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

(0) Obligation:

Clauses:

in(X, tree(X, X1, X2)).
in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)).
in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)).
less(0, s(X5)).
less(s(X), s(Y)) :- less(X, Y).

Query: in(a,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: in(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: in(T1, T2), SCOPE: 1, CLAUSE: 0 | in(T1, T2), SCOPE: 1, CLAUSE: 1 | in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | in(T1, tree(T6, T7, T8)), SCOPE: 1, CLAUSE: 1 | in(T1, tree(T6, T7, T8)), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: in(T1, T2), SCOPE: 1, CLAUSE: 1 | in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT2 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: in(T1, tree(T6, T7, T8)), SCOPE: 1, CLAUSE: 1 | in(T1, tree(T6, T7, T8)), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(less(T17, T14), in(T17, T15)) | in(T1, tree(T14, T15, T16)), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(less(T17, T14), in(T17, T15)), SCOPE: 2, CLAUSE: 3 | ','(less(T17, T14), in(T17, T15)), SCOPE: 2, CLAUSE: 4 | ?_2 | in(T1, tree(T14, T15, T16)), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(less(T17, T14), in(T17, T15)), SCOPE: 2, CLAUSE: 3\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(less(T17, T14), in(T17, T15)), SCOPE: 2, CLAUSE: 4 | ?_2 | in(T1, tree(T14, T15, T16)), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: in(0, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: in(0, T15), SCOPE: 3, CLAUSE: 0 | in(0, T15), SCOPE: 3, CLAUSE: 1 | in(0, T15), SCOPE: 3, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: in(0, T15), SCOPE: 3, CLAUSE: 0\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: in(0, T15), SCOPE: 3, CLAUSE: 1 | in(0, T15), SCOPE: 3, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: in(0, T15), SCOPE: 3, CLAUSE: 1\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: in(0, T15), SCOPE: 3, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(less(0, T49), in(0, T50))\n\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: less(0, T49)\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: in(0, T50)\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(0, T49), SCOPE: 4, CLAUSE: 3 | less(0, T49), SCOPE: 4, CLAUSE: 4\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(0, T49), SCOPE: 4, CLAUSE: 3\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(0, T49), SCOPE: 4, CLAUSE: 4\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(less(T71, 0), in(0, T73))\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: less(T71, 0)\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: in(0, T73)\n\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T71, 0), SCOPE: 5, CLAUSE: 3 | less(T71, 0), SCOPE: 5, CLAUSE: 4\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T71, 0), SCOPE: 5, CLAUSE: 4\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(less(T17, T14), in(T17, T15)), SCOPE: 2, CLAUSE: 4\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ?_2 | in(T1, tree(T14, T15, T16)), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(less(T89, T88), in(s(T89), T15))\n\nT15 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T89, T88)\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: in(s(T92), T15)\n\nT15 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T89, T88), SCOPE: 6, CLAUSE: 3 | less(T89, T88), SCOPE: 6, CLAUSE: 4\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T89, T88), SCOPE: 6, CLAUSE: 3\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: less(T89, T88), SCOPE: 6, CLAUSE: 4\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T106, T105)\n\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: in(T1, tree(T14, T15, T16)), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(less(T127, T130), in(T130, T129))\n\nT127 is ground\nT129 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(T127, T130)\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: in(T133, T129)\n\nT129 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T127, T130), SCOPE: 7, CLAUSE: 3 | less(T127, T130), SCOPE: 7, CLAUSE: 4\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(T127, T130), SCOPE: 7, CLAUSE: 3\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(T127, T130), SCOPE: 7, CLAUSE: 4\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: less(T145, T147)\n\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(T164, T161), in(T164, T162)) | in(T1, tree(T161, T162, T163)), SCOPE: 1, CLAUSE: 2\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: in(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT2 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, T2) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(less(T164, T161), in(T164, T162)), SCOPE: 8, CLAUSE: 3 | ','(less(T164, T161), in(T164, T162)), SCOPE: 8, CLAUSE: 4 | ?_8 | in(T1, tree(T161, T162, T163)), SCOPE: 1, CLAUSE: 2\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(less(T164, T161), in(T164, T162)), SCOPE: 8, CLAUSE: 3\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(less(T164, T161), in(T164, T162)), SCOPE: 8, CLAUSE: 4 | ?_8 | in(T1, tree(T161, T162, T163)), SCOPE: 1, CLAUSE: 2\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: in(0, T162)\n\nT162 is ground\nT163 is ground\nT169 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(s(T169), T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(less(T164, T161), in(T164, T162)), SCOPE: 8, CLAUSE: 4\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ?_8 | in(T1, tree(T161, T162, T163)), SCOPE: 1, CLAUSE: 2\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(less(T184, T183), in(s(T184), T162))\n\nT162 is ground\nT163 is ground\nT183 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(s(T183), T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: in(T1, tree(T161, T162, T163)), SCOPE: 1, CLAUSE: 2\n\nT161 is ground\nT162 is ground\nT163 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T161, T162, T163)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(less(T199, T202), in(T202, T201))\n\nT199 is ground\nT200 is ground\nT201 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T199, T200, T201)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(less(T210, T213), in(T213, T212))\n\nT210 is ground\nT211 is ground\nT212 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T210, T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(less(T210, T213), in(T213, T212)), SCOPE: 9, CLAUSE: 3 | ','(less(T210, T213), in(T213, T212)), SCOPE: 9, CLAUSE: 4\n\nT210 is ground\nT211 is ground\nT212 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T210, T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(less(T210, T213), in(T213, T212)), SCOPE: 9, CLAUSE: 3\n\nT210 is ground\nT211 is ground\nT212 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T210, T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(less(T210, T213), in(T213, T212)), SCOPE: 9, CLAUSE: 4\n\nT210 is ground\nT211 is ground\nT212 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(T210, T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: in(s(T219), T212)\n\nT211 is ground\nT212 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(0, T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(less(T230, T232), in(s(T232), T212))\n\nT211 is ground\nT212 is ground\nT230 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(s(T230), T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: less(T230, T232)\n\nT211 is ground\nT212 is ground\nT230 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T1, tree(s(T230), T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: in(s(T235), T212)\n\nT211 is ground\nT212 is ground\nT230 is ground\nX9 is free\nX10 is free\nX11 is free\nin(T236, tree(s(T230), T211, T212)) !=? in(X9, tree(X9, X10, X11))", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="EVAL with clause\nin(X9, tree(X9, X10, X11)).\nand substitutionT1 -> T6,\nX9 -> T6,\nX10 -> T7,\nX11 -> T8,\nT2 -> tree(T6, T7, T8)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 89 [arrowhead = none ];
89 -> 3;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 90 [arrowhead = none ];
90 -> 4;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 91 [arrowhead = none ];
91 -> 5;

92 [label="EVAL with clause\nin(X182, tree(X183, X184, X185)) :- ','(less(X182, X183), in(X182, X184)).\nand substitutionT1 -> T164,\nX182 -> T164,\nX183 -> T161,\nX184 -> T162,\nX185 -> T163,\nT2 -> tree(T161, T162, T163),\nT160 -> T164", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 92 [arrowhead = none ];
92 -> 64;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 93 [arrowhead = none ];
93 -> 65;

94 [label="ONLY EVAL with clause\nin(X16, tree(X17, X18, X19)) :- ','(less(X16, X17), in(X16, X18)).\nand substitutionT1 -> T17,\nX16 -> T17,\nT6 -> T14,\nX17 -> T14,\nT7 -> T15,\nX18 -> T15,\nT8 -> T16,\nX19 -> T16,\nT13 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 94 [arrowhead = none ];
94 -> 6;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 95 [arrowhead = none ];
95 -> 7;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 96 [arrowhead = none ];
96 -> 8;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 97 [arrowhead = none ];
97 -> 9;

98 [label="EVAL with clause\nless(0, s(X24)).\nand substitutionT17 -> 0,\nX24 -> T22,\nT14 -> s(T22)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 98 [arrowhead = none ];
98 -> 10;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 100 [arrowhead = none ];
100 -> 38;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 101 [arrowhead = none ];
101 -> 39;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 102 [arrowhead = none ];
102 -> 12;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 103 [arrowhead = none ];
103 -> 13;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 104 [arrowhead = none ];
104 -> 14;

105 [label="EVAL with clause\nin(X43, tree(X43, X44, X45)).\nand substitutionX43 -> 0,\nX44 -> T35,\nX45 -> T36,\nT15 -> tree(0, T35, T36)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 105 [arrowhead = none ];
105 -> 15;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 106 [arrowhead = none ];
106 -> 16;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 107 [arrowhead = none ];
107 -> 18;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 108 [arrowhead = none ];
108 -> 19;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 109 [arrowhead = none ];
109 -> 17;

110 [label="EVAL with clause\nin(X62, tree(X63, X64, X65)) :- ','(less(X62, X63), in(X62, X64)).\nand substitutionX62 -> 0,\nX63 -> T49,\nX64 -> T50,\nX65 -> T51,\nT15 -> tree(T49, T50, T51)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 111 [arrowhead = none ];
111 -> 21;

112 [label="EVAL with clause\nin(X91, tree(X92, X93, X94)) :- ','(less(X92, X91), in(X91, X94)).\nand substitutionX91 -> 0,\nX92 -> T71,\nX93 -> T72,\nX94 -> T73,\nT15 -> tree(T71, T72, T73)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 112 [arrowhead = none ];
112 -> 31;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 113 [arrowhead = none ];
113 -> 32;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 114 [arrowhead = none ];
114 -> 22;

115 [label="SPLIT 2\nnew knowledge:\nT49 is ground", fontsize=14, style = filled, fillcolor = violet];
20 -> 115 [arrowhead = none ];
115 -> 23;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 116 [arrowhead = none ];
116 -> 24;

117 [label="INSTANCE with matching:\nT15 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 117 [arrowhead = none , style=dashed];
117 -> 10[style=dashed];

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 118 [arrowhead = none ];
118 -> 25;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 119 [arrowhead = none ];
119 -> 26;

120 [label="EVAL with clause\nless(0, s(X74)).\nand substitutionX74 -> T60,\nT49 -> s(T60)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 120 [arrowhead = none ];
120 -> 27;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 121 [arrowhead = none ];
121 -> 28;

122 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 122 [arrowhead = none ];
122 -> 30;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 123 [arrowhead = none ];
123 -> 29;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 124 [arrowhead = none ];
124 -> 33;

125 [label="SPLIT 2\nnew knowledge:\nT71 is ground", fontsize=14, style = filled, fillcolor = violet];
31 -> 125 [arrowhead = none ];
125 -> 34;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 126 [arrowhead = none ];
126 -> 35;

127 [label="INSTANCE with matching:\nT15 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 127 [arrowhead = none , style=dashed];
127 -> 10[style=dashed];

128 [label="BACKTRACK\nfor clause: less(0, s(X5))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 128 [arrowhead = none ];
128 -> 36;

129 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 129 [arrowhead = none ];
129 -> 37;

130 [label="EVAL with clause\nless(s(X116), s(X117)) :- less(X116, X117).\nand substitutionX116 -> T89,\nT17 -> s(T89),\nX117 -> T88,\nT14 -> s(T88),\nT87 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 131 [arrowhead = none ];
131 -> 41;

132 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 132 [arrowhead = none ];
132 -> 52;

133 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 133 [arrowhead = none ];
133 -> 42;

134 [label="SPLIT 2\nnew knowledge:\nT92 is ground\nT88 is ground\nreplacements:T89 -> T92", fontsize=14, style = filled, fillcolor = violet];
40 -> 134 [arrowhead = none ];
134 -> 43;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 135 [arrowhead = none ];
135 -> 44;

136 [label="INSTANCE with matching:\nT1 -> s(T92)\nT2 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 136 [arrowhead = none , style=dashed];
136 -> 1[style=dashed];

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 137 [arrowhead = none ];
137 -> 45;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 138 [arrowhead = none ];
138 -> 46;

139 [label="EVAL with clause\nless(0, s(X126)).\nand substitutionT89 -> 0,\nX126 -> T99,\nT88 -> s(T99)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 139 [arrowhead = none ];
139 -> 47;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 140 [arrowhead = none ];
140 -> 48;

141 [label="EVAL with clause\nless(s(X131), s(X132)) :- less(X131, X132).\nand substitutionX131 -> T106,\nT89 -> s(T106),\nX132 -> T105,\nT88 -> s(T105),\nT104 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 141 [arrowhead = none ];
141 -> 50;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 142 [arrowhead = none ];
142 -> 51;

143 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 143 [arrowhead = none ];
143 -> 49;

144 [label="INSTANCE with matching:\nT89 -> T106\nT88 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 144 [arrowhead = none , style=dashed];
144 -> 42[style=dashed];

145 [label="ONLY EVAL with clause\nin(X151, tree(X152, X153, X154)) :- ','(less(X152, X151), in(X151, X154)).\nand substitutionT1 -> T130,\nX151 -> T130,\nT14 -> T127,\nX152 -> T127,\nT15 -> T128,\nX153 -> T128,\nT16 -> T129,\nX154 -> T129,\nT126 -> T130", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 145 [arrowhead = none ];
145 -> 53;

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 146 [arrowhead = none ];
146 -> 54;

147 [label="SPLIT 2\nnew knowledge:\nT127 is ground\nreplacements:T130 -> T133", fontsize=14, style = filled, fillcolor = violet];
53 -> 147 [arrowhead = none ];
147 -> 55;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 148 [arrowhead = none ];
148 -> 56;

149 [label="INSTANCE with matching:\nT1 -> T133\nT2 -> T129", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 149 [arrowhead = none , style=dashed];
149 -> 1[style=dashed];

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 150 [arrowhead = none ];
150 -> 57;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 151 [arrowhead = none ];
151 -> 58;

152 [label="EVAL with clause\nless(0, s(X163)).\nand substitutionT127 -> 0,\nX163 -> T140,\nT130 -> s(T140)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 152 [arrowhead = none ];
152 -> 59;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 153 [arrowhead = none ];
153 -> 60;

154 [label="EVAL with clause\nless(s(X168), s(X169)) :- less(X168, X169).\nand substitutionX168 -> T145,\nT127 -> s(T145),\nX169 -> T147,\nT130 -> s(T147),\nT146 -> T147", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 154 [arrowhead = none ];
154 -> 62;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 155 [arrowhead = none ];
155 -> 63;

156 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 156 [arrowhead = none ];
156 -> 61;

157 [label="INSTANCE with matching:\nT127 -> T145\nT130 -> T147", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 157 [arrowhead = none , style=dashed];
157 -> 54[style=dashed];

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 158 [arrowhead = none ];
158 -> 66;

159 [label="EVAL with clause\nin(X229, tree(X230, X231, X232)) :- ','(less(X230, X229), in(X229, X232)).\nand substitutionT1 -> T213,\nX229 -> T213,\nX230 -> T210,\nX231 -> T211,\nX232 -> T212,\nT2 -> tree(T210, T211, T212),\nT209 -> T213", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 159 [arrowhead = none ];
159 -> 77;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 160 [arrowhead = none ];
160 -> 78;

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 161 [arrowhead = none ];
161 -> 67;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 162 [arrowhead = none ];
162 -> 68;

163 [label="EVAL with clause\nless(0, s(X190)).\nand substitutionT164 -> 0,\nX190 -> T169,\nT161 -> s(T169)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 163 [arrowhead = none ];
163 -> 69;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 164 [arrowhead = none ];
164 -> 70;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 165 [arrowhead = none ];
165 -> 71;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 166 [arrowhead = none ];
166 -> 72;

167 [label="INSTANCE with matching:\nT15 -> T162", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 167 [arrowhead = none , style=dashed];
167 -> 10[style=dashed];

168 [label="EVAL with clause\nless(s(X205), s(X206)) :- less(X205, X206).\nand substitutionX205 -> T184,\nT164 -> s(T184),\nX206 -> T183,\nT161 -> s(T183),\nT182 -> T184", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 168 [arrowhead = none ];
168 -> 73;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 169 [arrowhead = none ];
169 -> 74;

170 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 170 [arrowhead = none ];
170 -> 75;

171 [label="INSTANCE with matching:\nT89 -> T184\nT88 -> T183\nT15 -> T162", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 171 [arrowhead = none , style=dashed];
171 -> 40[style=dashed];

172 [label="ONLY EVAL with clause\nin(X219, tree(X220, X221, X222)) :- ','(less(X220, X219), in(X219, X222)).\nand substitutionT1 -> T202,\nX219 -> T202,\nT161 -> T199,\nX220 -> T199,\nT162 -> T200,\nX221 -> T200,\nT163 -> T201,\nX222 -> T201,\nT198 -> T202", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 172 [arrowhead = none ];
172 -> 76;

173 [label="INSTANCE with matching:\nT127 -> T199\nT130 -> T202\nT129 -> T201", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 173 [arrowhead = none , style=dashed];
173 -> 53[style=dashed];

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 174 [arrowhead = none ];
174 -> 79;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 175 [arrowhead = none ];
175 -> 80;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 176 [arrowhead = none ];
176 -> 81;

177 [label="EVAL with clause\nless(0, s(X237)).\nand substitutionT210 -> 0,\nX237 -> T219,\nT213 -> s(T219),\nT218 -> T219", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 177 [arrowhead = none ];
177 -> 82;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 178 [arrowhead = none ];
178 -> 83;

179 [label="EVAL with clause\nless(s(X248), s(X249)) :- less(X248, X249).\nand substitutionX248 -> T230,\nT210 -> s(T230),\nX249 -> T232,\nT213 -> s(T232),\nT231 -> T232", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 179 [arrowhead = none ];
179 -> 84;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 180 [arrowhead = none ];
180 -> 85;

181 [label="INSTANCE with matching:\nT1 -> s(T219)\nT2 -> T212", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 181 [arrowhead = none , style=dashed];
181 -> 1[style=dashed];

182 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
84 -> 182 [arrowhead = none ];
182 -> 86;

183 [label="SPLIT 2\nnew knowledge:\nT230 is ground\nreplacements:T232 -> T235,\nT1 -> T236", fontsize=14, style = filled, fillcolor = violet];
84 -> 183 [arrowhead = none ];
183 -> 87;

184 [label="INSTANCE with matching:\nT127 -> T230\nT130 -> T232", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 184 [arrowhead = none , style=dashed];
184 -> 54[style=dashed];

185 [label="INSTANCE with matching:\nT1 -> s(T235)\nT2 -> T212", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 185 [arrowhead = none , style=dashed];
185 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

inA(tree(0, T35, T36)).
inA(tree(T49, T50, T51)) :- lessB(T49).
inA(tree(T49, T50, T51)) :- ','(lessB(T49), inA(T50)).
lessD(0, s(T99)).
lessD(s(T106), s(T105)) :- lessD(T106, T105).
lessE(0, s(T140)).
lessE(s(T145), s(T147)) :- lessE(T145, T147).
pF(T89, T88, T15) :- lessD(T89, T88).
pF(T92, T88, T15) :- ','(lessD(T92, T88), inG(s(T92), T15)).
pH(T127, T130, T129) :- lessE(T127, T130).
pH(T127, T133, T129) :- ','(lessE(T127, T133), inG(T133, T129)).
lessB(s(T60)).
inG(T6, tree(T6, T7, T8)).
inG(0, tree(s(T22), T15, T16)) :- inA(T15).
inG(s(T89), tree(s(T88), T15, T16)) :- pF(T89, T88, T15).
inG(T130, tree(T127, T128, T129)) :- pH(T127, T130, T129).
inG(0, tree(s(T169), T162, T163)) :- inA(T162).
inG(s(T184), tree(s(T183), T162, T163)) :- pF(T184, T183, T162).
inG(T202, tree(T199, T200, T201)) :- pH(T199, T202, T201).
inG(s(T219), tree(0, T211, T212)) :- inG(s(T219), T212).
inG(s(T232), tree(s(T230), T211, T212)) :- lessE(T230, T232).
inG(s(T235), tree(s(T230), T211, T212)) :- ','(lessE(T230, T235), inG(s(T235), T212)).

Query: inG(a,g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
inG_in: (f,b) (b,b)
inA_in: (b)
pF_in: (f,b,b) (b,b,b)
lessD_in: (f,b) (b,b)
pH_in: (b,b,b) (b,f,b)
lessE_in: (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(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:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(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:

ING_IN_AG(0, tree(s(T22), T15, T16)) → U11_AG(T22, T15, T16, inA_in_g(T15))
ING_IN_AG(0, tree(s(T22), T15, T16)) → INA_IN_G(T15)
INA_IN_G(tree(T49, T50, T51)) → U1_G(T49, T50, T51, lessB_in_g(T49))
INA_IN_G(tree(T49, T50, T51)) → LESSB_IN_G(T49)
U1_G(T49, T50, T51, lessB_out_g(T49)) → U2_G(T49, T50, T51, inA_in_g(T50))
U1_G(T49, T50, T51, lessB_out_g(T49)) → INA_IN_G(T50)
ING_IN_AG(s(T89), tree(s(T88), T15, T16)) → U12_AG(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
ING_IN_AG(s(T89), tree(s(T88), T15, T16)) → PF_IN_AGG(T89, T88, T15)
PF_IN_AGG(T89, T88, T15) → U5_AGG(T89, T88, T15, lessD_in_ag(T89, T88))
PF_IN_AGG(T89, T88, T15) → LESSD_IN_AG(T89, T88)
LESSD_IN_AG(s(T106), s(T105)) → U3_AG(T106, T105, lessD_in_ag(T106, T105))
LESSD_IN_AG(s(T106), s(T105)) → LESSD_IN_AG(T106, T105)
PF_IN_AGG(T92, T88, T15) → U6_AGG(T92, T88, T15, lessD_in_ag(T92, T88))
U6_AGG(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_AGG(T92, T88, T15, inG_in_gg(s(T92), T15))
U6_AGG(T92, T88, T15, lessD_out_ag(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(0, tree(s(T22), T15, T16)) → U11_GG(T22, T15, T16, inA_in_g(T15))
ING_IN_GG(0, tree(s(T22), T15, T16)) → INA_IN_G(T15)
ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → U12_GG(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
PF_IN_GGG(T89, T88, T15) → U5_GGG(T89, T88, T15, lessD_in_gg(T89, T88))
PF_IN_GGG(T89, T88, T15) → LESSD_IN_GG(T89, T88)
LESSD_IN_GG(s(T106), s(T105)) → U3_GG(T106, T105, lessD_in_gg(T106, T105))
LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)
PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T88, T15, lessD_in_gg(T92, T88))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_GGG(T92, T88, T15, inG_in_gg(s(T92), T15))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(T130, tree(T127, T128, T129)) → U13_GG(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
PH_IN_GGG(T127, T130, T129) → U8_GGG(T127, T130, T129, lessE_in_gg(T127, T130))
PH_IN_GGG(T127, T130, T129) → LESSE_IN_GG(T127, T130)
LESSE_IN_GG(s(T145), s(T147)) → U4_GG(T145, T147, lessE_in_gg(T145, T147))
LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)
PH_IN_GGG(T127, T133, T129) → U9_GGG(T127, T133, T129, lessE_in_gg(T127, T133))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_GGG(T127, T133, T129, inG_in_gg(T133, T129))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → ING_IN_GG(T133, T129)
ING_IN_GG(0, tree(s(T169), T162, T163)) → U14_GG(T169, T162, T163, inA_in_g(T162))
ING_IN_GG(s(T184), tree(s(T183), T162, T163)) → U15_GG(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
ING_IN_GG(T202, tree(T199, T200, T201)) → U16_GG(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
ING_IN_GG(s(T219), tree(0, T211, T212)) → U17_GG(T219, T211, T212, inG_in_gg(s(T219), T212))
ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
ING_IN_GG(s(T232), tree(s(T230), T211, T212)) → U18_GG(T232, T230, T211, T212, lessE_in_gg(T230, T232))
ING_IN_GG(s(T232), tree(s(T230), T211, T212)) → LESSE_IN_GG(T230, T232)
ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_GG(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → ING_IN_GG(s(T235), T212)
ING_IN_AG(T130, tree(T127, T128, T129)) → U13_AG(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
ING_IN_AG(T130, tree(T127, T128, T129)) → PH_IN_GAG(T127, T130, T129)
PH_IN_GAG(T127, T130, T129) → U8_GAG(T127, T130, T129, lessE_in_ga(T127, T130))
PH_IN_GAG(T127, T130, T129) → LESSE_IN_GA(T127, T130)
LESSE_IN_GA(s(T145), s(T147)) → U4_GA(T145, T147, lessE_in_ga(T145, T147))
LESSE_IN_GA(s(T145), s(T147)) → LESSE_IN_GA(T145, T147)
PH_IN_GAG(T127, T133, T129) → U9_GAG(T127, T133, T129, lessE_in_ga(T127, T133))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_GAG(T127, T133, T129, inG_in_ag(T133, T129))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → ING_IN_AG(T133, T129)
ING_IN_AG(0, tree(s(T169), T162, T163)) → U14_AG(T169, T162, T163, inA_in_g(T162))
ING_IN_AG(s(T184), tree(s(T183), T162, T163)) → U15_AG(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
ING_IN_AG(T202, tree(T199, T200, T201)) → U16_AG(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
ING_IN_AG(s(T219), tree(0, T211, T212)) → U17_AG(T219, T211, T212, inG_in_ag(s(T219), T212))
ING_IN_AG(s(T219), tree(0, T211, T212)) → ING_IN_AG(s(T219), T212)
ING_IN_AG(s(T232), tree(s(T230), T211, T212)) → U18_AG(T232, T230, T211, T212, lessE_in_ga(T230, T232))
ING_IN_AG(s(T232), tree(s(T230), T211, T212)) → LESSE_IN_GA(T230, T232)
ING_IN_AG(s(T235), tree(s(T230), T211, T212)) → U19_AG(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_AG(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → ING_IN_AG(s(T235), T212)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
ING_IN_AG(x1, x2)  =  ING_IN_AG(x2)
U11_AG(x1, x2, x3, x4)  =  U11_AG(x4)
INA_IN_G(x1)  =  INA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x4)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)
U12_AG(x1, x2, x3, x4, x5)  =  U12_AG(x5)
PF_IN_AGG(x1, x2, x3)  =  PF_IN_AGG(x2, x3)
U5_AGG(x1, x2, x3, x4)  =  U5_AGG(x4)
LESSD_IN_AG(x1, x2)  =  LESSD_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x3)
U6_AGG(x1, x2, x3, x4)  =  U6_AGG(x3, x4)
U7_AGG(x1, x2, x3, x4)  =  U7_AGG(x1, x4)
ING_IN_GG(x1, x2)  =  ING_IN_GG(x1, x2)
U11_GG(x1, x2, x3, x4)  =  U11_GG(x4)
U12_GG(x1, x2, x3, x4, x5)  =  U12_GG(x5)
PF_IN_GGG(x1, x2, x3)  =  PF_IN_GGG(x1, x2, x3)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x4)
LESSD_IN_GG(x1, x2)  =  LESSD_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x3, x4)
U7_GGG(x1, x2, x3, x4)  =  U7_GGG(x4)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x5)
PH_IN_GGG(x1, x2, x3)  =  PH_IN_GGG(x1, x2, x3)
U8_GGG(x1, x2, x3, x4)  =  U8_GGG(x4)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U9_GGG(x1, x2, x3, x4)  =  U9_GGG(x2, x3, x4)
U10_GGG(x1, x2, x3, x4)  =  U10_GGG(x4)
U14_GG(x1, x2, x3, x4)  =  U14_GG(x4)
U15_GG(x1, x2, x3, x4, x5)  =  U15_GG(x5)
U16_GG(x1, x2, x3, x4, x5)  =  U16_GG(x5)
U17_GG(x1, x2, x3, x4)  =  U17_GG(x4)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x4, x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x5)
U13_AG(x1, x2, x3, x4, x5)  =  U13_AG(x5)
PH_IN_GAG(x1, x2, x3)  =  PH_IN_GAG(x1, x3)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x4)
LESSE_IN_GA(x1, x2)  =  LESSE_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x3, x4)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x4)
U14_AG(x1, x2, x3, x4)  =  U14_AG(x4)
U15_AG(x1, x2, x3, x4, x5)  =  U15_AG(x5)
U16_AG(x1, x2, x3, x4, x5)  =  U16_AG(x5)
U17_AG(x1, x2, x3, x4)  =  U17_AG(x4)
U18_AG(x1, x2, x3, x4, x5)  =  U18_AG(x5)
U19_AG(x1, x2, x3, x4, x5)  =  U19_AG(x4, x5)
U20_AG(x1, x2, x3, x4, x5)  =  U20_AG(x5)

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

(6) Obligation:

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

ING_IN_AG(0, tree(s(T22), T15, T16)) → U11_AG(T22, T15, T16, inA_in_g(T15))
ING_IN_AG(0, tree(s(T22), T15, T16)) → INA_IN_G(T15)
INA_IN_G(tree(T49, T50, T51)) → U1_G(T49, T50, T51, lessB_in_g(T49))
INA_IN_G(tree(T49, T50, T51)) → LESSB_IN_G(T49)
U1_G(T49, T50, T51, lessB_out_g(T49)) → U2_G(T49, T50, T51, inA_in_g(T50))
U1_G(T49, T50, T51, lessB_out_g(T49)) → INA_IN_G(T50)
ING_IN_AG(s(T89), tree(s(T88), T15, T16)) → U12_AG(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
ING_IN_AG(s(T89), tree(s(T88), T15, T16)) → PF_IN_AGG(T89, T88, T15)
PF_IN_AGG(T89, T88, T15) → U5_AGG(T89, T88, T15, lessD_in_ag(T89, T88))
PF_IN_AGG(T89, T88, T15) → LESSD_IN_AG(T89, T88)
LESSD_IN_AG(s(T106), s(T105)) → U3_AG(T106, T105, lessD_in_ag(T106, T105))
LESSD_IN_AG(s(T106), s(T105)) → LESSD_IN_AG(T106, T105)
PF_IN_AGG(T92, T88, T15) → U6_AGG(T92, T88, T15, lessD_in_ag(T92, T88))
U6_AGG(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_AGG(T92, T88, T15, inG_in_gg(s(T92), T15))
U6_AGG(T92, T88, T15, lessD_out_ag(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(0, tree(s(T22), T15, T16)) → U11_GG(T22, T15, T16, inA_in_g(T15))
ING_IN_GG(0, tree(s(T22), T15, T16)) → INA_IN_G(T15)
ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → U12_GG(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
PF_IN_GGG(T89, T88, T15) → U5_GGG(T89, T88, T15, lessD_in_gg(T89, T88))
PF_IN_GGG(T89, T88, T15) → LESSD_IN_GG(T89, T88)
LESSD_IN_GG(s(T106), s(T105)) → U3_GG(T106, T105, lessD_in_gg(T106, T105))
LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)
PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T88, T15, lessD_in_gg(T92, T88))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_GGG(T92, T88, T15, inG_in_gg(s(T92), T15))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(T130, tree(T127, T128, T129)) → U13_GG(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
PH_IN_GGG(T127, T130, T129) → U8_GGG(T127, T130, T129, lessE_in_gg(T127, T130))
PH_IN_GGG(T127, T130, T129) → LESSE_IN_GG(T127, T130)
LESSE_IN_GG(s(T145), s(T147)) → U4_GG(T145, T147, lessE_in_gg(T145, T147))
LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)
PH_IN_GGG(T127, T133, T129) → U9_GGG(T127, T133, T129, lessE_in_gg(T127, T133))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_GGG(T127, T133, T129, inG_in_gg(T133, T129))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → ING_IN_GG(T133, T129)
ING_IN_GG(0, tree(s(T169), T162, T163)) → U14_GG(T169, T162, T163, inA_in_g(T162))
ING_IN_GG(s(T184), tree(s(T183), T162, T163)) → U15_GG(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
ING_IN_GG(T202, tree(T199, T200, T201)) → U16_GG(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
ING_IN_GG(s(T219), tree(0, T211, T212)) → U17_GG(T219, T211, T212, inG_in_gg(s(T219), T212))
ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
ING_IN_GG(s(T232), tree(s(T230), T211, T212)) → U18_GG(T232, T230, T211, T212, lessE_in_gg(T230, T232))
ING_IN_GG(s(T232), tree(s(T230), T211, T212)) → LESSE_IN_GG(T230, T232)
ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_GG(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → ING_IN_GG(s(T235), T212)
ING_IN_AG(T130, tree(T127, T128, T129)) → U13_AG(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
ING_IN_AG(T130, tree(T127, T128, T129)) → PH_IN_GAG(T127, T130, T129)
PH_IN_GAG(T127, T130, T129) → U8_GAG(T127, T130, T129, lessE_in_ga(T127, T130))
PH_IN_GAG(T127, T130, T129) → LESSE_IN_GA(T127, T130)
LESSE_IN_GA(s(T145), s(T147)) → U4_GA(T145, T147, lessE_in_ga(T145, T147))
LESSE_IN_GA(s(T145), s(T147)) → LESSE_IN_GA(T145, T147)
PH_IN_GAG(T127, T133, T129) → U9_GAG(T127, T133, T129, lessE_in_ga(T127, T133))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_GAG(T127, T133, T129, inG_in_ag(T133, T129))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → ING_IN_AG(T133, T129)
ING_IN_AG(0, tree(s(T169), T162, T163)) → U14_AG(T169, T162, T163, inA_in_g(T162))
ING_IN_AG(s(T184), tree(s(T183), T162, T163)) → U15_AG(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
ING_IN_AG(T202, tree(T199, T200, T201)) → U16_AG(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
ING_IN_AG(s(T219), tree(0, T211, T212)) → U17_AG(T219, T211, T212, inG_in_ag(s(T219), T212))
ING_IN_AG(s(T219), tree(0, T211, T212)) → ING_IN_AG(s(T219), T212)
ING_IN_AG(s(T232), tree(s(T230), T211, T212)) → U18_AG(T232, T230, T211, T212, lessE_in_ga(T230, T232))
ING_IN_AG(s(T232), tree(s(T230), T211, T212)) → LESSE_IN_GA(T230, T232)
ING_IN_AG(s(T235), tree(s(T230), T211, T212)) → U19_AG(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_AG(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → ING_IN_AG(s(T235), T212)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
ING_IN_AG(x1, x2)  =  ING_IN_AG(x2)
U11_AG(x1, x2, x3, x4)  =  U11_AG(x4)
INA_IN_G(x1)  =  INA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x4)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)
U12_AG(x1, x2, x3, x4, x5)  =  U12_AG(x5)
PF_IN_AGG(x1, x2, x3)  =  PF_IN_AGG(x2, x3)
U5_AGG(x1, x2, x3, x4)  =  U5_AGG(x4)
LESSD_IN_AG(x1, x2)  =  LESSD_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x3)
U6_AGG(x1, x2, x3, x4)  =  U6_AGG(x3, x4)
U7_AGG(x1, x2, x3, x4)  =  U7_AGG(x1, x4)
ING_IN_GG(x1, x2)  =  ING_IN_GG(x1, x2)
U11_GG(x1, x2, x3, x4)  =  U11_GG(x4)
U12_GG(x1, x2, x3, x4, x5)  =  U12_GG(x5)
PF_IN_GGG(x1, x2, x3)  =  PF_IN_GGG(x1, x2, x3)
U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x4)
LESSD_IN_GG(x1, x2)  =  LESSD_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x3, x4)
U7_GGG(x1, x2, x3, x4)  =  U7_GGG(x4)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x5)
PH_IN_GGG(x1, x2, x3)  =  PH_IN_GGG(x1, x2, x3)
U8_GGG(x1, x2, x3, x4)  =  U8_GGG(x4)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U9_GGG(x1, x2, x3, x4)  =  U9_GGG(x2, x3, x4)
U10_GGG(x1, x2, x3, x4)  =  U10_GGG(x4)
U14_GG(x1, x2, x3, x4)  =  U14_GG(x4)
U15_GG(x1, x2, x3, x4, x5)  =  U15_GG(x5)
U16_GG(x1, x2, x3, x4, x5)  =  U16_GG(x5)
U17_GG(x1, x2, x3, x4)  =  U17_GG(x4)
U18_GG(x1, x2, x3, x4, x5)  =  U18_GG(x5)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x4, x5)
U20_GG(x1, x2, x3, x4, x5)  =  U20_GG(x5)
U13_AG(x1, x2, x3, x4, x5)  =  U13_AG(x5)
PH_IN_GAG(x1, x2, x3)  =  PH_IN_GAG(x1, x3)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x4)
LESSE_IN_GA(x1, x2)  =  LESSE_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x3, x4)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x4)
U14_AG(x1, x2, x3, x4)  =  U14_AG(x4)
U15_AG(x1, x2, x3, x4, x5)  =  U15_AG(x5)
U16_AG(x1, x2, x3, x4, x5)  =  U16_AG(x5)
U17_AG(x1, x2, x3, x4)  =  U17_AG(x4)
U18_AG(x1, x2, x3, x4, x5)  =  U18_AG(x5)
U19_AG(x1, x2, x3, x4, x5)  =  U19_AG(x4, x5)
U20_AG(x1, x2, x3, x4, x5)  =  U20_AG(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 43 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSE_IN_GA(s(T145), s(T147)) → LESSE_IN_GA(T145, T147)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
LESSE_IN_GA(x1, x2)  =  LESSE_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESSE_IN_GA(s(T145), s(T147)) → LESSE_IN_GA(T145, T147)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LESSE_IN_GA(s(T145)) → LESSE_IN_GA(T145)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSE_IN_GA(s(T145)) → LESSE_IN_GA(T145)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSE_IN_GG(s(T145), s(T147)) → LESSE_IN_GG(T145, T147)
  The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
LESSD_IN_GG(x1, x2)  =  LESSD_IN_GG(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

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

LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSD_IN_GG(s(T106), s(T105)) → LESSD_IN_GG(T106, T105)
  The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

LESSD_IN_AG(s(T106), s(T105)) → LESSD_IN_AG(T106, T105)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
LESSD_IN_AG(x1, x2)  =  LESSD_IN_AG(x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

LESSD_IN_AG(s(T106), s(T105)) → LESSD_IN_AG(T106, T105)

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

LESSD_IN_AG(s(T105)) → LESSD_IN_AG(T105)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESSD_IN_AG(s(T105)) → LESSD_IN_AG(T105)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

U1_G(T49, T50, T51, lessB_out_g(T49)) → INA_IN_G(T50)
INA_IN_G(tree(T49, T50, T51)) → U1_G(T49, T50, T51, lessB_in_g(T49))

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
INA_IN_G(x1)  =  INA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x4)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

U1_G(T49, T50, T51, lessB_out_g(T49)) → INA_IN_G(T50)
INA_IN_G(tree(T49, T50, T51)) → U1_G(T49, T50, T51, lessB_in_g(T49))

The TRS R consists of the following rules:

lessB_in_g(s(T60)) → lessB_out_g(s(T60))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
INA_IN_G(x1)  =  INA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x4)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

U1_G(T50, lessB_out_g) → INA_IN_G(T50)
INA_IN_G(tree(T49, T50, T51)) → U1_G(T50, lessB_in_g(T49))

The TRS R consists of the following rules:

lessB_in_g(s(T60)) → lessB_out_g

The set Q consists of the following terms:

lessB_in_g(x0)

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

(42) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• INA_IN_G(tree(T49, T50, T51)) → U1_G(T50, lessB_in_g(T49))
  The graph contains the following edges 1 > 1

• U1_G(T50, lessB_out_g) → INA_IN_G(T50)
  The graph contains the following edges 1 >= 1

(43) YES

(44) Obligation:

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

ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T88, T15, lessD_in_gg(T92, T88))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
PH_IN_GGG(T127, T133, T129) → U9_GGG(T127, T133, T129, lessE_in_gg(T127, T133))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → ING_IN_GG(T133, T129)
ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → ING_IN_GG(s(T235), T212)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
ING_IN_GG(x1, x2)  =  ING_IN_GG(x1, x2)
PF_IN_GGG(x1, x2, x3)  =  PF_IN_GGG(x1, x2, x3)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x3, x4)
PH_IN_GGG(x1, x2, x3)  =  PH_IN_GGG(x1, x2, x3)
U9_GGG(x1, x2, x3, x4)  =  U9_GGG(x2, x3, x4)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x4, x5)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T88, T15, lessD_in_gg(T92, T88))
U6_GGG(T92, T88, T15, lessD_out_gg(T92, T88)) → ING_IN_GG(s(T92), T15)
ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
PH_IN_GGG(T127, T133, T129) → U9_GGG(T127, T133, T129, lessE_in_gg(T127, T133))
U9_GGG(T127, T133, T129, lessE_out_gg(T127, T133)) → ING_IN_GG(T133, T129)
ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_GG(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → ING_IN_GG(s(T235), T212)

The TRS R consists of the following rules:

lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
ING_IN_GG(x1, x2)  =  ING_IN_GG(x1, x2)
PF_IN_GGG(x1, x2, x3)  =  PF_IN_GGG(x1, x2, x3)
U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x3, x4)
PH_IN_GGG(x1, x2, x3)  =  PH_IN_GGG(x1, x2, x3)
U9_GGG(x1, x2, x3, x4)  =  U9_GGG(x2, x3, x4)
U19_GG(x1, x2, x3, x4, x5)  =  U19_GG(x1, x4, x5)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T15, lessD_in_gg(T92, T88))
U6_GGG(T92, T15, lessD_out_gg) → ING_IN_GG(s(T92), T15)
ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
PH_IN_GGG(T127, T133, T129) → U9_GGG(T133, T129, lessE_in_gg(T127, T133))
U9_GGG(T133, T129, lessE_out_gg) → ING_IN_GG(T133, T129)
ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T212, lessE_in_gg(T230, T235))
U19_GG(T235, T212, lessE_out_gg) → ING_IN_GG(s(T235), T212)

The TRS R consists of the following rules:

lessD_in_gg(0, s(T99)) → lessD_out_gg
lessD_in_gg(s(T106), s(T105)) → U3_gg(lessD_in_gg(T106, T105))
lessE_in_gg(0, s(T140)) → lessE_out_gg
lessE_in_gg(s(T145), s(T147)) → U4_gg(lessE_in_gg(T145, T147))
U3_gg(lessD_out_gg) → lessD_out_gg
U4_gg(lessE_out_gg) → lessE_out_gg

The set Q consists of the following terms:

lessD_in_gg(x0, x1)
lessE_in_gg(x0, x1)
U3_gg(x0)
U4_gg(x0)

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

(49) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• PF_IN_GGG(T92, T88, T15) → U6_GGG(T92, T15, lessD_in_gg(T92, T88))
  The graph contains the following edges 1 >= 1, 3 >= 2

• U6_GGG(T92, T15, lessD_out_gg) → ING_IN_GG(s(T92), T15)
  The graph contains the following edges 2 >= 2

• ING_IN_GG(s(T89), tree(s(T88), T15, T16)) → PF_IN_GGG(T89, T88, T15)
  The graph contains the following edges 1 > 1, 2 > 2, 2 > 3

• ING_IN_GG(s(T219), tree(0, T211, T212)) → ING_IN_GG(s(T219), T212)
  The graph contains the following edges 1 >= 1, 2 > 2

• PH_IN_GGG(T127, T133, T129) → U9_GGG(T133, T129, lessE_in_gg(T127, T133))
  The graph contains the following edges 2 >= 1, 3 >= 2

• U9_GGG(T133, T129, lessE_out_gg) → ING_IN_GG(T133, T129)
  The graph contains the following edges 1 >= 1, 2 >= 2

• U19_GG(T235, T212, lessE_out_gg) → ING_IN_GG(s(T235), T212)
  The graph contains the following edges 2 >= 2

• ING_IN_GG(T130, tree(T127, T128, T129)) → PH_IN_GGG(T127, T130, T129)
  The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3

• ING_IN_GG(s(T235), tree(s(T230), T211, T212)) → U19_GG(T235, T212, lessE_in_gg(T230, T235))
  The graph contains the following edges 1 > 1, 2 > 2

(50) YES

(51) Obligation:

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

ING_IN_AG(T130, tree(T127, T128, T129)) → PH_IN_GAG(T127, T130, T129)
PH_IN_GAG(T127, T133, T129) → U9_GAG(T127, T133, T129, lessE_in_ga(T127, T133))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → ING_IN_AG(T133, T129)
ING_IN_AG(s(T219), tree(0, T211, T212)) → ING_IN_AG(s(T219), T212)
ING_IN_AG(s(T235), tree(s(T230), T211, T212)) → U19_AG(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → ING_IN_AG(s(T235), T212)

The TRS R consists of the following rules:

inG_in_ag(T6, tree(T6, T7, T8)) → inG_out_ag(T6, tree(T6, T7, T8))
inG_in_ag(0, tree(s(T22), T15, T16)) → U11_ag(T22, T15, T16, inA_in_g(T15))
inA_in_g(tree(0, T35, T36)) → inA_out_g(tree(0, T35, T36))
inA_in_g(tree(T49, T50, T51)) → U1_g(T49, T50, T51, lessB_in_g(T49))
lessB_in_g(s(T60)) → lessB_out_g(s(T60))
U1_g(T49, T50, T51, lessB_out_g(T49)) → inA_out_g(tree(T49, T50, T51))
U1_g(T49, T50, T51, lessB_out_g(T49)) → U2_g(T49, T50, T51, inA_in_g(T50))
U2_g(T49, T50, T51, inA_out_g(T50)) → inA_out_g(tree(T49, T50, T51))
U11_ag(T22, T15, T16, inA_out_g(T15)) → inG_out_ag(0, tree(s(T22), T15, T16))
inG_in_ag(s(T89), tree(s(T88), T15, T16)) → U12_ag(T89, T88, T15, T16, pF_in_agg(T89, T88, T15))
pF_in_agg(T89, T88, T15) → U5_agg(T89, T88, T15, lessD_in_ag(T89, T88))
lessD_in_ag(0, s(T99)) → lessD_out_ag(0, s(T99))
lessD_in_ag(s(T106), s(T105)) → U3_ag(T106, T105, lessD_in_ag(T106, T105))
U3_ag(T106, T105, lessD_out_ag(T106, T105)) → lessD_out_ag(s(T106), s(T105))
U5_agg(T89, T88, T15, lessD_out_ag(T89, T88)) → pF_out_agg(T89, T88, T15)
pF_in_agg(T92, T88, T15) → U6_agg(T92, T88, T15, lessD_in_ag(T92, T88))
U6_agg(T92, T88, T15, lessD_out_ag(T92, T88)) → U7_agg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T6, tree(T6, T7, T8)) → inG_out_gg(T6, tree(T6, T7, T8))
inG_in_gg(0, tree(s(T22), T15, T16)) → U11_gg(T22, T15, T16, inA_in_g(T15))
U11_gg(T22, T15, T16, inA_out_g(T15)) → inG_out_gg(0, tree(s(T22), T15, T16))
inG_in_gg(s(T89), tree(s(T88), T15, T16)) → U12_gg(T89, T88, T15, T16, pF_in_ggg(T89, T88, T15))
pF_in_ggg(T89, T88, T15) → U5_ggg(T89, T88, T15, lessD_in_gg(T89, T88))
lessD_in_gg(0, s(T99)) → lessD_out_gg(0, s(T99))
lessD_in_gg(s(T106), s(T105)) → U3_gg(T106, T105, lessD_in_gg(T106, T105))
U3_gg(T106, T105, lessD_out_gg(T106, T105)) → lessD_out_gg(s(T106), s(T105))
U5_ggg(T89, T88, T15, lessD_out_gg(T89, T88)) → pF_out_ggg(T89, T88, T15)
pF_in_ggg(T92, T88, T15) → U6_ggg(T92, T88, T15, lessD_in_gg(T92, T88))
U6_ggg(T92, T88, T15, lessD_out_gg(T92, T88)) → U7_ggg(T92, T88, T15, inG_in_gg(s(T92), T15))
inG_in_gg(T130, tree(T127, T128, T129)) → U13_gg(T130, T127, T128, T129, pH_in_ggg(T127, T130, T129))
pH_in_ggg(T127, T130, T129) → U8_ggg(T127, T130, T129, lessE_in_gg(T127, T130))
lessE_in_gg(0, s(T140)) → lessE_out_gg(0, s(T140))
lessE_in_gg(s(T145), s(T147)) → U4_gg(T145, T147, lessE_in_gg(T145, T147))
U4_gg(T145, T147, lessE_out_gg(T145, T147)) → lessE_out_gg(s(T145), s(T147))
U8_ggg(T127, T130, T129, lessE_out_gg(T127, T130)) → pH_out_ggg(T127, T130, T129)
pH_in_ggg(T127, T133, T129) → U9_ggg(T127, T133, T129, lessE_in_gg(T127, T133))
U9_ggg(T127, T133, T129, lessE_out_gg(T127, T133)) → U10_ggg(T127, T133, T129, inG_in_gg(T133, T129))
inG_in_gg(0, tree(s(T169), T162, T163)) → U14_gg(T169, T162, T163, inA_in_g(T162))
U14_gg(T169, T162, T163, inA_out_g(T162)) → inG_out_gg(0, tree(s(T169), T162, T163))
inG_in_gg(s(T184), tree(s(T183), T162, T163)) → U15_gg(T184, T183, T162, T163, pF_in_ggg(T184, T183, T162))
U15_gg(T184, T183, T162, T163, pF_out_ggg(T184, T183, T162)) → inG_out_gg(s(T184), tree(s(T183), T162, T163))
inG_in_gg(T202, tree(T199, T200, T201)) → U16_gg(T202, T199, T200, T201, pH_in_ggg(T199, T202, T201))
U16_gg(T202, T199, T200, T201, pH_out_ggg(T199, T202, T201)) → inG_out_gg(T202, tree(T199, T200, T201))
inG_in_gg(s(T219), tree(0, T211, T212)) → U17_gg(T219, T211, T212, inG_in_gg(s(T219), T212))
inG_in_gg(s(T232), tree(s(T230), T211, T212)) → U18_gg(T232, T230, T211, T212, lessE_in_gg(T230, T232))
U18_gg(T232, T230, T211, T212, lessE_out_gg(T230, T232)) → inG_out_gg(s(T232), tree(s(T230), T211, T212))
inG_in_gg(s(T235), tree(s(T230), T211, T212)) → U19_gg(T235, T230, T211, T212, lessE_in_gg(T230, T235))
U19_gg(T235, T230, T211, T212, lessE_out_gg(T230, T235)) → U20_gg(T235, T230, T211, T212, inG_in_gg(s(T235), T212))
U20_gg(T235, T230, T211, T212, inG_out_gg(s(T235), T212)) → inG_out_gg(s(T235), tree(s(T230), T211, T212))
U17_gg(T219, T211, T212, inG_out_gg(s(T219), T212)) → inG_out_gg(s(T219), tree(0, T211, T212))
U10_ggg(T127, T133, T129, inG_out_gg(T133, T129)) → pH_out_ggg(T127, T133, T129)
U13_gg(T130, T127, T128, T129, pH_out_ggg(T127, T130, T129)) → inG_out_gg(T130, tree(T127, T128, T129))
U7_ggg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_ggg(T92, T88, T15)
U12_gg(T89, T88, T15, T16, pF_out_ggg(T89, T88, T15)) → inG_out_gg(s(T89), tree(s(T88), T15, T16))
U7_agg(T92, T88, T15, inG_out_gg(s(T92), T15)) → pF_out_agg(T92, T88, T15)
U12_ag(T89, T88, T15, T16, pF_out_agg(T89, T88, T15)) → inG_out_ag(s(T89), tree(s(T88), T15, T16))
inG_in_ag(T130, tree(T127, T128, T129)) → U13_ag(T130, T127, T128, T129, pH_in_gag(T127, T130, T129))
pH_in_gag(T127, T130, T129) → U8_gag(T127, T130, T129, lessE_in_ga(T127, T130))
lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))
U8_gag(T127, T130, T129, lessE_out_ga(T127, T130)) → pH_out_gag(T127, T130, T129)
pH_in_gag(T127, T133, T129) → U9_gag(T127, T133, T129, lessE_in_ga(T127, T133))
U9_gag(T127, T133, T129, lessE_out_ga(T127, T133)) → U10_gag(T127, T133, T129, inG_in_ag(T133, T129))
inG_in_ag(0, tree(s(T169), T162, T163)) → U14_ag(T169, T162, T163, inA_in_g(T162))
U14_ag(T169, T162, T163, inA_out_g(T162)) → inG_out_ag(0, tree(s(T169), T162, T163))
inG_in_ag(s(T184), tree(s(T183), T162, T163)) → U15_ag(T184, T183, T162, T163, pF_in_agg(T184, T183, T162))
U15_ag(T184, T183, T162, T163, pF_out_agg(T184, T183, T162)) → inG_out_ag(s(T184), tree(s(T183), T162, T163))
inG_in_ag(T202, tree(T199, T200, T201)) → U16_ag(T202, T199, T200, T201, pH_in_gag(T199, T202, T201))
U16_ag(T202, T199, T200, T201, pH_out_gag(T199, T202, T201)) → inG_out_ag(T202, tree(T199, T200, T201))
inG_in_ag(s(T219), tree(0, T211, T212)) → U17_ag(T219, T211, T212, inG_in_ag(s(T219), T212))
inG_in_ag(s(T232), tree(s(T230), T211, T212)) → U18_ag(T232, T230, T211, T212, lessE_in_ga(T230, T232))
U18_ag(T232, T230, T211, T212, lessE_out_ga(T230, T232)) → inG_out_ag(s(T232), tree(s(T230), T211, T212))
inG_in_ag(s(T235), tree(s(T230), T211, T212)) → U19_ag(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_ag(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → U20_ag(T235, T230, T211, T212, inG_in_ag(s(T235), T212))
U20_ag(T235, T230, T211, T212, inG_out_ag(s(T235), T212)) → inG_out_ag(s(T235), tree(s(T230), T211, T212))
U17_ag(T219, T211, T212, inG_out_ag(s(T219), T212)) → inG_out_ag(s(T219), tree(0, T211, T212))
U10_gag(T127, T133, T129, inG_out_ag(T133, T129)) → pH_out_gag(T127, T133, T129)
U13_ag(T130, T127, T128, T129, pH_out_gag(T127, T130, T129)) → inG_out_ag(T130, tree(T127, T128, T129))

The argument filtering Pi contains the following mapping:
inG_in_ag(x1, x2)  =  inG_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
inG_out_ag(x1, x2)  =  inG_out_ag
s(x1)  =  s(x1)
U11_ag(x1, x2, x3, x4)  =  U11_ag(x4)
inA_in_g(x1)  =  inA_in_g(x1)
0  =  0
inA_out_g(x1)  =  inA_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x4)
lessB_in_g(x1)  =  lessB_in_g(x1)
lessB_out_g(x1)  =  lessB_out_g
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U12_ag(x1, x2, x3, x4, x5)  =  U12_ag(x5)
pF_in_agg(x1, x2, x3)  =  pF_in_agg(x2, x3)
U5_agg(x1, x2, x3, x4)  =  U5_agg(x4)
lessD_in_ag(x1, x2)  =  lessD_in_ag(x2)
lessD_out_ag(x1, x2)  =  lessD_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
pF_out_agg(x1, x2, x3)  =  pF_out_agg(x1)
U6_agg(x1, x2, x3, x4)  =  U6_agg(x3, x4)
U7_agg(x1, x2, x3, x4)  =  U7_agg(x1, x4)
inG_in_gg(x1, x2)  =  inG_in_gg(x1, x2)
inG_out_gg(x1, x2)  =  inG_out_gg
U11_gg(x1, x2, x3, x4)  =  U11_gg(x4)
U12_gg(x1, x2, x3, x4, x5)  =  U12_gg(x5)
pF_in_ggg(x1, x2, x3)  =  pF_in_ggg(x1, x2, x3)
U5_ggg(x1, x2, x3, x4)  =  U5_ggg(x4)
lessD_in_gg(x1, x2)  =  lessD_in_gg(x1, x2)
lessD_out_gg(x1, x2)  =  lessD_out_gg
U3_gg(x1, x2, x3)  =  U3_gg(x3)
pF_out_ggg(x1, x2, x3)  =  pF_out_ggg
U6_ggg(x1, x2, x3, x4)  =  U6_ggg(x1, x3, x4)
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
pH_in_ggg(x1, x2, x3)  =  pH_in_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3, x4)  =  U8_ggg(x4)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
lessE_out_gg(x1, x2)  =  lessE_out_gg
U4_gg(x1, x2, x3)  =  U4_gg(x3)
pH_out_ggg(x1, x2, x3)  =  pH_out_ggg
U9_ggg(x1, x2, x3, x4)  =  U9_ggg(x2, x3, x4)
U10_ggg(x1, x2, x3, x4)  =  U10_ggg(x4)
U14_gg(x1, x2, x3, x4)  =  U14_gg(x4)
U15_gg(x1, x2, x3, x4, x5)  =  U15_gg(x5)
U16_gg(x1, x2, x3, x4, x5)  =  U16_gg(x5)
U17_gg(x1, x2, x3, x4)  =  U17_gg(x4)
U18_gg(x1, x2, x3, x4, x5)  =  U18_gg(x5)
U19_gg(x1, x2, x3, x4, x5)  =  U19_gg(x1, x4, x5)
U20_gg(x1, x2, x3, x4, x5)  =  U20_gg(x5)
U13_ag(x1, x2, x3, x4, x5)  =  U13_ag(x5)
pH_in_gag(x1, x2, x3)  =  pH_in_gag(x1, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
pH_out_gag(x1, x2, x3)  =  pH_out_gag
U9_gag(x1, x2, x3, x4)  =  U9_gag(x3, x4)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x4)
U15_ag(x1, x2, x3, x4, x5)  =  U15_ag(x5)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x5)
U17_ag(x1, x2, x3, x4)  =  U17_ag(x4)
U18_ag(x1, x2, x3, x4, x5)  =  U18_ag(x5)
U19_ag(x1, x2, x3, x4, x5)  =  U19_ag(x4, x5)
U20_ag(x1, x2, x3, x4, x5)  =  U20_ag(x5)
ING_IN_AG(x1, x2)  =  ING_IN_AG(x2)
PH_IN_GAG(x1, x2, x3)  =  PH_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x3, x4)
U19_AG(x1, x2, x3, x4, x5)  =  U19_AG(x4, x5)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

ING_IN_AG(T130, tree(T127, T128, T129)) → PH_IN_GAG(T127, T130, T129)
PH_IN_GAG(T127, T133, T129) → U9_GAG(T127, T133, T129, lessE_in_ga(T127, T133))
U9_GAG(T127, T133, T129, lessE_out_ga(T127, T133)) → ING_IN_AG(T133, T129)
ING_IN_AG(s(T219), tree(0, T211, T212)) → ING_IN_AG(s(T219), T212)
ING_IN_AG(s(T235), tree(s(T230), T211, T212)) → U19_AG(T235, T230, T211, T212, lessE_in_ga(T230, T235))
U19_AG(T235, T230, T211, T212, lessE_out_ga(T230, T235)) → ING_IN_AG(s(T235), T212)

The TRS R consists of the following rules:

lessE_in_ga(0, s(T140)) → lessE_out_ga(0, s(T140))
lessE_in_ga(s(T145), s(T147)) → U4_ga(T145, T147, lessE_in_ga(T145, T147))
U4_ga(T145, T147, lessE_out_ga(T145, T147)) → lessE_out_ga(s(T145), s(T147))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
lessE_in_ga(x1, x2)  =  lessE_in_ga(x1)
lessE_out_ga(x1, x2)  =  lessE_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ING_IN_AG(x1, x2)  =  ING_IN_AG(x2)
PH_IN_GAG(x1, x2, x3)  =  PH_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x3, x4)
U19_AG(x1, x2, x3, x4, x5)  =  U19_AG(x4, x5)

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) Obligation:

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

ING_IN_AG(tree(T127, T128, T129)) → PH_IN_GAG(T127, T129)
PH_IN_GAG(T127, T129) → U9_GAG(T129, lessE_in_ga(T127))
U9_GAG(T129, lessE_out_ga) → ING_IN_AG(T129)
ING_IN_AG(tree(0, T211, T212)) → ING_IN_AG(T212)
ING_IN_AG(tree(s(T230), T211, T212)) → U19_AG(T212, lessE_in_ga(T230))
U19_AG(T212, lessE_out_ga) → ING_IN_AG(T212)

The TRS R consists of the following rules:

lessE_in_ga(0) → lessE_out_ga
lessE_in_ga(s(T145)) → U4_ga(lessE_in_ga(T145))
U4_ga(lessE_out_ga) → lessE_out_ga

The set Q consists of the following terms:

lessE_in_ga(x0)
U4_ga(x0)

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

(56) 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:

• PH_IN_GAG(T127, T129) → U9_GAG(T129, lessE_in_ga(T127))
  The graph contains the following edges 2 >= 1

• U9_GAG(T129, lessE_out_ga) → ING_IN_AG(T129)
  The graph contains the following edges 1 >= 1

• ING_IN_AG(tree(T127, T128, T129)) → PH_IN_GAG(T127, T129)
  The graph contains the following edges 1 > 1, 1 > 2

• ING_IN_AG(tree(s(T230), T211, T212)) → U19_AG(T212, lessE_in_ga(T230))
  The graph contains the following edges 1 > 1

• ING_IN_AG(tree(0, T211, T212)) → ING_IN_AG(T212)
  The graph contains the following edges 1 > 1

• U19_AG(T212, lessE_out_ga) → ING_IN_AG(T212)
  The graph contains the following edges 1 >= 1

(57) YES