Left Termination of the query pattern minsort_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

minsort([], []).
minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))).
min1(M, .(X, L)) :- min2(X, M, L).
min2(X, X, []).
min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)).
min(X, Y, X) :- le(X, Y).
min(X, Y, Y) :- gt(X, Y).
remove(N, .(N, L), L).
remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).

Queries:

minsort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: minsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: minsort(T1, T2), SCOPE: 1, CLAUSE: 0 | minsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | minsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: minsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nminsort(T1, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: minsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(min1(T7, []), ','(remove(T7, [], X7), minsort(X7, T8)))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(min1(T7, []), ','(remove(T7, [], X7), minsort(X7, T8))), SCOPE: 2, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(min1(T16, T13), ','(remove(T16, T13, X20), minsort(X20, T17)))\n\nT13 is ground\nX20 is free; \nminsort(T13, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(min1(T16, T13), ','(remove(T16, T13, X20), minsort(X20, T17))), SCOPE: 3, CLAUSE: 2\n\nT13 is ground\nX20 is free; \nminsort(T13, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(min2(T25, T27, T26), ','(remove(T27, .(T25, T26), X20), minsort(X20, T28)))\n\nT25 is ground\nT26 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: min2(T25, T27, T26)\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(remove(T31, .(T25, T26), X20), minsort(X20, T32))\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 3 | min2(T25, T27, T26), SCOPE: 4, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(min(T48, T50, X59), min2(X59, T52, T51))\n\nT48 is ground\nT50 is ground\nT51 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: min(T48, T50, X59)\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: min2(T55, T52, T51)\n\nT51 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: min(T48, T50, X59), SCOPE: 5, CLAUSE: 5 | min(T48, T50, X59), SCOPE: 5, CLAUSE: 6\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: min(T48, T50, X59), SCOPE: 5, CLAUSE: 5\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: min(T48, T50, X59), SCOPE: 5, CLAUSE: 6\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: le(T70, T71)\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: le(T70, T71), SCOPE: 6, CLAUSE: 11 | le(T70, T71), SCOPE: 6, CLAUSE: 12 | le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T70, T71), SCOPE: 6, CLAUSE: 11\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T70, T71), SCOPE: 6, CLAUSE: 12 | le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: le(T82, T83)\n\nT82 is ground\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T70, T71), SCOPE: 6, CLAUSE: 12\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: gt(T97, T98)\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: gt(T97, T98), SCOPE: 7, CLAUSE: 9 | gt(T97, T98), SCOPE: 7, CLAUSE: 10\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: gt(T97, T98), SCOPE: 7, CLAUSE: 9\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: gt(T97, T98), SCOPE: 7, CLAUSE: 10\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: gt(T109, T110)\n\nT109 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: remove(T31, .(T25, T26), X20)\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minsort(T122, T32)\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 7 | remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 8\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 7\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 8\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(notEq(T145, T146), remove(T145, T147, X170))\n\nT145 is ground\nT146 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: notEq(T145, T146)\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: remove(T145, T147, X170)\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: notEq(T145, T146), SCOPE: 9, CLAUSE: 14 | notEq(T145, T146), SCOPE: 9, CLAUSE: 15 | notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: notEq(T145, T146), SCOPE: 9, CLAUSE: 14\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: notEq(T145, T146), SCOPE: 9, CLAUSE: 15 | notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: notEq(T160, T161)\n\nT160 is ground\nT161 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: notEq(T145, T146), SCOPE: 9, CLAUSE: 15\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 7 | remove(T145, T147, X170), SCOPE: 10, CLAUSE: 8\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 7\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 8\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(notEq(T192, T193), remove(T192, T194, X229))\n\nT192 is ground\nT193 is ground\nT194 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 85 [arrowhead = none ];
85 -> 2;

86 [label="EVAL with clause\nminsort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 86 [arrowhead = none ];
86 -> 3;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 87 [arrowhead = none ];
87 -> 4;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 88 [arrowhead = none ];
88 -> 5;

89 [label="EVAL with clause\nminsort(X17, .(X18, X19)) :- ','(min1(X18, X17), ','(remove(X18, X17, X20), minsort(X20, X19))).\nand substitution\nT1 -> T13\nX17 -> T13\nX18 -> T16\nX19 -> T17\nT2 -> .(T16, T17)\nT14 -> T16\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 89 [arrowhead = none ];
89 -> 10;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="EVAL with clause\nminsort(X4, .(X5, X6)) :- ','(min1(X5, X4), ','(remove(X5, X4, X7), minsort(X7, X6))).\nand substitution\nX4 -> []\nX5 -> T7\nX6 -> T8\nT2 -> .(T7, T8)\nT5 -> T7\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 91 [arrowhead = none ];
91 -> 6;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 92 [arrowhead = none ];
92 -> 7;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 93 [arrowhead = none ];
93 -> 8;

94 [label="BACKTRACK\nfor clause: min1(M, .(X, L)) :- min2(X, M, L)because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 94 [arrowhead = none ];
94 -> 9;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 95 [arrowhead = none ];
95 -> 12;

96 [label="EVAL with clause\nmin1(X30, .(X31, X32)) :- min2(X31, X30, X32).\nand substitution\nT16 -> T27\nX30 -> T27\nX31 -> T25\nX32 -> T26\nT13 -> .(T25, T26)\nT24 -> T27\nT17 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 96 [arrowhead = none ];
96 -> 13;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 97 [arrowhead = none ];
97 -> 14;

98 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 98 [arrowhead = none ];
98 -> 15;

99 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nreplacements:\nT27 -> T31\nT28 -> T32", fontsize=14, style = filled, fillcolor = violet];
13 -> 99 [arrowhead = none ];
99 -> 16;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 100 [arrowhead = none ];
100 -> 17;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 101 [arrowhead = none ];
101 -> 53;

102 [label="SPLIT 2\nnew knowledge:\nT122 is ground\nreplacements:\nX20 -> T122", fontsize=14, style = filled, fillcolor = violet];
16 -> 102 [arrowhead = none ];
102 -> 54;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 103 [arrowhead = none ];
103 -> 18;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="EVAL with clause\nmin2(X44, X44, []).\nand substitution\nT25 -> T39\nX44 -> T39\nT27 -> T39\nT26 -> []", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 105 [arrowhead = none ];
105 -> 20;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 106 [arrowhead = none ];
106 -> 21;

107 [label="EVAL with clause\nmin2(X55, X56, .(X57, X58)) :- ','(min(X55, X57, X59), min2(X59, X56, X58)).\nand substitution\nT25 -> T48\nX55 -> T48\nT27 -> T52\nX56 -> T52\nX57 -> T50\nX58 -> T51\nT26 -> .(T50, T51)\nT49 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 107 [arrowhead = none ];
107 -> 23;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 108 [arrowhead = none ];
108 -> 24;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 109 [arrowhead = none ];
109 -> 22;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nreplacements:\nX59 -> T55", fontsize=14, style = filled, fillcolor = violet];
23 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="INSTANCE with matching:\nT25 -> T55\nT27 -> T52\nT26 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 113 [arrowhead = none , style=dashed];
113 -> 15[style=dashed];

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 114 [arrowhead = none ];
114 -> 28;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 115 [arrowhead = none ];
115 -> 29;

116 [label="ONLY EVAL with clause\nmin(X80, X81, X80) :- le(X80, X81).\nand substitution\nT48 -> T70\nX80 -> T70\nT50 -> T71\nX81 -> T71\nX59 -> T70", fontsize=14, style = filled, fillcolor = white];
28 -> 116 [arrowhead = none ];
116 -> 30;

117 [label="ONLY EVAL with clause\nmin(X110, X111, X111) :- gt(X110, X111).\nand substitution\nT48 -> T97\nX110 -> T97\nT50 -> T98\nX111 -> T98\nX59 -> T98", fontsize=14, style = filled, fillcolor = white];
29 -> 117 [arrowhead = none ];
117 -> 44;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 118 [arrowhead = none ];
118 -> 31;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 119 [arrowhead = none ];
119 -> 32;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 120 [arrowhead = none ];
120 -> 33;

121 [label="EVAL with clause\nle(s(X92), s(X93)) :- le(X92, X93).\nand substitution\nX92 -> T82\nT70 -> s(T82)\nX93 -> T83\nT71 -> s(T83)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 122 [arrowhead = none ];
122 -> 35;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 123 [arrowhead = none ];
123 -> 36;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 124 [arrowhead = none ];
124 -> 37;

125 [label="INSTANCE with matching:\nT70 -> T82\nT71 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 125 [arrowhead = none , style=dashed];
125 -> 30[style=dashed];

126 [label="EVAL with clause\nle(0, s(X102)).\nand substitution\nT70 -> 0\nX102 -> T90\nT71 -> s(T90)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 126 [arrowhead = none ];
126 -> 38;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 127 [arrowhead = none ];
127 -> 39;

128 [label="EVAL with clause\nle(0, 0).\nand substitution\nT70 -> 0\nT71 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 128 [arrowhead = none ];
128 -> 41;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 129 [arrowhead = none ];
129 -> 42;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 131 [arrowhead = none ];
131 -> 43;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 132 [arrowhead = none ];
132 -> 45;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 133 [arrowhead = none ];
133 -> 46;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 134 [arrowhead = none ];
134 -> 47;

135 [label="EVAL with clause\ngt(s(X122), s(X123)) :- gt(X122, X123).\nand substitution\nX122 -> T109\nT97 -> s(T109)\nX123 -> T110\nT98 -> s(T110)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 135 [arrowhead = none ];
135 -> 48;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="EVAL with clause\ngt(s(X130), 0).\nand substitution\nX130 -> T115\nT97 -> s(T115)\nT98 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="INSTANCE with matching:\nT97 -> T109\nT98 -> T110", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 139 [arrowhead = none , style=dashed];
139 -> 44[style=dashed];

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 141 [arrowhead = none ];
141 -> 55;

142 [label="INSTANCE with matching:\nT1 -> T122\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 142 [arrowhead = none , style=dashed];
142 -> 1[style=dashed];

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 143 [arrowhead = none ];
143 -> 56;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 144 [arrowhead = none ];
144 -> 57;

145 [label="EVAL with clause\nremove(X150, .(X150, X151), X151).\nand substitution\nT31 -> T135\nX150 -> T135\nT25 -> T135\nT26 -> T136\nX151 -> T136\nX20 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 145 [arrowhead = none ];
145 -> 58;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 146 [arrowhead = none ];
146 -> 59;

147 [label="ONLY EVAL with clause\nremove(X166, .(X167, X168), .(X167, X169)) :- ','(notEq(X166, X167), remove(X166, X168, X169)).\nand substitution\nT31 -> T145\nX166 -> T145\nT25 -> T146\nX167 -> T146\nT26 -> T147\nX168 -> T147\nX169 -> X170\nX20 -> .(T146, X170)", fontsize=14, style = filled, fillcolor = white];
57 -> 147 [arrowhead = none ];
147 -> 61;

148 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 149 [arrowhead = none ];
149 -> 62;

150 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
61 -> 150 [arrowhead = none ];
150 -> 63;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 151 [arrowhead = none ];
151 -> 64;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 152 [arrowhead = none ];
152 -> 77;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 154 [arrowhead = none ];
154 -> 66;

155 [label="EVAL with clause\nnotEq(s(X183), s(X184)) :- notEq(X183, X184).\nand substitution\nX183 -> T160\nT145 -> s(T160)\nX184 -> T161\nT146 -> s(T161)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 155 [arrowhead = none ];
155 -> 67;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 156 [arrowhead = none ];
156 -> 68;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 157 [arrowhead = none ];
157 -> 69;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="INSTANCE with matching:\nT145 -> T160\nT146 -> T161", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 159 [arrowhead = none , style=dashed];
159 -> 62[style=dashed];

160 [label="EVAL with clause\nnotEq(s(X193), 0).\nand substitution\nX193 -> T168\nT145 -> s(T168)\nT146 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 160 [arrowhead = none ];
160 -> 71;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 161 [arrowhead = none ];
161 -> 72;

162 [label="EVAL with clause\nnotEq(0, s(X197)).\nand substitution\nT145 -> 0\nX197 -> T171\nT146 -> s(T171)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 162 [arrowhead = none ];
162 -> 74;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 163 [arrowhead = none ];
163 -> 75;

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

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 165 [arrowhead = none ];
165 -> 76;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 166 [arrowhead = none ];
166 -> 78;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 167 [arrowhead = none ];
167 -> 79;

168 [label="EVAL with clause\nremove(X211, .(X211, X212), X212).\nand substitution\nT145 -> T184\nX211 -> T184\nX212 -> T185\nT147 -> .(T184, T185)\nX170 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 169 [arrowhead = none ];
169 -> 81;

170 [label="EVAL with clause\nremove(X225, .(X226, X227), .(X226, X228)) :- ','(notEq(X225, X226), remove(X225, X227, X228)).\nand substitution\nT145 -> T192\nX225 -> T192\nX226 -> T193\nX227 -> T194\nT147 -> .(T193, T194)\nX228 -> X229\nX170 -> .(T193, X229)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 170 [arrowhead = none ];
170 -> 83;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 171 [arrowhead = none ];
171 -> 84;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 172 [arrowhead = none ];
172 -> 82;

173 [label="INSTANCE with matching:\nT145 -> T192\nT146 -> T193\nT147 -> T194\nX170 -> X229", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 173 [arrowhead = none , style=dashed];
173 -> 61[style=dashed];

}

(2) Obligation:

Clauses:

min215(T39, T39, []).
min215(T48, T52, .(T50, T51)) :- min25(T48, T50, X59).
min215(T48, T52, .(T50, T51)) :- ','(min25(T48, T50, T55), min215(T55, T52, T51)).
le30(s(T82), s(T83)) :- le30(T82, T83).
le30(0, s(T90)).
le30(0, 0).
gt44(s(T109), s(T110)) :- gt44(T109, T110).
gt44(s(T115), 0).
notEq62(s(T160), s(T161)) :- notEq62(T160, T161).
notEq62(s(T168), 0).
notEq62(0, s(T171)).
p61(T145, T146, T147, X170) :- notEq62(T145, T146).
p61(T184, T146, .(T184, T185), T185) :- notEq62(T184, T146).
p61(T192, T146, .(T193, T194), .(T193, X229)) :- ','(notEq62(T192, T146), p61(T192, T193, T194, X229)).
remove53(T135, T135, T136, T136).
remove53(T145, T146, T147, .(T146, X170)) :- p61(T145, T146, T147, X170).
min25(T70, T71, T70) :- le30(T70, T71).
min25(T97, T98, T98) :- gt44(T97, T98).
minsort1([], []).
minsort1(.(T25, T26), .(T27, T28)) :- min215(T25, T27, T26).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min215(T25, T31, T26), remove53(T31, T25, T26, X20)).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min215(T25, T31, T26), ','(remove53(T31, T25, T26, T122), minsort1(T122, T32))).

Queries:

minsort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort1_in: (b,f) (f,f)
min215_in: (b,f,b) (f,f,f) (b,f,f)
min25_in: (b,b,f) (f,f,f) (b,f,f)
le30_in: (b,b) (f,f) (b,f)
gt44_in: (b,b) (f,f) (b,f)
remove53_in: (f,b,b,f) (f,f,f,f)
p61_in: (f,b,b,f) (f,f,f,f)
notEq62_in: (f,b) (b,b) (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x3, x4, x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x2, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x2, x3, x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x2, x3, x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x1, x2, x3, x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x2, x3, x4, x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x1, x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(6) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x3, x4, x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x2, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x2, x3, x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x2, x3, x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x1, x2, x3, x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x2, x3, x4, x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x1, x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 61 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

NOTEQ62_IN_AA → NOTEQ62_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = NOTEQ62_IN_AA evaluates to t =NOTEQ62_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NOTEQ62_IN_AA to NOTEQ62_IN_AA.

(15) NO

(16) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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:

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

P61_IN_AAAA → U9_AAAA(notEq62_in_aa)
U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

The TRS R consists of the following rules:

notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_AAAA → U9_AAAA(notEq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P61_IN_AAAA → U9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAA → U9_AAAA(notEq62_out_aa)

(22) Obligation:

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

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
P61_IN_AAAA → U9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAA → U9_AAAA(notEq62_out_aa)

The TRS R consists of the following rules:

notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(23) NonTerminationProof (EQUIVALENT transformation)

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

s = P61_IN_AAAA evaluates to t =P61_IN_AAAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

P61_IN_AAAA → U9_AAAA(notEq62_out_aa)
with rule P61_IN_AAAA → U9_AAAA(notEq62_out_aa) at position [] and matcher [ ]

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
with rule U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

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


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

(24) NO

(25) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)

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

(30) 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:

• GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)
  The graph contains the following edges 1 > 1

(31) YES

(32) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)

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

(37) 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:

• LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)
  The graph contains the following edges 1 > 1

(38) YES

(39) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

MIN215_IN_GAA(T48) → U2_GAA(T48, min25_in_gaa(T48))
U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(44) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MIN215_IN_GAA(T48) → U2_GAA(T48, min25_in_gaa(T48)) at position [1] we obtained the following new rules [LPAR04]:

MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

(45) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(48) QReductionProof (EQUIVALENT transformation)

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

min25_in_gaa(x0)

(49) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U12_gaa(x₁, x₂) ) = x₁ + 2

POL( U2_GAA(x₁, x₂) ) = max{0, 2x₂ - 2}

POL( le30_in_ga(x₁) ) = 0

POL( s(x₁) ) = x₁ + 2

POL( U4_ga(x₁, x₂) ) = x₁ + 2

POL( 0 ) = 0

POL( le30_out_ga(x₁) ) = max{0, 2x₁ - 2}

POL( min25_out_gaa(x₁, x₂) ) = x₂ + 2

POL( U13_gaa(x₁, x₂) ) = x₂

POL( gt44_in_ga(x₁) ) = x₁

POL( U5_ga(x₁, x₂) ) = x₂ + 2

POL( gt44_out_ga(x₁, x₂) ) = x₂ + 2

POL( MIN215_IN_GAA(x₁) ) = 2x₁ + 2

The following usable rules [FROCOS05] were oriented:

U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

(51) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(54) QReductionProof (EQUIVALENT transformation)

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

U13_gaa(x0, x1)
gt44_in_ga(x0)
U5_ga(x0, x1)

(55) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
le30_in_ga(x0)
U4_ga(x0, x1)

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

(56) NonTerminationProof (EQUIVALENT transformation)

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

s = U2_GAA(0, U12_gaa(0, le30_in_ga(0))) evaluates to t =U2_GAA(0, U12_gaa(0, le30_in_ga(0)))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U2_GAA(0, U12_gaa(0, le30_in_ga(0))) → U2_GAA(0, U12_gaa(0, le30_out_ga(0)))
with rule le30_in_ga(0) → le30_out_ga(0) at position [1,1] and matcher [ ]

U2_GAA(0, U12_gaa(0, le30_out_ga(0))) → U2_GAA(0, min25_out_gaa(0, 0))
with rule U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70) at position [1] and matcher [T70 / 0]

U2_GAA(0, min25_out_gaa(0, 0)) → MIN215_IN_GAA(0)
with rule U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55) at position [] and matcher [T48 / 0, T55 / 0]

MIN215_IN_GAA(0) → U2_GAA(0, U12_gaa(0, le30_in_ga(0)))
with rule MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

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


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

(57) NO

(58) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(61) PiDPToQDPProof (SOUND transformation)

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

(62) Obligation:

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

GT44_IN_AA → GT44_IN_AA

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

(63) NonTerminationProof (EQUIVALENT transformation)

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

s = GT44_IN_AA evaluates to t =GT44_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT44_IN_AA to GT44_IN_AA.

(64) NO

(65) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

LE30_IN_AA → LE30_IN_AA

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

(70) NonTerminationProof (EQUIVALENT transformation)

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

s = LE30_IN_AA evaluates to t =LE30_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE30_IN_AA to LE30_IN_AA.

(71) NO

(72) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

MINSORT1_IN_AA → U15_AA(min215_in_aaa)
U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

The TRS R consists of the following rules:

min215_in_aaa → min215_out_aaa
min215_in_aaa → U1_aaa(min25_in_aaa)
min215_in_aaa → U2_aaa(min25_in_aaa)
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MINSORT1_IN_AA → U15_AA(min215_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

(78) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min215_in_aaa → min215_out_aaa
min215_in_aaa → U1_aaa(min25_in_aaa)
min215_in_aaa → U2_aaa(min25_in_aaa)
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(81) QReductionProof (EQUIVALENT transformation)

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

min215_in_aaa

(82) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(83) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

(84) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(87) QReductionProof (EQUIVALENT transformation)

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

remove53_in_aaaa

(88) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(89) NonTerminationProof (EQUIVALENT transformation)

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

s = MINSORT1_IN_AA evaluates to t =MINSORT1_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

MINSORT1_IN_AA → U15_AA(min215_out_aaa)
with rule MINSORT1_IN_AA → U15_AA(min215_out_aaa) at position [] and matcher [ ]

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
with rule U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa) at position [] and matcher [ ]

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
with rule U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

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


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

(90) NO

(91) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)

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

(92) UsableRulesProof (EQUIVALENT transformation)

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

(93) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(94) PiDPToQDPProof (EQUIVALENT transformation)

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

(95) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(96) 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:

• NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
  The graph contains the following edges 1 > 1, 2 > 2

(97) YES

(98) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(99) UsableRulesProof (EQUIVALENT transformation)

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

(100) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(101) PiDPToQDPProof (SOUND transformation)

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

(102) Obligation:

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

NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)

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

(103) 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:

• NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)
  The graph contains the following edges 1 > 1

(104) YES

(105) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)

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

(106) UsableRulesProof (EQUIVALENT transformation)

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

(107) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)

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

(108) PiDPToQDPProof (SOUND transformation)

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

(109) Obligation:

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

P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T146, T193, T194, notEq62_in_ag(T146))
U9_AGGA(T146, T193, T194, notEq62_out_ag(T146)) → P61_IN_AGGA(T193, T194)

The TRS R consists of the following rules:

notEq62_in_ag(s(T161)) → U6_ag(T161, notEq62_in_ag(T161))
notEq62_in_ag(0) → notEq62_out_ag(0)
notEq62_in_ag(s(T171)) → notEq62_out_ag(s(T171))
U6_ag(T161, notEq62_out_ag(T161)) → notEq62_out_ag(s(T161))

The set Q consists of the following terms:

notEq62_in_ag(x0)
U6_ag(x0, x1)

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

(110) 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:

• U9_AGGA(T146, T193, T194, notEq62_out_ag(T146)) → P61_IN_AGGA(T193, T194)
  The graph contains the following edges 2 >= 1, 3 >= 2

• P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T146, T193, T194, notEq62_in_ag(T146))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(111) YES

(112) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)

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

(113) UsableRulesProof (EQUIVALENT transformation)

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

(114) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(115) PiDPToQDPProof (EQUIVALENT transformation)

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

(116) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(117) 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:

• GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
  The graph contains the following edges 1 > 1, 2 > 2

(118) YES

(119) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)

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

(120) UsableRulesProof (EQUIVALENT transformation)

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

(121) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(122) PiDPToQDPProof (EQUIVALENT transformation)

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

(123) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(124) 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:

• LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
  The graph contains the following edges 1 > 1, 2 > 2

(125) YES

(126) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)

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

(127) UsableRulesProof (EQUIVALENT transformation)

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

(128) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)

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

(129) PiDPToQDPProof (SOUND transformation)

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

(130) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T48, T50, T51, min25_in_gga(T48, T50))
U2_GAG(T48, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gga(x0, x1)
U12_gga(x0, x1, x2)
U13_gga(x0, x1, x2)
le30_in_gg(x0, x1)
gt44_in_gg(x0, x1)
U4_gg(x0, x1, x2)
U5_gg(x0, x1, x2)

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

(131) 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:

• U2_GAG(T48, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)
  The graph contains the following edges 4 > 1, 3 >= 2

• MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T48, T50, T51, min25_in_gga(T48, T50))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(132) YES

(133) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort1_in: (b,f) (f,f)
min215_in: (b,f,b) (f,f,f) (b,f,f)
min25_in: (b,b,f) (f,f,f) (b,f,f)
le30_in: (b,b) (f,f) (b,f)
gt44_in: (b,b) (f,f) (b,f)
remove53_in: (f,b,b,f) (f,f,f,f)
p61_in: (f,b,b,f) (f,f,f,f)
notEq62_in: (f,b) (b,b) (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(134) Obligation:

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

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

(135) 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:

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(136) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(137) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 61 less nodes.

(138) Complex Obligation (AND)

(139) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(140) UsableRulesProof (EQUIVALENT transformation)

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

(141) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(142) PiDPToQDPProof (SOUND transformation)

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

(143) Obligation:

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

NOTEQ62_IN_AA → NOTEQ62_IN_AA

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

(144) NonTerminationProof (EQUIVALENT transformation)

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

s = NOTEQ62_IN_AA evaluates to t =NOTEQ62_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NOTEQ62_IN_AA to NOTEQ62_IN_AA.

(145) NO

(146) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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

(147) UsableRulesProof (EQUIVALENT transformation)

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

(148) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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

(149) PiDPToQDPProof (SOUND transformation)

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

(150) Obligation:

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

P61_IN_AAAA → U9_AAAA(notEq62_in_aa)
U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

The TRS R consists of the following rules:

notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(151) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_AAAA → U9_AAAA(notEq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P61_IN_AAAA → U9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAA → U9_AAAA(notEq62_out_aa)

(152) Obligation:

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

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
P61_IN_AAAA → U9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAA → U9_AAAA(notEq62_out_aa)

The TRS R consists of the following rules:

notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(153) NonTerminationProof (EQUIVALENT transformation)

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

s = P61_IN_AAAA evaluates to t =P61_IN_AAAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

P61_IN_AAAA → U9_AAAA(notEq62_out_aa)
with rule P61_IN_AAAA → U9_AAAA(notEq62_out_aa) at position [] and matcher [ ]

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
with rule U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

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


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

(154) NO

(155) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(156) UsableRulesProof (EQUIVALENT transformation)

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

(157) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(158) PiDPToQDPProof (SOUND transformation)

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

(159) Obligation:

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

GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)

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

(160) 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:

• GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)
  The graph contains the following edges 1 > 1

(161) YES

(162) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(163) UsableRulesProof (EQUIVALENT transformation)

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

(164) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(165) PiDPToQDPProof (SOUND transformation)

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

(166) Obligation:

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

LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)

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

(167) 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:

• LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)
  The graph contains the following edges 1 > 1

(168) YES

(169) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)

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

(170) UsableRulesProof (EQUIVALENT transformation)

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

(171) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)

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

(172) PiDPToQDPProof (SOUND transformation)

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

(173) Obligation:

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

MIN215_IN_GAA(T48) → U2_GAA(min25_in_gaa(T48))
U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(174) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MIN215_IN_GAA(T48) → U2_GAA(min25_in_gaa(T48)) at position [0] we obtained the following new rules [LPAR04]:

MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

(175) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(176) UsableRulesProof (EQUIVALENT transformation)

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

(177) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(178) QReductionProof (EQUIVALENT transformation)

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

min25_in_gaa(x0)

(179) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(MIN215_IN_GAA(x₁)) = 1 + x₁   
POL(U12_gaa(x₁, x₂)) = 1 + x₁   
POL(U13_gaa(x₁)) = x₁   
POL(U2_GAA(x₁)) = x₁   
POL(U4_ga(x₁)) = 0   
POL(U5_ga(x₁)) = 1 + x₁   
POL(gt44_in_ga(x₁)) = x₁   
POL(gt44_out_ga(x₁)) = 1 + x₁   
POL(le30_in_ga(x₁)) = 0   
POL(le30_out_ga) = 0   
POL(min25_out_gaa(x₁)) = 1 + x₁   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

(181) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(182) UsableRulesProof (EQUIVALENT transformation)

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

(183) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(184) QReductionProof (EQUIVALENT transformation)

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

U13_gaa(x0)
gt44_in_ga(x0)
U5_ga(x0)

(185) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
le30_in_ga(x0)
U4_ga(x0)

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

(186) NonTerminationProof (EQUIVALENT transformation)

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

s = U2_GAA(U12_gaa(T70, le30_in_ga(0))) evaluates to t =U2_GAA(U12_gaa(T70, le30_in_ga(T70)))

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

• Semiunifier: [T70 / 0]
• Matcher: [ ]

---

Rewriting sequence

U2_GAA(U12_gaa(0, le30_in_ga(0))) → U2_GAA(U12_gaa(0, le30_out_ga))
with rule le30_in_ga(0) → le30_out_ga at position [0,1] and matcher [ ]

U2_GAA(U12_gaa(0, le30_out_ga)) → U2_GAA(min25_out_gaa(0))
with rule U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70) at position [0] and matcher [T70 / 0]

U2_GAA(min25_out_gaa(0)) → MIN215_IN_GAA(0)
with rule U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55) at position [] and matcher [T55 / 0]

MIN215_IN_GAA(0) → U2_GAA(U12_gaa(0, le30_in_ga(0)))
with rule MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

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


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

(187) NO

(188) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(189) UsableRulesProof (EQUIVALENT transformation)

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

(190) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(191) PiDPToQDPProof (SOUND transformation)

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

(192) Obligation:

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

GT44_IN_AA → GT44_IN_AA

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

(193) NonTerminationProof (EQUIVALENT transformation)

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

s = GT44_IN_AA evaluates to t =GT44_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT44_IN_AA to GT44_IN_AA.

(194) NO

(195) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(196) UsableRulesProof (EQUIVALENT transformation)

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

(197) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(198) PiDPToQDPProof (SOUND transformation)

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

(199) Obligation:

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

LE30_IN_AA → LE30_IN_AA

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

(200) NonTerminationProof (EQUIVALENT transformation)

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

s = LE30_IN_AA evaluates to t =LE30_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE30_IN_AA to LE30_IN_AA.

(201) NO

(202) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(203) UsableRulesProof (EQUIVALENT transformation)

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

(204) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(205) PiDPToQDPProof (SOUND transformation)

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

(206) Obligation:

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

MINSORT1_IN_AA → U15_AA(min215_in_aaa)
U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

The TRS R consists of the following rules:

min215_in_aaa → min215_out_aaa
min215_in_aaa → U1_aaa(min25_in_aaa)
min215_in_aaa → U2_aaa(min25_in_aaa)
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa) → min215_out_aaa
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U3_gaa(min215_out_gaa) → min215_out_gaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(207) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MINSORT1_IN_AA → U15_AA(min215_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

(208) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min215_in_aaa → min215_out_aaa
min215_in_aaa → U1_aaa(min25_in_aaa)
min215_in_aaa → U2_aaa(min25_in_aaa)
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa) → min215_out_aaa
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U3_gaa(min215_out_gaa) → min215_out_gaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(209) UsableRulesProof (EQUIVALENT transformation)

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

(210) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(211) QReductionProof (EQUIVALENT transformation)

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

min215_in_aaa

(212) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(213) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

(214) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaa → remove53_out_aaaa
remove53_in_aaaa → U11_aaaa(p61_in_aaaa)
p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(215) UsableRulesProof (EQUIVALENT transformation)

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

(216) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(217) QReductionProof (EQUIVALENT transformation)

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

remove53_in_aaaa

(218) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AA → U15_AA(min215_out_aaa)
MINSORT1_IN_AA → U15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AA → U15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaa → U7_aaaa(notEq62_in_aa)
p61_in_aaaa → U8_aaaa(notEq62_in_aa)
p61_in_aaaa → U9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aa → U6_aa(notEq62_in_aa)
notEq62_in_aa → notEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaa → U12_aaa(le30_in_aa)
min25_in_aaa → U13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aa → U5_aa(gt44_in_aa)
gt44_in_aa → gt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aa → U4_aa(le30_in_aa)
le30_in_aa → le30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(219) NonTerminationProof (EQUIVALENT transformation)

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

s = MINSORT1_IN_AA evaluates to t =MINSORT1_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

MINSORT1_IN_AA → U15_AA(min215_out_aaa)
with rule MINSORT1_IN_AA → U15_AA(min215_out_aaa) at position [] and matcher [ ]

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
with rule U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa) at position [] and matcher [ ]

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
with rule U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

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


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

(220) NO

(221) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)

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

(222) UsableRulesProof (EQUIVALENT transformation)

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

(223) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(224) PiDPToQDPProof (EQUIVALENT transformation)

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

(225) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(226) 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:

• NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
  The graph contains the following edges 1 > 1, 2 > 2

(227) YES

(228) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(229) UsableRulesProof (EQUIVALENT transformation)

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

(230) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(231) PiDPToQDPProof (SOUND transformation)

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

(232) Obligation:

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

NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)

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

(233) 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:

• NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)
  The graph contains the following edges 1 > 1

(234) YES

(235) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)

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

(236) UsableRulesProof (EQUIVALENT transformation)

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

(237) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)

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

(238) PiDPToQDPProof (SOUND transformation)

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

(239) Obligation:

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

P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T193, T194, notEq62_in_ag(T146))
U9_AGGA(T193, T194, notEq62_out_ag) → P61_IN_AGGA(T193, T194)

The TRS R consists of the following rules:

notEq62_in_ag(s(T161)) → U6_ag(notEq62_in_ag(T161))
notEq62_in_ag(0) → notEq62_out_ag
notEq62_in_ag(s(T171)) → notEq62_out_ag
U6_ag(notEq62_out_ag) → notEq62_out_ag

The set Q consists of the following terms:

notEq62_in_ag(x0)
U6_ag(x0)

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

(240) 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:

• U9_AGGA(T193, T194, notEq62_out_ag) → P61_IN_AGGA(T193, T194)
  The graph contains the following edges 1 >= 1, 2 >= 2

• P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T193, T194, notEq62_in_ag(T146))
  The graph contains the following edges 2 > 1, 2 > 2

(241) YES

(242) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)

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

(243) UsableRulesProof (EQUIVALENT transformation)

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

(244) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(245) PiDPToQDPProof (EQUIVALENT transformation)

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

(246) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(247) 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:

• GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
  The graph contains the following edges 1 > 1, 2 > 2

(248) YES

(249) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)

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

(250) UsableRulesProof (EQUIVALENT transformation)

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

(251) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(252) PiDPToQDPProof (EQUIVALENT transformation)

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

(253) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(254) 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:

• LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
  The graph contains the following edges 1 > 1, 2 > 2

(255) YES

(256) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)

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

(257) UsableRulesProof (EQUIVALENT transformation)

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

(258) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)

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

(259) PiDPToQDPProof (SOUND transformation)

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

(260) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T51, min25_in_gga(T48, T50))
U2_GAG(T51, min25_out_gga(T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71) → U12_gga(T70, le30_in_gg(T70, T71))
min25_in_gga(T97, T98) → U13_gga(T98, gt44_in_gg(T97, T98))
U12_gga(T70, le30_out_gg) → min25_out_gga(T70)
U13_gga(T98, gt44_out_gg) → min25_out_gga(T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg
le30_in_gg(0, 0) → le30_out_gg
gt44_in_gg(s(T109), s(T110)) → U5_gg(gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg
U4_gg(le30_out_gg) → le30_out_gg
U5_gg(gt44_out_gg) → gt44_out_gg

The set Q consists of the following terms:

min25_in_gga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
le30_in_gg(x0, x1)
gt44_in_gg(x0, x1)
U4_gg(x0)
U5_gg(x0)

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

(261) 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:

• U2_GAG(T51, min25_out_gga(T55)) → MIN215_IN_GAG(T55, T51)
  The graph contains the following edges 2 > 1, 1 >= 2

• MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T51, min25_in_gga(T48, T50))
  The graph contains the following edges 2 > 1

(262) YES