 86;

226 [label="EVAL with clause\nless(s(X191), s(X192)) :- less(X191, X192).\nand substitutionX191 -> T152,\nT139 -> s(T152),\nX192 -> T153,\nT140 -> s(T153)", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 226 [arrowhead = none ];
226 -> 88;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 227 [arrowhead = none ];
227 -> 89;

228 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 228 [arrowhead = none ];
228 -> 87;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 229 [arrowhead = none ];
229 -> 90;

230 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 230 [arrowhead = none ];
230 -> 91;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 231 [arrowhead = none ];
231 -> 92;

232 [label="EVAL with clause\nless(0, s(X199)).\nand substitutionT152 -> 0,\nX199 -> T160,\nT153 -> s(T160)", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 232 [arrowhead = none ];
232 -> 93;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 233 [arrowhead = none ];
233 -> 94;

234 [label="EVAL with clause\nless(s(X204), s(X205)) :- less(X204, X205).\nand substitutionX204 -> T165,\nT152 -> s(T165),\nX205 -> T166,\nT153 -> s(T166)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 234 [arrowhead = none ];
234 -> 96;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 235 [arrowhead = none ];
235 -> 97;

236 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 236 [arrowhead = none ];
236 -> 95;

237 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 237 [arrowhead = none ];
237 -> 98;

238 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 238 [arrowhead = none ];
238 -> 99;

239 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 239 [arrowhead = none ];
239 -> 100;

240 [label="EVAL with clause\nless(0, s(X212)).\nand substitutionT165 -> 0,\nX212 -> T173,\nT166 -> s(T173)", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 240 [arrowhead = none ];
240 -> 101;

241 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 241 [arrowhead = none ];
241 -> 102;

242 [label="EVAL with clause\nless(s(X217), s(X218)) :- less(X217, X218).\nand substitutionX217 -> T178,\nT165 -> s(T178),\nX218 -> T179,\nT166 -> s(T179)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 242 [arrowhead = none ];
242 -> 104;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 243 [arrowhead = none ];
243 -> 105;

244 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 244 [arrowhead = none ];
244 -> 103;

245 [label="GENERALIZATION\nT182 <-- s(s(s(s(s(s(s(T178)))))))\nT183 <-- s(s(s(s(s(s(s(T179)))))))\n\nNew Knowledge:\nT182 is ground\nT183 is ground", fontsize=14, style = filled, fillcolor = violet];
104 -> 245 [arrowhead = none ];
245 -> 106;

246 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
106 -> 246 [arrowhead = none ];
246 -> 107;

247 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 247 [arrowhead = none ];
247 -> 108;

248 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 248 [arrowhead = none ];
248 -> 109;

249 [label="EVAL with clause\nless(0, s(X227)).\nand substitutionT178 -> 0,\nX227 -> T190,\nT179 -> s(T190)", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 249 [arrowhead = none ];
249 -> 110;

250 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 250 [arrowhead = none ];
250 -> 111;

251 [label="EVAL with clause\nless(s(X232), s(X233)) :- less(X232, X233).\nand substitutionX232 -> T195,\nT178 -> s(T195),\nX233 -> T196,\nT179 -> s(T196)", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 251 [arrowhead = none ];
251 -> 113;

252 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 252 [arrowhead = none ];
252 -> 114;

253 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 253 [arrowhead = none ];
253 -> 112;

254 [label="INSTANCE with matching:\nT178 -> T195\nT179 -> T196", fontsize=14, style = filled, fillcolor = lightgrey];
113 -> 254 [arrowhead = none , style=dashed];
254 -> 106[style=dashed];

255 [label="EVAL with clause\ninsert(X248, tree(X249, X250, X251), tree(X249, X250, X252)) :- ','(less(X249, X248), insert(X248, X251, X252)).\nand substitutionT66 -> T211,\nX248 -> T211,\nT67 -> T212,\nX249 -> T212,\nT68 -> T213,\nX250 -> T213,\nT69 -> T214,\nX251 -> T214,\nX252 -> T216,\nT3 -> tree(T212, T213, T216),\nT215 -> T216", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 255 [arrowhead = none ];
255 -> 116;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 256 [arrowhead = none ];
256 -> 117;

257 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
116 -> 257 [arrowhead = none ];
257 -> 118;

258 [label="SPLIT 2\nnew knowledge:\nT212 is ground\nT211 is ground", fontsize=14, style = filled, fillcolor = violet];
116 -> 258 [arrowhead = none ];
258 -> 119;

259 [label="INSTANCE with matching:\nT178 -> T212\nT179 -> T211", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 259 [arrowhead = none , style=dashed];
259 -> 106[style=dashed];

260 [label="INSTANCE with matching:\nT1 -> T211\nT2 -> T214\nT3 -> T216", fontsize=14, style = filled, fillcolor = lightgrey];
119 -> 260 [arrowhead = none , style=dashed];
260 -> 1[style=dashed];

261 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
120 -> 261 [arrowhead = none ];
261 -> 122;

262 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 262 [arrowhead = none ];
262 -> 123;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 263 [arrowhead = none ];
263 -> 124;

264 [label="EVAL with clause\nless(0, s(X273)).\nand substitutionT229 -> 0,\nX273 -> T238,\nT228 -> s(T238)", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 264 [arrowhead = none ];
264 -> 125;

265 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 265 [arrowhead = none ];
265 -> 126;

266 [label="EVAL with clause\nless(s(X280), s(X281)) :- less(X280, X281).\nand substitutionX280 -> T245,\nT229 -> s(T245),\nX281 -> T246,\nT228 -> s(T246)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 266 [arrowhead = none ];
266 -> 127;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 267 [arrowhead = none ];
267 -> 128;

268 [label="INSTANCE with matching:\nT1 -> s(T238)\nT2 -> T231\nT3 -> T233", fontsize=14, style = filled, fillcolor = lightgrey];
125 -> 268 [arrowhead = none , style=dashed];
268 -> 1[style=dashed];

269 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
127 -> 269 [arrowhead = none ];
269 -> 129;

270 [label="SPLIT 2\nnew knowledge:\nT245 is ground\nT246 is ground", fontsize=14, style = filled, fillcolor = violet];
127 -> 270 [arrowhead = none ];
270 -> 130;

271 [label="INSTANCE with matching:\nT178 -> T245\nT179 -> T246", fontsize=14, style = filled, fillcolor = lightgrey];
129 -> 271 [arrowhead = none , style=dashed];
271 -> 106[style=dashed];

272 [label="INSTANCE with matching:\nT1 -> s(T246)\nT2 -> T231\nT3 -> T233", fontsize=14, style = filled, fillcolor = lightgrey];
130 -> 272 [arrowhead = none , style=dashed];
272 -> 1[style=dashed];

}

(2) Obligation:

Triples:

lessC(s(X1)) :- lessC(X1).
pB(X1, X2, X3) :- lessC(X1).
pB(X1, X2, X3) :- ','(lesscC(X1), insertA(s(X1), X2, X3)).
lessE(s(X1), s(X2)) :- lessE(X1, X2).
insertA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- pB(X1, X2, X4).
insertA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) :- lessC(X1).
insertA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) :- ','(lesscC(X1), insertA(X1, X3, X4)).
insertA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) :- pB(X1, X3, X4).
insertA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertA(0, X2, X4).
insertA(s(s(s(s(s(s(s(s(X1)))))))), tree(s(s(s(s(s(s(s(s(X2)))))))), X3, X4), tree(s(s(s(s(s(s(s(s(X2)))))))), X5, X4)) :- lessE(X1, X2).
insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- ','(lesscD(X1, X2), insertA(s(X1), X3, X5)).
insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- lessE(X2, X1).
insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- ','(lesscE(X2, X1), insertA(X1, X4, X5)).
insertA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertA(s(X1), X3, X4).
insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- lessE(X2, X1).
insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- ','(lesscE(X2, X1), insertA(s(X1), X4, X5)).

Clauses:

insertcA(X1, void, tree(X1, void, void)).
insertcA(X1, tree(X1, X2, X3), tree(X1, X2, X3)).
insertcA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- qcB(X1, X2, X4).
insertcA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) :- ','(lesscC(X1), insertcA(X1, X3, X4)).
insertcA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) :- qcB(X1, X3, X4).
insertcA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertcA(0, X2, X4).
insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- ','(lesscD(X1, X2), insertcA(s(X1), X3, X5)).
insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- ','(lesscE(X2, X1), insertcA(X1, X4, X5)).
insertcA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertcA(s(X1), X3, X4).
insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- ','(lesscE(X2, X1), insertcA(s(X1), X4, X5)).
lesscC(s(X1)) :- lesscC(X1).
qcB(X1, X2, X3) :- ','(lesscC(X1), insertcA(s(X1), X2, X3)).
lesscE(0, s(X1)).
lesscE(s(X1), s(X2)) :- lesscE(X1, X2).
lesscD(0, s(X1)).
lesscD(s(0), s(s(X1))).
lesscD(s(s(0)), s(s(s(X1)))).
lesscD(s(s(s(0))), s(s(s(s(X1))))).
lesscD(s(s(s(s(0)))), s(s(s(s(s(X1)))))).
lesscD(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))).
lesscD(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))).
lesscD(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) :- lesscE(X1, X2).

Afs:

insertA(x1, x2, x3)  =  insertA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
insertA_in: (b,b,f)
pB_in: (b,b,f)
lessC_in: (b)
lesscC_in: (b)
lessE_in: (b,b)
lesscD_in: (b,b)
lesscE_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → U6_GGA(X1, X2, X3, X4, pB_in_gga(X1, X2, X4))
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → PB_IN_GGA(X1, X2, X4)
PB_IN_GGA(X1, X2, X3) → U2_GGA(X1, X2, X3, lessC_in_g(X1))
PB_IN_GGA(X1, X2, X3) → LESSC_IN_G(X1)
LESSC_IN_G(s(X1)) → U1_G(X1, lessC_in_g(X1))
LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)
PB_IN_GGA(X1, X2, X3) → U3_GGA(X1, X2, X3, lesscC_in_g(X1))
U3_GGA(X1, X2, X3, lesscC_out_g(X1)) → U4_GGA(X1, X2, X3, insertA_in_gga(s(X1), X2, X3))
U3_GGA(X1, X2, X3, lesscC_out_g(X1)) → INSERTA_IN_GGA(s(X1), X2, X3)
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → U7_GGA(X1, X2, X3, X4, lessC_in_g(X1))
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → LESSC_IN_G(X1)
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → U8_GGA(X1, X2, X3, X4, lesscC_in_g(X1))
U8_GGA(X1, X2, X3, X4, lesscC_out_g(X1)) → U9_GGA(X1, X2, X3, X4, insertA_in_gga(X1, X3, X4))
U8_GGA(X1, X2, X3, X4, lesscC_out_g(X1)) → INSERTA_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) → U10_GGA(X1, X2, X3, X4, pB_in_gga(X1, X3, X4))
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) → PB_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → U11_GGA(X1, X2, X3, X4, insertA_in_gga(0, X2, X4))
INSERTA_IN_GGA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → INSERTA_IN_GGA(0, X2, X4)
INSERTA_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), tree(s(s(s(s(s(s(s(s(X2)))))))), X3, X4), tree(s(s(s(s(s(s(s(s(X2)))))))), X5, X4)) → U12_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X1, X2))
INSERTA_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), tree(s(s(s(s(s(s(s(s(X2)))))))), X3, X4), tree(s(s(s(s(s(s(s(s(X2)))))))), X5, X4)) → LESSE_IN_GG(X1, X2)
LESSE_IN_GG(s(X1), s(X2)) → U5_GG(X1, X2, lessE_in_gg(X1, X2))
LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) → U13_GGA(X1, X2, X3, X4, X5, lesscD_in_gg(X1, X2))
U13_GGA(X1, X2, X3, X4, X5, lesscD_out_gg(X1, X2)) → U14_GGA(X1, X2, X3, X4, X5, insertA_in_gga(s(X1), X3, X5))
U13_GGA(X1, X2, X3, X4, X5, lesscD_out_gg(X1, X2)) → INSERTA_IN_GGA(s(X1), X3, X5)
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → U15_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X2, X1))
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → LESSE_IN_GG(X2, X1)
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → U16_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U16_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → U17_GGA(X1, X2, X3, X4, X5, insertA_in_gga(X1, X4, X5))
U16_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(X1, X4, X5)
INSERTA_IN_GGA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) → U18_GGA(X1, X2, X3, X4, insertA_in_gga(s(X1), X3, X4))
INSERTA_IN_GGA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) → INSERTA_IN_GGA(s(X1), X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → U19_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X2, X1))
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → LESSE_IN_GG(X2, X1)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → U20_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U20_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → U21_GGA(X1, X2, X3, X4, X5, insertA_in_gga(s(X1), X4, X5))
U20_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(s(X1), X4, X5)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

The argument filtering Pi contains the following mapping:
insertA_in_gga(x1, x2, x3)  =  insertA_in_gga(x1, x2)
s(x1)  =  s(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pB_in_gga(x1, x2, x3)  =  pB_in_gga(x1, x2)
lessC_in_g(x1)  =  lessC_in_g(x1)
lesscC_in_g(x1)  =  lesscC_in_g(x1)
U35_g(x1, x2)  =  U35_g(x1, x2)
lesscC_out_g(x1)  =  lesscC_out_g(x1)
0  =  0
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lesscD_in_gg(x1, x2)  =  lesscD_in_gg(x1, x2)
lesscD_out_gg(x1, x2)  =  lesscD_out_gg(x1, x2)
U39_gg(x1, x2, x3)  =  U39_gg(x1, x2, x3)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x1, x2, x3)
INSERTA_IN_GGA(x1, x2, x3)  =  INSERTA_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
PB_IN_GGA(x1, x2, x3)  =  PB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
LESSC_IN_G(x1)  =  LESSC_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x3, x5)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
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, x6)  =  U16_GGA(x1, x2, x3, x4, x6)
U17_GGA(x1, x2, x3, x4, x5, x6)  =  U17_GGA(x1, x2, x3, x4, x6)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5, x6)  =  U19_GGA(x1, x2, x3, x4, x6)
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)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → U6_GGA(X1, X2, X3, X4, pB_in_gga(X1, X2, X4))
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → PB_IN_GGA(X1, X2, X4)
PB_IN_GGA(X1, X2, X3) → U2_GGA(X1, X2, X3, lessC_in_g(X1))
PB_IN_GGA(X1, X2, X3) → LESSC_IN_G(X1)
LESSC_IN_G(s(X1)) → U1_G(X1, lessC_in_g(X1))
LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)
PB_IN_GGA(X1, X2, X3) → U3_GGA(X1, X2, X3, lesscC_in_g(X1))
U3_GGA(X1, X2, X3, lesscC_out_g(X1)) → U4_GGA(X1, X2, X3, insertA_in_gga(s(X1), X2, X3))
U3_GGA(X1, X2, X3, lesscC_out_g(X1)) → INSERTA_IN_GGA(s(X1), X2, X3)
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → U7_GGA(X1, X2, X3, X4, lessC_in_g(X1))
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → LESSC_IN_G(X1)
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → U8_GGA(X1, X2, X3, X4, lesscC_in_g(X1))
U8_GGA(X1, X2, X3, X4, lesscC_out_g(X1)) → U9_GGA(X1, X2, X3, X4, insertA_in_gga(X1, X3, X4))
U8_GGA(X1, X2, X3, X4, lesscC_out_g(X1)) → INSERTA_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) → U10_GGA(X1, X2, X3, X4, pB_in_gga(X1, X3, X4))
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) → PB_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → U11_GGA(X1, X2, X3, X4, insertA_in_gga(0, X2, X4))
INSERTA_IN_GGA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → INSERTA_IN_GGA(0, X2, X4)
INSERTA_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), tree(s(s(s(s(s(s(s(s(X2)))))))), X3, X4), tree(s(s(s(s(s(s(s(s(X2)))))))), X5, X4)) → U12_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X1, X2))
INSERTA_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), tree(s(s(s(s(s(s(s(s(X2)))))))), X3, X4), tree(s(s(s(s(s(s(s(s(X2)))))))), X5, X4)) → LESSE_IN_GG(X1, X2)
LESSE_IN_GG(s(X1), s(X2)) → U5_GG(X1, X2, lessE_in_gg(X1, X2))
LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) → U13_GGA(X1, X2, X3, X4, X5, lesscD_in_gg(X1, X2))
U13_GGA(X1, X2, X3, X4, X5, lesscD_out_gg(X1, X2)) → U14_GGA(X1, X2, X3, X4, X5, insertA_in_gga(s(X1), X3, X5))
U13_GGA(X1, X2, X3, X4, X5, lesscD_out_gg(X1, X2)) → INSERTA_IN_GGA(s(X1), X3, X5)
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → U15_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X2, X1))
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → LESSE_IN_GG(X2, X1)
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → U16_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U16_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → U17_GGA(X1, X2, X3, X4, X5, insertA_in_gga(X1, X4, X5))
U16_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(X1, X4, X5)
INSERTA_IN_GGA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) → U18_GGA(X1, X2, X3, X4, insertA_in_gga(s(X1), X3, X4))
INSERTA_IN_GGA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) → INSERTA_IN_GGA(s(X1), X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → U19_GGA(X1, X2, X3, X4, X5, lessE_in_gg(X2, X1))
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → LESSE_IN_GG(X2, X1)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → U20_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U20_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → U21_GGA(X1, X2, X3, X4, X5, insertA_in_gga(s(X1), X4, X5))
U20_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(s(X1), X4, X5)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

The argument filtering Pi contains the following mapping:
insertA_in_gga(x1, x2, x3)  =  insertA_in_gga(x1, x2)
s(x1)  =  s(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pB_in_gga(x1, x2, x3)  =  pB_in_gga(x1, x2)
lessC_in_g(x1)  =  lessC_in_g(x1)
lesscC_in_g(x1)  =  lesscC_in_g(x1)
U35_g(x1, x2)  =  U35_g(x1, x2)
lesscC_out_g(x1)  =  lesscC_out_g(x1)
0  =  0
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lesscD_in_gg(x1, x2)  =  lesscD_in_gg(x1, x2)
lesscD_out_gg(x1, x2)  =  lesscD_out_gg(x1, x2)
U39_gg(x1, x2, x3)  =  U39_gg(x1, x2, x3)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x1, x2, x3)
INSERTA_IN_GGA(x1, x2, x3)  =  INSERTA_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
PB_IN_GGA(x1, x2, x3)  =  PB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
LESSC_IN_G(x1)  =  LESSC_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x3, x5)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
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, x6)  =  U16_GGA(x1, x2, x3, x4, x6)
U17_GGA(x1, x2, x3, x4, x5, x6)  =  U17_GGA(x1, x2, x3, x4, x6)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5, x6)  =  U19_GGA(x1, x2, x3, x4, x6)
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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 21 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSE_IN_GG(s(X1), s(X2)) → LESSE_IN_GG(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSC_IN_G(s(X1)) → LESSC_IN_G(X1)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → PB_IN_GGA(X1, X2, X4)
PB_IN_GGA(X1, X2, X3) → U3_GGA(X1, X2, X3, lesscC_in_g(X1))
U3_GGA(X1, X2, X3, lesscC_out_g(X1)) → INSERTA_IN_GGA(s(X1), X2, X3)
INSERTA_IN_GGA(X1, tree(X1, X2, X3), tree(X1, X2, X4)) → U8_GGA(X1, X2, X3, X4, lesscC_in_g(X1))
U8_GGA(X1, X2, X3, X4, lesscC_out_g(X1)) → INSERTA_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3), tree(s(X1), X2, X4)) → PB_IN_GGA(X1, X3, X4)
INSERTA_IN_GGA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) → INSERTA_IN_GGA(0, X2, X4)
INSERTA_IN_GGA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) → U16_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U16_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(X1, X4, X5)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) → U13_GGA(X1, X2, X3, X4, X5, lesscD_in_gg(X1, X2))
U13_GGA(X1, X2, X3, X4, X5, lesscD_out_gg(X1, X2)) → INSERTA_IN_GGA(s(X1), X3, X5)
INSERTA_IN_GGA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) → INSERTA_IN_GGA(s(X1), X3, X4)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) → U20_GGA(X1, X2, X3, X4, X5, lesscE_in_gg(X2, X1))
U20_GGA(X1, X2, X3, X4, X5, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(s(X1), X4, X5)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
lesscC_in_g(x1)  =  lesscC_in_g(x1)
U35_g(x1, x2)  =  U35_g(x1, x2)
lesscC_out_g(x1)  =  lesscC_out_g(x1)
0  =  0
lesscD_in_gg(x1, x2)  =  lesscD_in_gg(x1, x2)
lesscD_out_gg(x1, x2)  =  lesscD_out_gg(x1, x2)
U39_gg(x1, x2, x3)  =  U39_gg(x1, x2, x3)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U38_gg(x1, x2, x3)  =  U38_gg(x1, x2, x3)
INSERTA_IN_GGA(x1, x2, x3)  =  INSERTA_IN_GGA(x1, x2)
PB_IN_GGA(x1, x2, x3)  =  PB_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x1, x2, x3, x4, x6)
U20_GGA(x1, x2, x3, x4, x5, x6)  =  U20_GGA(x1, x2, x3, x4, x6)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3)) → PB_IN_GGA(X1, X2)
PB_IN_GGA(X1, X2) → U3_GGA(X1, X2, lesscC_in_g(X1))
U3_GGA(X1, X2, lesscC_out_g(X1)) → INSERTA_IN_GGA(s(X1), X2)
INSERTA_IN_GGA(X1, tree(X1, X2, X3)) → U8_GGA(X1, X2, X3, lesscC_in_g(X1))
U8_GGA(X1, X2, X3, lesscC_out_g(X1)) → INSERTA_IN_GGA(X1, X3)
INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3)) → PB_IN_GGA(X1, X3)
INSERTA_IN_GGA(0, tree(s(X1), X2, X3)) → INSERTA_IN_GGA(0, X2)
INSERTA_IN_GGA(X1, tree(X2, X3, X4)) → U16_GGA(X1, X2, X3, X4, lesscE_in_gg(X2, X1))
U16_GGA(X1, X2, X3, X4, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(X1, X4)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4)) → U13_GGA(X1, X2, X3, X4, lesscD_in_gg(X1, X2))
U13_GGA(X1, X2, X3, X4, lesscD_out_gg(X1, X2)) → INSERTA_IN_GGA(s(X1), X3)
INSERTA_IN_GGA(s(X1), tree(0, X2, X3)) → INSERTA_IN_GGA(s(X1), X3)
INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4)) → U20_GGA(X1, X2, X3, X4, lesscE_in_gg(X2, X1))
U20_GGA(X1, X2, X3, X4, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(s(X1), X4)

The TRS R consists of the following rules:

lesscC_in_g(s(X1)) → U35_g(X1, lesscC_in_g(X1))
U35_g(X1, lesscC_out_g(X1)) → lesscC_out_g(s(X1))
lesscD_in_gg(0, s(X1)) → lesscD_out_gg(0, s(X1))
lesscD_in_gg(s(0), s(s(X1))) → lesscD_out_gg(s(0), s(s(X1)))
lesscD_in_gg(s(s(0)), s(s(s(X1)))) → lesscD_out_gg(s(s(0)), s(s(s(X1))))
lesscD_in_gg(s(s(s(0))), s(s(s(s(X1))))) → lesscD_out_gg(s(s(s(0))), s(s(s(s(X1)))))
lesscD_in_gg(s(s(s(s(0)))), s(s(s(s(s(X1)))))) → lesscD_out_gg(s(s(s(s(0)))), s(s(s(s(s(X1))))))
lesscD_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1))))))) → lesscD_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))))
lesscD_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1)))))))) → lesscD_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))))
lesscD_in_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2)))))))) → U39_gg(X1, X2, lesscE_in_gg(X1, X2))
lesscE_in_gg(0, s(X1)) → lesscE_out_gg(0, s(X1))
lesscE_in_gg(s(X1), s(X2)) → U38_gg(X1, X2, lesscE_in_gg(X1, X2))
U38_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscE_out_gg(s(X1), s(X2))
U39_gg(X1, X2, lesscE_out_gg(X1, X2)) → lesscD_out_gg(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(X2))))))))

The set Q consists of the following terms:

lesscC_in_g(x0)
U35_g(x0, x1)
lesscD_in_gg(x0, x1)
lesscE_in_gg(x0, x1)
U38_gg(x0, x1, x2)
U39_gg(x0, x1, x2)

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

(24) 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:

• PB_IN_GGA(X1, X2) → U3_GGA(X1, X2, lesscC_in_g(X1))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U3_GGA(X1, X2, lesscC_out_g(X1)) → INSERTA_IN_GGA(s(X1), X2)
  The graph contains the following edges 2 >= 2

• INSERTA_IN_GGA(s(X1), tree(0, X2, X3)) → INSERTA_IN_GGA(s(X1), X3)
  The graph contains the following edges 1 >= 1, 2 > 2

• INSERTA_IN_GGA(0, tree(s(X1), X2, X3)) → INSERTA_IN_GGA(0, X2)
  The graph contains the following edges 1 >= 1, 2 > 2

• U8_GGA(X1, X2, X3, lesscC_out_g(X1)) → INSERTA_IN_GGA(X1, X3)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• INSERTA_IN_GGA(X1, tree(X1, X2, X3)) → U8_GGA(X1, X2, X3, lesscC_in_g(X1))
  The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 2 > 3

• U16_GGA(X1, X2, X3, X4, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(X1, X4)
  The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

• INSERTA_IN_GGA(X1, tree(X2, X3, X4)) → U16_GGA(X1, X2, X3, X4, lesscE_in_gg(X2, X1))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

• U13_GGA(X1, X2, X3, X4, lesscD_out_gg(X1, X2)) → INSERTA_IN_GGA(s(X1), X3)
  The graph contains the following edges 3 >= 2

• U20_GGA(X1, X2, X3, X4, lesscE_out_gg(X2, X1)) → INSERTA_IN_GGA(s(X1), X4)
  The graph contains the following edges 4 >= 2

• INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4)) → U13_GGA(X1, X2, X3, X4, lesscD_in_gg(X1, X2))
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4

• INSERTA_IN_GGA(s(X1), tree(s(X2), X3, X4)) → U20_GGA(X1, X2, X3, X4, lesscE_in_gg(X2, X1))
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4

• INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3)) → PB_IN_GGA(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 1, 2 > 2

• INSERTA_IN_GGA(s(X1), tree(s(X1), X2, X3)) → PB_IN_GGA(X1, X3)
  The graph contains the following edges 1 > 1, 2 > 1, 2 > 2

(25) YES