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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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: ','(vl(T3, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))) | vlcnd(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(vl(T3, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 2, CLAUSE: 2 | ','(vl(T3, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 2, CLAUSE: 3 | ?_2 | vlcnd(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, ','(eq(X14, []), ','(select(X6, X14, X7), ','(vl(X8, X7), vlcnd(X8))))) | ','(vl(0, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 2, CLAUSE: 3 | ?_2 | vlcnd(0), SCOPE: 1, CLAUSE: 1\n\nX5 is free\nX6 is free; \nX7 is free; \nX8 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(vl(T3, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 2, CLAUSE: 3 | ?_2 | vlcnd(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; \nvl(T3, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X14, []), ','(select(X6, X14, X7), ','(vl(X8, X7), vlcnd(X8)))) | vlcnd(0), SCOPE: 1, CLAUSE: 1\n\nX6 is free\nX7 is free; \nX8 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(X14, []), ','(select(X6, X14, X7), ','(vl(X8, X7), vlcnd(X8))))\n\nX6 is free\nX7 is free; \nX8 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: vlcnd(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(eq(X14, []), ','(select(X6, X14, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 3, CLAUSE: 8\n\nX6 is free\nX7 is free; \nX8 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(select(X6, [], X7), ','(vl(X8, X7), vlcnd(X8)))\n\nX6 is free\nX7 is free; \nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: select(X6, [], X7)\n\nX6 is free\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(vl(X8, T5), vlcnd(X8))\n\nT5 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: select(X6, [], X7), SCOPE: 4, CLAUSE: 4 | select(X6, [], X7), SCOPE: 4, CLAUSE: 5\n\nX6 is free\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: select(X6, [], X7), SCOPE: 4, CLAUSE: 5\n\nX6 is free\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: vl(X8, T5)\n\nT5 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: vlcnd(T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: vl(X8, T5), SCOPE: 5, CLAUSE: 2 | vl(X8, T5), SCOPE: 5, CLAUSE: 3\n\nT5 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_5, eq(T11, [])) | vl(X8, T11), SCOPE: 5, CLAUSE: 3\n\nT11 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(T11, [])\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: eq(T11, []), SCOPE: 6, CLAUSE: 8\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: vlcnd(T7), SCOPE: 7, CLAUSE: 0 | vlcnd(T7), SCOPE: 7, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: vlcnd(T7), SCOPE: 7, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: vlcnd(T7), SCOPE: 7, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(vl(T23, X110), ','(select(X111, X110, X112), ','(vl(X113, X112), vlcnd(X113))))\n\nX110 is free\nX111 is free; \nX112 is free; \nX113 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: vl(T23, X110)\n\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(select(X111, T24, X112), ','(vl(X113, X112), vlcnd(X113)))\n\nX111 is free\nX112 is free; \nX113 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: vl(T23, X110), SCOPE: 8, CLAUSE: 2 | vl(T23, X110), SCOPE: 8, CLAUSE: 3\n\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(!_8, eq(X123, [])) | vl(T23, X110), SCOPE: 8, CLAUSE: 3\n\nX110 is free\nX123 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: vl(T23, X110), SCOPE: 8, CLAUSE: 3\n\nX110 is free\nvl(T23, X110) !=? vl(0, X122)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(X123, [])\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: eq(X123, []), SCOPE: 9, CLAUSE: 8\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T29, X147), vl(X147, X149))\n\nX149 is free\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T29, X147), vl(X147, X149)), SCOPE: 10, CLAUSE: 6 | ','(p(T29, X147), vl(X147, X149)), SCOPE: 10, CLAUSE: 7\n\nX149 is free\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(T29, X147), vl(X147, X149)), SCOPE: 10, CLAUSE: 6\n\nX149 is free\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(p(T29, X147), vl(X147, X149)), SCOPE: 10, CLAUSE: 7\n\nX149 is free\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: vl(0, X149)\n\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: vl(0, X149), SCOPE: 11, CLAUSE: 2 | vl(0, X149), SCOPE: 11, CLAUSE: 3\n\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_11, eq(X159, [])) | vl(0, X149), SCOPE: 11, CLAUSE: 3\n\nX149 is free\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(X159, [])\n\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(X159, []), SCOPE: 12, CLAUSE: 8\n\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: vl(T33, X149)\n\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: select(X111, T24, X112)\n\nX111 is free\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(vl(X113, T37), vlcnd(X113))\n\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: select(X111, T24, X112), SCOPE: 13, CLAUSE: 4 | select(X111, T24, X112), SCOPE: 13, CLAUSE: 5\n\nX111 is free\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: select(X111, T24, X112), SCOPE: 13, CLAUSE: 4\n\nX111 is free\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: select(X111, T24, X112), SCOPE: 13, CLAUSE: 5\n\nX111 is free\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: select(X212, T50, X213)\n\nX212 is free\nX213 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: vl(X113, T37)\n\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: vlcnd(T57)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: vl(X113, T37), SCOPE: 14, CLAUSE: 2 | vl(X113, T37), SCOPE: 14, CLAUSE: 3\n\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(!_14, eq(T63, [])) | vl(X113, T37), SCOPE: 14, CLAUSE: 3\n\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: eq(T63, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: eq(T63, []), SCOPE: 15, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(vl(T3, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; \nvl(T3, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ?_2 | vlcnd(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nvl(T3, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(p(T72, X278), ','(vl(X278, X280), ','(select(X6, .(X279, X280), X7), ','(vl(X8, X7), vlcnd(X8)))))\n\nT72 is ground\nX6 is free; \nX7 is free; \nX8 is free; \nX279 is free; \nX280 is free; \nX278 is free; \nvl(T72, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(p(T72, X278), ','(vl(X278, X280), ','(select(X6, .(X279, X280), X7), ','(vl(X8, X7), vlcnd(X8))))), SCOPE: 16, CLAUSE: 6 | ','(p(T72, X278), ','(vl(X278, X280), ','(select(X6, .(X279, X280), X7), ','(vl(X8, X7), vlcnd(X8))))), SCOPE: 16, CLAUSE: 7\n\nT72 is ground\nX6 is free; \nX7 is free; \nX8 is free; \nX279 is free; \nX280 is free; \nX278 is free; \nvl(T72, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(T72, X278), ','(vl(X278, X280), ','(select(X6, .(X279, X280), X7), ','(vl(X8, X7), vlcnd(X8))))), SCOPE: 16, CLAUSE: 7\n\nT72 is ground\nX6 is free; \nX7 is free; \nX8 is free; \nX279 is free; \nX280 is free; \nX278 is free; \nvl(T72, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(vl(T75, X280), ','(select(X6, .(X279, X280), X7), ','(vl(X8, X7), vlcnd(X8))))\n\nT75 is ground\nX6 is free; \nX7 is free; \nX8 is free; \nX279 is free; \nX280 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: vl(T75, X280)\n\nT75 is ground\nX280 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(select(X6, .(X279, T76), X7), ','(vl(X8, X7), vlcnd(X8)))\n\nX6 is free\nX7 is free; \nX8 is free; \nX279 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: vlcnd(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nvl(T3, X5) !=? vl(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="ONLY EVAL with clause\nvlcnd(X4) :- ','(vl(X4, X5), ','(select(X6, X5, X7), ','(vl(X8, X7), vlcnd(X8)))).\nand substitution\nT1 -> T3\nX4 -> T3", fontsize=14, style = filled, fillcolor = white];
2 -> 89 [arrowhead = none ];
89 -> 3;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 90 [arrowhead = none ];
90 -> 4;

91 [label="EVAL with clause\nvl(0, X13) :- ','(!_2, eq(X13, [])).\nand substitution\nT3 -> 0\nX5 -> X14\nX13 -> X14", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 91 [arrowhead = none ];
91 -> 5;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 92 [arrowhead = none ];
92 -> 6;

93 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 93 [arrowhead = none ];
93 -> 7;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 94 [arrowhead = none ];
94 -> 77;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 95 [arrowhead = none ];
95 -> 78;

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

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

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 98 [arrowhead = none ];
98 -> 10;

99 [label="ONLY EVAL with clause\nvlcnd(0).\nand substitution", fontsize=14, style = filled, fillcolor = white];
9 -> 99 [arrowhead = none ];
99 -> 75;

100 [label="ONLY EVAL with clause\neq(X47, X47).\nand substitution\nX14 -> []\nX47 -> []\nX48 -> []", fontsize=14, style = filled, fillcolor = white];
10 -> 100 [arrowhead = none ];
100 -> 11;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 101 [arrowhead = none ];
101 -> 12;

102 [label="SPLIT 2\nnew knowledge:\nT4 is ground\nT5 is ground\nreplacements:\nX6 -> T4\nX7 -> T5", fontsize=14, style = filled, fillcolor = violet];
11 -> 102 [arrowhead = none ];
102 -> 13;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 103 [arrowhead = none ];
103 -> 14;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 104 [arrowhead = none ];
104 -> 17;

105 [label="SPLIT 2\nreplacements:\nX8 -> T7", fontsize=14, style = filled, fillcolor = violet];
13 -> 105 [arrowhead = none ];
105 -> 18;

106 [label="BACKTRACK\nfor clause: select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 106 [arrowhead = none ];
106 -> 15;

107 [label="BACKTRACK\nfor clause: select(X, .(X, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 107 [arrowhead = none ];
107 -> 16;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 108 [arrowhead = none ];
108 -> 19;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 109 [arrowhead = none ];
109 -> 26;

110 [label="ONLY EVAL with clause\nvl(0, X83) :- ','(!_5, eq(X83, [])).\nand substitution\nX8 -> 0\nT5 -> T11\nX83 -> T11", fontsize=14, style = filled, fillcolor = white];
19 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 111 [arrowhead = none ];
111 -> 21;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 112 [arrowhead = none ];
112 -> 22;

113 [label="EVAL with clause\neq(X86, X86).\nand substitution\nT11 -> []\nX86 -> []\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 113 [arrowhead = none ];
113 -> 23;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 114 [arrowhead = none ];
114 -> 24;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 115 [arrowhead = none ];
115 -> 25;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 116 [arrowhead = none ];
116 -> 27;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 117 [arrowhead = none ];
117 -> 28;

118 [label="ONLY EVAL with clause\nvlcnd(X109) :- ','(vl(X109, X110), ','(select(X111, X110, X112), ','(vl(X113, X112), vlcnd(X113)))).\nand substitution\nT7 -> T23\nX109 -> T23\nT22 -> T23", fontsize=14, style = filled, fillcolor = white];
27 -> 118 [arrowhead = none ];
118 -> 29;

119 [label="EVAL with clause\nvlcnd(0).\nand substitution\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 119 [arrowhead = none ];
119 -> 72;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 120 [arrowhead = none ];
120 -> 73;

121 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 121 [arrowhead = none ];
121 -> 30;

122 [label="SPLIT 2\nreplacements:\nX110 -> T24", fontsize=14, style = filled, fillcolor = violet];
29 -> 122 [arrowhead = none ];
122 -> 31;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 123 [arrowhead = none ];
123 -> 32;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 124 [arrowhead = none ];
124 -> 53;

125 [label="SPLIT 2\nreplacements:\nX111 -> T36\nX112 -> T37", fontsize=14, style = filled, fillcolor = violet];
31 -> 125 [arrowhead = none ];
125 -> 54;

126 [label="EVAL with clause\nvl(0, X122) :- ','(!_8, eq(X122, [])).\nand substitution\nT23 -> 0\nX110 -> X123\nX122 -> X123", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 126 [arrowhead = none ];
126 -> 33;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 127 [arrowhead = none ];
127 -> 34;

128 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 128 [arrowhead = none ];
128 -> 35;

129 [label="ONLY EVAL with clause\nvl(X144, .(X145, X146)) :- ','(p(X144, X147), vl(X147, X146)).\nand substitution\nT23 -> T29\nX144 -> T29\nX145 -> X148\nX146 -> X149\nX110 -> .(X148, X149)\nT28 -> T29", fontsize=14, style = filled, fillcolor = white];
34 -> 129 [arrowhead = none ];
129 -> 39;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 130 [arrowhead = none ];
130 -> 36;

131 [label="ONLY EVAL with clause\neq(X129, X129).\nand substitution\nX123 -> []\nX129 -> []\nX130 -> []", fontsize=14, style = filled, fillcolor = white];
36 -> 131 [arrowhead = none ];
131 -> 37;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 132 [arrowhead = none ];
132 -> 38;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 133 [arrowhead = none ];
133 -> 40;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 134 [arrowhead = none ];
134 -> 41;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 135 [arrowhead = none ];
135 -> 42;

136 [label="EVAL with clause\np(0, 0).\nand substitution\nT29 -> 0\nX147 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 136 [arrowhead = none ];
136 -> 43;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 137 [arrowhead = none ];
137 -> 44;

138 [label="EVAL with clause\np(s(X168), X168).\nand substitution\nX168 -> T33\nT29 -> s(T33)\nX147 -> T33\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 139 [arrowhead = none ];
139 -> 52;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 140 [arrowhead = none ];
140 -> 45;

141 [label="ONLY EVAL with clause\nvl(0, X158) :- ','(!_11, eq(X158, [])).\nand substitution\nX149 -> X159\nX158 -> X159", fontsize=14, style = filled, fillcolor = white];
45 -> 141 [arrowhead = none ];
141 -> 46;

142 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 142 [arrowhead = none ];
142 -> 47;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 143 [arrowhead = none ];
143 -> 48;

144 [label="ONLY EVAL with clause\neq(X164, X164).\nand substitution\nX159 -> []\nX164 -> []\nX165 -> []", fontsize=14, style = filled, fillcolor = white];
48 -> 144 [arrowhead = none ];
144 -> 49;

145 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 145 [arrowhead = none ];
145 -> 50;

146 [label="INSTANCE with matching:\nT23 -> T33\nX110 -> X149", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 146 [arrowhead = none , style=dashed];
146 -> 30[style=dashed];

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 147 [arrowhead = none ];
147 -> 55;

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 148 [arrowhead = none ];
148 -> 63;

149 [label="SPLIT 2\nreplacements:\nX113 -> T57", fontsize=14, style = filled, fillcolor = violet];
54 -> 149 [arrowhead = none ];
149 -> 64;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 150 [arrowhead = none ];
150 -> 56;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 151 [arrowhead = none ];
151 -> 57;

152 [label="EVAL with clause\nselect(X208, .(X209, X210), .(X209, X211)) :- select(X208, X210, X211).\nand substitution\nX111 -> X212\nX208 -> X212\nX209 -> T48\nX210 -> T50\nT24 -> .(T48, T50)\nX211 -> X213\nX112 -> .(T48, X213)\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 152 [arrowhead = none ];
152 -> 58;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 153 [arrowhead = none ];
153 -> 59;

154 [label="EVAL with clause\nselect(X230, .(X230, X231), X231).\nand substitution\nX111 -> X232\nX230 -> X232\nX231 -> T55\nT24 -> .(X232, T55)\nX112 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 154 [arrowhead = none ];
154 -> 60;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 155 [arrowhead = none ];
155 -> 61;

156 [label="INSTANCE with matching:\nX111 -> X212\nT24 -> T50\nX112 -> X213", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 156 [arrowhead = none , style=dashed];
156 -> 53[style=dashed];

157 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 157 [arrowhead = none ];
157 -> 62;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 158 [arrowhead = none ];
158 -> 65;

159 [label="INSTANCE with matching:\nT7 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 159 [arrowhead = none , style=dashed];
159 -> 18[style=dashed];

160 [label="ONLY EVAL with clause\nvl(0, X239) :- ','(!_14, eq(X239, [])).\nand substitution\nX113 -> 0\nT37 -> T63\nX239 -> T63\nT62 -> T63", fontsize=14, style = filled, fillcolor = white];
65 -> 160 [arrowhead = none ];
160 -> 66;

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

162 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 162 [arrowhead = none ];
162 -> 68;

163 [label="EVAL with clause\neq(X242, X242).\nand substitution\nT63 -> []\nX242 -> []\nT66 -> []", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 163 [arrowhead = none ];
163 -> 69;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 164 [arrowhead = none ];
164 -> 70;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 165 [arrowhead = none ];
165 -> 71;

166 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 166 [arrowhead = none ];
166 -> 74;

167 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="ONLY EVAL with clause\nvl(X275, .(X276, X277)) :- ','(p(X275, X278), vl(X278, X277)).\nand substitution\nT3 -> T72\nX275 -> T72\nX276 -> X279\nX277 -> X280\nX5 -> .(X279, X280)", fontsize=14, style = filled, fillcolor = white];
77 -> 168 [arrowhead = none ];
168 -> 79;

169 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 169 [arrowhead = none ];
169 -> 86;

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 170 [arrowhead = none ];
170 -> 80;

171 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (vl(T72, X5), vl(0, X13))", fontsize=14, style = filled, fillcolor = white];
80 -> 171 [arrowhead = none ];
171 -> 81;

172 [label="EVAL with clause\np(s(X285), X285).\nand substitution\nX285 -> T75\nT72 -> s(T75)\nX278 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 172 [arrowhead = none ];
172 -> 82;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 173 [arrowhead = none ];
173 -> 83;

174 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 174 [arrowhead = none ];
174 -> 84;

175 [label="SPLIT 2\nreplacements:\nX280 -> T76", fontsize=14, style = filled, fillcolor = violet];
82 -> 175 [arrowhead = none ];
175 -> 85;

176 [label="INSTANCE with matching:\nT23 -> T75\nX110 -> X280", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 176 [arrowhead = none , style=dashed];
176 -> 30[style=dashed];

177 [label="INSTANCE with matching:\nX111 -> X6\nT24 -> .(X279, T76)\nX112 -> X7\nX113 -> X8", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 177 [arrowhead = none , style=dashed];
177 -> 31[style=dashed];

178 [label="BACKTRACK\nfor clause: vlcnd(0)\nwith clash: (vl(T3, X5), vl(0, X13))", fontsize=14, style = filled, fillcolor = white];
86 -> 178 [arrowhead = none ];
178 -> 87;

}

(2) Obligation:

Triples:

vl30(s(T33), .(X148, X149)) :- vl30(T33, X149).
select53(X212, .(T48, T50), .(T48, X213)) :- select53(X212, T50, X213).
vlcnd18(T23) :- vl30(T23, X110).
vlcnd18(T23) :- ','(vlc30(T23, T24), p31(X111, T24, X112, X113)).
p31(X111, T24, X112, X113) :- select53(X111, T24, X112).
p31(T36, T24, T37, T57) :- ','(selectc53(T36, T24, T37), ','(vlc63(T57, T37), vlcnd18(T57))).
vlcnd1(0) :- ','(selectc12(T4, T5), ','(vlc17(T7, T5), vlcnd18(T7))).
vlcnd1(s(T75)) :- vl30(T75, X280).
vlcnd1(s(T75)) :- ','(vlc30(T75, T76), p31(X6, .(X279, T76), X7, X8)).

Clauses:

vlc30(0, []).
vlc30(0, .(X148, [])).
vlc30(s(T33), .(X148, X149)) :- vlc30(T33, X149).
selectc53(X212, .(T48, T50), .(T48, X213)) :- selectc53(X212, T50, X213).
selectc53(X232, .(X232, T55), T55).
vlcndc18(T23) :- ','(vlc30(T23, T24), qc31(X111, T24, X112, X113)).
vlcndc18(0).
qc31(T36, T24, T37, T57) :- ','(selectc53(T36, T24, T37), ','(vlc63(T57, T37), vlcndc18(T57))).
vlc17(0, []).
vlc63(0, []).

Afs:

vlcnd1(x1)  =  vlcnd1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

vl30(s(T33), .(X148, X149)) :- vl30(T33, X149).
select53(X212, .(T48, T50), .(T48, X213)) :- select53(X212, T50, X213).
vlcnd18(T23) :- vl30(T23, X110).
vlcnd18(T23) :- ','(vlc30(T23, T24), p31(X111, T24, X112, X113)).
p31(X111, T24, X112, X113) :- select53(X111, T24, X112).
p31(T36, T24, T37, T57) :- ','(selectc53(T36, T24, T37), ','(vlc63(T57, T37), vlcnd18(T57))).
vlcnd1(s(T75)) :- vl30(T75, X280).
vlcnd1(s(T75)) :- ','(vlc30(T75, T76), p31(X6, .(X279, T76), X7, X8)).

Clauses:

vlc30(0, []).
vlc30(0, .(X148, [])).
vlc30(s(T33), .(X148, X149)) :- vlc30(T33, X149).
selectc53(X212, .(T48, T50), .(T48, X213)) :- selectc53(X212, T50, X213).
selectc53(X232, .(X232, T55), T55).
vlcndc18(T23) :- ','(vlc30(T23, T24), qc31(X111, T24, X112, X113)).
vlcndc18(0).
qc31(T36, T24, T37, T57) :- ','(selectc53(T36, T24, T37), ','(vlc63(T57, T37), vlcndc18(T57))).
vlc17(0, []).
vlc63(0, []).

Afs:

vlcnd1(x1)  =  vlcnd1(x1)

(5) TriplesToPiDPProof (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)
vl30_in: (b,f)
vlc30_in: (b,f)
p31_in: (f,b,f,f)
select53_in: (f,b,f)
selectc53_in: (f,b,f)
vlcnd18_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

VLCND1_IN_G(s(T75)) → U10_G(T75, vl30_in_ga(T75, X280))
VLCND1_IN_G(s(T75)) → VL30_IN_GA(T75, X280)
VL30_IN_GA(s(T33), .(X148, X149)) → U1_GA(T33, X148, X149, vl30_in_ga(T33, X149))
VL30_IN_GA(s(T33), .(X148, X149)) → VL30_IN_GA(T33, X149)
VLCND1_IN_G(s(T75)) → U11_G(T75, vlc30_in_ga(T75, T76))
U11_G(T75, vlc30_out_ga(T75, T76)) → U12_G(T75, p31_in_agaa(X6, .(X279, T76), X7, X8))
U11_G(T75, vlc30_out_ga(T75, T76)) → P31_IN_AGAA(X6, .(X279, T76), X7, X8)
P31_IN_AGAA(X111, T24, X112, X113) → U6_AGAA(X111, T24, X112, X113, select53_in_aga(X111, T24, X112))
P31_IN_AGAA(X111, T24, X112, X113) → SELECT53_IN_AGA(X111, T24, X112)
SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → U2_AGA(X212, T48, T50, X213, select53_in_aga(X212, T50, X213))
SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → SELECT53_IN_AGA(X212, T50, X213)
P31_IN_AGAA(T36, T24, T37, T57) → U7_AGAA(T36, T24, T37, T57, selectc53_in_aga(T36, T24, T37))
U7_AGAA(T36, T24, T37, T57, selectc53_out_aga(T36, T24, T37)) → U8_AGAA(T36, T24, T37, T57, vlc63_in_ag(T57, T37))
U8_AGAA(T36, T24, T37, T57, vlc63_out_ag(T57, T37)) → U9_AGAA(T36, T24, T37, T57, vlcnd18_in_g(T57))
U8_AGAA(T36, T24, T37, T57, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)
VLCND18_IN_G(T23) → U3_G(T23, vl30_in_ga(T23, X110))
VLCND18_IN_G(T23) → VL30_IN_GA(T23, X110)
VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23, T24))
U4_G(T23, vlc30_out_ga(T23, T24)) → U5_G(T23, p31_in_agaa(X111, T24, X112, X113))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(X111, T24, X112, X113)

The TRS R consists of the following rules:

vlc30_in_ga(0, []) → vlc30_out_ga(0, [])
vlc30_in_ga(0, .(X148, [])) → vlc30_out_ga(0, .(X148, []))
vlc30_in_ga(s(T33), .(X148, X149)) → U14_ga(T33, X148, X149, vlc30_in_ga(T33, X149))
U14_ga(T33, X148, X149, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X148, X149))
selectc53_in_aga(X212, .(T48, T50), .(T48, X213)) → U15_aga(X212, T48, T50, X213, selectc53_in_aga(X212, T50, X213))
selectc53_in_aga(X232, .(X232, T55), T55) → selectc53_out_aga(X232, .(X232, T55), T55)
U15_aga(X212, T48, T50, X213, selectc53_out_aga(X212, T50, X213)) → selectc53_out_aga(X212, .(T48, T50), .(T48, X213))
vlc63_in_ag(0, []) → vlc63_out_ag(0, [])

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
vl30_in_ga(x1, x2)  =  vl30_in_ga(x1)
.(x1, x2)  =  .(x2)
vlc30_in_ga(x1, x2)  =  vlc30_in_ga(x1)
0  =  0
vlc30_out_ga(x1, x2)  =  vlc30_out_ga(x1, x2)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
p31_in_agaa(x1, x2, x3, x4)  =  p31_in_agaa(x2)
select53_in_aga(x1, x2, x3)  =  select53_in_aga(x2)
selectc53_in_aga(x1, x2, x3)  =  selectc53_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc53_out_aga(x1, x2, x3)  =  selectc53_out_aga(x2, x3)
vlc63_in_ag(x1, x2)  =  vlc63_in_ag(x2)
[]  =  []
vlc63_out_ag(x1, x2)  =  vlc63_out_ag(x1, x2)
vlcnd18_in_g(x1)  =  vlcnd18_in_g(x1)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
VL30_IN_GA(x1, x2)  =  VL30_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
P31_IN_AGAA(x1, x2, x3, x4)  =  P31_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
SELECT53_IN_AGA(x1, x2, x3)  =  SELECT53_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, 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)
VLCND18_IN_G(x1)  =  VLCND18_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

VLCND1_IN_G(s(T75)) → U10_G(T75, vl30_in_ga(T75, X280))
VLCND1_IN_G(s(T75)) → VL30_IN_GA(T75, X280)
VL30_IN_GA(s(T33), .(X148, X149)) → U1_GA(T33, X148, X149, vl30_in_ga(T33, X149))
VL30_IN_GA(s(T33), .(X148, X149)) → VL30_IN_GA(T33, X149)
VLCND1_IN_G(s(T75)) → U11_G(T75, vlc30_in_ga(T75, T76))
U11_G(T75, vlc30_out_ga(T75, T76)) → U12_G(T75, p31_in_agaa(X6, .(X279, T76), X7, X8))
U11_G(T75, vlc30_out_ga(T75, T76)) → P31_IN_AGAA(X6, .(X279, T76), X7, X8)
P31_IN_AGAA(X111, T24, X112, X113) → U6_AGAA(X111, T24, X112, X113, select53_in_aga(X111, T24, X112))
P31_IN_AGAA(X111, T24, X112, X113) → SELECT53_IN_AGA(X111, T24, X112)
SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → U2_AGA(X212, T48, T50, X213, select53_in_aga(X212, T50, X213))
SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → SELECT53_IN_AGA(X212, T50, X213)
P31_IN_AGAA(T36, T24, T37, T57) → U7_AGAA(T36, T24, T37, T57, selectc53_in_aga(T36, T24, T37))
U7_AGAA(T36, T24, T37, T57, selectc53_out_aga(T36, T24, T37)) → U8_AGAA(T36, T24, T37, T57, vlc63_in_ag(T57, T37))
U8_AGAA(T36, T24, T37, T57, vlc63_out_ag(T57, T37)) → U9_AGAA(T36, T24, T37, T57, vlcnd18_in_g(T57))
U8_AGAA(T36, T24, T37, T57, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)
VLCND18_IN_G(T23) → U3_G(T23, vl30_in_ga(T23, X110))
VLCND18_IN_G(T23) → VL30_IN_GA(T23, X110)
VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23, T24))
U4_G(T23, vlc30_out_ga(T23, T24)) → U5_G(T23, p31_in_agaa(X111, T24, X112, X113))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(X111, T24, X112, X113)

The TRS R consists of the following rules:

vlc30_in_ga(0, []) → vlc30_out_ga(0, [])
vlc30_in_ga(0, .(X148, [])) → vlc30_out_ga(0, .(X148, []))
vlc30_in_ga(s(T33), .(X148, X149)) → U14_ga(T33, X148, X149, vlc30_in_ga(T33, X149))
U14_ga(T33, X148, X149, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X148, X149))
selectc53_in_aga(X212, .(T48, T50), .(T48, X213)) → U15_aga(X212, T48, T50, X213, selectc53_in_aga(X212, T50, X213))
selectc53_in_aga(X232, .(X232, T55), T55) → selectc53_out_aga(X232, .(X232, T55), T55)
U15_aga(X212, T48, T50, X213, selectc53_out_aga(X212, T50, X213)) → selectc53_out_aga(X212, .(T48, T50), .(T48, X213))
vlc63_in_ag(0, []) → vlc63_out_ag(0, [])

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
vl30_in_ga(x1, x2)  =  vl30_in_ga(x1)
.(x1, x2)  =  .(x2)
vlc30_in_ga(x1, x2)  =  vlc30_in_ga(x1)
0  =  0
vlc30_out_ga(x1, x2)  =  vlc30_out_ga(x1, x2)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
p31_in_agaa(x1, x2, x3, x4)  =  p31_in_agaa(x2)
select53_in_aga(x1, x2, x3)  =  select53_in_aga(x2)
selectc53_in_aga(x1, x2, x3)  =  selectc53_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc53_out_aga(x1, x2, x3)  =  selectc53_out_aga(x2, x3)
vlc63_in_ag(x1, x2)  =  vlc63_in_ag(x2)
[]  =  []
vlc63_out_ag(x1, x2)  =  vlc63_out_ag(x1, x2)
vlcnd18_in_g(x1)  =  vlcnd18_in_g(x1)
VLCND1_IN_G(x1)  =  VLCND1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
VL30_IN_GA(x1, x2)  =  VL30_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
P31_IN_AGAA(x1, x2, x3, x4)  =  P31_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4, x5)  =  U6_AGAA(x2, x5)
SELECT53_IN_AGA(x1, x2, x3)  =  SELECT53_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, 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)
VLCND18_IN_G(x1)  =  VLCND18_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 13 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → SELECT53_IN_AGA(X212, T50, X213)

The TRS R consists of the following rules:

vlc30_in_ga(0, []) → vlc30_out_ga(0, [])
vlc30_in_ga(0, .(X148, [])) → vlc30_out_ga(0, .(X148, []))
vlc30_in_ga(s(T33), .(X148, X149)) → U14_ga(T33, X148, X149, vlc30_in_ga(T33, X149))
U14_ga(T33, X148, X149, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X148, X149))
selectc53_in_aga(X212, .(T48, T50), .(T48, X213)) → U15_aga(X212, T48, T50, X213, selectc53_in_aga(X212, T50, X213))
selectc53_in_aga(X232, .(X232, T55), T55) → selectc53_out_aga(X232, .(X232, T55), T55)
U15_aga(X212, T48, T50, X213, selectc53_out_aga(X212, T50, X213)) → selectc53_out_aga(X212, .(T48, T50), .(T48, X213))
vlc63_in_ag(0, []) → vlc63_out_ag(0, [])

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
vlc30_in_ga(x1, x2)  =  vlc30_in_ga(x1)
0  =  0
vlc30_out_ga(x1, x2)  =  vlc30_out_ga(x1, x2)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
selectc53_in_aga(x1, x2, x3)  =  selectc53_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc53_out_aga(x1, x2, x3)  =  selectc53_out_aga(x2, x3)
vlc63_in_ag(x1, x2)  =  vlc63_in_ag(x2)
[]  =  []
vlc63_out_ag(x1, x2)  =  vlc63_out_ag(x1, x2)
SELECT53_IN_AGA(x1, x2, x3)  =  SELECT53_IN_AGA(x2)

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:

SELECT53_IN_AGA(X212, .(T48, T50), .(T48, X213)) → SELECT53_IN_AGA(X212, T50, X213)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SELECT53_IN_AGA(x1, x2, x3)  =  SELECT53_IN_AGA(x2)

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:

SELECT53_IN_AGA(.(T50)) → SELECT53_IN_AGA(T50)

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:

• SELECT53_IN_AGA(.(T50)) → SELECT53_IN_AGA(T50)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

VL30_IN_GA(s(T33), .(X148, X149)) → VL30_IN_GA(T33, X149)

The TRS R consists of the following rules:

vlc30_in_ga(0, []) → vlc30_out_ga(0, [])
vlc30_in_ga(0, .(X148, [])) → vlc30_out_ga(0, .(X148, []))
vlc30_in_ga(s(T33), .(X148, X149)) → U14_ga(T33, X148, X149, vlc30_in_ga(T33, X149))
U14_ga(T33, X148, X149, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X148, X149))
selectc53_in_aga(X212, .(T48, T50), .(T48, X213)) → U15_aga(X212, T48, T50, X213, selectc53_in_aga(X212, T50, X213))
selectc53_in_aga(X232, .(X232, T55), T55) → selectc53_out_aga(X232, .(X232, T55), T55)
U15_aga(X212, T48, T50, X213, selectc53_out_aga(X212, T50, X213)) → selectc53_out_aga(X212, .(T48, T50), .(T48, X213))
vlc63_in_ag(0, []) → vlc63_out_ag(0, [])

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
vlc30_in_ga(x1, x2)  =  vlc30_in_ga(x1)
0  =  0
vlc30_out_ga(x1, x2)  =  vlc30_out_ga(x1, x2)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
selectc53_in_aga(x1, x2, x3)  =  selectc53_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc53_out_aga(x1, x2, x3)  =  selectc53_out_aga(x2, x3)
vlc63_in_ag(x1, x2)  =  vlc63_in_ag(x2)
[]  =  []
vlc63_out_ag(x1, x2)  =  vlc63_out_ag(x1, x2)
VL30_IN_GA(x1, x2)  =  VL30_IN_GA(x1)

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:

VL30_IN_GA(s(T33), .(X148, X149)) → VL30_IN_GA(T33, X149)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
VL30_IN_GA(x1, x2)  =  VL30_IN_GA(x1)

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:

VL30_IN_GA(s(T33)) → VL30_IN_GA(T33)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• VL30_IN_GA(s(T33)) → VL30_IN_GA(T33)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23, T24))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(X111, T24, X112, X113)
P31_IN_AGAA(T36, T24, T37, T57) → U7_AGAA(T36, T24, T37, T57, selectc53_in_aga(T36, T24, T37))
U7_AGAA(T36, T24, T37, T57, selectc53_out_aga(T36, T24, T37)) → U8_AGAA(T36, T24, T37, T57, vlc63_in_ag(T57, T37))
U8_AGAA(T36, T24, T37, T57, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)

The TRS R consists of the following rules:

vlc30_in_ga(0, []) → vlc30_out_ga(0, [])
vlc30_in_ga(0, .(X148, [])) → vlc30_out_ga(0, .(X148, []))
vlc30_in_ga(s(T33), .(X148, X149)) → U14_ga(T33, X148, X149, vlc30_in_ga(T33, X149))
U14_ga(T33, X148, X149, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X148, X149))
selectc53_in_aga(X212, .(T48, T50), .(T48, X213)) → U15_aga(X212, T48, T50, X213, selectc53_in_aga(X212, T50, X213))
selectc53_in_aga(X232, .(X232, T55), T55) → selectc53_out_aga(X232, .(X232, T55), T55)
U15_aga(X212, T48, T50, X213, selectc53_out_aga(X212, T50, X213)) → selectc53_out_aga(X212, .(T48, T50), .(T48, X213))
vlc63_in_ag(0, []) → vlc63_out_ag(0, [])

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
vlc30_in_ga(x1, x2)  =  vlc30_in_ga(x1)
0  =  0
vlc30_out_ga(x1, x2)  =  vlc30_out_ga(x1, x2)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
selectc53_in_aga(x1, x2, x3)  =  selectc53_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc53_out_aga(x1, x2, x3)  =  selectc53_out_aga(x2, x3)
vlc63_in_ag(x1, x2)  =  vlc63_in_ag(x2)
[]  =  []
vlc63_out_ag(x1, x2)  =  vlc63_out_ag(x1, x2)
P31_IN_AGAA(x1, x2, x3, x4)  =  P31_IN_AGAA(x2)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x5)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x2, x3, x5)
VLCND18_IN_G(x1)  =  VLCND18_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x1, x2)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(T24, selectc53_out_aga(T24, T37)) → U8_AGAA(T24, T37, vlc63_in_ag(T37))
U8_AGAA(T24, T37, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)

The TRS R consists of the following rules:

vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))
selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc63_in_ag([]) → vlc63_out_ag(0, [])

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(x0, x1)
vlc63_in_ag(x0)

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

(26) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U7_AGAA(T24, selectc53_out_aga(T24, T37)) → U8_AGAA(T24, T37, vlc63_in_ag(T37)) at position [2] we obtained the following new rules [LPAR04]:

U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))

(27) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U8_AGAA(T24, T37, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))

The TRS R consists of the following rules:

vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))
selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc63_in_ag([]) → vlc63_out_ag(0, [])

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(x0, x1)
vlc63_in_ag(x0)

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

(28) 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.

(29) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U8_AGAA(T24, T37, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(x0, x1)
vlc63_in_ag(x0)

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

(30) 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].

vlc63_in_ag(x0)

(31) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U8_AGAA(T24, T37, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57)
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(32) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_AGAA(T24, T37, vlc63_out_ag(T57, T37)) → VLCND18_IN_G(T57) we obtained the following new rules [LPAR04]:

U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

(33) Obligation:

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

VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23))
U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(34) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule VLCND18_IN_G(T23) → U4_G(T23, vlc30_in_ga(T23)) we obtained the following new rules [LPAR04]:

VLCND18_IN_G(0) → U4_G(0, vlc30_in_ga(0))

(35) Obligation:

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

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_in_ga(0))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))
vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
vlc30_in_ga(s(T33)) → U14_ga(T33, vlc30_in_ga(T33))
U14_ga(T33, vlc30_out_ga(T33, X149)) → vlc30_out_ga(s(T33), .(X149))

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(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:

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_in_ga(0))

The TRS R consists of the following rules:

vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

vlc30_in_ga(x0)
U14_ga(x0, x1)
selectc53_in_aga(x0)
U15_aga(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].

U14_ga(x0, x1)

(39) Obligation:

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

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_in_ga(0))

The TRS R consists of the following rules:

vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

vlc30_in_ga(x0)
selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(40) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule VLCND18_IN_G(0) → U4_G(0, vlc30_in_ga(0)) at position [1] we obtained the following new rules [LPAR04]:

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))

(41) Obligation:

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

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))

The TRS R consists of the following rules:

vlc30_in_ga(0) → vlc30_out_ga(0, [])
vlc30_in_ga(0) → vlc30_out_ga(0, .([]))
selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

vlc30_in_ga(x0)
selectc53_in_aga(x0)
U15_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:

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

vlc30_in_ga(x0)
selectc53_in_aga(x0)
U15_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].

vlc30_in_ga(x0)

(45) Obligation:

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

U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24)
P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_G(T23, vlc30_out_ga(T23, T24)) → P31_IN_AGAA(T24) we obtained the following new rules [LPAR04]:

U4_G(0, vlc30_out_ga(0, [])) → P31_IN_AGAA([])
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))

(47) Obligation:

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

P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, [])) → P31_IN_AGAA([])
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(48) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P31_IN_AGAA(T24) → U7_AGAA(T24, selectc53_in_aga(T24)) at position [1] we obtained the following new rules [LPAR04]:

P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), U15_aga(x0, selectc53_in_aga(x0)))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))

(49) Obligation:

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

U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, []))
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, [])) → P31_IN_AGAA([])
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), U15_aga(x0, selectc53_in_aga(x0)))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_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:

U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), U15_aga(x0, selectc53_in_aga(x0)))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_aga(x0, x1)

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

(52) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), U15_aga(x0, selectc53_in_aga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

P31_IN_AGAA(.(.(x0))) → U7_AGAA(.(.(x0)), U15_aga(.(x0), U15_aga(x0, selectc53_in_aga(x0))))
P31_IN_AGAA(.(.(x0))) → U7_AGAA(.(.(x0)), U15_aga(.(x0), selectc53_out_aga(.(x0), x0)))

(53) Obligation:

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

U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))
P31_IN_AGAA(.(.(x0))) → U7_AGAA(.(.(x0)), U15_aga(.(x0), U15_aga(x0, selectc53_in_aga(x0))))
P31_IN_AGAA(.(.(x0))) → U7_AGAA(.(.(x0)), U15_aga(.(x0), selectc53_out_aga(.(x0), x0)))

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_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:

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

The TRS R consists of the following rules:

selectc53_in_aga(.(T50)) → U15_aga(T50, selectc53_in_aga(T50))
selectc53_in_aga(.(T55)) → selectc53_out_aga(.(T55), T55)
U15_aga(T50, selectc53_out_aga(T50, X213)) → selectc53_out_aga(.(T50), .(X213))

The set Q consists of the following terms:

selectc53_in_aga(x0)
U15_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:

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

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

selectc53_in_aga(x0)
U15_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].

selectc53_in_aga(x0)
U15_aga(x0, x1)

(59) Obligation:

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

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(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 P31_IN_AGAA(.(x0)) → U7_AGAA(.(x0), selectc53_out_aga(.(x0), x0)) we obtained the following new rules [LPAR04]:

P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_out_aga(.([]), []))

(61) Obligation:

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

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, []))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_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 U7_AGAA(y0, selectc53_out_aga(y0, [])) → U8_AGAA(y0, [], vlc63_out_ag(0, [])) we obtained the following new rules [LPAR04]:

U7_AGAA(.([]), selectc53_out_aga(.([]), [])) → U8_AGAA(.([]), [], vlc63_out_ag(0, []))

(63) Obligation:

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

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_out_aga(.([]), []))
U7_AGAA(.([]), selectc53_out_aga(.([]), [])) → U8_AGAA(.([]), [], vlc63_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 U8_AGAA(z0, [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0) we obtained the following new rules [LPAR04]:

U8_AGAA(.([]), [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

(65) Obligation:

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

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_out_aga(.([]), []))
U7_AGAA(.([]), selectc53_out_aga(.([]), [])) → U8_AGAA(.([]), [], vlc63_out_ag(0, []))
U8_AGAA(.([]), [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)

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

(66) 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 = U4_G(0, vlc30_out_ga(0, .([]))) evaluates to t =U4_G(0, vlc30_out_ga(0, .([])))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([]))
with rule U4_G(0, vlc30_out_ga(0, .([]))) → P31_IN_AGAA(.([])) at position [] and matcher [ ]

P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_out_aga(.([]), []))
with rule P31_IN_AGAA(.([])) → U7_AGAA(.([]), selectc53_out_aga(.([]), [])) at position [] and matcher [ ]

U7_AGAA(.([]), selectc53_out_aga(.([]), [])) → U8_AGAA(.([]), [], vlc63_out_ag(0, []))
with rule U7_AGAA(.([]), selectc53_out_aga(.([]), [])) → U8_AGAA(.([]), [], vlc63_out_ag(0, [])) at position [] and matcher [ ]

U8_AGAA(.([]), [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0)
with rule U8_AGAA(.([]), [], vlc63_out_ag(0, [])) → VLCND18_IN_G(0) at position [] and matcher [ ]

VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(0, .([])))
with rule VLCND18_IN_G(0) → U4_G(0, vlc30_out_ga(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.

(67) NO