Left Termination of the query pattern lessleaves_in_2(g, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

lessleaves(nil, cons(X1, X2)).
lessleaves(cons(U, V), cons(W, Z)) :- ','(app(U, V, U1), ','(app(W, Z, W1), lessleaves(U1, W1))).
app(nil, Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, cons(U, Z)) :- ','(head(X, U), ','(tail(X, V), app(V, Y, Z))).
head([], X3).
head(.(X, X4), X).
tail([], []).
tail(.(X5, X), X).
eq(X, X).

Queries:

lessleaves(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: lessleaves(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: lessleaves(T1, T2), SCOPE: 1, CLAUSE: 0 | lessleaves(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: lessleaves(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: lessleaves(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\nX10 is free\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(T5, T6, X20), ','(app(T7, T8, X21), lessleaves(X20, X21)))\n\nT5 is ground\nT6 is ground\nT7 is ground\nT8 is ground\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\nX16 is free\nX17 is free; \nX18 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(T5, T6, X20)\n\nT5 is ground\nT6 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(T7, T8, X21), lessleaves(T9, X21))\n\nT7 is ground\nT8 is ground\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(T5, T6, X20), SCOPE: 2, CLAUSE: 2 | app(T5, T6, X20), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT5 is ground\nT6 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_2, eq(T10, X26)) | app(nil, T10, X20), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT10 is ground\nX20 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app(T5, T6, X20), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX20 is free; \nX24 is free; \nX25 is free; \napp(T5, T6, X20) !=? app(nil, X24, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: eq(T10, X26)\n\nT10 is ground\nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: eq(T10, X26), SCOPE: 3, CLAUSE: 8\n\nT10 is ground\nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T12, X38), ','(tail(T12, X37), app(X37, T13, X39)))\n\nT12 is ground\nT13 is ground\nX20 is free; \nX24 is free; \nX25 is free; \nX38 is free; \nX39 is free; \nX37 is free; \napp(T12, T13, X20) !=? app(nil, X24, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T12, X38), ','(tail(T12, X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 4 | ','(head(T12, X38), ','(tail(T12, X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT12 is ground\nT13 is ground\nX20 is free; \nX24 is free; \nX25 is free; \nX38 is free; \nX39 is free; \nX37 is free; \napp(T12, T13, X20) !=? app(nil, X24, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail([], X37), app(X37, T13, X39)) | ','(head([], X38), ','(tail([], X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT13 is ground\nX20 is free; \nX24 is free; \nX25 is free; \nX38 is free; \nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T12, X38), ','(tail(T12, X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT12 is ground\nT13 is ground\nX20 is free; \nX24 is free; \nX25 is free; \nX38 is free; \nX39 is free; \nX37 is free; \nX41 is free; \napp(T12, T13, X20) !=? app(nil, X24, X25)\nhead(T12, X38) !=? head([], X41)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail([], X37), app(X37, T13, X39)), SCOPE: 5, CLAUSE: 6 | ','(tail([], X37), app(X37, T13, X39)), SCOPE: 5, CLAUSE: 7 | ','(head([], X38), ','(tail([], X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT13 is ground\nX38 is free; \nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app([], T13, X39) | ','(tail([], X37), app(X37, T13, X39)), SCOPE: 5, CLAUSE: 7 | ','(head([], X38), ','(tail([], X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT13 is ground\nX38 is free; \nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app([], T13, X39)\n\nT13 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail([], X37), app(X37, T13, X39)), SCOPE: 5, CLAUSE: 7 | ','(head([], X38), ','(tail([], X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT13 is ground\nX38 is free; \nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app([], T13, X39), SCOPE: 6, CLAUSE: 2 | app([], T13, X39), SCOPE: 6, CLAUSE: 3 | ?_6\n\nT13 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app([], T13, X39), SCOPE: 6, CLAUSE: 3\n\nT13 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head([], X54), ','(tail([], X53), app(X53, T14, X55)))\n\nT14 is ground\nX54 is free; \nX55 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head([], X54), ','(tail([], X53), app(X53, T14, X55))), SCOPE: 7, CLAUSE: 4 | ','(head([], X54), ','(tail([], X53), app(X53, T14, X55))), SCOPE: 7, CLAUSE: 5\n\nT14 is ground\nX54 is free; \nX55 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail([], X53), app(X53, T14, X55)) | ','(head([], X54), ','(tail([], X53), app(X53, T14, X55))), SCOPE: 7, CLAUSE: 5\n\nT14 is ground\nX54 is free; \nX55 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail([], X53), app(X53, T14, X55)), SCOPE: 8, CLAUSE: 6 | ','(tail([], X53), app(X53, T14, X55)), SCOPE: 8, CLAUSE: 7 | ','(head([], X54), ','(tail([], X53), app(X53, T14, X55))), SCOPE: 7, CLAUSE: 5\n\nT14 is ground\nX54 is free; \nX55 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app([], T14, X55) | ','(tail([], X53), app(X53, T14, X55)), SCOPE: 8, CLAUSE: 7 | ','(head([], X54), ','(tail([], X53), app(X53, T14, X55))), SCOPE: 7, CLAUSE: 5\n\nT14 is ground\nX54 is free; \nX55 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head([], X38), ','(tail([], X37), app(X37, T13, X39))), SCOPE: 4, CLAUSE: 5\n\nT13 is ground\nX38 is free; \nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(tail(.(T15, T16), X37), app(X37, T13, X39))\n\nT13 is ground\nT15 is ground\nT16 is ground\nX20 is free; \nX24 is free; \nX25 is free; \nX38 is free; \nX39 is free; \nX37 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX20 is free\nX24 is free; \nX25 is free; \nX38 is free; \nX41 is free; \nX65 is free; \nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(tail(.(T15, T16), X37), app(X37, T13, X39)), SCOPE: 9, CLAUSE: 6 | ','(tail(.(T15, T16), X37), app(X37, T13, X39)), SCOPE: 9, CLAUSE: 7\n\nT13 is ground\nT15 is ground\nT16 is ground\nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(tail(.(T15, T16), X37), app(X37, T13, X39)), SCOPE: 9, CLAUSE: 7\n\nT13 is ground\nT15 is ground\nT16 is ground\nX39 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(T18, T13, X39)\n\nT13 is ground\nT18 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T7, T8, X21)\n\nT7 is ground\nT8 is ground\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: lessleaves(T9, T19)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T7, T8, X21), SCOPE: 10, CLAUSE: 2 | app(T7, T8, X21), SCOPE: 10, CLAUSE: 3 | ?_10\n\nT7 is ground\nT8 is ground\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(T7, T8, X21), SCOPE: 10, CLAUSE: 2 | app(T7, T8, X21), SCOPE: 10, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_10, eq(T20, X77)) | app(nil, T20, X21), SCOPE: 10, CLAUSE: 3\n\nT20 is ground\nX21 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app(T7, T8, X21), SCOPE: 10, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX21 is free; \nX75 is free; \nX76 is free; \napp(T7, T8, X21) !=? app(nil, X75, X76)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(T20, X77)\n\nT20 is ground\nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: eq(T20, X77), SCOPE: 11, CLAUSE: 8\n\nT20 is ground\nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: lessleaves(T9, T19), SCOPE: 12, CLAUSE: 0 | lessleaves(T9, T19), SCOPE: 12, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: lessleaves(T9, T19), SCOPE: 12, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: lessleaves(T9, T19), SCOPE: 12, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\nX88 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(app(T28, T29, X98), ','(app(T30, T31, X99), lessleaves(X98, X99)))\n\nX98 is free\nX99 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\nX94 is free\nX95 is free; \nX96 is free; \nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T28, T29, X98)\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(app(T33, T34, X99), lessleaves(T32, X99))\n\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app(T28, T29, X98), SCOPE: 13, CLAUSE: 2 | app(T28, T29, X98), SCOPE: 13, CLAUSE: 3 | ?_13\n\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(!_13, eq(T36, X104)) | app(T28, T29, X98), SCOPE: 13, CLAUSE: 3 | ?_13\n\nX98 is free\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app(T28, T29, X98), SCOPE: 13, CLAUSE: 3\n\nX98 is free\nX102 is free; \nX103 is free; \napp(T28, T29, X98) !=? app(nil, X102, X103)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: eq(T36, X104)\n\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: eq(T36, X104), SCOPE: 14, CLAUSE: 8\n\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117)))\n\nX98 is free\nX102 is free; \nX103 is free; \nX116 is free; \nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 4 | ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(tail([], X115), app(X115, T42, X117)) | ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; \nX119 is free; \nhead(T40, X116) !=? head([], X119)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(tail([], X115), app(X115, T42, X117)), SCOPE: 16, CLAUSE: 6 | ','(tail([], X115), app(X115, T42, X117)), SCOPE: 16, CLAUSE: 7 | ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app([], T42, X117) | ','(tail([], X115), app(X115, T42, X117)), SCOPE: 16, CLAUSE: 7 | ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app([], T42, X117)\n\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(tail([], X115), app(X115, T42, X117)), SCOPE: 16, CLAUSE: 7 | ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: app([], T42, X117), SCOPE: 17, CLAUSE: 2 | app([], T42, X117), SCOPE: 17, CLAUSE: 3 | ?_17\n\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app([], T42, X117), SCOPE: 17, CLAUSE: 3\n\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(head([], X132), ','(tail([], X131), app(X131, T44, X133)))\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 4 | ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(tail([], X131), app(X131, T44, X133)) | ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(tail([], X131), app(X131, T44, X133)), SCOPE: 19, CLAUSE: 6 | ','(tail([], X131), app(X131, T44, X133)), SCOPE: 19, CLAUSE: 7 | ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: app([], T44, X133) | ','(tail([], X131), app(X131, T44, X133)), SCOPE: 19, CLAUSE: 7 | ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: app([], T44, X133)\n\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail([], X131), app(X131, T44, X133)), SCOPE: 19, CLAUSE: 7 | ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(head([], X132), ','(tail([], X131), app(X131, T44, X133))), SCOPE: 18, CLAUSE: 5\n\nX132 is free\nX133 is free; \nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(head(T40, X116), ','(tail(T40, X115), app(X115, T41, X117))), SCOPE: 15, CLAUSE: 5\n\nX116 is free\nX117 is free; \nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(tail(.(T47, T48), X115), app(X115, T49, X117))\n\nX117 is free\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\nX116 is free\nX145 is free; \nX146 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(tail(.(T47, T48), X115), app(X115, T49, X117)), SCOPE: 20, CLAUSE: 6 | ','(tail(.(T47, T48), X115), app(X115, T49, X117)), SCOPE: 20, CLAUSE: 7\n\nX117 is free\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(tail(.(T47, T48), X115), app(X115, T49, X117)), SCOPE: 20, CLAUSE: 7\n\nX117 is free\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: app(T52, T53, X117)\n\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(tail(.(T56, T57), X115), app(X115, T58, X117))\n\nX116 is free\nX117 is free; \nX115 is free; \nX119 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\nX116 is free\nX119 is free; \nX153 is free; \nX154 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(tail(.(T56, T57), X115), app(X115, T58, X117)), SCOPE: 21, CLAUSE: 6 | ','(tail(.(T56, T57), X115), app(X115, T58, X117)), SCOPE: 21, CLAUSE: 7\n\nX117 is free\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(tail(.(T56, T57), X115), app(X115, T58, X117)), SCOPE: 21, CLAUSE: 7\n\nX117 is free\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: app(T61, T62, X117)\n\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: app(T33, T34, X99)\n\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: lessleaves(T64, T63)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 101 [arrowhead = none ];
101 -> 2;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 102 [arrowhead = none ];
102 -> 3;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 103 [arrowhead = none ];
103 -> 4;

104 [label="EVAL with clause\nlessleaves(nil, cons(X10, X11)).\nand substitution\nT1 -> nil\nX10 -> T3\nX11 -> T4\nT2 -> cons(T3, T4)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 104 [arrowhead = none ];
104 -> 5;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 105 [arrowhead = none ];
105 -> 6;

106 [label="EVAL with clause\nlessleaves(cons(X16, X17), cons(X18, X19)) :- ','(app(X16, X17, X20), ','(app(X18, X19, X21), lessleaves(X20, X21))).\nand substitution\nX16 -> T5\nX17 -> T6\nT1 -> cons(T5, T6)\nX18 -> T7\nX19 -> T8\nT2 -> cons(T7, T8)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 106 [arrowhead = none ];
106 -> 8;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 107 [arrowhead = none ];
107 -> 9;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 108 [arrowhead = none ];
108 -> 7;

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 109 [arrowhead = none ];
109 -> 10;

110 [label="SPLIT 2\nreplacements:\nX20 -> T9", fontsize=14, style = filled, fillcolor = violet];
8 -> 110 [arrowhead = none ];
110 -> 11;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 111 [arrowhead = none ];
111 -> 12;

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 112 [arrowhead = none ];
112 -> 41;

113 [label="SPLIT 2\nreplacements:\nX21 -> T19", fontsize=14, style = filled, fillcolor = violet];
11 -> 113 [arrowhead = none ];
113 -> 42;

114 [label="EVAL with clause\napp(nil, X24, X25) :- ','(!, eq(X24, X25)).\nand substitution\nT5 -> nil\nT6 -> T10\nX24 -> T10\nX20 -> X26\nX25 -> X26", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 114 [arrowhead = none ];
114 -> 13;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 115 [arrowhead = none ];
115 -> 14;

116 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 116 [arrowhead = none ];
116 -> 15;

117 [label="EVAL with clause\napp(X33, X34, cons(X35, X36)) :- ','(head(X33, X35), ','(tail(X33, X37), app(X37, X34, X36))).\nand substitution\nT5 -> T12\nX33 -> T12\nT6 -> T13\nX34 -> T13\nX35 -> X38\nX36 -> X39\nX20 -> cons(X38, X39)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 117 [arrowhead = none ];
117 -> 19;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 118 [arrowhead = none ];
118 -> 16;

119 [label="EVAL with clause\neq(X28, X28).\nand substitution\nT10 -> T11\nX28 -> T11\nX26 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 119 [arrowhead = none ];
119 -> 17;

120 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 120 [arrowhead = none ];
120 -> 18;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 121 [arrowhead = none ];
121 -> 20;

122 [label="EVAL with clause\nhead([], X41).\nand substitution\nT12 -> []\nX38 -> X42\nX41 -> X42", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 122 [arrowhead = none ];
122 -> 21;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 123 [arrowhead = none ];
123 -> 22;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 124 [arrowhead = none ];
124 -> 23;

125 [label="EVAL with clause\nhead(.(X65, X66), X65).\nand substitution\nX65 -> T15\nX66 -> T16\nT12 -> .(T15, T16)\nX38 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 125 [arrowhead = none ];
125 -> 36;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 126 [arrowhead = none ];
126 -> 37;

127 [label="EVAL with clause\ntail([], []).\nand substitution\nX37 -> []", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 127 [arrowhead = none ];
127 -> 24;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 128 [arrowhead = none ];
128 -> 25;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 129 [arrowhead = none ];
129 -> 26;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 130 [arrowhead = none ];
130 -> 27;

131 [label="BACKTRACK\nfor clause: tail(.(X5, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
26 -> 131 [arrowhead = none ];
131 -> 34;

132 [label="BACKTRACK\nfor clause: app(nil, Y, Z) :- ','(!, eq(Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 132 [arrowhead = none ];
132 -> 28;

133 [label="EVAL with clause\napp(X49, X50, cons(X51, X52)) :- ','(head(X49, X51), ','(tail(X49, X53), app(X53, X50, X52))).\nand substitution\nX49 -> []\nT13 -> T14\nX50 -> T14\nX51 -> X54\nX52 -> X55\nX39 -> cons(X54, X55)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 133 [arrowhead = none ];
133 -> 29;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 134 [arrowhead = none ];
134 -> 30;

135 [label="EVAL with clause\nhead([], X57).\nand substitution\nX54 -> X58\nX57 -> X58", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 135 [arrowhead = none ];
135 -> 31;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 136 [arrowhead = none ];
136 -> 32;

137 [label="EVAL with clause\ntail([], []).\nand substitution\nX53 -> []", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 137 [arrowhead = none ];
137 -> 33;

138 [label="INSTANCE with matching:\nT13 -> T14\nX39 -> X55\nX37 -> X53\nX38 -> X54", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 138 [arrowhead = none , style=dashed];
138 -> 24[style=dashed];

139 [label="BACKTRACK\nfor clause: head(.(X, X4), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 139 [arrowhead = none ];
139 -> 35;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 140 [arrowhead = none ];
140 -> 38;

141 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 141 [arrowhead = none ];
141 -> 39;

142 [label="EVAL with clause\ntail(.(X69, X70), X70).\nand substitution\nT15 -> T17\nX69 -> T17\nT16 -> T18\nX70 -> T18\nX37 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 142 [arrowhead = none ];
142 -> 40;

143 [label="INSTANCE with matching:\nT5 -> T18\nT6 -> T13\nX20 -> X39", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 143 [arrowhead = none , style=dashed];
143 -> 10[style=dashed];

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 144 [arrowhead = none ];
144 -> 43;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 145 [arrowhead = none ];
145 -> 52;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 146 [arrowhead = none ];
146 -> 44;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 147 [arrowhead = none ];
147 -> 45;

148 [label="EVAL with clause\napp(nil, X75, X76) :- ','(!, eq(X75, X76)).\nand substitution\nT7 -> nil\nT8 -> T20\nX75 -> T20\nX21 -> X77\nX76 -> X77", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 148 [arrowhead = none ];
148 -> 46;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 149 [arrowhead = none ];
149 -> 47;

150 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 150 [arrowhead = none ];
150 -> 48;

151 [label="INSTANCE with matching:\nT5 -> T7\nT6 -> T8\nX20 -> X21\nX24 -> X75\nX25 -> X76", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 151 [arrowhead = none , style=dashed];
151 -> 14[style=dashed];

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 152 [arrowhead = none ];
152 -> 49;

153 [label="EVAL with clause\neq(X79, X79).\nand substitution\nT20 -> T21\nX79 -> T21\nX77 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 153 [arrowhead = none ];
153 -> 50;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 154 [arrowhead = none ];
154 -> 51;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 155 [arrowhead = none ];
155 -> 53;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 156 [arrowhead = none ];
156 -> 54;

157 [label="EVAL with clause\nlessleaves(nil, cons(X88, X89)).\nand substitution\nT9 -> nil\nX88 -> T22\nX89 -> T23\nT19 -> cons(T22, T23)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 157 [arrowhead = none ];
157 -> 55;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 158 [arrowhead = none ];
158 -> 56;

159 [label="EVAL with clause\nlessleaves(cons(X94, X95), cons(X96, X97)) :- ','(app(X94, X95, X98), ','(app(X96, X97, X99), lessleaves(X98, X99))).\nand substitution\nX94 -> T28\nX95 -> T29\nT9 -> cons(T28, T29)\nX96 -> T30\nX97 -> T31\nT19 -> cons(T30, T31)\nT24 -> T28\nT25 -> T29\nT26 -> T30\nT27 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 159 [arrowhead = none ];
159 -> 58;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 160 [arrowhead = none ];
160 -> 59;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 161 [arrowhead = none ];
161 -> 57;

162 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 162 [arrowhead = none ];
162 -> 60;

163 [label="SPLIT 2\nreplacements:\nX98 -> T32\nT30 -> T33\nT31 -> T34", fontsize=14, style = filled, fillcolor = violet];
58 -> 163 [arrowhead = none ];
163 -> 61;

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 164 [arrowhead = none ];
164 -> 62;

165 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 165 [arrowhead = none ];
165 -> 99;

166 [label="SPLIT 2\nreplacements:\nX99 -> T63\nT32 -> T64", fontsize=14, style = filled, fillcolor = violet];
61 -> 166 [arrowhead = none ];
166 -> 100;

167 [label="EVAL with clause\napp(nil, X102, X103) :- ','(!, eq(X102, X103)).\nand substitution\nT28 -> nil\nT29 -> T36\nX102 -> T36\nX98 -> X104\nX103 -> X104\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 167 [arrowhead = none ];
167 -> 63;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 168 [arrowhead = none ];
168 -> 64;

169 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 169 [arrowhead = none ];
169 -> 65;

170 [label="EVAL with clause\napp(X111, X112, cons(X113, X114)) :- ','(head(X111, X113), ','(tail(X111, X115), app(X115, X112, X114))).\nand substitution\nT28 -> T40\nX111 -> T40\nT29 -> T41\nX112 -> T41\nX113 -> X116\nX114 -> X117\nX98 -> cons(X116, X117)\nT38 -> T40\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 170 [arrowhead = none ];
170 -> 69;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 171 [arrowhead = none ];
171 -> 66;

172 [label="EVAL with clause\neq(X106, X106).\nand substitution\nT36 -> T37\nX106 -> T37\nX104 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 172 [arrowhead = none ];
172 -> 67;

173 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 173 [arrowhead = none ];
173 -> 68;

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 174 [arrowhead = none ];
174 -> 70;

175 [label="EVAL with clause\nhead([], X119).\nand substitution\nT40 -> []\nX116 -> X120\nX119 -> X120\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 175 [arrowhead = none ];
175 -> 71;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 176 [arrowhead = none ];
176 -> 72;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 177 [arrowhead = none ];
177 -> 73;

178 [label="EVAL with clause\nhead(.(X153, X154), X153).\nand substitution\nX153 -> T56\nX154 -> T57\nT40 -> .(T56, T57)\nX116 -> T56\nT54 -> T56\nT55 -> T57\nT41 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 178 [arrowhead = none ];
178 -> 94;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 179 [arrowhead = none ];
179 -> 95;

180 [label="EVAL with clause\ntail([], []).\nand substitution\nX115 -> []", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 180 [arrowhead = none ];
180 -> 74;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 181 [arrowhead = none ];
181 -> 75;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 182 [arrowhead = none ];
182 -> 76;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 183 [arrowhead = none ];
183 -> 77;

184 [label="BACKTRACK\nfor clause: tail(.(X5, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
76 -> 184 [arrowhead = none ];
184 -> 88;

185 [label="BACKTRACK\nfor clause: app(nil, Y, Z) :- ','(!, eq(Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
77 -> 185 [arrowhead = none ];
185 -> 78;

186 [label="EVAL with clause\napp(X127, X128, cons(X129, X130)) :- ','(head(X127, X129), ','(tail(X127, X131), app(X131, X128, X130))).\nand substitution\nX127 -> []\nT42 -> T44\nX128 -> T44\nX129 -> X132\nX130 -> X133\nX117 -> cons(X132, X133)\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 186 [arrowhead = none ];
186 -> 79;

187 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 187 [arrowhead = none ];
187 -> 80;

188 [label="EVAL with clause\nhead([], X135).\nand substitution\nX132 -> X136\nX135 -> X136", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 188 [arrowhead = none ];
188 -> 81;

189 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 189 [arrowhead = none ];
189 -> 82;

190 [label="EVAL with clause\ntail([], []).\nand substitution\nX131 -> []", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 190 [arrowhead = none ];
190 -> 83;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 191 [arrowhead = none ];
191 -> 84;

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 192 [arrowhead = none ];
192 -> 85;

193 [label="INSTANCE with matching:\nT42 -> T44\nX117 -> X133", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 193 [arrowhead = none , style=dashed];
193 -> 75[style=dashed];

194 [label="BACKTRACK\nfor clause: tail(.(X5, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
85 -> 194 [arrowhead = none ];
194 -> 86;

195 [label="BACKTRACK\nfor clause: head(.(X, X4), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
86 -> 195 [arrowhead = none ];
195 -> 87;

196 [label="EVAL with clause\nhead(.(X145, X146), X145).\nand substitution\nX145 -> T47\nX146 -> T48\nT40 -> .(T47, T48)\nX116 -> T47\nT45 -> T47\nT46 -> T48\nT41 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 196 [arrowhead = none ];
196 -> 89;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 197 [arrowhead = none ];
197 -> 90;

198 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
89 -> 198 [arrowhead = none ];
198 -> 91;

199 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
91 -> 199 [arrowhead = none ];
199 -> 92;

200 [label="EVAL with clause\ntail(.(X149, X150), X150).\nand substitution\nT47 -> T50\nX149 -> T50\nT48 -> T52\nX150 -> T52\nX115 -> T52\nT51 -> T52\nT49 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 200 [arrowhead = none ];
200 -> 93;

201 [label="INSTANCE with matching:\nT28 -> T52\nT29 -> T53\nX98 -> X117", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 201 [arrowhead = none , style=dashed];
201 -> 60[style=dashed];

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 202 [arrowhead = none ];
202 -> 96;

203 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
96 -> 203 [arrowhead = none ];
203 -> 97;

204 [label="EVAL with clause\ntail(.(X157, X158), X158).\nand substitution\nT56 -> T59\nX157 -> T59\nT57 -> T61\nX158 -> T61\nX115 -> T61\nT60 -> T61\nT58 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 204 [arrowhead = none ];
204 -> 98;

205 [label="INSTANCE with matching:\nT28 -> T61\nT29 -> T62\nX98 -> X117", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 205 [arrowhead = none , style=dashed];
205 -> 60[style=dashed];

206 [label="INSTANCE with matching:\nT28 -> T33\nT29 -> T34\nX98 -> X99", fontsize=14, style = filled, fillcolor = lightgrey];
99 -> 206 [arrowhead = none , style=dashed];
206 -> 60[style=dashed];

207 [label="INSTANCE with matching:\nT9 -> T64\nT19 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 207 [arrowhead = none , style=dashed];
207 -> 42[style=dashed];

}

(2) Obligation:

Clauses:

app24(T14, cons(X58, X55), X37, X38) :- app24(T14, X55, X53, X54).
app10(nil, T11, T11).
app10([], T14, cons(X42, cons(X58, X55))) :- app24(T14, X55, X53, X54).
app10(.(T17, T18), T13, cons(T17, X39)) :- app10(T18, T13, X39).
app14([], T14, cons(X42, cons(X58, X55))) :- app24(T14, X55, X53, X54).
app14(.(T17, T18), T13, cons(T17, X39)) :- app10(T18, T13, X39).
app75(T44, cons(X136, X133)) :- app75(T44, X133).
app60(nil, T37, T37).
app60([], T44, cons(X120, cons(X136, X133))) :- app75(T44, X133).
app60(.(T50, T52), T53, cons(T50, X117)) :- app60(T52, T53, X117).
app60(.(T59, T61), T62, cons(T59, X117)) :- app60(T61, T62, X117).
lessleaves42(nil, cons(T22, T23)).
lessleaves42(cons(T28, T29), cons(T30, T31)) :- app60(T28, T29, X98).
lessleaves42(cons(T28, T29), cons(T33, T34)) :- ','(app60(T28, T29, T32), app60(T33, T34, X99)).
lessleaves42(cons(T28, T29), cons(T33, T34)) :- ','(app60(T28, T29, T64), ','(app60(T33, T34, T63), lessleaves42(T64, T63))).
app41(nil, T21, T21).
app41(T7, T8, X21) :- app14(T7, T8, X21).
lessleaves1(nil, cons(T3, T4)).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- app10(T5, T6, X20).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- ','(app10(T5, T6, T9), app41(T7, T8, X21)).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- ','(app10(T5, T6, nil), app41(T7, T8, cons(T22, T23))).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- ','(app10(T5, T6, cons(T28, T29)), ','(app41(T7, T8, cons(T30, T31)), app60(T28, T29, X98))).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- ','(app10(T5, T6, cons(T28, T29)), ','(app41(T7, T8, cons(T33, T34)), ','(app60(T28, T29, T32), app60(T33, T34, X99)))).
lessleaves1(cons(T5, T6), cons(T7, T8)) :- ','(app10(T5, T6, cons(T28, T29)), ','(app41(T7, T8, cons(T33, T34)), ','(app60(T28, T29, T64), ','(app60(T33, T34, T63), lessleaves42(T64, T63))))).

Queries:

lessleaves1(g,g).

(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:
lessleaves1_in: (b,b)
app10_in: (b,b,f) (b,b,b)
app24_in: (b,f,f,f) (b,b,f,f)
app41_in: (b,b,f)
app14_in: (b,b,f)
app60_in: (f,f,f)
app75_in: (f,f)
lessleaves42_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)

(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:

LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U17_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, X20)
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U2_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → U1_GAAA(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U3_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U18_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → APP41_IN_GGA(T7, T8, X21)
APP41_IN_GGA(T7, T8, X21) → U16_GGA(T7, T8, X21, app14_in_gga(T7, T8, X21))
APP41_IN_GGA(T7, T8, X21) → APP14_IN_GGA(T7, T8, X21)
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U4_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U5_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U20_GG(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGG(T5, T6, nil)
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → U2_GGG(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GGAA(T14, X55, X53, X54)
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → U1_GGAA(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → U3_GGG(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → APP41_IN_GGA(T7, T8, cons(T22, T23))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U22_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, cons(T28, T29))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → APP41_IN_GGA(T7, T8, cons(T30, T31))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_GG(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → APP60_IN_AAA(T28, T29, X98)
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → U7_AAA(T44, X120, X136, X133, app75_in_aa(T44, X133))
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → APP75_IN_AA(T44, X133)
APP75_IN_AA(T44, cons(X136, X133)) → U6_AA(T44, X136, X133, app75_in_aa(T44, X133))
APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → U8_AAA(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)
APP60_IN_AAA(.(T59, T61), T62, cons(T59, X117)) → U9_AAA(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → APP60_IN_AAA(T28, T29, T32)
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_GG(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_GG(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_GG(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → U10_AA(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → APP60_IN_AAA(T28, T29, X98)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U11_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_AA(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_AA(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES1_IN_GG(x1, x2)  =  LESSLEAVES1_IN_GG(x1, x2)
U17_GG(x1, x2, x3, x4, x5)  =  U17_GG(x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x6)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x3, x4, x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x5)
APP41_IN_GGA(x1, x2, x3)  =  APP41_IN_GGA(x1, x2)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)
APP14_IN_GGA(x1, x2, x3)  =  APP14_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x3, x4, x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5)  =  U2_GGG(x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x6)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x5)
U21_GG(x1, x2, x3, x4, x5)  =  U21_GG(x5)
U22_GG(x1, x2, x3, x4, x5)  =  U22_GG(x3, x4, x5)
U23_GG(x1, x2, x3, x4, x5, x6, x7)  =  U23_GG(x7)
U24_GG(x1, x2, x3, x4, x5)  =  U24_GG(x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U25_GG(x1, x2, x3, x4, x5, x6, x7)  =  U25_GG(x7)
U26_GG(x1, x2, x3, x4, x5, x6, x7)  =  U26_GG(x7)
U27_GG(x1, x2, x3, x4, x5)  =  U27_GG(x5)
U28_GG(x1, x2, x3, x4, x5, x6, x7)  =  U28_GG(x7)
U29_GG(x1, x2, x3, x4, x5, x6)  =  U29_GG(x6)
U30_GG(x1, x2, x3, x4, x5)  =  U30_GG(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)

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

(6) Obligation:

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

LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U17_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, X20)
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U2_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → U1_GAAA(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U3_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U18_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → APP41_IN_GGA(T7, T8, X21)
APP41_IN_GGA(T7, T8, X21) → U16_GGA(T7, T8, X21, app14_in_gga(T7, T8, X21))
APP41_IN_GGA(T7, T8, X21) → APP14_IN_GGA(T7, T8, X21)
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U4_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U5_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U20_GG(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGG(T5, T6, nil)
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → U2_GGG(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GGAA(T14, X55, X53, X54)
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → U1_GGAA(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → U3_GGG(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → APP41_IN_GGA(T7, T8, cons(T22, T23))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U22_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, cons(T28, T29))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → APP41_IN_GGA(T7, T8, cons(T30, T31))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_GG(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → APP60_IN_AAA(T28, T29, X98)
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → U7_AAA(T44, X120, X136, X133, app75_in_aa(T44, X133))
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → APP75_IN_AA(T44, X133)
APP75_IN_AA(T44, cons(X136, X133)) → U6_AA(T44, X136, X133, app75_in_aa(T44, X133))
APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → U8_AAA(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)
APP60_IN_AAA(.(T59, T61), T62, cons(T59, X117)) → U9_AAA(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → APP60_IN_AAA(T28, T29, T32)
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_GG(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_GG(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_GG(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → U10_AA(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → APP60_IN_AAA(T28, T29, X98)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U11_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_AA(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_AA(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES1_IN_GG(x1, x2)  =  LESSLEAVES1_IN_GG(x1, x2)
U17_GG(x1, x2, x3, x4, x5)  =  U17_GG(x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x6)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x3, x4, x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x5)
APP41_IN_GGA(x1, x2, x3)  =  APP41_IN_GGA(x1, x2)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)
APP14_IN_GGA(x1, x2, x3)  =  APP14_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x3, x4, x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5)  =  U2_GGG(x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x6)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x5)
U21_GG(x1, x2, x3, x4, x5)  =  U21_GG(x5)
U22_GG(x1, x2, x3, x4, x5)  =  U22_GG(x3, x4, x5)
U23_GG(x1, x2, x3, x4, x5, x6, x7)  =  U23_GG(x7)
U24_GG(x1, x2, x3, x4, x5)  =  U24_GG(x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U25_GG(x1, x2, x3, x4, x5, x6, x7)  =  U25_GG(x7)
U26_GG(x1, x2, x3, x4, x5, x6, x7)  =  U26_GG(x7)
U27_GG(x1, x2, x3, x4, x5)  =  U27_GG(x5)
U28_GG(x1, x2, x3, x4, x5, x6, x7)  =  U28_GG(x7)
U29_GG(x1, x2, x3, x4, x5, x6)  =  U29_GG(x6)
U30_GG(x1, x2, x3, x4, x5)  =  U30_GG(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 51 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP75_IN_AA → APP75_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = APP75_IN_AA evaluates to t =APP75_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP75_IN_AA to APP75_IN_AA.

(15) FALSE

(16) Obligation:

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

APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP60_IN_AAA → APP60_IN_AAA

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

(21) NonTerminationProof (EQUIVALENT transformation)

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

s = APP60_IN_AAA evaluates to t =APP60_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP60_IN_AAA to APP60_IN_AAA.

(22) FALSE

(23) Obligation:

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

LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(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:

LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(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:

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U7_aaa(app75_in_aa)
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U7_aaa(app75_out_aa(T44)) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa
U9_aaa(app60_out_aaa) → app60_out_aaa
app75_in_aa → U6_aa(app75_in_aa)
U6_aa(app75_out_aa(T44)) → app75_out_aa(T44)

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(28) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U7_aaa(app75_out_aa(T44)) → app60_out_aaa

Used ordering: Polynomial interpretation [POLO]:

POL(LESSLEAVES42_IN_AA) = 0   
POL(U13_AA(x₁)) = x₁   
POL(U14_AA(x₁)) = 2·x₁   
POL(U6_aa(x₁)) = x₁   
POL(U7_aaa(x₁)) = x₁   
POL(U8_aaa(x₁)) = 2·x₁   
POL(U9_aaa(x₁)) = 2·x₁   
POL(app60_in_aaa) = 0   
POL(app60_out_aaa) = 0   
POL(app75_in_aa) = 0   
POL(app75_out_aa(x₁)) = 1 + 2·x₁   

(29) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U7_aaa(app75_in_aa)
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U8_aaa(app60_out_aaa) → app60_out_aaa
U9_aaa(app60_out_aaa) → app60_out_aaa
app75_in_aa → U6_aa(app75_in_aa)
U6_aa(app75_out_aa(T44)) → app75_out_aa(T44)

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(30) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

app60_in_aaa → U7_aaa(app75_in_aa)

Used ordering: Polynomial interpretation [POLO]:

POL(LESSLEAVES42_IN_AA) = 1   
POL(U13_AA(x₁)) = x₁   
POL(U14_AA(x₁)) = x₁   
POL(U6_aa(x₁)) = x₁   
POL(U7_aaa(x₁)) = x₁   
POL(U8_aaa(x₁)) = x₁   
POL(U9_aaa(x₁)) = x₁   
POL(app60_in_aaa) = 1   
POL(app60_out_aaa) = 1   
POL(app75_in_aa) = 0   
POL(app75_out_aa(x₁)) = x₁   

(31) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U8_aaa(app60_out_aaa) → app60_out_aaa
U9_aaa(app60_out_aaa) → app60_out_aaa
app75_in_aa → U6_aa(app75_in_aa)
U6_aa(app75_out_aa(T44)) → app75_out_aa(T44)

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(34) QReductionProof (EQUIVALENT transformation)

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

U7_aaa(x0)
app75_in_aa
U6_aa(x0)

(35) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(36) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))

(37) Obligation:

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

U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(38) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
U13_AA(app60_out_aaa) → U14_AA(U8_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(U9_aaa(app60_in_aaa))

(39) Obligation:

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

U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
U13_AA(app60_out_aaa) → U14_AA(U8_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(U9_aaa(app60_in_aaa))

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(40) NonTerminationProof (EQUIVALENT transformation)

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

s = LESSLEAVES42_IN_AA evaluates to t =LESSLEAVES42_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
with rule LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa) at position [] and matcher [ ]

U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
with rule U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa) at position [] and matcher [ ]

U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
with rule U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

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


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

(41) FALSE

(42) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)

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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)

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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55)) → APP24_IN_GGAA(T14, X55)

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

(47) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP24_IN_GGAA(T14, cons(X58, X55)) → APP24_IN_GGAA(T14, X55)
  The graph contains the following edges 1 >= 1, 2 > 2

(48) TRUE

(49) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)

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

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

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

(52) PiDPToQDPProof (EQUIVALENT transformation)

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

(53) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

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

(54) 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:

• APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(55) TRUE

(56) Obligation:

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

APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)

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

(57) UsableRulesProof (EQUIVALENT transformation)

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

(58) Obligation:

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

APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)

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

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

APP24_IN_GAAA(T14) → APP24_IN_GAAA(T14)

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

(61) NonTerminationProof (EQUIVALENT transformation)

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

s = APP24_IN_GAAA(T14) evaluates to t =APP24_IN_GAAA(T14)

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP24_IN_GAAA(T14) to APP24_IN_GAAA(T14).

(62) FALSE

(63) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga
U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)

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

(64) UsableRulesProof (EQUIVALENT transformation)

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

(65) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)

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

(66) PiDPToQDPProof (SOUND transformation)

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

(67) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13) → APP10_IN_GGA(T18, T13)

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

(68) 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:

• APP10_IN_GGA(.(T17, T18), T13) → APP10_IN_GGA(T18, T13)
  The graph contains the following edges 1 > 1, 2 >= 2

(69) TRUE

(70) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
lessleaves1_in: (b,b)
app10_in: (b,b,f) (b,b,b)
app24_in: (b,f,f,f) (b,b,f,f)
app41_in: (b,b,f)
app14_in: (b,b,f)
app60_in: (f,f,f)
app75_in: (f,f)
lessleaves42_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(71) Obligation:

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

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)

(72) 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:

LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U17_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, X20)
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U2_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → U1_GAAA(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U3_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U18_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → APP41_IN_GGA(T7, T8, X21)
APP41_IN_GGA(T7, T8, X21) → U16_GGA(T7, T8, X21, app14_in_gga(T7, T8, X21))
APP41_IN_GGA(T7, T8, X21) → APP14_IN_GGA(T7, T8, X21)
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U4_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U5_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U20_GG(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGG(T5, T6, nil)
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → U2_GGG(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GGAA(T14, X55, X53, X54)
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → U1_GGAA(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → U3_GGG(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → APP41_IN_GGA(T7, T8, cons(T22, T23))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U22_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, cons(T28, T29))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → APP41_IN_GGA(T7, T8, cons(T30, T31))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_GG(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → APP60_IN_AAA(T28, T29, X98)
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → U7_AAA(T44, X120, X136, X133, app75_in_aa(T44, X133))
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → APP75_IN_AA(T44, X133)
APP75_IN_AA(T44, cons(X136, X133)) → U6_AA(T44, X136, X133, app75_in_aa(T44, X133))
APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → U8_AAA(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)
APP60_IN_AAA(.(T59, T61), T62, cons(T59, X117)) → U9_AAA(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → APP60_IN_AAA(T28, T29, T32)
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_GG(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_GG(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_GG(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → U10_AA(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → APP60_IN_AAA(T28, T29, X98)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U11_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_AA(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_AA(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES1_IN_GG(x1, x2)  =  LESSLEAVES1_IN_GG(x1, x2)
U17_GG(x1, x2, x3, x4, x5)  =  U17_GG(x1, x2, x3, x4, x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x1, x2, x3, x4, x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x2, x3, x4, x5)
APP41_IN_GGA(x1, x2, x3)  =  APP41_IN_GGA(x1, x2)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
APP14_IN_GGA(x1, x2, x3)  =  APP14_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x1, x2, x3, x4, x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5)  =  U2_GGG(x1, x2, x3, x4, x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x1, x2, x3, x6)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
U21_GG(x1, x2, x3, x4, x5)  =  U21_GG(x1, x2, x3, x4, x5)
U22_GG(x1, x2, x3, x4, x5)  =  U22_GG(x1, x2, x3, x4, x5)
U23_GG(x1, x2, x3, x4, x5, x6, x7)  =  U23_GG(x1, x2, x3, x4, x7)
U24_GG(x1, x2, x3, x4, x5)  =  U24_GG(x1, x2, x3, x4, x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U25_GG(x1, x2, x3, x4, x5, x6, x7)  =  U25_GG(x1, x2, x3, x4, x7)
U26_GG(x1, x2, x3, x4, x5, x6, x7)  =  U26_GG(x1, x2, x3, x4, x7)
U27_GG(x1, x2, x3, x4, x5)  =  U27_GG(x1, x2, x3, x4, x5)
U28_GG(x1, x2, x3, x4, x5, x6, x7)  =  U28_GG(x1, x2, x3, x4, x7)
U29_GG(x1, x2, x3, x4, x5, x6)  =  U29_GG(x1, x2, x3, x4, x6)
U30_GG(x1, x2, x3, x4, x5)  =  U30_GG(x1, x2, x3, x4, x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)

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

(73) Obligation:

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

LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U17_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, X20)
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U2_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP10_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → U1_GAAA(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U3_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U18_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
U18_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → APP41_IN_GGA(T7, T8, X21)
APP41_IN_GGA(T7, T8, X21) → U16_GGA(T7, T8, X21, app14_in_gga(T7, T8, X21))
APP41_IN_GGA(T7, T8, X21) → APP14_IN_GGA(T7, T8, X21)
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → U4_GGA(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
APP14_IN_GGA([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GAAA(T14, X55, X53, X54)
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → U5_GGA(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
APP14_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U20_GG(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGG(T5, T6, nil)
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → U2_GGG(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
APP10_IN_GGG([], T14, cons(X42, cons(X58, X55))) → APP24_IN_GGAA(T14, X55, X53, X54)
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → U1_GGAA(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → U3_GGG(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_GG(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U20_GG(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → APP41_IN_GGA(T7, T8, cons(T22, T23))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → U22_GG(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
LESSLEAVES1_IN_GG(cons(T5, T6), cons(T7, T8)) → APP10_IN_GGA(T5, T6, cons(T28, T29))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → APP41_IN_GGA(T7, T8, cons(T30, T31))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_GG(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
U23_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → APP60_IN_AAA(T28, T29, X98)
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → U7_AAA(T44, X120, X136, X133, app75_in_aa(T44, X133))
APP60_IN_AAA([], T44, cons(X120, cons(X136, X133))) → APP75_IN_AA(T44, X133)
APP75_IN_AA(T44, cons(X136, X133)) → U6_AA(T44, X136, X133, app75_in_aa(T44, X133))
APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → U8_AAA(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)
APP60_IN_AAA(.(T59, T61), T62, cons(T59, X117)) → U9_AAA(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U22_GG(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_GG(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → APP60_IN_AAA(T28, T29, T32)
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_GG(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U26_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
U25_GG(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_GG(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_GG(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U28_GG(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_GG(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
U29_GG(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → U10_AA(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T30, T31)) → APP60_IN_AAA(T28, T29, X98)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U11_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_AA(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U11_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → APP60_IN_AAA(T33, T34, X99)
LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → APP60_IN_AAA(T33, T34, T63)
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_AA(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES1_IN_GG(x1, x2)  =  LESSLEAVES1_IN_GG(x1, x2)
U17_GG(x1, x2, x3, x4, x5)  =  U17_GG(x1, x2, x3, x4, x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x1, x2, x3, x4, x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x2, x3, x4, x5)
APP41_IN_GGA(x1, x2, x3)  =  APP41_IN_GGA(x1, x2)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
APP14_IN_GGA(x1, x2, x3)  =  APP14_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x1, x2, x3, x4, x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5)  =  U2_GGG(x1, x2, x3, x4, x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x1, x2, x3, x6)
U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
U21_GG(x1, x2, x3, x4, x5)  =  U21_GG(x1, x2, x3, x4, x5)
U22_GG(x1, x2, x3, x4, x5)  =  U22_GG(x1, x2, x3, x4, x5)
U23_GG(x1, x2, x3, x4, x5, x6, x7)  =  U23_GG(x1, x2, x3, x4, x7)
U24_GG(x1, x2, x3, x4, x5)  =  U24_GG(x1, x2, x3, x4, x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA
U6_AA(x1, x2, x3, x4)  =  U6_AA(x4)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U25_GG(x1, x2, x3, x4, x5, x6, x7)  =  U25_GG(x1, x2, x3, x4, x7)
U26_GG(x1, x2, x3, x4, x5, x6, x7)  =  U26_GG(x1, x2, x3, x4, x7)
U27_GG(x1, x2, x3, x4, x5)  =  U27_GG(x1, x2, x3, x4, x5)
U28_GG(x1, x2, x3, x4, x5, x6, x7)  =  U28_GG(x1, x2, x3, x4, x7)
U29_GG(x1, x2, x3, x4, x5, x6)  =  U29_GG(x1, x2, x3, x4, x6)
U30_GG(x1, x2, x3, x4, x5)  =  U30_GG(x1, x2, x3, x4, x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)

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

(74) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 51 less nodes.

(75) Complex Obligation (AND)

(76) Obligation:

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

APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

APP75_IN_AA(T44, cons(X136, X133)) → APP75_IN_AA(T44, X133)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP75_IN_AA(x1, x2)  =  APP75_IN_AA

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

APP75_IN_AA → APP75_IN_AA

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

(81) NonTerminationProof (EQUIVALENT transformation)

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

s = APP75_IN_AA evaluates to t =APP75_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP75_IN_AA to APP75_IN_AA.

(82) FALSE

(83) Obligation:

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

APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

APP60_IN_AAA(.(T50, T52), T53, cons(T50, X117)) → APP60_IN_AAA(T52, T53, X117)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
APP60_IN_AAA(x1, x2, x3)  =  APP60_IN_AAA

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

APP60_IN_AAA → APP60_IN_AAA

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

(88) NonTerminationProof (EQUIVALENT transformation)

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

s = APP60_IN_AAA evaluates to t =APP60_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP60_IN_AAA to APP60_IN_AAA.

(89) FALSE

(90) Obligation:

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

LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)

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

(91) UsableRulesProof (EQUIVALENT transformation)

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

(92) Obligation:

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

LESSLEAVES42_IN_AA(cons(T28, T29), cons(T33, T34)) → U13_AA(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_AA(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_AA(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_AA(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → LESSLEAVES42_IN_AA(T64, T63)

The TRS R consists of the following rules:

app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
.(x1, x2)  =  .(x1, x2)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
LESSLEAVES42_IN_AA(x1, x2)  =  LESSLEAVES42_IN_AA
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5, x6)  =  U14_AA(x6)

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

(93) PiDPToQDPProof (SOUND transformation)

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

(94) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U7_aaa(app75_in_aa)
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U7_aaa(app75_out_aa(T44)) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa
U9_aaa(app60_out_aaa) → app60_out_aaa
app75_in_aa → U6_aa(app75_in_aa)
U6_aa(app75_out_aa(T44)) → app75_out_aa(T44)

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(95) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

app60_in_aaa → U7_aaa(app75_in_aa)
U7_aaa(app75_out_aa(T44)) → app60_out_aaa

Used ordering: Polynomial interpretation [POLO]:

POL(LESSLEAVES42_IN_AA) = 1   
POL(U13_AA(x₁)) = x₁   
POL(U14_AA(x₁)) = x₁   
POL(U6_aa(x₁)) = x₁   
POL(U7_aaa(x₁)) = x₁   
POL(U8_aaa(x₁)) = x₁   
POL(U9_aaa(x₁)) = x₁   
POL(app60_in_aaa) = 1   
POL(app60_out_aaa) = 1   
POL(app75_in_aa) = 0   
POL(app75_out_aa(x₁)) = 2 + x₁   

(96) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U8_aaa(app60_out_aaa) → app60_out_aaa
U9_aaa(app60_out_aaa) → app60_out_aaa
app75_in_aa → U6_aa(app75_in_aa)
U6_aa(app75_out_aa(T44)) → app75_out_aa(T44)

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(97) UsableRulesProof (EQUIVALENT transformation)

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

(98) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U7_aaa(x0)
U8_aaa(x0)
U9_aaa(x0)
app75_in_aa
U6_aa(x0)

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

(99) QReductionProof (EQUIVALENT transformation)

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

U7_aaa(x0)
app75_in_aa
U6_aa(x0)

(100) Obligation:

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

LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa)
U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(101) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESSLEAVES42_IN_AA → U13_AA(app60_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))

(102) Obligation:

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

U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa)
U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(103) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U13_AA(app60_out_aaa) → U14_AA(app60_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
U13_AA(app60_out_aaa) → U14_AA(U8_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(U9_aaa(app60_in_aaa))

(104) Obligation:

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

U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
LESSLEAVES42_IN_AA → U13_AA(U8_aaa(app60_in_aaa))
LESSLEAVES42_IN_AA → U13_AA(U9_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
U13_AA(app60_out_aaa) → U14_AA(U8_aaa(app60_in_aaa))
U13_AA(app60_out_aaa) → U14_AA(U9_aaa(app60_in_aaa))

The TRS R consists of the following rules:

app60_in_aaa → app60_out_aaa
app60_in_aaa → U8_aaa(app60_in_aaa)
app60_in_aaa → U9_aaa(app60_in_aaa)
U9_aaa(app60_out_aaa) → app60_out_aaa
U8_aaa(app60_out_aaa) → app60_out_aaa

The set Q consists of the following terms:

app60_in_aaa
U8_aaa(x0)
U9_aaa(x0)

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

(105) NonTerminationProof (EQUIVALENT transformation)

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

s = LESSLEAVES42_IN_AA evaluates to t =LESSLEAVES42_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa)
with rule LESSLEAVES42_IN_AA → U13_AA(app60_out_aaa) at position [] and matcher [ ]

U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa)
with rule U13_AA(app60_out_aaa) → U14_AA(app60_out_aaa) at position [] and matcher [ ]

U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA
with rule U14_AA(app60_out_aaa) → LESSLEAVES42_IN_AA

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


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

(106) FALSE

(107) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)

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

(108) UsableRulesProof (EQUIVALENT transformation)

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

(109) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GGAA(T14, X55, X53, X54)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP24_IN_GGAA(x1, x2, x3, x4)  =  APP24_IN_GGAA(x1, x2)

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

(110) PiDPToQDPProof (SOUND transformation)

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

(111) Obligation:

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

APP24_IN_GGAA(T14, cons(X58, X55)) → APP24_IN_GGAA(T14, X55)

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

(112) 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:

• APP24_IN_GGAA(T14, cons(X58, X55)) → APP24_IN_GGAA(T14, X55)
  The graph contains the following edges 1 >= 1, 2 > 2

(113) TRUE

(114) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP10_IN_GGG(x1, x2, x3)  =  APP10_IN_GGG(x1, x2, x3)

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

(115) UsableRulesProof (EQUIVALENT transformation)

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

(116) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

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

(117) PiDPToQDPProof (EQUIVALENT transformation)

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

(118) Obligation:

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

APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)

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

(119) 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:

• APP10_IN_GGG(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGG(T18, T13, X39)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(120) TRUE

(121) Obligation:

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

APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)

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

(122) UsableRulesProof (EQUIVALENT transformation)

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

(123) Obligation:

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

APP24_IN_GAAA(T14, cons(X58, X55), X37, X38) → APP24_IN_GAAA(T14, X55, X53, X54)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APP24_IN_GAAA(x1, x2, x3, x4)  =  APP24_IN_GAAA(x1)

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

(124) PiDPToQDPProof (SOUND transformation)

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

(125) Obligation:

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

APP24_IN_GAAA(T14) → APP24_IN_GAAA(T14)

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

(126) NonTerminationProof (EQUIVALENT transformation)

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

s = APP24_IN_GAAA(T14) evaluates to t =APP24_IN_GAAA(T14)

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP24_IN_GAAA(T14) to APP24_IN_GAAA(T14).

(127) FALSE

(128) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)

The TRS R consists of the following rules:

lessleaves1_in_gg(nil, cons(T3, T4)) → lessleaves1_out_gg(nil, cons(T3, T4))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U17_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, X20))
app10_in_gga(nil, T11, T11) → app10_out_gga(nil, T11, T11)
app10_in_gga([], T14, cons(X42, cons(X58, X55))) → U2_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
app24_in_gaaa(T14, cons(X58, X55), X37, X38) → U1_gaaa(T14, X58, X55, X37, X38, app24_in_gaaa(T14, X55, X53, X54))
U1_gaaa(T14, X58, X55, X37, X38, app24_out_gaaa(T14, X55, X53, X54)) → app24_out_gaaa(T14, cons(X58, X55), X37, X38)
U2_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app10_out_gga([], T14, cons(X42, cons(X58, X55)))
app10_in_gga(.(T17, T18), T13, cons(T17, X39)) → U3_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U3_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app10_out_gga(.(T17, T18), T13, cons(T17, X39))
U17_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, X20)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U18_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, T9))
U18_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, T9)) → U19_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, X21))
app41_in_gga(nil, T21, T21) → app41_out_gga(nil, T21, T21)
app41_in_gga(T7, T8, X21) → U16_gga(T7, T8, X21, app14_in_gga(T7, T8, X21))
app14_in_gga([], T14, cons(X42, cons(X58, X55))) → U4_gga(T14, X42, X58, X55, app24_in_gaaa(T14, X55, X53, X54))
U4_gga(T14, X42, X58, X55, app24_out_gaaa(T14, X55, X53, X54)) → app14_out_gga([], T14, cons(X42, cons(X58, X55)))
app14_in_gga(.(T17, T18), T13, cons(T17, X39)) → U5_gga(T17, T18, T13, X39, app10_in_gga(T18, T13, X39))
U5_gga(T17, T18, T13, X39, app10_out_gga(T18, T13, X39)) → app14_out_gga(.(T17, T18), T13, cons(T17, X39))
U16_gga(T7, T8, X21, app14_out_gga(T7, T8, X21)) → app41_out_gga(T7, T8, X21)
U19_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, X21)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U20_gg(T5, T6, T7, T8, app10_in_ggg(T5, T6, nil))
app10_in_ggg(nil, T11, T11) → app10_out_ggg(nil, T11, T11)
app10_in_ggg([], T14, cons(X42, cons(X58, X55))) → U2_ggg(T14, X42, X58, X55, app24_in_ggaa(T14, X55, X53, X54))
app24_in_ggaa(T14, cons(X58, X55), X37, X38) → U1_ggaa(T14, X58, X55, X37, X38, app24_in_ggaa(T14, X55, X53, X54))
U1_ggaa(T14, X58, X55, X37, X38, app24_out_ggaa(T14, X55, X53, X54)) → app24_out_ggaa(T14, cons(X58, X55), X37, X38)
U2_ggg(T14, X42, X58, X55, app24_out_ggaa(T14, X55, X53, X54)) → app10_out_ggg([], T14, cons(X42, cons(X58, X55)))
app10_in_ggg(.(T17, T18), T13, cons(T17, X39)) → U3_ggg(T17, T18, T13, X39, app10_in_ggg(T18, T13, X39))
U3_ggg(T17, T18, T13, X39, app10_out_ggg(T18, T13, X39)) → app10_out_ggg(.(T17, T18), T13, cons(T17, X39))
U20_gg(T5, T6, T7, T8, app10_out_ggg(T5, T6, nil)) → U21_gg(T5, T6, T7, T8, app41_in_gga(T7, T8, cons(T22, T23)))
U21_gg(T5, T6, T7, T8, app41_out_gga(T7, T8, cons(T22, T23))) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
lessleaves1_in_gg(cons(T5, T6), cons(T7, T8)) → U22_gg(T5, T6, T7, T8, app10_in_gga(T5, T6, cons(T28, T29)))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U23_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T30, T31)))
U23_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T30, T31))) → U24_gg(T5, T6, T7, T8, app60_in_aaa(T28, T29, X98))
app60_in_aaa(nil, T37, T37) → app60_out_aaa(nil, T37, T37)
app60_in_aaa([], T44, cons(X120, cons(X136, X133))) → U7_aaa(T44, X120, X136, X133, app75_in_aa(T44, X133))
app75_in_aa(T44, cons(X136, X133)) → U6_aa(T44, X136, X133, app75_in_aa(T44, X133))
U6_aa(T44, X136, X133, app75_out_aa(T44, X133)) → app75_out_aa(T44, cons(X136, X133))
U7_aaa(T44, X120, X136, X133, app75_out_aa(T44, X133)) → app60_out_aaa([], T44, cons(X120, cons(X136, X133)))
app60_in_aaa(.(T50, T52), T53, cons(T50, X117)) → U8_aaa(T50, T52, T53, X117, app60_in_aaa(T52, T53, X117))
app60_in_aaa(.(T59, T61), T62, cons(T59, X117)) → U9_aaa(T59, T61, T62, X117, app60_in_aaa(T61, T62, X117))
U9_aaa(T59, T61, T62, X117, app60_out_aaa(T61, T62, X117)) → app60_out_aaa(.(T59, T61), T62, cons(T59, X117))
U8_aaa(T50, T52, T53, X117, app60_out_aaa(T52, T53, X117)) → app60_out_aaa(.(T50, T52), T53, cons(T50, X117))
U24_gg(T5, T6, T7, T8, app60_out_aaa(T28, T29, X98)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U22_gg(T5, T6, T7, T8, app10_out_gga(T5, T6, cons(T28, T29))) → U25_gg(T5, T6, T7, T8, T28, T29, app41_in_gga(T7, T8, cons(T33, T34)))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U26_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T32))
U26_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T32)) → U27_gg(T5, T6, T7, T8, app60_in_aaa(T33, T34, X99))
U27_gg(T5, T6, T7, T8, app60_out_aaa(T33, T34, X99)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))
U25_gg(T5, T6, T7, T8, T28, T29, app41_out_gga(T7, T8, cons(T33, T34))) → U28_gg(T5, T6, T7, T8, T33, T34, app60_in_aaa(T28, T29, T64))
U28_gg(T5, T6, T7, T8, T33, T34, app60_out_aaa(T28, T29, T64)) → U29_gg(T5, T6, T7, T8, T64, app60_in_aaa(T33, T34, T63))
U29_gg(T5, T6, T7, T8, T64, app60_out_aaa(T33, T34, T63)) → U30_gg(T5, T6, T7, T8, lessleaves42_in_aa(T64, T63))
lessleaves42_in_aa(nil, cons(T22, T23)) → lessleaves42_out_aa(nil, cons(T22, T23))
lessleaves42_in_aa(cons(T28, T29), cons(T30, T31)) → U10_aa(T28, T29, T30, T31, app60_in_aaa(T28, T29, X98))
U10_aa(T28, T29, T30, T31, app60_out_aaa(T28, T29, X98)) → lessleaves42_out_aa(cons(T28, T29), cons(T30, T31))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U11_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T32))
U11_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T32)) → U12_aa(T28, T29, T33, T34, app60_in_aaa(T33, T34, X99))
U12_aa(T28, T29, T33, T34, app60_out_aaa(T33, T34, X99)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
lessleaves42_in_aa(cons(T28, T29), cons(T33, T34)) → U13_aa(T28, T29, T33, T34, app60_in_aaa(T28, T29, T64))
U13_aa(T28, T29, T33, T34, app60_out_aaa(T28, T29, T64)) → U14_aa(T28, T29, T33, T34, T64, app60_in_aaa(T33, T34, T63))
U14_aa(T28, T29, T33, T34, T64, app60_out_aaa(T33, T34, T63)) → U15_aa(T28, T29, T33, T34, lessleaves42_in_aa(T64, T63))
U15_aa(T28, T29, T33, T34, lessleaves42_out_aa(T64, T63)) → lessleaves42_out_aa(cons(T28, T29), cons(T33, T34))
U30_gg(T5, T6, T7, T8, lessleaves42_out_aa(T64, T63)) → lessleaves1_out_gg(cons(T5, T6), cons(T7, T8))

The argument filtering Pi contains the following mapping:
lessleaves1_in_gg(x1, x2)  =  lessleaves1_in_gg(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
lessleaves1_out_gg(x1, x2)  =  lessleaves1_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4, x5)  =  U17_gg(x1, x2, x3, x4, x5)
app10_in_gga(x1, x2, x3)  =  app10_in_gga(x1, x2)
app10_out_gga(x1, x2, x3)  =  app10_out_gga(x1, x2)
[]  =  []
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
app24_in_gaaa(x1, x2, x3, x4)  =  app24_in_gaaa(x1)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
app24_out_gaaa(x1, x2, x3, x4)  =  app24_out_gaaa(x1)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x1, x2, x3, x4, x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x2, x3, x4, x5)
app41_in_gga(x1, x2, x3)  =  app41_in_gga(x1, x2)
app41_out_gga(x1, x2, x3)  =  app41_out_gga(x1, x2)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
app14_in_gga(x1, x2, x3)  =  app14_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
app14_out_gga(x1, x2, x3)  =  app14_out_gga(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x1, x2, x3, x4, x5)
app10_in_ggg(x1, x2, x3)  =  app10_in_ggg(x1, x2, x3)
app10_out_ggg(x1, x2, x3)  =  app10_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4, x5)  =  U2_ggg(x1, x2, x3, x4, x5)
app24_in_ggaa(x1, x2, x3, x4)  =  app24_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6)  =  U1_ggaa(x1, x2, x3, x6)
app24_out_ggaa(x1, x2, x3, x4)  =  app24_out_ggaa(x1, x2)
U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x1, x2, x3, x4, x5)
U21_gg(x1, x2, x3, x4, x5)  =  U21_gg(x1, x2, x3, x4, x5)
U22_gg(x1, x2, x3, x4, x5)  =  U22_gg(x1, x2, x3, x4, x5)
U23_gg(x1, x2, x3, x4, x5, x6, x7)  =  U23_gg(x1, x2, x3, x4, x7)
U24_gg(x1, x2, x3, x4, x5)  =  U24_gg(x1, x2, x3, x4, x5)
app60_in_aaa(x1, x2, x3)  =  app60_in_aaa
app60_out_aaa(x1, x2, x3)  =  app60_out_aaa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
app75_in_aa(x1, x2)  =  app75_in_aa
U6_aa(x1, x2, x3, x4)  =  U6_aa(x4)
app75_out_aa(x1, x2)  =  app75_out_aa(x1)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U25_gg(x1, x2, x3, x4, x5, x6, x7)  =  U25_gg(x1, x2, x3, x4, x7)
U26_gg(x1, x2, x3, x4, x5, x6, x7)  =  U26_gg(x1, x2, x3, x4, x7)
U27_gg(x1, x2, x3, x4, x5)  =  U27_gg(x1, x2, x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7)  =  U28_gg(x1, x2, x3, x4, x7)
U29_gg(x1, x2, x3, x4, x5, x6)  =  U29_gg(x1, x2, x3, x4, x6)
U30_gg(x1, x2, x3, x4, x5)  =  U30_gg(x1, x2, x3, x4, x5)
lessleaves42_in_aa(x1, x2)  =  lessleaves42_in_aa
lessleaves42_out_aa(x1, x2)  =  lessleaves42_out_aa
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5, x6)  =  U14_aa(x6)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)

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

(129) UsableRulesProof (EQUIVALENT transformation)

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

(130) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13, cons(T17, X39)) → APP10_IN_GGA(T18, T13, X39)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
APP10_IN_GGA(x1, x2, x3)  =  APP10_IN_GGA(x1, x2)

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

(131) PiDPToQDPProof (SOUND transformation)

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

(132) Obligation:

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

APP10_IN_GGA(.(T17, T18), T13) → APP10_IN_GGA(T18, T13)

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

(133) 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:

• APP10_IN_GGA(.(T17, T18), T13) → APP10_IN_GGA(T18, T13)
  The graph contains the following edges 1 > 1, 2 >= 2

(134) TRUE