Left Termination proof of /tmp/tmpGhyiGC/insert-bbf.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇔)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔)

        ↳29 YES

(0) Obligation:

Clauses:

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

Queries:

insert(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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\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\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: insert(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\ninsert(T12, tree(T12, T13, T14), T3) !=? insert(X32, tree(X33, X34, X35), tree(X33, X36, X35))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(less(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="18: ','(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="20: 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(T33, T33)\n\nT33 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(T45, T45), insert(T45, T47, T49))\n\nT45 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T45, T45)\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: insert(T45, T47, T49)\n\nT45 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(less(T58, T58), insert(T58, T60, T62))\n\nT58 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(less(T58, T58), insert(T58, T60, T62)), SCOPE: 4, CLAUSE: 4 | ','(less(T58, T58), insert(T58, T60, T62)), SCOPE: 4, CLAUSE: 5\n\nT58 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(less(T58, T58), insert(T58, T60, T62)), SCOPE: 4, CLAUSE: 5\n\nT58 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(less(T65, T65), insert(s(T65), T60, T62))\n\nT60 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(less(T72, T73), insert(T72, T74, T77)) | insert(T72, tree(T73, T74, T75), T3), SCOPE: 1, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nT74 is ground\nT75 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: insert(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\ninsert(T1, T2, T3) !=? insert(X3, void, tree(X3, void, void))\ninsert(T1, T2, T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))\ninsert(T1, T2, T3) !=? insert(X141, tree(X142, X143, X144), tree(X142, X145, X144))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(less(T72, T73), insert(T72, T74, T77)), SCOPE: 5, CLAUSE: 4 | ','(less(T72, T73), insert(T72, T74, T77)), SCOPE: 5, CLAUSE: 5 | ?_5 | insert(T72, tree(T73, T74, T75), T3), SCOPE: 1, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nT74 is ground\nT75 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(less(T72, T73), insert(T72, T74, T77)), SCOPE: 5, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nT74 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(less(T72, T73), insert(T72, T74, T77)), SCOPE: 5, CLAUSE: 5 | ?_5 | insert(T72, tree(T73, T74, T75), T3), SCOPE: 1, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nT74 is ground\nT75 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: insert(0, T74, T77)\n\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(less(T72, T73), insert(T72, T74, T77)), SCOPE: 5, CLAUSE: 5\n\nT72 is ground\nT73 is ground\nT74 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_5 | insert(T72, tree(T73, T74, T75), T3), SCOPE: 1, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nT74 is ground\nT75 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(less(T91, T92), insert(s(T91), T74, T77))\n\nT74 is ground\nT91 is ground\nT92 is ground\ninsert(s(T91), tree(s(T92), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(T91, T92)\n\nT91 is ground\nT92 is ground\ninsert(s(T91), tree(s(T92), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: insert(s(T91), T74, T77)\n\nT74 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T91, T92), SCOPE: 6, CLAUSE: 4 | less(T91, T92), SCOPE: 6, CLAUSE: 5\n\nT91 is ground\nT92 is ground\ninsert(s(T91), tree(s(T92), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T91, T92), SCOPE: 6, CLAUSE: 4\n\nT91 is ground\nT92 is ground\ninsert(s(T91), tree(s(T92), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(T91, T92), SCOPE: 6, CLAUSE: 5\n\nT91 is ground\nT92 is ground\ninsert(s(T91), tree(s(T92), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T106, T107)\n\nT106 is ground\nT107 is ground\ninsert(s(s(T106)), tree(s(s(T107)), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(T106, T107), SCOPE: 7, CLAUSE: 4 | less(T106, T107), SCOPE: 7, CLAUSE: 5\n\nT106 is ground\nT107 is ground\ninsert(s(s(T106)), tree(s(s(T107)), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T106, T107), SCOPE: 7, CLAUSE: 4\n\nT106 is ground\nT107 is ground\ninsert(s(s(T106)), tree(s(s(T107)), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T106, T107), SCOPE: 7, CLAUSE: 5\n\nT106 is ground\nT107 is ground\ninsert(s(s(T106)), tree(s(s(T107)), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T119, T120)\n\nT119 is ground\nT120 is ground\ninsert(s(s(s(T119))), tree(s(s(s(T120))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: less(T119, T120), SCOPE: 8, CLAUSE: 4 | less(T119, T120), SCOPE: 8, CLAUSE: 5\n\nT119 is ground\nT120 is ground\ninsert(s(s(s(T119))), tree(s(s(s(T120))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(T119, T120), SCOPE: 8, CLAUSE: 4\n\nT119 is ground\nT120 is ground\ninsert(s(s(s(T119))), tree(s(s(s(T120))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: less(T119, T120), SCOPE: 8, CLAUSE: 5\n\nT119 is ground\nT120 is ground\ninsert(s(s(s(T119))), tree(s(s(s(T120))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T132, T133)\n\nT132 is ground\nT133 is ground\ninsert(s(s(s(s(T132)))), tree(s(s(s(s(T133)))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: less(T132, T133), SCOPE: 9, CLAUSE: 4 | less(T132, T133), SCOPE: 9, CLAUSE: 5\n\nT132 is ground\nT133 is ground\ninsert(s(s(s(s(T132)))), tree(s(s(s(s(T133)))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T132, T133), SCOPE: 9, CLAUSE: 4\n\nT132 is ground\nT133 is ground\ninsert(s(s(s(s(T132)))), tree(s(s(s(s(T133)))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(T132, T133), SCOPE: 9, CLAUSE: 5\n\nT132 is ground\nT133 is ground\ninsert(s(s(s(s(T132)))), tree(s(s(s(s(T133)))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T145, T146)\n\nT145 is ground\nT146 is ground\ninsert(s(s(s(s(s(T145))))), tree(s(s(s(s(s(T146))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: less(T145, T146), SCOPE: 10, CLAUSE: 4 | less(T145, T146), SCOPE: 10, CLAUSE: 5\n\nT145 is ground\nT146 is ground\ninsert(s(s(s(s(s(T145))))), tree(s(s(s(s(s(T146))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: less(T145, T146), SCOPE: 10, CLAUSE: 4\n\nT145 is ground\nT146 is ground\ninsert(s(s(s(s(s(T145))))), tree(s(s(s(s(s(T146))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: less(T145, T146), SCOPE: 10, CLAUSE: 5\n\nT145 is ground\nT146 is ground\ninsert(s(s(s(s(s(T145))))), tree(s(s(s(s(s(T146))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T158, T159)\n\nT158 is ground\nT159 is ground\ninsert(s(s(s(s(s(s(T158)))))), tree(s(s(s(s(s(s(T159)))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: less(T158, T159), SCOPE: 11, CLAUSE: 4 | less(T158, T159), SCOPE: 11, CLAUSE: 5\n\nT158 is ground\nT159 is ground\ninsert(s(s(s(s(s(s(T158)))))), tree(s(s(s(s(s(s(T159)))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: less(T158, T159), SCOPE: 11, CLAUSE: 4\n\nT158 is ground\nT159 is ground\ninsert(s(s(s(s(s(s(T158)))))), tree(s(s(s(s(s(s(T159)))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: less(T158, T159), SCOPE: 11, CLAUSE: 5\n\nT158 is ground\nT159 is ground\ninsert(s(s(s(s(s(s(T158)))))), tree(s(s(s(s(s(s(T159)))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T171, T172)\n\nT171 is ground\nT172 is ground\ninsert(s(s(s(s(s(s(s(T171))))))), tree(s(s(s(s(s(s(s(T172))))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: less(T171, T172), SCOPE: 12, CLAUSE: 4 | less(T171, T172), SCOPE: 12, CLAUSE: 5\n\nT171 is ground\nT172 is ground\ninsert(s(s(s(s(s(s(s(T171))))))), tree(s(s(s(s(s(s(s(T172))))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: less(T171, T172), SCOPE: 12, CLAUSE: 4\n\nT171 is ground\nT172 is ground\ninsert(s(s(s(s(s(s(s(T171))))))), tree(s(s(s(s(s(s(s(T172))))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(T171, T172), SCOPE: 12, CLAUSE: 5\n\nT171 is ground\nT172 is ground\ninsert(s(s(s(s(s(s(s(T171))))))), tree(s(s(s(s(s(s(s(T172))))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", 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(T184, T185)\n\nT184 is ground\nT185 is ground\ninsert(s(s(s(s(s(s(s(s(T184)))))))), tree(s(s(s(s(s(s(s(s(T185)))))))), T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: less(T184, T185)\n\nT184 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: less(T184, T185), SCOPE: 13, CLAUSE: 4 | less(T184, T185), SCOPE: 13, CLAUSE: 5\n\nT184 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: less(T184, T185), SCOPE: 13, CLAUSE: 4\n\nT184 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: less(T184, T185), SCOPE: 13, CLAUSE: 5\n\nT184 is ground\nT185 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(T201, T202)\n\nT201 is ground\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: insert(T72, tree(T73, T74, T75), T3), SCOPE: 1, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nT74 is ground\nT75 is ground\ninsert(T72, tree(T73, T74, T75), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(less(T218, T217), insert(T217, T220, T222))\n\nT217 is ground\nT218 is ground\nT220 is ground\ninsert(T217, tree(T218, T219, T220), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: less(T218, T217)\n\nT217 is ground\nT218 is ground\ninsert(T217, tree(T218, T219, T220), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: insert(T217, T220, T222)\n\nT217 is ground\nT220 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(less(T235, T234), insert(T234, T237, T239))\n\nT234 is ground\nT235 is ground\nT237 is ground\ninsert(T234, tree(T235, T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(less(T235, T234), insert(T234, T237, T239)), SCOPE: 14, CLAUSE: 4 | ','(less(T235, T234), insert(T234, T237, T239)), SCOPE: 14, CLAUSE: 5\n\nT234 is ground\nT235 is ground\nT237 is ground\ninsert(T234, tree(T235, T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(less(T235, T234), insert(T234, T237, T239)), SCOPE: 14, CLAUSE: 4\n\nT234 is ground\nT235 is ground\nT237 is ground\ninsert(T234, tree(T235, T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(less(T235, T234), insert(T234, T237, T239)), SCOPE: 14, CLAUSE: 5\n\nT234 is ground\nT235 is ground\nT237 is ground\ninsert(T234, tree(T235, T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: insert(s(T244), T237, T239)\n\nT237 is ground\nT244 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: ','(less(T251, T252), insert(s(T252), T237, T239))\n\nT237 is ground\nT251 is ground\nT252 is ground\ninsert(s(T252), tree(s(T251), T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: less(T251, T252)\n\nT251 is ground\nT252 is ground\ninsert(s(T252), tree(s(T251), T236, T237), T3) !=? insert(X21, tree(X21, X22, X23), tree(X21, X22, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: insert(s(T252), T237, T239)\n\nT237 is ground\nT252 is ground", 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 substitution\nT1 -> 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 = lightblue];
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(X21, tree(X21, X22, X23), tree(X21, X22, X23)).\nand substitution\nT1 -> T12\nX21 -> T12\nX22 -> T13\nX23 -> T14\nT2 -> tree(T12, T13, T14)\nT3 -> tree(T12, T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 135 [arrowhead = none ];
135 -> 9;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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 = white];
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 = white];
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(X141, tree(X142, X143, X144), tree(X142, X145, X144)) :- ','(less(X141, X142), insert(X141, X143, X145)).\nand substitution\nT1 -> T72\nX141 -> T72\nX142 -> T73\nX143 -> T74\nX144 -> T75\nT2 -> tree(T73, T74, T75)\nX145 -> T77\nT3 -> tree(T73, T77, T75)\nT76 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 141 [arrowhead = none ];
141 -> 37;

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

143 [label="EVAL with clause\ninsert(X32, tree(X33, X34, X35), tree(X33, X36, X35)) :- ','(less(X32, X33), insert(X32, X34, X36)).\nand substitution\nT12 -> T19\nX32 -> T19\nX33 -> T19\nT13 -> T20\nX34 -> T20\nT14 -> T21\nX35 -> T21\nX36 -> 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 = lightblue];
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(X113, tree(X114, X115, X116), tree(X114, X115, X117)) :- ','(less(X114, X113), insert(X113, X116, X117)).\nand substitution\nT12 -> T58\nX113 -> T58\nX114 -> T58\nT13 -> T59\nX115 -> T59\nT14 -> T60\nX116 -> T60\nX117 -> T62\nT3 -> tree(T58, T59, T62)\nT61 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 146 [arrowhead = none ];
146 -> 31;

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

148 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = white];
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(X51), s(X52)) :- less(X51, X52).\nand substitution\nX51 -> T28\nT19 -> s(T28)\nX52 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 151 [arrowhead = none ];
151 -> 18;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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", 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 = white];
22 -> 158 [arrowhead = none ];
158 -> 23;

159 [label="EVAL with clause\nless(s(X68), s(X69)) :- less(X68, X69).\nand substitution\nX68 -> T33\nT28 -> s(T33)\nX69 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 159 [arrowhead = none ];
159 -> 24;

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

161 [label="INSTANCE with matching:\nT28 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 161 [arrowhead = none , style=dashed];
161 -> 20[style=dashed];

162 [label="EVAL with clause\ninsert(X88, tree(X89, X90, X91), tree(X89, X90, X92)) :- ','(less(X89, X88), insert(X88, X91, X92)).\nand substitution\nT19 -> T45\nX88 -> T45\nX89 -> T45\nT20 -> T46\nX90 -> T46\nT21 -> T47\nX91 -> T47\nX92 -> T49\nT3 -> tree(T45, T46, T49)\nT48 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 162 [arrowhead = none ];
162 -> 27;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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", fontsize=14, style = filled, fillcolor = violet];
27 -> 165 [arrowhead = none ];
165 -> 30;

166 [label="INSTANCE with matching:\nT28 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 166 [arrowhead = none , style=dashed];
166 -> 20[style=dashed];

167 [label="INSTANCE with matching:\nT1 -> T45\nT2 -> T47\nT3 -> T49", 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 = white];
33 -> 169 [arrowhead = none ];
169 -> 34;

170 [label="EVAL with clause\nless(s(X128), s(X129)) :- less(X128, X129).\nand substitution\nX128 -> T65\nT58 -> s(T65)\nX129 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 170 [arrowhead = none ];
170 -> 35;

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

172 [label="INSTANCE with matching:\nT28 -> T65\nT20 -> T60\nT23 -> T62", 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(X338, tree(X339, X340, X341), tree(X339, X340, X342)) :- ','(less(X339, X338), insert(X338, X341, X342)).\nand substitution\nT1 -> T234\nX338 -> T234\nX339 -> T235\nX340 -> T236\nX341 -> T237\nT2 -> tree(T235, T236, T237)\nX342 -> T239\nT3 -> tree(T235, T236, T239)\nT238 -> T239", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 174 [arrowhead = none ];
174 -> 120;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X155)).\nand substitution\nT72 -> 0\nX155 -> T82\nT73 -> s(T82)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 178 [arrowhead = none ];
178 -> 42;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 -> T74\nT3 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 182 [arrowhead = none , style=dashed];
182 -> 1[style=dashed];

183 [label="EVAL with clause\nless(s(X167), s(X168)) :- less(X167, X168).\nand substitution\nX167 -> T91\nT72 -> s(T91)\nX168 -> T92\nT73 -> s(T92)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 183 [arrowhead = none ];
183 -> 46;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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", 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(T91)\nT2 -> T74\nT3 -> T77", 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(X179)).\nand substitution\nT91 -> 0\nX179 -> T101\nT92 -> s(T101)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 192 [arrowhead = none ];
192 -> 53;

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

194 [label="EVAL with clause\nless(s(X185), s(X186)) :- less(X185, X186).\nand substitution\nX185 -> T106\nT91 -> s(T106)\nX186 -> T107\nT92 -> s(T107)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 194 [arrowhead = none ];
194 -> 56;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X195)).\nand substitution\nT106 -> 0\nX195 -> T114\nT107 -> s(T114)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 200 [arrowhead = none ];
200 -> 61;

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

202 [label="EVAL with clause\nless(s(X201), s(X202)) :- less(X201, X202).\nand substitution\nX201 -> T119\nT106 -> s(T119)\nX202 -> T120\nT107 -> s(T120)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 202 [arrowhead = none ];
202 -> 64;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X211)).\nand substitution\nT119 -> 0\nX211 -> T127\nT120 -> s(T127)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 208 [arrowhead = none ];
208 -> 69;

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

210 [label="EVAL with clause\nless(s(X217), s(X218)) :- less(X217, X218).\nand substitution\nX217 -> T132\nT119 -> s(T132)\nX218 -> T133\nT120 -> s(T133)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 210 [arrowhead = none ];
210 -> 72;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X227)).\nand substitution\nT132 -> 0\nX227 -> T140\nT133 -> s(T140)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 216 [arrowhead = none ];
216 -> 77;

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

218 [label="EVAL with clause\nless(s(X233), s(X234)) :- less(X233, X234).\nand substitution\nX233 -> T145\nT132 -> s(T145)\nX234 -> T146\nT133 -> s(T146)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 218 [arrowhead = none ];
218 -> 80;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X243)).\nand substitution\nT145 -> 0\nX243 -> T153\nT146 -> s(T153)", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 224 [arrowhead = none ];
224 -> 85;

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

226 [label="EVAL with clause\nless(s(X249), s(X250)) :- less(X249, X250).\nand substitution\nX249 -> T158\nT145 -> s(T158)\nX250 -> T159\nT146 -> s(T159)", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 226 [arrowhead = none ];
226 -> 88;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X259)).\nand substitution\nT158 -> 0\nX259 -> T166\nT159 -> s(T166)", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 232 [arrowhead = none ];
232 -> 93;

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

234 [label="EVAL with clause\nless(s(X265), s(X266)) :- less(X265, X266).\nand substitution\nX265 -> T171\nT158 -> s(T171)\nX266 -> T172\nT159 -> s(T172)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 234 [arrowhead = none ];
234 -> 96;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X275)).\nand substitution\nT171 -> 0\nX275 -> T179\nT172 -> s(T179)", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 240 [arrowhead = none ];
240 -> 101;

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

242 [label="EVAL with clause\nless(s(X281), s(X282)) :- less(X281, X282).\nand substitution\nX281 -> T184\nT171 -> s(T184)\nX282 -> T185\nT172 -> s(T185)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 242 [arrowhead = none ];
242 -> 104;

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

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

245 [label="GENERALIZATION\nT188 <-- s(s(s(s(s(s(s(T184)))))))\nT189 <-- s(s(s(s(s(s(s(T185)))))))\n\nNew Knowledge:\nT188 is ground\nT189 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(X293)).\nand substitution\nT184 -> 0\nX293 -> T196\nT185 -> s(T196)", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 249 [arrowhead = none ];
249 -> 110;

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

251 [label="EVAL with clause\nless(s(X299), s(X300)) :- less(X299, X300).\nand substitution\nX299 -> T201\nT184 -> s(T201)\nX300 -> T202\nT185 -> s(T202)", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 251 [arrowhead = none ];
251 -> 113;

252 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT184 -> T201\nT185 -> T202", fontsize=14, style = filled, fillcolor = lightgrey];
113 -> 254 [arrowhead = none , style=dashed];
254 -> 106[style=dashed];

255 [label="EVAL with clause\ninsert(X317, tree(X318, X319, X320), tree(X318, X319, X321)) :- ','(less(X318, X317), insert(X317, X320, X321)).\nand substitution\nT72 -> T217\nX317 -> T217\nT73 -> T218\nX318 -> T218\nT74 -> T219\nX319 -> T219\nT75 -> T220\nX320 -> T220\nX321 -> T222\nT3 -> tree(T218, T219, T222)\nT221 -> T222", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 255 [arrowhead = none ];
255 -> 116;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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", fontsize=14, style = filled, fillcolor = violet];
116 -> 258 [arrowhead = none ];
258 -> 119;

259 [label="INSTANCE with matching:\nT184 -> T218\nT185 -> T217", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 259 [arrowhead = none , style=dashed];
259 -> 106[style=dashed];

260 [label="INSTANCE with matching:\nT1 -> T217\nT2 -> T220\nT3 -> T222", 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(X352)).\nand substitution\nT235 -> 0\nX352 -> T244\nT234 -> s(T244)", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 264 [arrowhead = none ];
264 -> 125;

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

266 [label="EVAL with clause\nless(s(X360), s(X361)) :- less(X360, X361).\nand substitution\nX360 -> T251\nT235 -> s(T251)\nX361 -> T252\nT234 -> s(T252)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 266 [arrowhead = none ];
266 -> 127;

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

268 [label="INSTANCE with matching:\nT1 -> s(T244)\nT2 -> T237\nT3 -> T239", 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", fontsize=14, style = filled, fillcolor = violet];
127 -> 270 [arrowhead = none ];
270 -> 130;

271 [label="INSTANCE with matching:\nT184 -> T251\nT185 -> T252", fontsize=14, style = filled, fillcolor = lightgrey];
129 -> 271 [arrowhead = none , style=dashed];
271 -> 106[style=dashed];

272 [label="INSTANCE with matching:\nT1 -> s(T252)\nT2 -> T237\nT3 -> T239", fontsize=14, style = filled, fillcolor = lightgrey];
130 -> 272 [arrowhead = none , style=dashed];
272 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

less20(s(T33)) :- less20(T33).
p18(T28, T20, T23) :- less20(T28).
p18(T28, T20, T23) :- ','(less20(T28), insert1(s(T28), T20, T23)).
less106(0, s(T196)).
less106(s(T201), s(T202)) :- less106(T201, T202).
less48(0, s(T101)).
less48(s(0), s(s(T114))).
less48(s(s(0)), s(s(s(T127)))).
less48(s(s(s(0))), s(s(s(s(T140))))).
less48(s(s(s(s(0)))), s(s(s(s(s(T153)))))).
less48(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))).
less48(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))).
less48(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) :- less106(T184, T185).
insert1(T5, void, tree(T5, void, void)).
insert1(T12, tree(T12, T13, T14), tree(T12, T13, T14)).
insert1(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) :- p18(T28, T20, T23).
insert1(T45, tree(T45, T46, T47), tree(T45, T46, T49)) :- less20(T45).
insert1(T45, tree(T45, T46, T47), tree(T45, T46, T49)) :- ','(less20(T45), insert1(T45, T47, T49)).
insert1(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) :- p18(T65, T60, T62).
insert1(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) :- insert1(0, T74, T77).
insert1(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) :- less48(T91, T92).
insert1(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) :- ','(less48(T91, T92), insert1(s(T91), T74, T77)).
insert1(T217, tree(T218, T219, T220), tree(T218, T219, T222)) :- less106(T218, T217).
insert1(T217, tree(T218, T219, T220), tree(T218, T219, T222)) :- ','(less106(T218, T217), insert1(T217, T220, T222)).
insert1(s(T244), tree(0, T236, T237), tree(0, T236, T239)) :- insert1(s(T244), T237, T239).
insert1(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) :- less106(T251, T252).
insert1(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) :- ','(less106(T251, T252), insert1(s(T252), T237, T239)).

Queries:

insert1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
insert1_in: (b,b,f)
p18_in: (b,b,f)
less20_in: (b)
less48_in: (b,b)
less106_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

(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:

INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_GGA(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → P18_IN_GGA(T28, T20, T23)
P18_IN_GGA(T28, T20, T23) → U2_GGA(T28, T20, T23, less20_in_g(T28))
P18_IN_GGA(T28, T20, T23) → LESS20_IN_G(T28)
LESS20_IN_G(s(T33)) → U1_G(T33, less20_in_g(T33))
LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)
U2_GGA(T28, T20, T23, less20_out_g(T28)) → U3_GGA(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
U2_GGA(T28, T20, T23, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20, T23)
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_GGA(T45, T46, T47, T49, less20_in_g(T45))
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → LESS20_IN_G(T45)
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → U8_GGA(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47, T49)
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_GGA(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → P18_IN_GGA(T65, T60, T62)
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_GGA(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → INSERT1_IN_GGA(0, T74, T77)
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_GGA(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → LESS48_IN_GG(T91, T92)
LESS48_IN_GG(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_GG(T184, T185, less106_in_gg(T184, T185))
LESS48_IN_GG(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → LESS106_IN_GG(T184, T185)
LESS106_IN_GG(s(T201), s(T202)) → U4_GG(T201, T202, less106_in_gg(T201, T202))
LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_GGA(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74, T77)
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_GGA(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → LESS106_IN_GG(T218, T217)
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_GGA(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220, T222)
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_GGA(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → INSERT1_IN_GGA(s(T244), T237, T239)
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_GGA(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → LESS106_IN_GG(T251, T252)
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_GGA(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237, T239)

The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

The argument filtering Pi contains the following mapping:
insert1_in_gga(x1, x2, x3) = insert1_in_gga(x1, x2)
void = void
insert1_out_gga(x1, x2, x3) = insert1_out_gga(x1, x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
s(x1) = s(x1)
U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5)
p18_in_gga(x1, x2, x3) = p18_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
less20_in_g(x1) = less20_in_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
less20_out_g(x1) = less20_out_g(x1)
p18_out_gga(x1, x2, x3) = p18_out_gga(x1, x2)
U3_gga(x1, x2, x3, x4) = U3_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)
0 = 0
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x2, x3, x5)
U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6)
less48_in_gg(x1, x2) = less48_in_gg(x1, x2)
less48_out_gg(x1, x2) = less48_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
less106_in_gg(x1, x2) = less106_in_gg(x1, x2)
less106_out_gg(x1, x2) = less106_out_gg(x1, x2)
U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6)
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) = U15_gga(x1, x2, x3, x5)
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)
INSERT1_IN_GGA(x1, x2, x3) = INSERT1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x2, x3, x5)
P18_IN_GGA(x1, x2, x3) = P18_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
LESS20_IN_G(x1) = LESS20_IN_G(x1)
U1_G(x1, x2) = U1_G(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_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, x6) = U11_GGA(x1, x2, x3, x4, x6)
LESS48_IN_GG(x1, x2) = LESS48_IN_GG(x1, x2)
U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3)
LESS106_IN_GG(x1, x2) = LESS106_IN_GG(x1, x2)
U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3)
U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6)
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) = U15_GGA(x1, x2, x3, x5)
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)

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

(6) Obligation:

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

INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_GGA(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → P18_IN_GGA(T28, T20, T23)
P18_IN_GGA(T28, T20, T23) → U2_GGA(T28, T20, T23, less20_in_g(T28))
P18_IN_GGA(T28, T20, T23) → LESS20_IN_G(T28)
LESS20_IN_G(s(T33)) → U1_G(T33, less20_in_g(T33))
LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)
U2_GGA(T28, T20, T23, less20_out_g(T28)) → U3_GGA(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
U2_GGA(T28, T20, T23, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20, T23)
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_GGA(T45, T46, T47, T49, less20_in_g(T45))
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → LESS20_IN_G(T45)
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → U8_GGA(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47, T49)
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_GGA(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → P18_IN_GGA(T65, T60, T62)
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_GGA(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → INSERT1_IN_GGA(0, T74, T77)
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_GGA(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → LESS48_IN_GG(T91, T92)
LESS48_IN_GG(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_GG(T184, T185, less106_in_gg(T184, T185))
LESS48_IN_GG(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → LESS106_IN_GG(T184, T185)
LESS106_IN_GG(s(T201), s(T202)) → U4_GG(T201, T202, less106_in_gg(T201, T202))
LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_GGA(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74, T77)
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_GGA(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → LESS106_IN_GG(T218, T217)
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_GGA(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220, T222)
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_GGA(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → INSERT1_IN_GGA(s(T244), T237, T239)
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_GGA(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → LESS106_IN_GG(T251, T252)
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_GGA(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237, T239)

The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 18 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)

The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

(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:

LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)

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:

LESS106_IN_GG(s(T201), s(T202)) → LESS106_IN_GG(T201, T202)
The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)

The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

(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:

LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)

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:

LESS20_IN_G(s(T33)) → LESS20_IN_G(T33)
The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → P18_IN_GGA(T28, T20, T23)
P18_IN_GGA(T28, T20, T23) → U2_GGA(T28, T20, T23, less20_in_g(T28))
U2_GGA(T28, T20, T23, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20, T23)
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_GGA(T45, T46, T47, T49, less20_in_g(T45))
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47, T49)
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → P18_IN_GGA(T65, T60, T62)
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → INSERT1_IN_GGA(0, T74, T77)
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_GGA(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220, T222)
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_GGA(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74, T77)
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → INSERT1_IN_GGA(s(T244), T237, T239)
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_GGA(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237, T239)

The TRS R consists of the following rules:

insert1_in_gga(T5, void, tree(T5, void, void)) → insert1_out_gga(T5, void, tree(T5, void, void))
insert1_in_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14)) → insert1_out_gga(T12, tree(T12, T13, T14), tree(T12, T13, T14))
insert1_in_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → U6_gga(T28, T20, T21, T23, p18_in_gga(T28, T20, T23))
p18_in_gga(T28, T20, T23) → U2_gga(T28, T20, T23, less20_in_g(T28))
less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U2_gga(T28, T20, T23, less20_out_g(T28)) → p18_out_gga(T28, T20, T23)
U2_gga(T28, T20, T23, less20_out_g(T28)) → U3_gga(T28, T20, T23, insert1_in_gga(s(T28), T20, T23))
insert1_in_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_gga(T45, T46, T47, T49, less20_in_g(T45))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U7_gga(T45, T46, T47, T49, less20_out_g(T45)) → U8_gga(T45, T46, T47, T49, insert1_in_gga(T45, T47, T49))
insert1_in_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → U9_gga(T65, T59, T60, T62, p18_in_gga(T65, T60, T62))
U9_gga(T65, T59, T60, T62, p18_out_gga(T65, T60, T62)) → insert1_out_gga(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62))
insert1_in_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → U10_gga(T82, T74, T75, T77, insert1_in_gga(0, T74, T77))
insert1_in_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_gga(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U11_gga(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → U12_gga(T91, T92, T74, T75, T77, insert1_in_gga(s(T91), T74, T77))
insert1_in_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_gga(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U13_gga(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → U14_gga(T217, T218, T219, T220, T222, insert1_in_gga(T217, T220, T222))
insert1_in_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → U15_gga(T244, T236, T237, T239, insert1_in_gga(s(T244), T237, T239))
insert1_in_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_gga(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U16_gga(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → U17_gga(T252, T251, T236, T237, T239, insert1_in_gga(s(T252), T237, T239))
U17_gga(T252, T251, T236, T237, T239, insert1_out_gga(s(T252), T237, T239)) → insert1_out_gga(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239))
U15_gga(T244, T236, T237, T239, insert1_out_gga(s(T244), T237, T239)) → insert1_out_gga(s(T244), tree(0, T236, T237), tree(0, T236, T239))
U14_gga(T217, T218, T219, T220, T222, insert1_out_gga(T217, T220, T222)) → insert1_out_gga(T217, tree(T218, T219, T220), tree(T218, T219, T222))
U12_gga(T91, T92, T74, T75, T77, insert1_out_gga(s(T91), T74, T77)) → insert1_out_gga(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75))
U10_gga(T82, T74, T75, T77, insert1_out_gga(0, T74, T77)) → insert1_out_gga(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75))
U8_gga(T45, T46, T47, T49, insert1_out_gga(T45, T47, T49)) → insert1_out_gga(T45, tree(T45, T46, T47), tree(T45, T46, T49))
U3_gga(T28, T20, T23, insert1_out_gga(s(T28), T20, T23)) → p18_out_gga(T28, T20, T23)
U6_gga(T28, T20, T21, T23, p18_out_gga(T28, T20, T23)) → insert1_out_gga(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21))

The argument filtering Pi contains the following mapping:
insert1_in_gga(x1, x2, x3) = insert1_in_gga(x1, x2)
void = void
insert1_out_gga(x1, x2, x3) = insert1_out_gga(x1, x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
s(x1) = s(x1)
U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x2, x3, x5)
p18_in_gga(x1, x2, x3) = p18_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
less20_in_g(x1) = less20_in_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
less20_out_g(x1) = less20_out_g(x1)
p18_out_gga(x1, x2, x3) = p18_out_gga(x1, x2)
U3_gga(x1, x2, x3, x4) = U3_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)
0 = 0
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x2, x3, x5)
U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6)
less48_in_gg(x1, x2) = less48_in_gg(x1, x2)
less48_out_gg(x1, x2) = less48_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
less106_in_gg(x1, x2) = less106_in_gg(x1, x2)
less106_out_gg(x1, x2) = less106_out_gg(x1, x2)
U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6)
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) = U15_gga(x1, x2, x3, x5)
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)
INSERT1_IN_GGA(x1, x2, x3) = INSERT1_IN_GGA(x1, x2)
P18_IN_GGA(x1, x2, x3) = P18_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5) = U7_GGA(x1, x2, x3, x5)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
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)

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:

INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21), tree(s(T28), T23, T21)) → P18_IN_GGA(T28, T20, T23)
P18_IN_GGA(T28, T20, T23) → U2_GGA(T28, T20, T23, less20_in_g(T28))
U2_GGA(T28, T20, T23, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20, T23)
INSERT1_IN_GGA(T45, tree(T45, T46, T47), tree(T45, T46, T49)) → U7_GGA(T45, T46, T47, T49, less20_in_g(T45))
U7_GGA(T45, T46, T47, T49, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47, T49)
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60), tree(s(T65), T59, T62)) → P18_IN_GGA(T65, T60, T62)
INSERT1_IN_GGA(0, tree(s(T82), T74, T75), tree(s(T82), T77, T75)) → INSERT1_IN_GGA(0, T74, T77)
INSERT1_IN_GGA(T217, tree(T218, T219, T220), tree(T218, T219, T222)) → U13_GGA(T217, T218, T219, T220, T222, less106_in_gg(T218, T217))
U13_GGA(T217, T218, T219, T220, T222, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220, T222)
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75), tree(s(T92), T77, T75)) → U11_GGA(T91, T92, T74, T75, T77, less48_in_gg(T91, T92))
U11_GGA(T91, T92, T74, T75, T77, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74, T77)
INSERT1_IN_GGA(s(T244), tree(0, T236, T237), tree(0, T236, T239)) → INSERT1_IN_GGA(s(T244), T237, T239)
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237), tree(s(T251), T236, T239)) → U16_GGA(T252, T251, T236, T237, T239, less106_in_gg(T251, T252))
U16_GGA(T252, T251, T236, T237, T239, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237, T239)

The TRS R consists of the following rules:

less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
s(x1) = s(x1)
less20_in_g(x1) = less20_in_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
less20_out_g(x1) = less20_out_g(x1)
0 = 0
less48_in_gg(x1, x2) = less48_in_gg(x1, x2)
less48_out_gg(x1, x2) = less48_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
less106_in_gg(x1, x2) = less106_in_gg(x1, x2)
less106_out_gg(x1, x2) = less106_out_gg(x1, x2)
U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3)
INSERT1_IN_GGA(x1, x2, x3) = INSERT1_IN_GGA(x1, x2)
P18_IN_GGA(x1, x2, x3) = P18_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5) = U7_GGA(x1, x2, x3, x5)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
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)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21)) → P18_IN_GGA(T28, T20)
P18_IN_GGA(T28, T20) → U2_GGA(T28, T20, less20_in_g(T28))
U2_GGA(T28, T20, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20)
INSERT1_IN_GGA(T45, tree(T45, T46, T47)) → U7_GGA(T45, T46, T47, less20_in_g(T45))
U7_GGA(T45, T46, T47, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47)
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60)) → P18_IN_GGA(T65, T60)
INSERT1_IN_GGA(0, tree(s(T82), T74, T75)) → INSERT1_IN_GGA(0, T74)
INSERT1_IN_GGA(T217, tree(T218, T219, T220)) → U13_GGA(T217, T218, T219, T220, less106_in_gg(T218, T217))
U13_GGA(T217, T218, T219, T220, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220)
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75)) → U11_GGA(T91, T92, T74, T75, less48_in_gg(T91, T92))
U11_GGA(T91, T92, T74, T75, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74)
INSERT1_IN_GGA(s(T244), tree(0, T236, T237)) → INSERT1_IN_GGA(s(T244), T237)
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237)) → U16_GGA(T252, T251, T236, T237, less106_in_gg(T251, T252))
U16_GGA(T252, T251, T236, T237, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237)

The TRS R consists of the following rules:

less20_in_g(s(T33)) → U1_g(T33, less20_in_g(T33))
less106_in_gg(0, s(T196)) → less106_out_gg(0, s(T196))
less106_in_gg(s(T201), s(T202)) → U4_gg(T201, T202, less106_in_gg(T201, T202))
less48_in_gg(0, s(T101)) → less48_out_gg(0, s(T101))
less48_in_gg(s(0), s(s(T114))) → less48_out_gg(s(0), s(s(T114)))
less48_in_gg(s(s(0)), s(s(s(T127)))) → less48_out_gg(s(s(0)), s(s(s(T127))))
less48_in_gg(s(s(s(0))), s(s(s(s(T140))))) → less48_out_gg(s(s(s(0))), s(s(s(s(T140)))))
less48_in_gg(s(s(s(s(0)))), s(s(s(s(s(T153)))))) → less48_out_gg(s(s(s(s(0)))), s(s(s(s(s(T153))))))
less48_in_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166))))))) → less48_out_gg(s(s(s(s(s(0))))), s(s(s(s(s(s(T166)))))))
less48_in_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179)))))))) → less48_out_gg(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(T179))))))))
less48_in_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185)))))))) → U5_gg(T184, T185, less106_in_gg(T184, T185))
U1_g(T33, less20_out_g(T33)) → less20_out_g(s(T33))
U4_gg(T201, T202, less106_out_gg(T201, T202)) → less106_out_gg(s(T201), s(T202))
U5_gg(T184, T185, less106_out_gg(T184, T185)) → less48_out_gg(s(s(s(s(s(s(s(T184))))))), s(s(s(s(s(s(s(T185))))))))

The set Q consists of the following terms:

less20_in_g(x0)
less106_in_gg(x0, x1)
less48_in_gg(x0, x1)
U1_g(x0, x1)
U4_gg(x0, x1, x2)
U5_gg(x0, x1, x2)

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:

P18_IN_GGA(T28, T20) → U2_GGA(T28, T20, less20_in_g(T28))
The graph contains the following edges 1 >= 1, 2 >= 2
U2_GGA(T28, T20, less20_out_g(T28)) → INSERT1_IN_GGA(s(T28), T20)
The graph contains the following edges 2 >= 2
INSERT1_IN_GGA(s(T244), tree(0, T236, T237)) → INSERT1_IN_GGA(s(T244), T237)
The graph contains the following edges 1 >= 1, 2 > 2
INSERT1_IN_GGA(0, tree(s(T82), T74, T75)) → INSERT1_IN_GGA(0, T74)
The graph contains the following edges 1 >= 1, 2 > 2
U7_GGA(T45, T46, T47, less20_out_g(T45)) → INSERT1_IN_GGA(T45, T47)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
INSERT1_IN_GGA(T45, tree(T45, T46, T47)) → U7_GGA(T45, T46, T47, less20_in_g(T45))
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 2 > 3
U13_GGA(T217, T218, T219, T220, less106_out_gg(T218, T217)) → INSERT1_IN_GGA(T217, T220)
The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2
INSERT1_IN_GGA(T217, tree(T218, T219, T220)) → U13_GGA(T217, T218, T219, T220, less106_in_gg(T218, T217))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
U11_GGA(T91, T92, T74, T75, less48_out_gg(T91, T92)) → INSERT1_IN_GGA(s(T91), T74)
The graph contains the following edges 3 >= 2
U16_GGA(T252, T251, T236, T237, less106_out_gg(T251, T252)) → INSERT1_IN_GGA(s(T252), T237)
The graph contains the following edges 4 >= 2
INSERT1_IN_GGA(s(T91), tree(s(T92), T74, T75)) → U11_GGA(T91, T92, T74, T75, less48_in_gg(T91, T92))
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4
INSERT1_IN_GGA(s(T252), tree(s(T251), T236, T237)) → U16_GGA(T252, T251, T236, T237, less106_in_gg(T251, T252))
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4
INSERT1_IN_GGA(s(T28), tree(s(T28), T20, T21)) → P18_IN_GGA(T28, T20)
The graph contains the following edges 1 > 1, 2 > 1, 2 > 2
INSERT1_IN_GGA(s(T65), tree(s(T65), T59, T60)) → P18_IN_GGA(T65, T60)
The graph contains the following edges 1 > 1, 2 > 1, 2 > 2

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES