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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="9: 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="19: 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="24: 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="35: 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="49: 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="54: t(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 58 [arrowhead = none ];
58 -> 2;

59 [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 -> 59 [arrowhead = none ];
59 -> 3;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 60 [arrowhead = none ];
60 -> 4;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 61 [arrowhead = none ];
61 -> 5;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 62 [arrowhead = none ];
62 -> 6;

63 [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 -> 63 [arrowhead = none ];
63 -> 7;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 64 [arrowhead = none ];
64 -> 8;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 65 [arrowhead = none ];
65 -> 45;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 66 [arrowhead = none ];
66 -> 46;

67 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 67 [arrowhead = none ];
67 -> 9;

68 [label="SPLIT 2\nnew knowledge:\nT8 is ground\nreplacements:X32 -> T10", fontsize=14, style = filled, fillcolor = violet];
7 -> 68 [arrowhead = none ];
68 -> 10;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 69 [arrowhead = none ];
69 -> 11;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
10 -> 70 [arrowhead = none ];
70 -> 19;

71 [label="SPLIT 2\nreplacements:X5 -> T19,\nX31 -> T20,\nX6 -> T21", fontsize=14, style = filled, fillcolor = violet];
10 -> 71 [arrowhead = none ];
71 -> 20;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 72 [arrowhead = none ];
72 -> 12;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 73 [arrowhead = none ];
73 -> 13;

74 [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 -> 74 [arrowhead = none ];
74 -> 14;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 75 [arrowhead = none ];
75 -> 15;

76 [label="EVAL with clause\nll(0, []).\nand substitutionT8 -> 0,\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 76 [arrowhead = none ];
76 -> 16;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 77 [arrowhead = none ];
77 -> 17;

78 [label="INSTANCE with matching:\nT8 -> T16\nX32 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 78 [arrowhead = none , style=dashed];
78 -> 9[style=dashed];

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 79 [arrowhead = none ];
79 -> 18;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 81 [arrowhead = none ];
81 -> 35;

82 [label="SPLIT 2\nnew knowledge:\nT61 is ground\nreplacements:X7 -> T61", fontsize=14, style = filled, fillcolor = violet];
20 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 84 [arrowhead = none ];
84 -> 23;

85 [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 -> 85 [arrowhead = none ];
85 -> 24;

86 [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 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 89 [arrowhead = none ];
89 -> 27;

90 [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 -> 90 [arrowhead = none ];
90 -> 28;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 91 [arrowhead = none ];
91 -> 29;

92 [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 -> 92 [arrowhead = none ];
92 -> 30;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 93 [arrowhead = none ];
93 -> 31;

94 [label="INSTANCE with matching:\nX131 -> X168\nT34 -> T47\nX133 -> X169", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 94 [arrowhead = none , style=dashed];
94 -> 24[style=dashed];

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 95 [arrowhead = none ];
95 -> 32;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 96 [arrowhead = none ];
96 -> 34;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 97 [arrowhead = none ];
97 -> 37;

98 [label="INSTANCE with matching:\nT1 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 98 [arrowhead = none , style=dashed];
98 -> 1[style=dashed];

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 99 [arrowhead = none ];
99 -> 38;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 100 [arrowhead = none ];
100 -> 39;

101 [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 -> 101 [arrowhead = none ];
101 -> 40;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 102 [arrowhead = none ];
102 -> 41;

103 [label="EVAL with clause\nll(0, []).\nand substitutionX7 -> 0,\nT21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 103 [arrowhead = none ];
103 -> 42;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 104 [arrowhead = none ];
104 -> 43;

105 [label="INSTANCE with matching:\nX7 -> X221\nT21 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 105 [arrowhead = none , style=dashed];
105 -> 35[style=dashed];

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 106 [arrowhead = none ];
106 -> 44;

107 [label="EVAL with clause\nll(0, []).\nand substitutionT3 -> 0,\nX4 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 108 [arrowhead = none ];
108 -> 48;

109 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 109 [arrowhead = none ];
109 -> 54;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 110 [arrowhead = none ];
110 -> 49;

111 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nT79 is ground\nreplacements:X5 -> T78,\nX6 -> T79", fontsize=14, style = filled, fillcolor = violet];
47 -> 111 [arrowhead = none ];
111 -> 50;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 112 [arrowhead = none ];
112 -> 51;

113 [label="INSTANCE with matching:\nT21 -> T79", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 113 [arrowhead = none , style=dashed];
113 -> 20[style=dashed];

114 [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 -> 114 [arrowhead = none ];
114 -> 52;

115 [label="BACKTRACK\nfor clause: select(X, .(X, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 115 [arrowhead = none ];
115 -> 53;

116 [label="EVAL with clause\nt(0).\nand substitutionT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 116 [arrowhead = none ];
116 -> 55;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 117 [arrowhead = none ];
117 -> 56;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 118 [arrowhead = none ];
118 -> 57;

}

(2) Obligation:

Clauses:

llA(s(T16), .(X66, X67)) :- llA(T16, X67).
llA(0, []).
selectB(X168, .(T45, T47), .(T45, X169)) :- selectB(X168, T47, X169).
selectB(T54, .(T54, T55), T55).
llC(s(X221), .(T72, T74)) :- llC(X221, T74).
llC(0, []).
pD(X7, T21) :- llC(X7, T21).
pD(T61, T21) :- ','(llC(T61, T21), tE(T61)).
selectF(X131, X132, T34, .(X132, X133)) :- selectB(X131, T34, X133).
selectF(X193, X193, T58, T58).
tE(s(T8)) :- llA(T8, X32).
tE(s(T8)) :- ','(llA(T8, T10), selectF(X5, X31, T10, X6)).
tE(s(T8)) :- ','(llA(T8, T10), ','(selectF(T19, T20, T10, T21), pD(X7, T21))).
tE(0).

Query: tE(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
tE_in: (b)
llA_in: (b,f)
selectF_in: (f,f,b,f)
selectB_in: (f,b,f)
pD_in: (f,b)
llC_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

TE_IN_G(s(T8)) → U8_G(T8, llA_in_ga(T8, X32))
TE_IN_G(s(T8)) → LLA_IN_GA(T8, X32)
LLA_IN_GA(s(T16), .(X66, X67)) → U1_GA(T16, X66, X67, llA_in_ga(T16, X67))
LLA_IN_GA(s(T16), .(X66, X67)) → LLA_IN_GA(T16, X67)
TE_IN_G(s(T8)) → U9_G(T8, llA_in_ga(T8, T10))
U9_G(T8, llA_out_ga(T8, T10)) → U10_G(T8, selectF_in_aaga(X5, X31, T10, X6))
U9_G(T8, llA_out_ga(T8, T10)) → SELECTF_IN_AAGA(X5, X31, T10, X6)
SELECTF_IN_AAGA(X131, X132, T34, .(X132, X133)) → U7_AAGA(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
SELECTF_IN_AAGA(X131, X132, T34, .(X132, X133)) → SELECTB_IN_AGA(X131, T34, X133)
SELECTB_IN_AGA(X168, .(T45, T47), .(T45, X169)) → U2_AGA(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
SELECTB_IN_AGA(X168, .(T45, T47), .(T45, X169)) → SELECTB_IN_AGA(X168, T47, X169)
U9_G(T8, llA_out_ga(T8, T10)) → U11_G(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_G(T8, pD_in_ag(X7, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → PD_IN_AG(X7, T21)
PD_IN_AG(X7, T21) → U4_AG(X7, T21, llC_in_ag(X7, T21))
PD_IN_AG(X7, T21) → LLC_IN_AG(X7, T21)
LLC_IN_AG(s(X221), .(T72, T74)) → U3_AG(X221, T72, T74, llC_in_ag(X221, T74))
LLC_IN_AG(s(X221), .(T72, T74)) → LLC_IN_AG(X221, T74)
PD_IN_AG(T61, T21) → U5_AG(T61, T21, llC_in_ag(T61, T21))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → U6_AG(T61, T21, tE_in_g(T61))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → TE_IN_G(T61)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
TE_IN_G(x1)  =  TE_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x2)
LLA_IN_GA(x1, x2)  =  LLA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
SELECTF_IN_AAGA(x1, x2, x3, x4)  =  SELECTF_IN_AAGA(x3)
U7_AAGA(x1, x2, x3, x4, x5)  =  U7_AAGA(x5)
SELECTB_IN_AGA(x1, x2, x3)  =  SELECTB_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U11_G(x1, x2)  =  U11_G(x2)
U12_G(x1, x2)  =  U12_G(x2)
PD_IN_AG(x1, x2)  =  PD_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x3)
LLC_IN_AG(x1, x2)  =  LLC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U6_AG(x1, x2, x3)  =  U6_AG(x1, x3)

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

(6) Obligation:

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

TE_IN_G(s(T8)) → U8_G(T8, llA_in_ga(T8, X32))
TE_IN_G(s(T8)) → LLA_IN_GA(T8, X32)
LLA_IN_GA(s(T16), .(X66, X67)) → U1_GA(T16, X66, X67, llA_in_ga(T16, X67))
LLA_IN_GA(s(T16), .(X66, X67)) → LLA_IN_GA(T16, X67)
TE_IN_G(s(T8)) → U9_G(T8, llA_in_ga(T8, T10))
U9_G(T8, llA_out_ga(T8, T10)) → U10_G(T8, selectF_in_aaga(X5, X31, T10, X6))
U9_G(T8, llA_out_ga(T8, T10)) → SELECTF_IN_AAGA(X5, X31, T10, X6)
SELECTF_IN_AAGA(X131, X132, T34, .(X132, X133)) → U7_AAGA(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
SELECTF_IN_AAGA(X131, X132, T34, .(X132, X133)) → SELECTB_IN_AGA(X131, T34, X133)
SELECTB_IN_AGA(X168, .(T45, T47), .(T45, X169)) → U2_AGA(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
SELECTB_IN_AGA(X168, .(T45, T47), .(T45, X169)) → SELECTB_IN_AGA(X168, T47, X169)
U9_G(T8, llA_out_ga(T8, T10)) → U11_G(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_G(T8, pD_in_ag(X7, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → PD_IN_AG(X7, T21)
PD_IN_AG(X7, T21) → U4_AG(X7, T21, llC_in_ag(X7, T21))
PD_IN_AG(X7, T21) → LLC_IN_AG(X7, T21)
LLC_IN_AG(s(X221), .(T72, T74)) → U3_AG(X221, T72, T74, llC_in_ag(X221, T74))
LLC_IN_AG(s(X221), .(T72, T74)) → LLC_IN_AG(X221, T74)
PD_IN_AG(T61, T21) → U5_AG(T61, T21, llC_in_ag(T61, T21))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → U6_AG(T61, T21, tE_in_g(T61))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → TE_IN_G(T61)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
TE_IN_G(x1)  =  TE_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x2)
LLA_IN_GA(x1, x2)  =  LLA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
SELECTF_IN_AAGA(x1, x2, x3, x4)  =  SELECTF_IN_AAGA(x3)
U7_AAGA(x1, x2, x3, x4, x5)  =  U7_AAGA(x5)
SELECTB_IN_AGA(x1, x2, x3)  =  SELECTB_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U11_G(x1, x2)  =  U11_G(x2)
U12_G(x1, x2)  =  U12_G(x2)
PD_IN_AG(x1, x2)  =  PD_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x3)
LLC_IN_AG(x1, x2)  =  LLC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U6_AG(x1, x2, x3)  =  U6_AG(x1, x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 13 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LLC_IN_AG(s(X221), .(T72, T74)) → LLC_IN_AG(X221, T74)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
LLC_IN_AG(x1, x2)  =  LLC_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:

LLC_IN_AG(s(X221), .(T72, T74)) → LLC_IN_AG(X221, T74)

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

LLC_IN_AG(.(T74)) → LLC_IN_AG(T74)

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:

• LLC_IN_AG(.(T74)) → LLC_IN_AG(T74)
  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(X168, .(T45, T47), .(T45, X169)) → SELECTB_IN_AGA(X168, T47, X169)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
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(X168, .(T45, T47), .(T45, X169)) → SELECTB_IN_AGA(X168, T47, X169)

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(.(T47)) → SELECTB_IN_AGA(T47)

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(.(T47)) → SELECTB_IN_AGA(T47)
  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(T16), .(X66, X67)) → LLA_IN_GA(T16, X67)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
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(T16), .(X66, X67)) → LLA_IN_GA(T16, X67)

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(T16)) → LLA_IN_GA(T16)

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(T16)) → LLA_IN_GA(T16)
  The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

TE_IN_G(s(T8)) → U9_G(T8, llA_in_ga(T8, T10))
U9_G(T8, llA_out_ga(T8, T10)) → U11_G(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → PD_IN_AG(X7, T21)
PD_IN_AG(T61, T21) → U5_AG(T61, T21, llC_in_ag(T61, T21))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → TE_IN_G(T61)

The TRS R consists of the following rules:

tE_in_g(s(T8)) → U8_g(T8, llA_in_ga(T8, X32))
llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U8_g(T8, llA_out_ga(T8, X32)) → tE_out_g(s(T8))
tE_in_g(s(T8)) → U9_g(T8, llA_in_ga(T8, T10))
U9_g(T8, llA_out_ga(T8, T10)) → U10_g(T8, selectF_in_aaga(X5, X31, T10, X6))
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
U10_g(T8, selectF_out_aaga(X5, X31, T10, X6)) → tE_out_g(s(T8))
U9_g(T8, llA_out_ga(T8, T10)) → U11_g(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_g(T8, selectF_out_aaga(T19, T20, T10, T21)) → U12_g(T8, pD_in_ag(X7, T21))
pD_in_ag(X7, T21) → U4_ag(X7, T21, llC_in_ag(X7, T21))
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
U4_ag(X7, T21, llC_out_ag(X7, T21)) → pD_out_ag(X7, T21)
pD_in_ag(T61, T21) → U5_ag(T61, T21, llC_in_ag(T61, T21))
U5_ag(T61, T21, llC_out_ag(T61, T21)) → U6_ag(T61, T21, tE_in_g(T61))
tE_in_g(0) → tE_out_g(0)
U6_ag(T61, T21, tE_out_g(T61)) → pD_out_ag(T61, T21)
U12_g(T8, pD_out_ag(X7, T21)) → tE_out_g(s(T8))

The argument filtering Pi contains the following mapping:
tE_in_g(x1)  =  tE_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
tE_out_g(x1)  =  tE_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
pD_in_ag(x1, x2)  =  pD_in_ag(x2)
U4_ag(x1, x2, x3)  =  U4_ag(x3)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
pD_out_ag(x1, x2)  =  pD_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U6_ag(x1, x2, x3)  =  U6_ag(x1, x3)
TE_IN_G(x1)  =  TE_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x2)
U11_G(x1, x2)  =  U11_G(x2)
PD_IN_AG(x1, x2)  =  PD_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

TE_IN_G(s(T8)) → U9_G(T8, llA_in_ga(T8, T10))
U9_G(T8, llA_out_ga(T8, T10)) → U11_G(T8, selectF_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectF_out_aaga(T19, T20, T10, T21)) → PD_IN_AG(X7, T21)
PD_IN_AG(T61, T21) → U5_AG(T61, T21, llC_in_ag(T61, T21))
U5_AG(T61, T21, llC_out_ag(T61, T21)) → TE_IN_G(T61)

The TRS R consists of the following rules:

llA_in_ga(s(T16), .(X66, X67)) → U1_ga(T16, X66, X67, llA_in_ga(T16, X67))
llA_in_ga(0, []) → llA_out_ga(0, [])
selectF_in_aaga(X131, X132, T34, .(X132, X133)) → U7_aaga(X131, X132, T34, X133, selectB_in_aga(X131, T34, X133))
selectF_in_aaga(X193, X193, T58, T58) → selectF_out_aaga(X193, X193, T58, T58)
llC_in_ag(s(X221), .(T72, T74)) → U3_ag(X221, T72, T74, llC_in_ag(X221, T74))
llC_in_ag(0, []) → llC_out_ag(0, [])
U1_ga(T16, X66, X67, llA_out_ga(T16, X67)) → llA_out_ga(s(T16), .(X66, X67))
U7_aaga(X131, X132, T34, X133, selectB_out_aga(X131, T34, X133)) → selectF_out_aaga(X131, X132, T34, .(X132, X133))
U3_ag(X221, T72, T74, llC_out_ag(X221, T74)) → llC_out_ag(s(X221), .(T72, T74))
selectB_in_aga(X168, .(T45, T47), .(T45, X169)) → U2_aga(X168, T45, T47, X169, selectB_in_aga(X168, T47, X169))
selectB_in_aga(T54, .(T54, T55), T55) → selectB_out_aga(T54, .(T54, T55), T55)
U2_aga(X168, T45, T47, X169, selectB_out_aga(X168, T47, X169)) → selectB_out_aga(X168, .(T45, T47), .(T45, X169))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
llA_in_ga(x1, x2)  =  llA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
llA_out_ga(x1, x2)  =  llA_out_ga(x2)
.(x1, x2)  =  .(x2)
selectF_in_aaga(x1, x2, x3, x4)  =  selectF_in_aaga(x3)
U7_aaga(x1, x2, x3, x4, x5)  =  U7_aaga(x5)
selectB_in_aga(x1, x2, x3)  =  selectB_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
selectB_out_aga(x1, x2, x3)  =  selectB_out_aga(x3)
selectF_out_aaga(x1, x2, x3, x4)  =  selectF_out_aaga(x4)
llC_in_ag(x1, x2)  =  llC_in_ag(x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
[]  =  []
llC_out_ag(x1, x2)  =  llC_out_ag(x1)
TE_IN_G(x1)  =  TE_IN_G(x1)
U9_G(x1, x2)  =  U9_G(x2)
U11_G(x1, x2)  =  U11_G(x2)
PD_IN_AG(x1, x2)  =  PD_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x3)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

TE_IN_G(s(T8)) → U9_G(llA_in_ga(T8))
U9_G(llA_out_ga(T10)) → U11_G(selectF_in_aaga(T10))
U11_G(selectF_out_aaga(T21)) → PD_IN_AG(T21)
PD_IN_AG(T21) → U5_AG(llC_in_ag(T21))
U5_AG(llC_out_ag(T61)) → TE_IN_G(T61)

The TRS R consists of the following rules:

llA_in_ga(s(T16)) → U1_ga(llA_in_ga(T16))
llA_in_ga(0) → llA_out_ga([])
selectF_in_aaga(T34) → U7_aaga(selectB_in_aga(T34))
selectF_in_aaga(T58) → selectF_out_aaga(T58)
llC_in_ag(.(T74)) → U3_ag(llC_in_ag(T74))
llC_in_ag([]) → llC_out_ag(0)
U1_ga(llA_out_ga(X67)) → llA_out_ga(.(X67))
U7_aaga(selectB_out_aga(X133)) → selectF_out_aaga(.(X133))
U3_ag(llC_out_ag(X221)) → llC_out_ag(s(X221))
selectB_in_aga(.(T47)) → U2_aga(selectB_in_aga(T47))
selectB_in_aga(.(T55)) → selectB_out_aga(T55)
U2_aga(selectB_out_aga(X169)) → selectB_out_aga(.(X169))

The set Q consists of the following terms:

llA_in_ga(x0)
selectF_in_aaga(x0)
llC_in_ag(x0)
U1_ga(x0)
U7_aaga(x0)
U3_ag(x0)
selectB_in_aga(x0)
U2_aga(x0)

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

(35) MRRProof (EQUIVALENT transformation)

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

TE_IN_G(s(T8)) → U9_G(llA_in_ga(T8))
U9_G(llA_out_ga(T10)) → U11_G(selectF_in_aaga(T10))
U11_G(selectF_out_aaga(T21)) → PD_IN_AG(T21)
PD_IN_AG(T21) → U5_AG(llC_in_ag(T21))
U5_AG(llC_out_ag(T61)) → TE_IN_G(T61)

Strictly oriented rules of the TRS R:

llA_in_ga(s(T16)) → U1_ga(llA_in_ga(T16))
llA_in_ga(0) → llA_out_ga([])
selectF_in_aaga(T34) → U7_aaga(selectB_in_aga(T34))
selectF_in_aaga(T58) → selectF_out_aaga(T58)
llC_in_ag(.(T74)) → U3_ag(llC_in_ag(T74))
llC_in_ag([]) → llC_out_ag(0)
U1_ga(llA_out_ga(X67)) → llA_out_ga(.(X67))
U7_aaga(selectB_out_aga(X133)) → selectF_out_aaga(.(X133))
U3_ag(llC_out_ag(X221)) → llC_out_ag(s(X221))
selectB_in_aga(.(T47)) → U2_aga(selectB_in_aga(T47))
selectB_in_aga(.(T55)) → selectB_out_aga(T55)
U2_aga(selectB_out_aga(X169)) → selectB_out_aga(.(X169))

Used ordering: Knuth-Bendix order [KBO] with precedence:

PD_INAG1 > selectF_inaaga1 > U9_G1 > llC_inag1 > U7_aaga1 > s₁ > selectF_outaaga1 > U5_AG1 > selectB_inaga1 > U11_G1 > 0 > TE_ING1 > U2_aga1 > selectB_outaga1 > [] > llA_inga1 > U3_ag1 > llC_outag1 > .₁ > U1_ga1 > llA_outga1

and weight map:

0=2
[]=1
llA_in_ga_1=7
s_1=6
U1_ga_1=6
llA_out_ga_1=8
selectF_in_aaga_1=9
U7_aaga_1=6
selectB_in_aga_1=2
selectF_out_aaga_1=7
llC_in_ag_1=9
._1=6
U3_ag_1=6
llC_out_ag_1=7
selectB_out_aga_1=7
U2_aga_1=6
TE_IN_G_1=7
U9_G_1=5
U11_G_1=4
PD_IN_AG_1=10
U5_AG_1=1

The variable weight is 1

(36) Obligation:

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

llA_in_ga(x0)
selectF_in_aaga(x0)
llC_in_ag(x0)
U1_ga(x0)
U7_aaga(x0)
U3_ag(x0)
selectB_in_aga(x0)
U2_aga(x0)

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

(37) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(38) YES