Left Termination of the query pattern vlcnd_in_1(g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

vlcnd(N) :- ','(vl(N, Xs), ','(select(X1, Xs, Ys), ','(vl(M, Ys), vlcnd(M)))).
vlcnd(0).
vl(0, L) :- ','(!, eq(L, [])).
vl(N, .(X2, Xs)) :- ','(p(N, P), vl(P, Xs)).
select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys).
select(X, .(X, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

vlcnd(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: vlcnd(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: vlcnd(T1), SCOPE: 1, CLAUSE: 0 | vlcnd(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: vlcnd(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: vlcnd(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(vl(T2, X6), ','(select(X7, X6, X8), ','(vl(X9, X8), vlcnd(X9))))\n\nT2 is ground\nX6 is free; \nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: vl(T2, X6)\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(select(X7, T3, X8), ','(vl(X9, X8), vlcnd(X9)))\n\nX7 is free\nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: vl(T2, X6), SCOPE: 2, CLAUSE: 2 | vl(T2, X6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_2, eq(X12, [])) | vl(0, X6), SCOPE: 2, CLAUSE: 3 | ?_2\n\nX6 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: vl(T2, X6), SCOPE: 2, CLAUSE: 3\n\nT2 is ground\nX6 is free; \nX11 is free; \nvl(T2, X6) !=? vl(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(X12, [])\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(X12, []), SCOPE: 3, CLAUSE: 8\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T4, X22), vl(X22, X24))\n\nT4 is ground\nX6 is free; \nX11 is free; \nX24 is free; \nX22 is free; \nvl(T4, X6) !=? vl(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T4, X22), vl(X22, X24)), SCOPE: 4, CLAUSE: 6 | ','(p(T4, X22), vl(X22, X24)), SCOPE: 4, CLAUSE: 7\n\nT4 is ground\nX6 is free; \nX11 is free; \nX24 is free; \nX22 is free; \nvl(T4, X6) !=? vl(0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T4, X22), vl(X22, X24)), SCOPE: 4, CLAUSE: 7\n\nT4 is ground\nX6 is free; \nX11 is free; \nX24 is free; \nX22 is free; \nvl(T4, X6) !=? vl(0, X11)\np(T4, X22) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: vl(T5, X24)\n\nT5 is ground\nX6 is free; \nX11 is free; \nX24 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\nX6 is free\nX11 is free; \nX22 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: select(X7, T3, X8)\n\nX7 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(vl(X9, T7), vlcnd(X9))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: select(X7, T3, X8), SCOPE: 5, CLAUSE: 4 | select(X7, T3, X8), SCOPE: 5, CLAUSE: 5\n\nX7 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: select(X7, T3, X8), SCOPE: 5, CLAUSE: 4\n\nX7 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: select(X7, T3, X8), SCOPE: 5, CLAUSE: 5\n\nX7 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: select(X39, T10, X40)\n\nX39 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\nX7 is free\nX8 is free; \nX35 is free; \nX36 is free; \nX37 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX7 is free\nX8 is free; \nX43 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: vl(X9, T7)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: vlcnd(T12)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: vl(X9, T7), SCOPE: 6, CLAUSE: 2 | vl(X9, T7), SCOPE: 6, CLAUSE: 3 | ?_6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(!_6, eq(T14, [])) | vl(X9, T7), SCOPE: 6, CLAUSE: 3 | ?_6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: eq(T14, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(T14, []), SCOPE: 7, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: vlcnd(T12), SCOPE: 8, CLAUSE: 0 | vlcnd(T12), SCOPE: 8, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: vlcnd(T12), SCOPE: 8, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: vlcnd(T12), SCOPE: 8, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(vl(T17, X53), ','(select(X54, X53, X55), ','(vl(X56, X55), vlcnd(X56))))\n\nX53 is free\nX54 is free; \nX55 is free; \nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: vl(T17, X53)\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(select(X54, T18, X55), ','(vl(X56, X55), vlcnd(X56)))\n\nX54 is free\nX55 is free; \nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: vl(T17, X53), SCOPE: 9, CLAUSE: 2 | vl(T17, X53), SCOPE: 9, CLAUSE: 3 | ?_9\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_9, eq(X59, [])) | vl(T17, X53), SCOPE: 9, CLAUSE: 3 | ?_9\n\nX53 is free\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: vl(T17, X53), SCOPE: 9, CLAUSE: 3\n\nX53 is free\nX58 is free; \nvl(T17, X53) !=? vl(0, X58)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(X59, [])\n\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: eq(X59, []), SCOPE: 10, CLAUSE: 8\n\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T20, X69), vl(X69, X71))\n\nX53 is free\nX58 is free; \nX71 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T20, X69), vl(X69, X71)), SCOPE: 11, CLAUSE: 6 | ','(p(T20, X69), vl(X69, X71)), SCOPE: 11, CLAUSE: 7\n\nX71 is free\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: vl(0, X71) | ','(p(T20, X69), vl(X69, X71)), SCOPE: 11, CLAUSE: 7\n\nX71 is free\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T20, X69), vl(X69, X71)), SCOPE: 11, CLAUSE: 7\n\nX71 is free\nX69 is free; \np(T20, X69) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: vl(0, X71)\n\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T20, X69), vl(X69, X71)), SCOPE: 11, CLAUSE: 7\n\nX71 is free\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: vl(0, X71), SCOPE: 12, CLAUSE: 2 | vl(0, X71), SCOPE: 12, CLAUSE: 3 | ?_12\n\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(!_12, eq(X74, [])) | vl(0, X71), SCOPE: 12, CLAUSE: 3 | ?_12\n\nX71 is free\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: eq(X74, [])\n\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: eq(X74, []), SCOPE: 13, CLAUSE: 8\n\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: vl(T22, X71)\n\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\nX69 is free\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: vl(T24, X71)\n\nX71 is free\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\nX69 is free\nX81 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", 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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 74 [arrowhead = none ];
74 -> 2;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 75 [arrowhead = none ];
75 -> 3;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 76 [arrowhead = none ];
76 -> 4;

77 [label="EVAL with clause\nvlcnd(X5) :- ','(vl(X5, X6), ','(select(X7, X6, X8), ','(vl(X9, X8), vlcnd(X9)))).\nand substitution\nT1 -> T2\nX5 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 77 [arrowhead = none ];
77 -> 5;

78 [label="EVAL with clause\nvlcnd(0).\nand substitution\nT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 78 [arrowhead = none ];
78 -> 71;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 79 [arrowhead = none ];
79 -> 72;

80 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
5 -> 80 [arrowhead = none ];
80 -> 6;

81 [label="SPLIT 2\nreplacements:\nX6 -> T3", fontsize=14, style = filled, fillcolor = violet];
5 -> 81 [arrowhead = none ];
81 -> 7;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 82 [arrowhead = none ];
82 -> 8;

83 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="SPLIT 2\nreplacements:\nX7 -> T6\nX8 -> T7", fontsize=14, style = filled, fillcolor = violet];
7 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="EVAL with clause\nvl(0, X11) :- ','(!, eq(X11, [])).\nand substitution\nT2 -> 0\nX6 -> X12\nX11 -> X12", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 85 [arrowhead = none ];
85 -> 9;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 86 [arrowhead = none ];
86 -> 10;

87 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 87 [arrowhead = none ];
87 -> 11;

88 [label="EVAL with clause\nvl(X19, .(X20, X21)) :- ','(p(X19, X22), vl(X22, X21)).\nand substitution\nT2 -> T4\nX19 -> T4\nX20 -> X23\nX21 -> X24\nX6 -> .(X23, X24)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 88 [arrowhead = none ];
88 -> 15;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="EVAL with clause\neq(X14, X14).\nand substitution\nX12 -> []\nX14 -> []\nX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 91 [arrowhead = none ];
91 -> 14;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 92 [arrowhead = none ];
92 -> 16;

93 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (vl(T4, X6), vl(0, X11))", fontsize=14, style = filled, fillcolor = white];
16 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="EVAL with clause\np(s(X26), X26).\nand substitution\nX26 -> T5\nT4 -> s(T5)\nX22 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 94 [arrowhead = none ];
94 -> 18;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 95 [arrowhead = none ];
95 -> 19;

96 [label="INSTANCE with matching:\nT2 -> T5\nX6 -> X24", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 96 [arrowhead = none , style=dashed];
96 -> 6[style=dashed];

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 97 [arrowhead = none ];
97 -> 22;

98 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 98 [arrowhead = none ];
98 -> 30;

99 [label="SPLIT 2\nreplacements:\nX9 -> T12", fontsize=14, style = filled, fillcolor = violet];
21 -> 99 [arrowhead = none ];
99 -> 31;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 100 [arrowhead = none ];
100 -> 23;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 101 [arrowhead = none ];
101 -> 24;

102 [label="EVAL with clause\nselect(X35, .(X36, X37), .(X36, X38)) :- select(X35, X37, X38).\nand substitution\nX7 -> X39\nX35 -> X39\nX36 -> T8\nX37 -> T10\nT3 -> .(T8, T10)\nX38 -> X40\nX8 -> .(T8, X40)\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 103 [arrowhead = none ];
103 -> 26;

104 [label="EVAL with clause\nselect(X43, .(X43, X44), X44).\nand substitution\nX7 -> X45\nX43 -> X45\nX44 -> T11\nT3 -> .(X45, T11)\nX8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 104 [arrowhead = none ];
104 -> 27;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 105 [arrowhead = none ];
105 -> 28;

106 [label="INSTANCE with matching:\nX7 -> X39\nT3 -> T10\nX8 -> X40", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 106 [arrowhead = none , style=dashed];
106 -> 20[style=dashed];

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 107 [arrowhead = none ];
107 -> 29;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 108 [arrowhead = none ];
108 -> 32;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 109 [arrowhead = none ];
109 -> 39;

110 [label="EVAL with clause\nvl(0, X47) :- ','(!, eq(X47, [])).\nand substitution\nX9 -> 0\nT7 -> T14\nX47 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 111 [arrowhead = none ];
111 -> 34;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 112 [arrowhead = none ];
112 -> 35;

113 [label="EVAL with clause\neq(X49, X49).\nand substitution\nT14 -> []\nX49 -> []\nT15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 113 [arrowhead = none ];
113 -> 36;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 114 [arrowhead = none ];
114 -> 37;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 115 [arrowhead = none ];
115 -> 38;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 116 [arrowhead = none ];
116 -> 40;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="EVAL with clause\nvlcnd(X52) :- ','(vl(X52, X53), ','(select(X54, X53, X55), ','(vl(X56, X55), vlcnd(X56)))).\nand substitution\nT12 -> T17\nX52 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 118 [arrowhead = none ];
118 -> 42;

119 [label="EVAL with clause\nvlcnd(0).\nand substitution\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 119 [arrowhead = none ];
119 -> 68;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 120 [arrowhead = none ];
120 -> 69;

121 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
42 -> 121 [arrowhead = none ];
121 -> 43;

122 [label="SPLIT 2\nreplacements:\nX53 -> T18", fontsize=14, style = filled, fillcolor = violet];
42 -> 122 [arrowhead = none ];
122 -> 44;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 123 [arrowhead = none ];
123 -> 45;

124 [label="INSTANCE with matching:\nX7 -> X54\nT3 -> T18\nX8 -> X55\nX9 -> X56", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 124 [arrowhead = none , style=dashed];
124 -> 7[style=dashed];

125 [label="EVAL with clause\nvl(0, X58) :- ','(!, eq(X58, [])).\nand substitution\nT17 -> 0\nX53 -> X59\nX58 -> X59", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="EVAL with clause\nvl(X66, .(X67, X68)) :- ','(p(X66, X69), vl(X69, X68)).\nand substitution\nT17 -> T20\nX66 -> T20\nX67 -> X70\nX68 -> X71\nX53 -> .(X70, X71)\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 128 [arrowhead = none ];
128 -> 52;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="EVAL with clause\neq(X61, X61).\nand substitution\nX59 -> []\nX61 -> []\nX62 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 130 [arrowhead = none ];
130 -> 50;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 131 [arrowhead = none ];
131 -> 51;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 132 [arrowhead = none ];
132 -> 53;

133 [label="EVAL with clause\np(0, 0).\nand substitution\nT20 -> 0\nX69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 133 [arrowhead = none ];
133 -> 54;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 134 [arrowhead = none ];
134 -> 55;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 135 [arrowhead = none ];
135 -> 56;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 136 [arrowhead = none ];
136 -> 57;

137 [label="EVAL with clause\np(s(X81), X81).\nand substitution\nX81 -> T24\nT20 -> s(T24)\nX69 -> T24\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 137 [arrowhead = none ];
137 -> 66;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 138 [arrowhead = none ];
138 -> 67;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 139 [arrowhead = none ];
139 -> 58;

140 [label="EVAL with clause\np(s(X79), X79).\nand substitution\nX79 -> T22\nT20 -> s(T22)\nX69 -> T22\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 140 [arrowhead = none ];
140 -> 64;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 141 [arrowhead = none ];
141 -> 65;

142 [label="EVAL with clause\nvl(0, X73) :- ','(!, eq(X73, [])).\nand substitution\nX71 -> X74\nX73 -> X74", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 142 [arrowhead = none ];
142 -> 59;

143 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 143 [arrowhead = none ];
143 -> 60;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 144 [arrowhead = none ];
144 -> 61;

145 [label="EVAL with clause\neq(X76, X76).\nand substitution\nX74 -> []\nX76 -> []\nX77 -> []", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 145 [arrowhead = none ];
145 -> 62;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 146 [arrowhead = none ];
146 -> 63;

147 [label="INSTANCE with matching:\nT17 -> T22\nX53 -> X71", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 147 [arrowhead = none , style=dashed];
147 -> 43[style=dashed];

148 [label="INSTANCE with matching:\nT17 -> T24\nX53 -> X71", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 148 [arrowhead = none , style=dashed];
148 -> 43[style=dashed];

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 149 [arrowhead = none ];
149 -> 70;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 150 [arrowhead = none ];
150 -> 73;

}

(2) Obligation:

Clauses:

vl6(0, []).
vl6(s(T5), .(X23, X24)) :- vl6(T5, X24).
select20(X39, .(T8, T10), .(T8, X40)) :- select20(X39, T10, X40).
select20(X45, .(X45, T11), T11).
p7(X7, T3, X8, X9) :- select20(X7, T3, X8).
p7(T6, T3, T7, X9) :- ','(select20(T6, T3, T7), vl30(X9, T7)).
p7(T6, T3, T7, T17) :- ','(select20(T6, T3, T7), ','(vl30(T17, T7), vl43(T17, X53))).
p7(T6, T3, T7, T17) :- ','(select20(T6, T3, T7), ','(vl30(T17, T7), ','(vl43(T17, T18), p7(X54, T18, X55, X56)))).
p7(T6, T3, T7, 0) :- ','(select20(T6, T3, T7), vl30(0, T7)).
vl43(0, []).
vl43(0, .(X70, [])).
vl43(s(T22), .(X70, X71)) :- vl43(T22, X71).
vl43(s(T24), .(X70, X71)) :- vl43(T24, X71).
vl30(0, []).
vlcnd1(T2) :- vl6(T2, X6).
vlcnd1(T2) :- ','(vl6(T2, T3), select20(X7, T3, X8)).
vlcnd1(T2) :- ','(vl6(T2, T3), ','(select20(T6, T3, T7), vl30(X9, T7))).
vlcnd1(T2) :- ','(vl6(T2, T3), ','(select20(T6, T3, T7), ','(vl30(T17, T7), vl43(T17, X53)))).
vlcnd1(T2) :- ','(vl6(T2, T3), ','(select20(T6, T3, T7), ','(vl30(T17, T7), ','(vl43(T17, T18), p7(X54, T18, X55, X56))))).
vlcnd1(T2) :- ','(vl6(T2, T3), ','(select20(T6, T3, T7), vl30(0, T7))).
vlcnd1(0).

Queries:

vlcnd1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
vlcnd1_in: (b)
vl6_in: (b,f)
select20_in: (f,b,f)
vl43_in: (b,f)
p7_in: (f,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)

(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:

VLCND1_IN_G(T2) → U15_G(T2, vl6_in_ga(T2, X6))
VLCND1_IN_G(T2) → VL6_IN_GA(T2, X6)
VL6_IN_GA(s(T5), .(X23, X24)) → U1_GA(T5, X23, X24, vl6_in_ga(T5, X24))
VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)
VLCND1_IN_G(T2) → U16_G(T2, vl6_in_ga(T2, T3))
U16_G(T2, vl6_out_ga(T2, T3)) → U17_G(T2, select20_in_aga(X7, T3, X8))
U16_G(T2, vl6_out_ga(T2, T3)) → SELECT20_IN_AGA(X7, T3, X8)
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → U2_AGA(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)
U16_G(T2, vl6_out_ga(T2, T3)) → U18_G(T2, select20_in_aga(T6, T3, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → U19_G(T2, vl30_in_ag(X9, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U20_G(T2, vl30_in_ag(T17, T7))
U20_G(T2, vl30_out_ag(T17, T7)) → U21_G(T2, vl43_in_ga(T17, X53))
U20_G(T2, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
VL43_IN_GA(s(T22), .(X70, X71)) → U13_GA(T22, X70, X71, vl43_in_ga(T22, X71))
VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)
VL43_IN_GA(s(T24), .(X70, X71)) → U14_GA(T24, X70, X71, vl43_in_ga(T24, X71))
U20_G(T2, vl30_out_ag(T17, T7)) → U22_G(T2, vl43_in_ga(T17, T18))
U22_G(T2, vl43_out_ga(T17, T18)) → U23_G(T2, p7_in_agaa(X54, T18, X55, X56))
U22_G(T2, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(X7, T3, X8, X9) → U3_AGAA(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
P7_IN_AGAA(X7, T3, X8, X9) → SELECT20_IN_AGA(X7, T3, X8)
P7_IN_AGAA(T6, T3, T7, X9) → U4_AGAA(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_AGAA(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(T17, T7)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_AGAA(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(T6, T3, T7, 0) → U11_AGAA(T6, T3, T7, select20_in_aga(T6, T3, T7))
P7_IN_AGAA(T6, T3, T7, 0) → SELECT20_IN_AGA(T6, T3, T7)
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_AGAA(T6, T3, T7, vl30_in_gg(0, T7))
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U24_G(T2, vl30_in_gg(0, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U15_G(x1, x2)  =  U15_G(x1, x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
VL30_IN_AG(x1, x2)  =  VL30_IN_AG(x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
U22_G(x1, x2)  =  U22_G(x1, x2)
U23_G(x1, x2)  =  U23_G(x1, x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5)  =  U3_AGAA(x2, x5)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x2, x5)
U5_AGAA(x1, x2, x3, x4, x5)  =  U5_AGAA(x2, x3, x5)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x3, x5)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x2, x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x3, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x2, x3, x5)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x2, x4)
U12_AGAA(x1, x2, x3, x4)  =  U12_AGAA(x2, x3, x4)
VL30_IN_GG(x1, x2)  =  VL30_IN_GG(x1, x2)
U24_G(x1, x2)  =  U24_G(x1, x2)

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

(6) Obligation:

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

VLCND1_IN_G(T2) → U15_G(T2, vl6_in_ga(T2, X6))
VLCND1_IN_G(T2) → VL6_IN_GA(T2, X6)
VL6_IN_GA(s(T5), .(X23, X24)) → U1_GA(T5, X23, X24, vl6_in_ga(T5, X24))
VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)
VLCND1_IN_G(T2) → U16_G(T2, vl6_in_ga(T2, T3))
U16_G(T2, vl6_out_ga(T2, T3)) → U17_G(T2, select20_in_aga(X7, T3, X8))
U16_G(T2, vl6_out_ga(T2, T3)) → SELECT20_IN_AGA(X7, T3, X8)
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → U2_AGA(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)
U16_G(T2, vl6_out_ga(T2, T3)) → U18_G(T2, select20_in_aga(T6, T3, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → U19_G(T2, vl30_in_ag(X9, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U20_G(T2, vl30_in_ag(T17, T7))
U20_G(T2, vl30_out_ag(T17, T7)) → U21_G(T2, vl43_in_ga(T17, X53))
U20_G(T2, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
VL43_IN_GA(s(T22), .(X70, X71)) → U13_GA(T22, X70, X71, vl43_in_ga(T22, X71))
VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)
VL43_IN_GA(s(T24), .(X70, X71)) → U14_GA(T24, X70, X71, vl43_in_ga(T24, X71))
U20_G(T2, vl30_out_ag(T17, T7)) → U22_G(T2, vl43_in_ga(T17, T18))
U22_G(T2, vl43_out_ga(T17, T18)) → U23_G(T2, p7_in_agaa(X54, T18, X55, X56))
U22_G(T2, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(X7, T3, X8, X9) → U3_AGAA(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
P7_IN_AGAA(X7, T3, X8, X9) → SELECT20_IN_AGA(X7, T3, X8)
P7_IN_AGAA(T6, T3, T7, X9) → U4_AGAA(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_AGAA(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(T17, T7)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_AGAA(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(T6, T3, T7, 0) → U11_AGAA(T6, T3, T7, select20_in_aga(T6, T3, T7))
P7_IN_AGAA(T6, T3, T7, 0) → SELECT20_IN_AGA(T6, T3, T7)
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_AGAA(T6, T3, T7, vl30_in_gg(0, T7))
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U24_G(T2, vl30_in_gg(0, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U15_G(x1, x2)  =  U15_G(x1, x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
VL30_IN_AG(x1, x2)  =  VL30_IN_AG(x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
U22_G(x1, x2)  =  U22_G(x1, x2)
U23_G(x1, x2)  =  U23_G(x1, x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5)  =  U3_AGAA(x2, x5)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x2, x5)
U5_AGAA(x1, x2, x3, x4, x5)  =  U5_AGAA(x2, x3, x5)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x3, x5)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x2, x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x3, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x2, x3, x5)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x2, x4)
U12_AGAA(x1, x2, x3, x4)  =  U12_AGAA(x2, x3, x4)
VL30_IN_GG(x1, x2)  =  VL30_IN_GG(x1, x2)
U24_G(x1, x2)  =  U24_G(x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 33 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

VL43_IN_GA(s(T22)) → VL43_IN_GA(T22)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• VL43_IN_GA(s(T22)) → VL43_IN_GA(T22)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)

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:

SELECT20_IN_AGA(.(T10)) → SELECT20_IN_AGA(T10)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• SELECT20_IN_AGA(.(T10)) → SELECT20_IN_AGA(T10)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x3, x5)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)

The TRS R consists of the following rules:

select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
vl30_in_ag(0, []) → vl30_out_ag(0, [])
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x3, x5)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U6_AGAA(T3, select20_out_aga(T3, T7)) → U7_AGAA(T3, T7, vl30_in_ag(T7))
U7_AGAA(T3, T7, vl30_out_ag(T17, T7)) → U9_AGAA(T3, T7, vl43_in_ga(T17))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
vl30_in_ag([]) → vl30_out_ag(0, [])
vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
vl43_in_ga(s(T22)) → U13_ga(T22, vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(T24, vl43_in_ga(T24))
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))
U13_ga(T22, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X71))
U14_ga(T24, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X71))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U6_AGAA(T3, select20_out_aga(T3, T7)) → U7_AGAA(T3, T7, vl30_in_ag(T7)) at position [2] we obtained the following new rules [LPAR04]:

U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))

(29) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U7_AGAA(T3, T7, vl30_out_ag(T17, T7)) → U9_AGAA(T3, T7, vl43_in_ga(T17))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
vl30_in_ag([]) → vl30_out_ag(0, [])
vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
vl43_in_ga(s(T22)) → U13_ga(T22, vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(T24, vl43_in_ga(T24))
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))
U13_ga(T22, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X71))
U14_ga(T24, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X71))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(30) 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.

(31) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U7_AGAA(T3, T7, vl30_out_ag(T17, T7)) → U9_AGAA(T3, T7, vl43_in_ga(T17))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
vl43_in_ga(s(T22)) → U13_ga(T22, vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(T24, vl43_in_ga(T24))
U14_ga(T24, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X71))
U13_ga(T22, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X71))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(32) 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].

vl30_in_ag(x0)

(33) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U7_AGAA(T3, T7, vl30_out_ag(T17, T7)) → U9_AGAA(T3, T7, vl43_in_ga(T17))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
vl43_in_ga(s(T22)) → U13_ga(T22, vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(T24, vl43_in_ga(T24))
U14_ga(T24, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X71))
U13_ga(T22, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X71))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(34) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_AGAA(T3, T7, vl30_out_ag(T17, T7)) → U9_AGAA(T3, T7, vl43_in_ga(T17)) we obtained the following new rules [LPAR04]:

U7_AGAA(z0, [], vl30_out_ag(0, [])) → U9_AGAA(z0, [], vl43_in_ga(0))

(35) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(z0, [], vl30_out_ag(0, [])) → U9_AGAA(z0, [], vl43_in_ga(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
vl43_in_ga(s(T22)) → U13_ga(T22, vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(T24, vl43_in_ga(T24))
U14_ga(T24, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X71))
U13_ga(T22, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X71))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(36) 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.

(37) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(z0, [], vl30_out_ag(0, [])) → U9_AGAA(z0, [], vl43_in_ga(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)
U13_ga(x0, x1)
U14_ga(x0, x1)

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

(38) 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_ga(x0, x1)
U14_ga(x0, x1)

(39) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(z0, [], vl30_out_ag(0, [])) → U9_AGAA(z0, [], vl43_in_ga(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)

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

(40) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U7_AGAA(z0, [], vl30_out_ag(0, [])) → U9_AGAA(z0, [], vl43_in_ga(0)) at position [2] we obtained the following new rules [LPAR04]:

U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

(41) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(0, [])
vl43_in_ga(0) → vl43_out_ga(0, .([]))
select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)

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

(42) 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.

(43) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl43_in_ga(x0)
U2_aga(x0, x1)

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

(44) 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].

vl43_in_ga(x0)

(45) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18)
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(T3, T7, vl43_out_ga(T17, T18)) → P7_IN_AGAA(T18) we obtained the following new rules [LPAR04]:

U9_AGAA(z0, [], vl43_out_ga(0, [])) → P7_IN_AGAA([])
U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))

(47) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
U9_AGAA(z0, [], vl43_out_ga(0, [])) → P7_IN_AGAA([])
U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(48) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P7_IN_AGAA(T3) → U6_AGAA(T3, select20_in_aga(T3)) at position [1] we obtained the following new rules [LPAR04]:

P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), U2_aga(x0, select20_in_aga(x0)))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))

(49) Obligation:

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

U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
U9_AGAA(z0, [], vl43_out_ga(0, [])) → P7_IN_AGAA([])
U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), U2_aga(x0, select20_in_aga(x0)))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(51) Obligation:

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

U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), U2_aga(x0, select20_in_aga(x0)))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(52) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), U2_aga(x0, select20_in_aga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

P7_IN_AGAA(.(.(x0))) → U6_AGAA(.(.(x0)), U2_aga(.(x0), U2_aga(x0, select20_in_aga(x0))))
P7_IN_AGAA(.(.(x0))) → U6_AGAA(.(.(x0)), U2_aga(.(x0), select20_out_aga(.(x0), x0)))

(53) Obligation:

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

U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))
P7_IN_AGAA(.(.(x0))) → U6_AGAA(.(.(x0)), U2_aga(.(x0), U2_aga(x0, select20_in_aga(x0))))
P7_IN_AGAA(.(.(x0))) → U6_AGAA(.(.(x0)), U2_aga(.(x0), select20_out_aga(.(x0), x0)))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(55) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(T10, select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(.(T11), T11)
U2_aga(T10, select20_out_aga(T10, X40)) → select20_out_aga(.(T10), .(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(56) 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.

(57) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

R is empty.
The set Q consists of the following terms:

select20_in_aga(x0)
U2_aga(x0, x1)

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

(58) 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].

select20_in_aga(x0)
U2_aga(x0, x1)

(59) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))

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

(60) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P7_IN_AGAA(.(x0)) → U6_AGAA(.(x0), select20_out_aga(.(x0), x0)) we obtained the following new rules [LPAR04]:

P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))

(61) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, []))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))

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

(62) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_AGAA(y0, select20_out_aga(y0, [])) → U7_AGAA(y0, [], vl30_out_ag(0, [])) we obtained the following new rules [LPAR04]:

U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, []))

(63) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([])))
P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))
U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, []))

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

(64) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_AGAA(y0, [], vl30_out_ag(0, [])) → U9_AGAA(y0, [], vl43_out_ga(0, .([]))) we obtained the following new rules [LPAR04]:

U7_AGAA(.([]), [], vl30_out_ag(0, [])) → U9_AGAA(.([]), [], vl43_out_ga(0, .([])))

(65) Obligation:

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

U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))
U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, []))
U7_AGAA(.([]), [], vl30_out_ag(0, [])) → U9_AGAA(.([]), [], vl43_out_ga(0, .([])))

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

(66) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(z0, [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([])) we obtained the following new rules [LPAR04]:

U9_AGAA(.([]), [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))

(67) Obligation:

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

P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))
U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, []))
U7_AGAA(.([]), [], vl30_out_ag(0, [])) → U9_AGAA(.([]), [], vl43_out_ga(0, .([])))
U9_AGAA(.([]), [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))

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

(68) 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 = U6_AGAA(.([]), select20_out_aga(.([]), [])) evaluates to t =U6_AGAA(.([]), select20_out_aga(.([]), []))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, []))
with rule U6_AGAA(.([]), select20_out_aga(.([]), [])) → U7_AGAA(.([]), [], vl30_out_ag(0, [])) at position [] and matcher [ ]

U7_AGAA(.([]), [], vl30_out_ag(0, [])) → U9_AGAA(.([]), [], vl43_out_ga(0, .([])))
with rule U7_AGAA(.([]), [], vl30_out_ag(0, [])) → U9_AGAA(.([]), [], vl43_out_ga(0, .([]))) at position [] and matcher [ ]

U9_AGAA(.([]), [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([]))
with rule U9_AGAA(.([]), [], vl43_out_ga(0, .([]))) → P7_IN_AGAA(.([])) at position [] and matcher [ ]

P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))
with rule P7_IN_AGAA(.([])) → U6_AGAA(.([]), select20_out_aga(.([]), []))

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.

(69) FALSE

(70) Obligation:

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

VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g(x1)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x2, x3)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x2, x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x2, x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x2, x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x2, x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x2, x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x2, x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x2, x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x2, x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)

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

(71) UsableRulesProof (EQUIVALENT transformation)

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

(72) Obligation:

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

VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)

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

(73) PiDPToQDPProof (SOUND transformation)

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

(74) Obligation:

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

VL6_IN_GA(s(T5)) → VL6_IN_GA(T5)

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

(75) 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:

• VL6_IN_GA(s(T5)) → VL6_IN_GA(T5)
  The graph contains the following edges 1 > 1

(76) TRUE

(77) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
vlcnd1_in: (b)
vl6_in: (b,f)
select20_in: (f,b,f)
vl43_in: (b,f)
p7_in: (f,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(78) Obligation:

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

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)

(79) 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:

VLCND1_IN_G(T2) → U15_G(T2, vl6_in_ga(T2, X6))
VLCND1_IN_G(T2) → VL6_IN_GA(T2, X6)
VL6_IN_GA(s(T5), .(X23, X24)) → U1_GA(T5, X23, X24, vl6_in_ga(T5, X24))
VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)
VLCND1_IN_G(T2) → U16_G(T2, vl6_in_ga(T2, T3))
U16_G(T2, vl6_out_ga(T2, T3)) → U17_G(T2, select20_in_aga(X7, T3, X8))
U16_G(T2, vl6_out_ga(T2, T3)) → SELECT20_IN_AGA(X7, T3, X8)
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → U2_AGA(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)
U16_G(T2, vl6_out_ga(T2, T3)) → U18_G(T2, select20_in_aga(T6, T3, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → U19_G(T2, vl30_in_ag(X9, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U20_G(T2, vl30_in_ag(T17, T7))
U20_G(T2, vl30_out_ag(T17, T7)) → U21_G(T2, vl43_in_ga(T17, X53))
U20_G(T2, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
VL43_IN_GA(s(T22), .(X70, X71)) → U13_GA(T22, X70, X71, vl43_in_ga(T22, X71))
VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)
VL43_IN_GA(s(T24), .(X70, X71)) → U14_GA(T24, X70, X71, vl43_in_ga(T24, X71))
U20_G(T2, vl30_out_ag(T17, T7)) → U22_G(T2, vl43_in_ga(T17, T18))
U22_G(T2, vl43_out_ga(T17, T18)) → U23_G(T2, p7_in_agaa(X54, T18, X55, X56))
U22_G(T2, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(X7, T3, X8, X9) → U3_AGAA(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
P7_IN_AGAA(X7, T3, X8, X9) → SELECT20_IN_AGA(X7, T3, X8)
P7_IN_AGAA(T6, T3, T7, X9) → U4_AGAA(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_AGAA(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(T17, T7)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_AGAA(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(T6, T3, T7, 0) → U11_AGAA(T6, T3, T7, select20_in_aga(T6, T3, T7))
P7_IN_AGAA(T6, T3, T7, 0) → SELECT20_IN_AGA(T6, T3, T7)
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_AGAA(T6, T3, T7, vl30_in_gg(0, T7))
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U24_G(T2, vl30_in_gg(0, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U15_G(x1, x2)  =  U15_G(x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
VL30_IN_AG(x1, x2)  =  VL30_IN_AG(x2)
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1, x2)  =  U21_G(x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x4)
U22_G(x1, x2)  =  U22_G(x2)
U23_G(x1, x2)  =  U23_G(x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5)  =  U3_AGAA(x5)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x5)
U5_AGAA(x1, x2, x3, x4, x5)  =  U5_AGAA(x3, x5)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x3, x5)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x3, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x3, x5)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x4)
U12_AGAA(x1, x2, x3, x4)  =  U12_AGAA(x3, x4)
VL30_IN_GG(x1, x2)  =  VL30_IN_GG(x1, x2)
U24_G(x1, x2)  =  U24_G(x2)

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

(80) Obligation:

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

VLCND1_IN_G(T2) → U15_G(T2, vl6_in_ga(T2, X6))
VLCND1_IN_G(T2) → VL6_IN_GA(T2, X6)
VL6_IN_GA(s(T5), .(X23, X24)) → U1_GA(T5, X23, X24, vl6_in_ga(T5, X24))
VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)
VLCND1_IN_G(T2) → U16_G(T2, vl6_in_ga(T2, T3))
U16_G(T2, vl6_out_ga(T2, T3)) → U17_G(T2, select20_in_aga(X7, T3, X8))
U16_G(T2, vl6_out_ga(T2, T3)) → SELECT20_IN_AGA(X7, T3, X8)
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → U2_AGA(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)
U16_G(T2, vl6_out_ga(T2, T3)) → U18_G(T2, select20_in_aga(T6, T3, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → U19_G(T2, vl30_in_ag(X9, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U20_G(T2, vl30_in_ag(T17, T7))
U20_G(T2, vl30_out_ag(T17, T7)) → U21_G(T2, vl43_in_ga(T17, X53))
U20_G(T2, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
VL43_IN_GA(s(T22), .(X70, X71)) → U13_GA(T22, X70, X71, vl43_in_ga(T22, X71))
VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)
VL43_IN_GA(s(T24), .(X70, X71)) → U14_GA(T24, X70, X71, vl43_in_ga(T24, X71))
U20_G(T2, vl30_out_ag(T17, T7)) → U22_G(T2, vl43_in_ga(T17, T18))
U22_G(T2, vl43_out_ga(T17, T18)) → U23_G(T2, p7_in_agaa(X54, T18, X55, X56))
U22_G(T2, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(X7, T3, X8, X9) → U3_AGAA(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
P7_IN_AGAA(X7, T3, X8, X9) → SELECT20_IN_AGA(X7, T3, X8)
P7_IN_AGAA(T6, T3, T7, X9) → U4_AGAA(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_AGAA(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U4_AGAA(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(X9, T7)
P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → VL30_IN_AG(T17, T7)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → VL43_IN_GA(T17, X53)
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_AGAA(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)
P7_IN_AGAA(T6, T3, T7, 0) → U11_AGAA(T6, T3, T7, select20_in_aga(T6, T3, T7))
P7_IN_AGAA(T6, T3, T7, 0) → SELECT20_IN_AGA(T6, T3, T7)
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_AGAA(T6, T3, T7, vl30_in_gg(0, T7))
U11_AGAA(T6, T3, T7, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)
U18_G(T2, select20_out_aga(T6, T3, T7)) → U24_G(T2, vl30_in_gg(0, T7))
U18_G(T2, select20_out_aga(T6, T3, T7)) → VL30_IN_GG(0, T7)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U15_G(x1, x2)  =  U15_G(x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
VL30_IN_AG(x1, x2)  =  VL30_IN_AG(x2)
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1, x2)  =  U21_G(x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x4)
U22_G(x1, x2)  =  U22_G(x2)
U23_G(x1, x2)  =  U23_G(x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5)  =  U3_AGAA(x5)
U4_AGAA(x1, x2, x3, x4, x5)  =  U4_AGAA(x5)
U5_AGAA(x1, x2, x3, x4, x5)  =  U5_AGAA(x3, x5)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x3, x5)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x3, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x3, x5)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x4)
U12_AGAA(x1, x2, x3, x4)  =  U12_AGAA(x3, x4)
VL30_IN_GG(x1, x2)  =  VL30_IN_GG(x1, x2)
U24_G(x1, x2)  =  U24_G(x2)

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

(81) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 33 less nodes.

(82) Complex Obligation (AND)

(83) Obligation:

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

VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

VL43_IN_GA(s(T22), .(X70, X71)) → VL43_IN_GA(T22, X71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
VL43_IN_GA(x1, x2)  =  VL43_IN_GA(x1)

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

VL43_IN_GA(s(T22)) → VL43_IN_GA(T22)

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

(88) 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:

• VL43_IN_GA(s(T22)) → VL43_IN_GA(T22)
  The graph contains the following edges 1 > 1

(89) TRUE

(90) Obligation:

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

SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)

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

(91) UsableRulesProof (EQUIVALENT transformation)

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

(92) Obligation:

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

SELECT20_IN_AGA(X39, .(T8, T10), .(T8, X40)) → SELECT20_IN_AGA(X39, T10, X40)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SELECT20_IN_AGA(x1, x2, x3)  =  SELECT20_IN_AGA(x2)

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

(93) PiDPToQDPProof (SOUND transformation)

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

(94) Obligation:

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

SELECT20_IN_AGA(.(T10)) → SELECT20_IN_AGA(T10)

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

(95) 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:

• SELECT20_IN_AGA(.(T10)) → SELECT20_IN_AGA(T10)
  The graph contains the following edges 1 > 1

(96) TRUE

(97) Obligation:

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

P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x3, x5)

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

(98) UsableRulesProof (EQUIVALENT transformation)

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

(99) Obligation:

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

P7_IN_AGAA(T6, T3, T7, T17) → U6_AGAA(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_AGAA(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_AGAA(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_AGAA(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_AGAA(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_AGAA(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → P7_IN_AGAA(X54, T18, X55, X56)

The TRS R consists of the following rules:

select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
vl30_in_ag(0, []) → vl30_out_ag(0, [])
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
P7_IN_AGAA(x1, x2, x3, x4)  =  P7_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x3, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x3, x5)

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

(100) PiDPToQDPProof (SOUND transformation)

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

(101) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(T11)
vl30_in_ag([]) → vl30_out_ag(0)
vl43_in_ga(0) → vl43_out_ga([])
vl43_in_ga(0) → vl43_out_ga(.([]))
vl43_in_ga(s(T22)) → U13_ga(vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(vl43_in_ga(T24))
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))
U13_ga(vl43_out_ga(X71)) → vl43_out_ga(.(X71))
U14_ga(vl43_out_ga(X71)) → vl43_out_ga(.(X71))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)
U13_ga(x0)
U14_ga(x0)

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

(102) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

vl43_in_ga(s(T22)) → U13_ga(vl43_in_ga(T22))
vl43_in_ga(s(T24)) → U14_ga(vl43_in_ga(T24))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁)) = 2·x₁   
POL(0) = 0   
POL(P7_IN_AGAA(x₁)) = 2·x₁   
POL(U13_ga(x₁)) = 2·x₁   
POL(U14_ga(x₁)) = 2·x₁   
POL(U2_aga(x₁)) = 2·x₁   
POL(U6_AGAA(x₁)) = 2·x₁   
POL(U7_AGAA(x₁, x₂)) = x₁ + x₂   
POL(U9_AGAA(x₁, x₂)) = x₁ + 2·x₂   
POL([]) = 0   
POL(s(x₁)) = 1 + 2·x₁   
POL(select20_in_aga(x₁)) = x₁   
POL(select20_out_aga(x₁)) = x₁   
POL(vl30_in_ag(x₁)) = x₁   
POL(vl30_out_ag(x₁)) = 2·x₁   
POL(vl43_in_ga(x₁)) = x₁   
POL(vl43_out_ga(x₁)) = x₁   

(103) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga([])
vl43_in_ga(0) → vl43_out_ga(.([]))
U14_ga(vl43_out_ga(X71)) → vl43_out_ga(.(X71))
U13_ga(vl43_out_ga(X71)) → vl43_out_ga(.(X71))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(T11)
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)
U13_ga(x0)
U14_ga(x0)

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

(104) 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.

(105) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga([])
vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(T11)
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)
U13_ga(x0)
U14_ga(x0)

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

(106) 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_ga(x0)
U14_ga(x0)

(107) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga([])
vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(T11)
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)

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

(108) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

vl43_in_ga(0) → vl43_out_ga([])

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁)) = 1 + x₁   
POL(0) = 0   
POL(P7_IN_AGAA(x₁)) = 2·x₁   
POL(U2_aga(x₁)) = 1 + x₁   
POL(U6_AGAA(x₁)) = 2·x₁   
POL(U7_AGAA(x₁, x₂)) = x₁ + x₂   
POL(U9_AGAA(x₁, x₂)) = x₁ + x₂   
POL([]) = 0   
POL(select20_in_aga(x₁)) = x₁   
POL(select20_out_aga(x₁)) = 1 + x₁   
POL(vl30_in_ag(x₁)) = 2 + x₁   
POL(vl30_out_ag(x₁)) = 2 + 2·x₁   
POL(vl43_in_ga(x₁)) = 2 + 2·x₁   
POL(vl43_out_ga(x₁)) = 2·x₁   

(109) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))
select20_in_aga(.(T11)) → select20_out_aga(T11)
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)

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

(110) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

select20_in_aga(.(T10)) → U2_aga(select20_in_aga(T10))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁)) = 1 + 2·x₁   
POL(0) = 0   
POL(P7_IN_AGAA(x₁)) = 2·x₁   
POL(U2_aga(x₁)) = 2·x₁   
POL(U6_AGAA(x₁)) = x₁   
POL(U7_AGAA(x₁, x₂)) = 1 + x₁ + x₂   
POL(U9_AGAA(x₁, x₂)) = x₁ + 2·x₂   
POL([]) = 0   
POL(select20_in_aga(x₁)) = 2·x₁   
POL(select20_out_aga(x₁)) = 2 + 2·x₁   
POL(vl30_in_ag(x₁)) = 1 + x₁   
POL(vl30_out_ag(x₁)) = 1 + 2·x₁   
POL(vl43_in_ga(x₁)) = 1 + x₁   
POL(vl43_out_ga(x₁)) = x₁   

(111) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T11)) → select20_out_aga(T11)
U2_aga(select20_out_aga(X40)) → select20_out_aga(.(X40))

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)

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

(112) 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.

(113) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T11)) → select20_out_aga(T11)

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)
U2_aga(x0)

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

(114) 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].

U2_aga(x0)

(115) Obligation:

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

P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3))
U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T11)) → select20_out_aga(T11)

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)

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

(116) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P7_IN_AGAA(T3) → U6_AGAA(select20_in_aga(T3)) at position [0] we obtained the following new rules [LPAR04]:

P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))

(117) Obligation:

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

U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)
select20_in_aga(.(T11)) → select20_out_aga(T11)

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)

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

(118) 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.

(119) Obligation:

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

U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)

The set Q consists of the following terms:

select20_in_aga(x0)
vl30_in_ag(x0)
vl43_in_ga(x0)

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

(120) 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].

select20_in_aga(x0)

(121) Obligation:

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

U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7))
U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)

The set Q consists of the following terms:

vl30_in_ag(x0)
vl43_in_ga(x0)

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

(122) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U6_AGAA(select20_out_aga(T7)) → U7_AGAA(T7, vl30_in_ag(T7)) at position [1] we obtained the following new rules [LPAR04]:

U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))

(123) Obligation:

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

U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))
vl30_in_ag([]) → vl30_out_ag(0)

The set Q consists of the following terms:

vl30_in_ag(x0)
vl43_in_ga(x0)

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

(124) 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.

(125) Obligation:

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

U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))

The set Q consists of the following terms:

vl30_in_ag(x0)
vl43_in_ga(x0)

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

(126) 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].

vl30_in_ag(x0)

(127) Obligation:

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

U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17))
U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))

The set Q consists of the following terms:

vl43_in_ga(x0)

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

(128) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U7_AGAA(T7, vl30_out_ag(T17)) → U9_AGAA(T7, vl43_in_ga(T17)) at position [1] we obtained the following new rules [LPAR04]:

U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))

(129) Obligation:

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

U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))

The TRS R consists of the following rules:

vl43_in_ga(0) → vl43_out_ga(.([]))

The set Q consists of the following terms:

vl43_in_ga(x0)

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

(130) 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.

(131) Obligation:

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

U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))

R is empty.
The set Q consists of the following terms:

vl43_in_ga(x0)

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

(132) 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].

vl43_in_ga(x0)

(133) Obligation:

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

U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18)
P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))

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

(134) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(T7, vl43_out_ga(T18)) → P7_IN_AGAA(T18) we obtained the following new rules [LPAR04]:

U9_AGAA(z0, vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))

(135) Obligation:

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

P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0))
U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))
U9_AGAA(z0, vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))

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

(136) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule P7_IN_AGAA(.(x0)) → U6_AGAA(select20_out_aga(x0)) we obtained the following new rules [LPAR04]:

P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([]))

(137) Obligation:

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

U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([])))
U9_AGAA(z0, vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([]))

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

(138) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_AGAA(y0, vl30_out_ag(0)) → U9_AGAA(y0, vl43_out_ga(.([]))) we obtained the following new rules [LPAR04]:

U7_AGAA([], vl30_out_ag(0)) → U9_AGAA([], vl43_out_ga(.([])))

(139) Obligation:

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

U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
U9_AGAA(z0, vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))
P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([]))
U7_AGAA([], vl30_out_ag(0)) → U9_AGAA([], vl43_out_ga(.([])))

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

(140) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(z0, vl43_out_ga(.([]))) → P7_IN_AGAA(.([])) we obtained the following new rules [LPAR04]:

U9_AGAA([], vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))

(141) Obligation:

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

U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([]))
U7_AGAA([], vl30_out_ag(0)) → U9_AGAA([], vl43_out_ga(.([])))
U9_AGAA([], vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))

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

(142) 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 = U7_AGAA([], vl30_out_ag(0)) evaluates to t =U7_AGAA([], vl30_out_ag(0))

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U7_AGAA([], vl30_out_ag(0)) → U9_AGAA([], vl43_out_ga(.([])))
with rule U7_AGAA([], vl30_out_ag(0)) → U9_AGAA([], vl43_out_ga(.([]))) at position [] and matcher [ ]

U9_AGAA([], vl43_out_ga(.([]))) → P7_IN_AGAA(.([]))
with rule U9_AGAA([], vl43_out_ga(.([]))) → P7_IN_AGAA(.([])) at position [] and matcher [ ]

P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([]))
with rule P7_IN_AGAA(.([])) → U6_AGAA(select20_out_aga([])) at position [] and matcher [ ]

U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))
with rule U6_AGAA(select20_out_aga([])) → U7_AGAA([], vl30_out_ag(0))

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.

(143) FALSE

(144) Obligation:

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

VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)

The TRS R consists of the following rules:

vlcnd1_in_g(T2) → U15_g(T2, vl6_in_ga(T2, X6))
vl6_in_ga(0, []) → vl6_out_ga(0, [])
vl6_in_ga(s(T5), .(X23, X24)) → U1_ga(T5, X23, X24, vl6_in_ga(T5, X24))
U1_ga(T5, X23, X24, vl6_out_ga(T5, X24)) → vl6_out_ga(s(T5), .(X23, X24))
U15_g(T2, vl6_out_ga(T2, X6)) → vlcnd1_out_g(T2)
vlcnd1_in_g(T2) → U16_g(T2, vl6_in_ga(T2, T3))
U16_g(T2, vl6_out_ga(T2, T3)) → U17_g(T2, select20_in_aga(X7, T3, X8))
select20_in_aga(X39, .(T8, T10), .(T8, X40)) → U2_aga(X39, T8, T10, X40, select20_in_aga(X39, T10, X40))
select20_in_aga(X45, .(X45, T11), T11) → select20_out_aga(X45, .(X45, T11), T11)
U2_aga(X39, T8, T10, X40, select20_out_aga(X39, T10, X40)) → select20_out_aga(X39, .(T8, T10), .(T8, X40))
U17_g(T2, select20_out_aga(X7, T3, X8)) → vlcnd1_out_g(T2)
U16_g(T2, vl6_out_ga(T2, T3)) → U18_g(T2, select20_in_aga(T6, T3, T7))
U18_g(T2, select20_out_aga(T6, T3, T7)) → U19_g(T2, vl30_in_ag(X9, T7))
vl30_in_ag(0, []) → vl30_out_ag(0, [])
U19_g(T2, vl30_out_ag(X9, T7)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U20_g(T2, vl30_in_ag(T17, T7))
U20_g(T2, vl30_out_ag(T17, T7)) → U21_g(T2, vl43_in_ga(T17, X53))
vl43_in_ga(0, []) → vl43_out_ga(0, [])
vl43_in_ga(0, .(X70, [])) → vl43_out_ga(0, .(X70, []))
vl43_in_ga(s(T22), .(X70, X71)) → U13_ga(T22, X70, X71, vl43_in_ga(T22, X71))
vl43_in_ga(s(T24), .(X70, X71)) → U14_ga(T24, X70, X71, vl43_in_ga(T24, X71))
U14_ga(T24, X70, X71, vl43_out_ga(T24, X71)) → vl43_out_ga(s(T24), .(X70, X71))
U13_ga(T22, X70, X71, vl43_out_ga(T22, X71)) → vl43_out_ga(s(T22), .(X70, X71))
U21_g(T2, vl43_out_ga(T17, X53)) → vlcnd1_out_g(T2)
U20_g(T2, vl30_out_ag(T17, T7)) → U22_g(T2, vl43_in_ga(T17, T18))
U22_g(T2, vl43_out_ga(T17, T18)) → U23_g(T2, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(X7, T3, X8, X9) → U3_agaa(X7, T3, X8, X9, select20_in_aga(X7, T3, X8))
U3_agaa(X7, T3, X8, X9, select20_out_aga(X7, T3, X8)) → p7_out_agaa(X7, T3, X8, X9)
p7_in_agaa(T6, T3, T7, X9) → U4_agaa(T6, T3, T7, X9, select20_in_aga(T6, T3, T7))
U4_agaa(T6, T3, T7, X9, select20_out_aga(T6, T3, T7)) → U5_agaa(T6, T3, T7, X9, vl30_in_ag(X9, T7))
U5_agaa(T6, T3, T7, X9, vl30_out_ag(X9, T7)) → p7_out_agaa(T6, T3, T7, X9)
p7_in_agaa(T6, T3, T7, T17) → U6_agaa(T6, T3, T7, T17, select20_in_aga(T6, T3, T7))
U6_agaa(T6, T3, T7, T17, select20_out_aga(T6, T3, T7)) → U7_agaa(T6, T3, T7, T17, vl30_in_ag(T17, T7))
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U8_agaa(T6, T3, T7, T17, vl43_in_ga(T17, X53))
U8_agaa(T6, T3, T7, T17, vl43_out_ga(T17, X53)) → p7_out_agaa(T6, T3, T7, T17)
U7_agaa(T6, T3, T7, T17, vl30_out_ag(T17, T7)) → U9_agaa(T6, T3, T7, T17, vl43_in_ga(T17, T18))
U9_agaa(T6, T3, T7, T17, vl43_out_ga(T17, T18)) → U10_agaa(T6, T3, T7, T17, p7_in_agaa(X54, T18, X55, X56))
p7_in_agaa(T6, T3, T7, 0) → U11_agaa(T6, T3, T7, select20_in_aga(T6, T3, T7))
U11_agaa(T6, T3, T7, select20_out_aga(T6, T3, T7)) → U12_agaa(T6, T3, T7, vl30_in_gg(0, T7))
vl30_in_gg(0, []) → vl30_out_gg(0, [])
U12_agaa(T6, T3, T7, vl30_out_gg(0, T7)) → p7_out_agaa(T6, T3, T7, 0)
U10_agaa(T6, T3, T7, T17, p7_out_agaa(X54, T18, X55, X56)) → p7_out_agaa(T6, T3, T7, T17)
U23_g(T2, p7_out_agaa(X54, T18, X55, X56)) → vlcnd1_out_g(T2)
U18_g(T2, select20_out_aga(T6, T3, T7)) → U24_g(T2, vl30_in_gg(0, T7))
U24_g(T2, vl30_out_gg(0, T7)) → vlcnd1_out_g(T2)
vlcnd1_in_g(0) → vlcnd1_out_g(0)

The argument filtering Pi contains the following mapping:
vlcnd1_in_g(x1)  =  vlcnd1_in_g(x1)
U15_g(x1, x2)  =  U15_g(x2)
vl6_in_ga(x1, x2)  =  vl6_in_ga(x1)
0  =  0
vl6_out_ga(x1, x2)  =  vl6_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
vlcnd1_out_g(x1)  =  vlcnd1_out_g
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
select20_in_aga(x1, x2, x3)  =  select20_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
select20_out_aga(x1, x2, x3)  =  select20_out_aga(x3)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
vl30_in_ag(x1, x2)  =  vl30_in_ag(x2)
[]  =  []
vl30_out_ag(x1, x2)  =  vl30_out_ag(x1)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
vl43_in_ga(x1, x2)  =  vl43_in_ga(x1)
vl43_out_ga(x1, x2)  =  vl43_out_ga(x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
p7_in_agaa(x1, x2, x3, x4)  =  p7_in_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5)  =  U3_agaa(x5)
p7_out_agaa(x1, x2, x3, x4)  =  p7_out_agaa(x3)
U4_agaa(x1, x2, x3, x4, x5)  =  U4_agaa(x5)
U5_agaa(x1, x2, x3, x4, x5)  =  U5_agaa(x3, x5)
U6_agaa(x1, x2, x3, x4, x5)  =  U6_agaa(x5)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x3, x5)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x3, x5)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x3, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x3, x5)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U12_agaa(x1, x2, x3, x4)  =  U12_agaa(x3, x4)
vl30_in_gg(x1, x2)  =  vl30_in_gg(x1, x2)
vl30_out_gg(x1, x2)  =  vl30_out_gg
U24_g(x1, x2)  =  U24_g(x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)

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

(145) UsableRulesProof (EQUIVALENT transformation)

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

(146) Obligation:

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

VL6_IN_GA(s(T5), .(X23, X24)) → VL6_IN_GA(T5, X24)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
VL6_IN_GA(x1, x2)  =  VL6_IN_GA(x1)

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

(147) PiDPToQDPProof (SOUND transformation)

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

(148) Obligation:

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

VL6_IN_GA(s(T5)) → VL6_IN_GA(T5)

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

(149) 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:

• VL6_IN_GA(s(T5)) → VL6_IN_GA(T5)
  The graph contains the following edges 1 > 1

(150) TRUE