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

(0) Obligation:

Clauses:

t(N) :- ','(ll(N, Xs), ','(select(X1, Xs, Xs1), ','(ll(M, Xs1), t(M)))).
t(0).
ll(s(N), .(X, Xs)) :- ll(N, Xs).
ll(0, []).
select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys).
select(X, .(X, Xs), Xs).

Query: t(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="C: t(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: t(T1), SCOPE: 1, CLAUSE: 0 | t(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))) | t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))), SCOPE: 2, CLAUSE: 2 | ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))), SCOPE: 2, CLAUSE: 3 | ?_2 | t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX4 is free\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))), SCOPE: 2, CLAUSE: 3 | ?_2 | t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX4 is free\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(ll(T8, X32), ','(select(X5, .(X31, X32), X6), ','(ll(X7, X6), t(X7))))\n\nT8 is ground\nX5 is free\nX6 is free\nX7 is free\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="A: ll(T8, X32)\n\nT8 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(select(X5, .(X31, T10), X6), ','(ll(X7, X6), t(X7)))\n\nX5 is free\nX6 is free\nX7 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ll(T8, X32), SCOPE: 3, CLAUSE: 2 | ll(T8, X32), SCOPE: 3, CLAUSE: 3\n\nT8 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ll(T8, X32), SCOPE: 3, CLAUSE: 2\n\nT8 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ll(T8, X32), SCOPE: 3, CLAUSE: 3\n\nT8 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ll(T16, X67)\n\nT16 is ground\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="D: select(X5, .(X31, T10), X6)\n\nX5 is free\nX6 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(ll(X7, T21), t(X7))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: select(X5, .(X31, T10), X6), SCOPE: 4, CLAUSE: 4 | select(X5, .(X31, T10), X6), SCOPE: 4, CLAUSE: 5\n\nX5 is free\nX6 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: select(X5, .(X31, T10), X6), SCOPE: 4, CLAUSE: 4\n\nX5 is free\nX6 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: select(X5, .(X31, T10), X6), SCOPE: 4, CLAUSE: 5\n\nX5 is free\nX6 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="B: select(X131, T34, X133)\n\nX131 is free\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: select(X131, T34, X133), SCOPE: 5, CLAUSE: 4 | select(X131, T34, X133), SCOPE: 5, CLAUSE: 5\n\nX131 is free\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: select(X131, T34, X133), SCOPE: 5, CLAUSE: 4\n\nX131 is free\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: select(X131, T34, X133), SCOPE: 5, CLAUSE: 5\n\nX131 is free\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: select(X168, T47, X169)\n\nX168 is free\nX169 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="E: ll(X7, T21)\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: t(T61)\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ll(X7, T21), SCOPE: 6, CLAUSE: 2 | ll(X7, T21), SCOPE: 6, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ll(X7, T21), SCOPE: 6, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ll(X7, T21), SCOPE: 6, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ll(X221, T74)\n\nX221 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(ll(T3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX4 is free\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ?_2 | t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(select(X5, [], X6), ','(ll(X7, X6), t(X7)))\n\nX5 is free\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="F: select(X5, [], X6)\n\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(ll(X7, T79), t(X7))\n\nT79 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: select(X5, [], X6), SCOPE: 7, CLAUSE: 4 | select(X5, [], X6), SCOPE: 7, CLAUSE: 5\n\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: select(X5, [], X6), SCOPE: 7, CLAUSE: 5\n\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="G: ll(X7, T79)\n\nT79 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: t(T82)\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ll(X7, T79), SCOPE: 8, CLAUSE: 2 | ll(X7, T79), SCOPE: 8, CLAUSE: 3\n\nT79 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ll(X7, T79), SCOPE: 8, CLAUSE: 2\n\nT79 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ll(X7, T79), SCOPE: 8, CLAUSE: 3\n\nT79 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ll(X276, T94)\n\nT94 is ground\nX276 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 68 [arrowhead = none ];
68 -> 2;

69 [label="ONLY EVAL with clause\nt(X3) :- ','(ll(X3, X4), ','(select(X5, X4, X6), ','(ll(X7, X6), t(X7)))).\nand substitutionT1 -> T3,\nX3 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 3;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 70 [arrowhead = none ];
70 -> 4;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 71 [arrowhead = none ];
71 -> 5;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 72 [arrowhead = none ];
72 -> 6;

73 [label="EVAL with clause\nll(s(X28), .(X29, X30)) :- ll(X28, X30).\nand substitutionX28 -> T8,\nT3 -> s(T8),\nX29 -> X31,\nX30 -> X32,\nX4 -> .(X31, X32)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 73 [arrowhead = none ];
73 -> 7;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 74 [arrowhead = none ];
74 -> 8;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 75 [arrowhead = none ];
75 -> 45;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 76 [arrowhead = none ];
76 -> 46;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 77 [arrowhead = none ];
77 -> 9;

78 [label="SPLIT 2\nnew knowledge:\nT8 is ground\nreplacements:X32 -> T10", fontsize=14, style = filled, fillcolor = violet];
7 -> 78 [arrowhead = none ];
78 -> 10;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 79 [arrowhead = none ];
79 -> 11;

80 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
10 -> 80 [arrowhead = none ];
80 -> 19;

81 [label="SPLIT 2\nreplacements:X5 -> T19,\nX31 -> T20,\nX6 -> T21", fontsize=14, style = filled, fillcolor = violet];
10 -> 81 [arrowhead = none ];
81 -> 20;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 82 [arrowhead = none ];
82 -> 12;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 83 [arrowhead = none ];
83 -> 13;

84 [label="EVAL with clause\nll(s(X63), .(X64, X65)) :- ll(X63, X65).\nand substitutionX63 -> T16,\nT8 -> s(T16),\nX64 -> X66,\nX65 -> X67,\nX32 -> .(X66, X67)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="EVAL with clause\nll(0, []).\nand substitutionT8 -> 0,\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="INSTANCE with matching:\nT8 -> T16\nX32 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 88 [arrowhead = none , style=dashed];
88 -> 9[style=dashed];

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 89 [arrowhead = none ];
89 -> 18;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 90 [arrowhead = none ];
90 -> 21;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 91 [arrowhead = none ];
91 -> 35;

92 [label="SPLIT 2\nnew knowledge:\nT61 is ground\nreplacements:X7 -> T61", fontsize=14, style = filled, fillcolor = violet];
20 -> 92 [arrowhead = none ];
92 -> 36;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 93 [arrowhead = none ];
93 -> 22;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 94 [arrowhead = none ];
94 -> 23;

95 [label="ONLY EVAL with clause\nselect(X127, .(X128, X129), .(X128, X130)) :- select(X127, X129, X130).\nand substitutionX5 -> X131,\nX127 -> X131,\nX31 -> X132,\nX128 -> X132,\nT10 -> T34,\nX129 -> T34,\nX130 -> X133,\nX6 -> .(X132, X133),\nT33 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 95 [arrowhead = none ];
95 -> 24;

96 [label="ONLY EVAL with clause\nselect(X191, .(X191, X192), X192).\nand substitutionX5 -> X193,\nX191 -> X193,\nX31 -> X193,\nT10 -> T58,\nX192 -> T58,\nX6 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 96 [arrowhead = none ];
96 -> 33;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 97 [arrowhead = none ];
97 -> 25;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 98 [arrowhead = none ];
98 -> 26;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 99 [arrowhead = none ];
99 -> 27;

100 [label="EVAL with clause\nselect(X164, .(X165, X166), .(X165, X167)) :- select(X164, X166, X167).\nand substitutionX131 -> X168,\nX164 -> X168,\nX165 -> T45,\nX166 -> T47,\nT34 -> .(T45, T47),\nX167 -> X169,\nX133 -> .(T45, X169),\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 100 [arrowhead = none ];
100 -> 28;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 101 [arrowhead = none ];
101 -> 29;

102 [label="EVAL with clause\nselect(X182, .(X182, X183), X183).\nand substitutionX131 -> T54,\nX182 -> T54,\nX184 -> T54,\nX183 -> T55,\nT34 -> .(T54, T55),\nX133 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 102 [arrowhead = none ];
102 -> 30;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 103 [arrowhead = none ];
103 -> 31;

104 [label="INSTANCE with matching:\nX131 -> X168\nT34 -> T47\nX133 -> X169", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 104 [arrowhead = none , style=dashed];
104 -> 24[style=dashed];

105 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 105 [arrowhead = none ];
105 -> 32;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 106 [arrowhead = none ];
106 -> 34;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 107 [arrowhead = none ];
107 -> 37;

108 [label="INSTANCE with matching:\nT1 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 108 [arrowhead = none , style=dashed];
108 -> 1[style=dashed];

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 109 [arrowhead = none ];
109 -> 38;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 110 [arrowhead = none ];
110 -> 39;

111 [label="EVAL with clause\nll(s(X218), .(X219, X220)) :- ll(X218, X220).\nand substitutionX218 -> X221,\nX7 -> s(X221),\nX219 -> T72,\nX220 -> T74,\nT21 -> .(T72, T74),\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 111 [arrowhead = none ];
111 -> 40;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 112 [arrowhead = none ];
112 -> 41;

113 [label="EVAL with clause\nll(0, []).\nand substitutionX7 -> 0,\nT21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 113 [arrowhead = none ];
113 -> 42;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 114 [arrowhead = none ];
114 -> 43;

115 [label="INSTANCE with matching:\nX7 -> X221\nT21 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 115 [arrowhead = none , style=dashed];
115 -> 35[style=dashed];

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 116 [arrowhead = none ];
116 -> 44;

117 [label="EVAL with clause\nll(0, []).\nand substitutionT3 -> 0,\nX4 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 117 [arrowhead = none ];
117 -> 47;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 118 [arrowhead = none ];
118 -> 48;

119 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 119 [arrowhead = none ];
119 -> 64;

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 120 [arrowhead = none ];
120 -> 49;

121 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nT79 is ground\nreplacements:X5 -> T78,\nX6 -> T79", fontsize=14, style = filled, fillcolor = violet];
47 -> 121 [arrowhead = none ];
121 -> 50;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 122 [arrowhead = none ];
122 -> 51;

123 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 123 [arrowhead = none ];
123 -> 54;

124 [label="SPLIT 2\nnew knowledge:\nT82 is ground\nT79 is ground\nreplacements:X7 -> T82", fontsize=14, style = filled, fillcolor = violet];
50 -> 124 [arrowhead = none ];
124 -> 55;

125 [label="BACKTRACK\nfor clause: select(X, .(Y, Xs), .(Y, Ys)) :- select(X, Xs, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="BACKTRACK\nfor clause: select(X, .(X, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 126 [arrowhead = none ];
126 -> 53;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 127 [arrowhead = none ];
127 -> 56;

128 [label="INSTANCE with matching:\nT1 -> T82", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 128 [arrowhead = none , style=dashed];
128 -> 1[style=dashed];

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 129 [arrowhead = none ];
129 -> 57;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 130 [arrowhead = none ];
130 -> 58;

131 [label="EVAL with clause\nll(s(X273), .(X274, X275)) :- ll(X273, X275).\nand substitutionX273 -> X276,\nX7 -> s(X276),\nX274 -> T93,\nX275 -> T94,\nT79 -> .(T93, T94)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 131 [arrowhead = none ];
131 -> 59;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 132 [arrowhead = none ];
132 -> 60;

133 [label="EVAL with clause\nll(0, []).\nand substitutionX7 -> 0,\nT79 -> []", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 133 [arrowhead = none ];
133 -> 61;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 134 [arrowhead = none ];
134 -> 62;

135 [label="INSTANCE with matching:\nX7 -> X276\nT79 -> T94", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 135 [arrowhead = none , style=dashed];
135 -> 54[style=dashed];

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 136 [arrowhead = none ];
136 -> 63;

137 [label="EVAL with clause\nt(0).\nand substitutionT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 137 [arrowhead = none ];
137 -> 65;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 138 [arrowhead = none ];
138 -> 66;

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 139 [arrowhead = none ];
139 -> 67;

}

(2) Obligation:

Triples:

llA(s(X1), .(X2, X3)) :- llA(X1, X3).
selectB(X1, .(X2, X3), .(X2, X4)) :- selectB(X1, X3, X4).
llE(s(X1), .(X2, X3)) :- llE(X1, X3).
llG(s(X1), .(X2, X3)) :- llG(X1, X3).
tC(s(X1)) :- llA(X1, X2).
tC(s(X1)) :- ','(llcA(X1, X2), selectB(X3, X2, X4)).
tC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), llE(X6, X5))).
tC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), ','(llcE(X6, X5), tC(X6)))).
tC(0) :- ','(selectcF(X1, X2), llG(X3, X2)).
tC(0) :- ','(selectcF(X1, X2), ','(llcG(X3, X2), tC(X3))).

Clauses:

llcA(s(X1), .(X2, X3)) :- llcA(X1, X3).
llcA(0, []).
selectcB(X1, .(X2, X3), .(X2, X4)) :- selectcB(X1, X3, X4).
selectcB(X1, .(X1, X2), X2).
tcC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), ','(llcE(X6, X5), tcC(X6)))).
tcC(0) :- ','(selectcF(X1, X2), ','(llcG(X3, X2), tcC(X3))).
tcC(0).
llcE(s(X1), .(X2, X3)) :- llcE(X1, X3).
llcE(0, []).
llcG(s(X1), .(X2, X3)) :- llcG(X1, X3).
llcG(0, []).
selectcD(X1, X2, X3, .(X2, X4)) :- selectcB(X1, X3, X4).
selectcD(X1, X1, X2, X2).

Afs:

tC(x1)  =  tC(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

llA(s(X1), .(X2, X3)) :- llA(X1, X3).
selectB(X1, .(X2, X3), .(X2, X4)) :- selectB(X1, X3, X4).
llE(s(X1), .(X2, X3)) :- llE(X1, X3).
llG(s(X1), .(X2, X3)) :- llG(X1, X3).
tC(s(X1)) :- llA(X1, X2).
tC(s(X1)) :- ','(llcA(X1, X2), selectB(X3, X2, X4)).
tC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), llE(X6, X5))).
tC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), ','(llcE(X6, X5), tC(X6)))).

Clauses:

llcA(s(X1), .(X2, X3)) :- llcA(X1, X3).
llcA(0, []).
selectcB(X1, .(X2, X3), .(X2, X4)) :- selectcB(X1, X3, X4).
selectcB(X1, .(X1, X2), X2).
tcC(s(X1)) :- ','(llcA(X1, X2), ','(selectcD(X3, X4, X2, X5), ','(llcE(X6, X5), tcC(X6)))).
tcC(0).
llcE(s(X1), .(X2, X3)) :- llcE(X1, X3).
llcE(0, []).
llcG(s(X1), .(X2, X3)) :- llcG(X1, X3).
llcG(0, []).
selectcD(X1, X2, X3, .(X2, X4)) :- selectcB(X1, X3, X4).
selectcD(X1, X1, X2, X2).

Afs:

tC(x1)  =  tC(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
tC_in: (b)
llA_in: (b,f)
llcA_in: (b,f)
selectB_in: (f,b,f)
selectcD_in: (f,f,b,f)
selectcB_in: (f,b,f)
llE_in: (f,b)
llcE_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TC_IN_G(s(X1)) → U5_G(X1, llA_in_ga(X1, X2))
TC_IN_G(s(X1)) → LLA_IN_GA(X1, X2)
LLA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, llA_in_ga(X1, X3))
LLA_IN_GA(s(X1), .(X2, X3)) → LLA_IN_GA(X1, X3)
TC_IN_G(s(X1)) → U6_G(X1, llcA_in_ga(X1, X2))
U6_G(X1, llcA_out_ga(X1, X2)) → U7_G(X1, selectB_in_aga(X3, X2, X4))
U6_G(X1, llcA_out_ga(X1, X2)) → SELECTB_IN_AGA(X3, X2, X4)
SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → U2_AGA(X1, X2, X3, X4, selectB_in_aga(X1, X3, X4))
SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_AGA(X1, X3, X4)
U6_G(X1, llcA_out_ga(X1, X2)) → U8_G(X1, selectcD_in_aaga(X3, X4, X2, X5))
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → U9_G(X1, llE_in_ag(X6, X5))
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → LLE_IN_AG(X6, X5)
LLE_IN_AG(s(X1), .(X2, X3)) → U3_AG(X1, X2, X3, llE_in_ag(X1, X3))
LLE_IN_AG(s(X1), .(X2, X3)) → LLE_IN_AG(X1, X3)
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → U10_G(X1, llcE_in_ag(X6, X5))
U10_G(X1, llcE_out_ag(X6, X5)) → U11_G(X1, tC_in_g(X6))
U10_G(X1, llcE_out_ag(X6, X5)) → TC_IN_G(X6)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
tC_in_g(x1)  =  tC_in_g(x1)
s(x1)  =  s(x1)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llE_in_ag(x1, x2)  =  llE_in_ag(x2)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
TC_IN_G(x1)  =  TC_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
LLA_IN_GA(x1, x2)  =  LLA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U6_G(x1, x2)  =  U6_G(x1, x2)
U7_G(x1, x2)  =  U7_G(x1, x2)
SELECTB_IN_AGA(x1, x2, x3)  =  SELECTB_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
LLE_IN_AG(x1, x2)  =  LLE_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_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:

TC_IN_G(s(X1)) → U5_G(X1, llA_in_ga(X1, X2))
TC_IN_G(s(X1)) → LLA_IN_GA(X1, X2)
LLA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, llA_in_ga(X1, X3))
LLA_IN_GA(s(X1), .(X2, X3)) → LLA_IN_GA(X1, X3)
TC_IN_G(s(X1)) → U6_G(X1, llcA_in_ga(X1, X2))
U6_G(X1, llcA_out_ga(X1, X2)) → U7_G(X1, selectB_in_aga(X3, X2, X4))
U6_G(X1, llcA_out_ga(X1, X2)) → SELECTB_IN_AGA(X3, X2, X4)
SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → U2_AGA(X1, X2, X3, X4, selectB_in_aga(X1, X3, X4))
SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_AGA(X1, X3, X4)
U6_G(X1, llcA_out_ga(X1, X2)) → U8_G(X1, selectcD_in_aaga(X3, X4, X2, X5))
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → U9_G(X1, llE_in_ag(X6, X5))
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → LLE_IN_AG(X6, X5)
LLE_IN_AG(s(X1), .(X2, X3)) → U3_AG(X1, X2, X3, llE_in_ag(X1, X3))
LLE_IN_AG(s(X1), .(X2, X3)) → LLE_IN_AG(X1, X3)
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → U10_G(X1, llcE_in_ag(X6, X5))
U10_G(X1, llcE_out_ag(X6, X5)) → U11_G(X1, tC_in_g(X6))
U10_G(X1, llcE_out_ag(X6, X5)) → TC_IN_G(X6)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
tC_in_g(x1)  =  tC_in_g(x1)
s(x1)  =  s(x1)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llE_in_ag(x1, x2)  =  llE_in_ag(x2)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
TC_IN_G(x1)  =  TC_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
LLA_IN_GA(x1, x2)  =  LLA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U6_G(x1, x2)  =  U6_G(x1, x2)
U7_G(x1, x2)  =  U7_G(x1, x2)
SELECTB_IN_AGA(x1, x2, x3)  =  SELECTB_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
LLE_IN_AG(x1, x2)  =  LLE_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_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 10 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LLE_IN_AG(s(X1), .(X2, X3)) → LLE_IN_AG(X1, X3)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
LLE_IN_AG(x1, x2)  =  LLE_IN_AG(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:

LLE_IN_AG(s(X1), .(X2, X3)) → LLE_IN_AG(X1, X3)

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

LLE_IN_AG(.(X3)) → LLE_IN_AG(X3)

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:

• LLE_IN_AG(.(X3)) → LLE_IN_AG(X3)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_AGA(X1, X3, X4)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
SELECTB_IN_AGA(x1, x2, x3)  =  SELECTB_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:

SELECTB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_AGA(X1, X3, X4)

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

SELECTB_IN_AGA(.(X3)) → SELECTB_IN_AGA(X3)

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:

• SELECTB_IN_AGA(.(X3)) → SELECTB_IN_AGA(X3)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

LLA_IN_GA(s(X1), .(X2, X3)) → LLA_IN_GA(X1, X3)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
LLA_IN_GA(x1, x2)  =  LLA_IN_GA(x1)

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:

LLA_IN_GA(s(X1), .(X2, X3)) → LLA_IN_GA(X1, X3)

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

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:

LLA_IN_GA(s(X1)) → LLA_IN_GA(X1)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LLA_IN_GA(s(X1)) → LLA_IN_GA(X1)
  The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

TC_IN_G(s(X1)) → U6_G(X1, llcA_in_ga(X1, X2))
U6_G(X1, llcA_out_ga(X1, X2)) → U8_G(X1, selectcD_in_aaga(X3, X4, X2, X5))
U8_G(X1, selectcD_out_aaga(X3, X4, X2, X5)) → U10_G(X1, llcE_in_ag(X6, X5))
U10_G(X1, llcE_out_ag(X6, X5)) → TC_IN_G(X6)

The TRS R consists of the following rules:

llcA_in_ga(s(X1), .(X2, X3)) → U13_ga(X1, X2, X3, llcA_in_ga(X1, X3))
llcA_in_ga(0, []) → llcA_out_ga(0, [])
U13_ga(X1, X2, X3, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X2, X3))
selectcD_in_aaga(X1, X2, X3, .(X2, X4)) → U21_aaga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X2, X3), .(X2, X4)) → U14_aga(X1, X2, X3, X4, selectcB_in_aga(X1, X3, X4))
selectcB_in_aga(X1, .(X1, X2), X2) → selectcB_out_aga(X1, .(X1, X2), X2)
U14_aga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcB_out_aga(X1, .(X2, X3), .(X2, X4))
U21_aaga(X1, X2, X3, X4, selectcB_out_aga(X1, X3, X4)) → selectcD_out_aaga(X1, X2, X3, .(X2, X4))
selectcD_in_aaga(X1, X1, X2, X2) → selectcD_out_aaga(X1, X1, X2, X2)
llcE_in_ag(s(X1), .(X2, X3)) → U19_ag(X1, X2, X3, llcE_in_ag(X1, X3))
llcE_in_ag(0, []) → llcE_out_ag(0, [])
U19_ag(X1, X2, X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llcA_in_ga(x1, x2)  =  llcA_in_ga(x1)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
0  =  0
llcA_out_ga(x1, x2)  =  llcA_out_ga(x1, x2)
selectcD_in_aaga(x1, x2, x3, x4)  =  selectcD_in_aaga(x3)
U21_aaga(x1, x2, x3, x4, x5)  =  U21_aaga(x3, x5)
selectcB_in_aga(x1, x2, x3)  =  selectcB_in_aga(x2)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x3, x5)
selectcB_out_aga(x1, x2, x3)  =  selectcB_out_aga(x2, x3)
selectcD_out_aaga(x1, x2, x3, x4)  =  selectcD_out_aaga(x3, x4)
llcE_in_ag(x1, x2)  =  llcE_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llcE_out_ag(x1, x2)  =  llcE_out_ag(x1, x2)
TC_IN_G(x1)  =  TC_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

TC_IN_G(s(X1)) → U6_G(X1, llcA_in_ga(X1))
U6_G(X1, llcA_out_ga(X1, X2)) → U8_G(X1, selectcD_in_aaga(X2))
U8_G(X1, selectcD_out_aaga(X2, X5)) → U10_G(X1, llcE_in_ag(X5))
U10_G(X1, llcE_out_ag(X6, X5)) → TC_IN_G(X6)

The TRS R consists of the following rules:

llcA_in_ga(s(X1)) → U13_ga(X1, llcA_in_ga(X1))
llcA_in_ga(0) → llcA_out_ga(0, [])
U13_ga(X1, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X3))
selectcD_in_aaga(X3) → U21_aaga(X3, selectcB_in_aga(X3))
selectcB_in_aga(.(X3)) → U14_aga(X3, selectcB_in_aga(X3))
selectcB_in_aga(.(X2)) → selectcB_out_aga(.(X2), X2)
U14_aga(X3, selectcB_out_aga(X3, X4)) → selectcB_out_aga(.(X3), .(X4))
U21_aaga(X3, selectcB_out_aga(X3, X4)) → selectcD_out_aaga(X3, .(X4))
selectcD_in_aaga(X2) → selectcD_out_aaga(X2, X2)
llcE_in_ag(.(X3)) → U19_ag(X3, llcE_in_ag(X3))
llcE_in_ag([]) → llcE_out_ag(0, [])
U19_ag(X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X3))

The set Q consists of the following terms:

llcA_in_ga(x0)
U13_ga(x0, x1)
selectcD_in_aaga(x0)
selectcB_in_aga(x0)
U14_aga(x0, x1)
U21_aaga(x0, x1)
llcE_in_ag(x0)
U19_ag(x0, x1)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U6_G(X1, llcA_out_ga(X1, X2)) → U8_G(X1, selectcD_in_aaga(X2))

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

POL(.(x₁)) = 1 + x₁   
POL(0) = 0   
POL(TC_IN_G(x₁)) = x₁   
POL(U10_G(x₁, x₂)) = x₂   
POL(U13_ga(x₁, x₂)) = 1 + x₂   
POL(U14_aga(x₁, x₂)) = 1 + x₂   
POL(U19_ag(x₁, x₂)) = 1 + x₂   
POL(U21_aaga(x₁, x₂)) = x₂   
POL(U6_G(x₁, x₂)) = 1 + x₂   
POL(U8_G(x₁, x₂)) = x₂   
POL([]) = 0   
POL(llcA_in_ga(x₁)) = x₁   
POL(llcA_out_ga(x₁, x₂)) = x₂   
POL(llcE_in_ag(x₁)) = x₁   
POL(llcE_out_ag(x₁, x₂)) = x₁   
POL(s(x₁)) = 1 + x₁   
POL(selectcB_in_aga(x₁)) = x₁   
POL(selectcB_out_aga(x₁, x₂)) = 1 + x₂   
POL(selectcD_in_aaga(x₁)) = x₁   
POL(selectcD_out_aaga(x₁, x₂)) = x₂   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

llcA_in_ga(s(X1)) → U13_ga(X1, llcA_in_ga(X1))
llcA_in_ga(0) → llcA_out_ga(0, [])
selectcD_in_aaga(X3) → U21_aaga(X3, selectcB_in_aga(X3))
selectcD_in_aaga(X2) → selectcD_out_aaga(X2, X2)
llcE_in_ag(.(X3)) → U19_ag(X3, llcE_in_ag(X3))
llcE_in_ag([]) → llcE_out_ag(0, [])
U13_ga(X1, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X3))
selectcB_in_aga(.(X3)) → U14_aga(X3, selectcB_in_aga(X3))
selectcB_in_aga(.(X2)) → selectcB_out_aga(.(X2), X2)
U21_aaga(X3, selectcB_out_aga(X3, X4)) → selectcD_out_aaga(X3, .(X4))
U14_aga(X3, selectcB_out_aga(X3, X4)) → selectcB_out_aga(.(X3), .(X4))
U19_ag(X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X3))

(34) Obligation:

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

TC_IN_G(s(X1)) → U6_G(X1, llcA_in_ga(X1))
U8_G(X1, selectcD_out_aaga(X2, X5)) → U10_G(X1, llcE_in_ag(X5))
U10_G(X1, llcE_out_ag(X6, X5)) → TC_IN_G(X6)

The TRS R consists of the following rules:

llcA_in_ga(s(X1)) → U13_ga(X1, llcA_in_ga(X1))
llcA_in_ga(0) → llcA_out_ga(0, [])
U13_ga(X1, llcA_out_ga(X1, X3)) → llcA_out_ga(s(X1), .(X3))
selectcD_in_aaga(X3) → U21_aaga(X3, selectcB_in_aga(X3))
selectcB_in_aga(.(X3)) → U14_aga(X3, selectcB_in_aga(X3))
selectcB_in_aga(.(X2)) → selectcB_out_aga(.(X2), X2)
U14_aga(X3, selectcB_out_aga(X3, X4)) → selectcB_out_aga(.(X3), .(X4))
U21_aaga(X3, selectcB_out_aga(X3, X4)) → selectcD_out_aaga(X3, .(X4))
selectcD_in_aaga(X2) → selectcD_out_aaga(X2, X2)
llcE_in_ag(.(X3)) → U19_ag(X3, llcE_in_ag(X3))
llcE_in_ag([]) → llcE_out_ag(0, [])
U19_ag(X3, llcE_out_ag(X1, X3)) → llcE_out_ag(s(X1), .(X3))

The set Q consists of the following terms:

llcA_in_ga(x0)
U13_ga(x0, x1)
selectcD_in_aaga(x0)
selectcB_in_aga(x0)
U14_aga(x0, x1)
U21_aaga(x0, x1)
llcE_in_ag(x0)
U19_ag(x0, x1)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(36) TRUE