Left Termination of the query pattern t_in_1(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).

Queries:

t(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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, X70)\n\nT16 is ground\nX70 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(X137, T34, X139)\n\nX137 is free\nX139 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: select(X137, T34, X139), SCOPE: 5, CLAUSE: 4 | select(X137, T34, X139), SCOPE: 5, CLAUSE: 5\n\nX137 is free\nX139 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: select(X137, T34, X139), SCOPE: 5, CLAUSE: 4\n\nX137 is free\nX139 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: select(X137, T34, X139), SCOPE: 5, CLAUSE: 5\n\nX137 is free\nX139 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: select(X174, T47, X175)\n\nX174 is free\nX175 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(T58)\n\nT58 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(X233, T71)\n\nX233 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, T76), t(X7))\n\nT76 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 substitution\nT1 -> T3\nX3 -> T3", fontsize=14, style = filled, fillcolor = white];
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 substitution\nX28 -> 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 = lightblue];
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\nreplacements:\nX32 -> 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:\nX5 -> 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(X66), .(X67, X68)) :- ll(X66, X68).\nand substitution\nX66 -> T16\nT8 -> s(T16)\nX67 -> X69\nX68 -> X70\nX32 -> .(X69, X70)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 74 [arrowhead = none ];
74 -> 14;

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

76 [label="EVAL with clause\nll(0, []).\nand substitution\nT8 -> 0\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 76 [arrowhead = none ];
76 -> 16;

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

78 [label="INSTANCE with matching:\nT8 -> T16\nX32 -> X70", 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:\nT58 is ground\nreplacements:\nX7 -> T58", 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(X133, .(X134, X135), .(X134, X136)) :- select(X133, X135, X136).\nand substitution\nX5 -> X137\nX133 -> X137\nX31 -> X138\nX134 -> X138\nT10 -> T34\nX135 -> T34\nX136 -> X139\nX6 -> .(X138, X139)\nT33 -> T34", fontsize=14, style = filled, fillcolor = white];
22 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="ONLY EVAL with clause\nselect(X203, .(X203, X204), X204).\nand substitution\nX5 -> X205\nX203 -> X205\nX31 -> X205\nT10 -> T55\nX204 -> T55\nX6 -> T55", fontsize=14, style = filled, fillcolor = white];
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(X170, .(X171, X172), .(X171, X173)) :- select(X170, X172, X173).\nand substitution\nX137 -> X174\nX170 -> X174\nX171 -> T45\nX172 -> T47\nT34 -> .(T45, T47)\nX173 -> X175\nX139 -> .(T45, X175)\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 90 [arrowhead = none ];
90 -> 28;

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

92 [label="EVAL with clause\nselect(X192, .(X192, X193), X193).\nand substitution\nX137 -> X194\nX192 -> X194\nX193 -> T52\nT34 -> .(X194, T52)\nX139 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 92 [arrowhead = none ];
92 -> 30;

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

94 [label="INSTANCE with matching:\nX137 -> X174\nT34 -> T47\nX139 -> X175", 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 -> T58", 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(X230), .(X231, X232)) :- ll(X230, X232).\nand substitution\nX230 -> X233\nX7 -> s(X233)\nX231 -> T69\nX232 -> T71\nT21 -> .(T69, T71)\nT70 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 101 [arrowhead = none ];
101 -> 40;

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

103 [label="EVAL with clause\nll(0, []).\nand substitution\nX7 -> 0\nT21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 103 [arrowhead = none ];
103 -> 42;

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

105 [label="INSTANCE with matching:\nX7 -> X233\nT21 -> T71", 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 substitution\nT3 -> 0\nX4 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT75 is ground\nT76 is ground\nreplacements:\nX5 -> T75\nX6 -> T76", 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 -> T76", 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 = white];
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 = white];
52 -> 115 [arrowhead = none ];
115 -> 53;

116 [label="EVAL with clause\nt(0).\nand substitution\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 116 [arrowhead = none ];
116 -> 55;

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

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

}

(2) Obligation:

Triples:

ll9(s(T16), .(X69, X70)) :- ll9(T16, X70).
select24(X174, .(T45, T47), .(T45, X175)) :- select24(X174, T47, X175).
ll35(s(X233), .(T69, T71)) :- ll35(X233, T71).
p20(X7, T21) :- ll35(X7, T21).
p20(T58, T21) :- ','(llc35(T58, T21), t1(T58)).
t1(s(T8)) :- ll9(T8, X32).
t1(s(T8)) :- ','(llc9(T8, T34), select24(X137, T34, X139)).
t1(s(T8)) :- ','(llc9(T8, T10), ','(selectc19(T19, T20, T10, T21), p20(X7, T21))).
t1(0) :- ','(selectc49(T75, T76), p20(X7, T76)).

Clauses:

llc9(s(T16), .(X69, X70)) :- llc9(T16, X70).
llc9(0, []).
selectc24(X174, .(T45, T47), .(T45, X175)) :- selectc24(X174, T47, X175).
selectc24(X194, .(X194, T52), T52).
tc1(s(T8)) :- ','(llc9(T8, T10), ','(selectc19(T19, T20, T10, T21), qc20(X7, T21))).
tc1(0) :- ','(selectc49(T75, T76), qc20(X7, T76)).
tc1(0).
llc35(s(X233), .(T69, T71)) :- llc35(X233, T71).
llc35(0, []).
qc20(T58, T21) :- ','(llc35(T58, T21), tc1(T58)).
selectc19(X137, X138, T34, .(X138, X139)) :- selectc24(X137, T34, X139).
selectc19(X205, X205, T55, T55).

Afs:

t1(x1)  =  t1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

ll9(s(T16), .(X69, X70)) :- ll9(T16, X70).
select24(X174, .(T45, T47), .(T45, X175)) :- select24(X174, T47, X175).
ll35(s(X233), .(T69, T71)) :- ll35(X233, T71).
p20(X7, T21) :- ll35(X7, T21).
p20(T58, T21) :- ','(llc35(T58, T21), t1(T58)).
t1(s(T8)) :- ll9(T8, X32).
t1(s(T8)) :- ','(llc9(T8, T34), select24(X137, T34, X139)).
t1(s(T8)) :- ','(llc9(T8, T10), ','(selectc19(T19, T20, T10, T21), p20(X7, T21))).

Clauses:

llc9(s(T16), .(X69, X70)) :- llc9(T16, X70).
llc9(0, []).
selectc24(X174, .(T45, T47), .(T45, X175)) :- selectc24(X174, T47, X175).
selectc24(X194, .(X194, T52), T52).
tc1(s(T8)) :- ','(llc9(T8, T10), ','(selectc19(T19, T20, T10, T21), qc20(X7, T21))).
tc1(0).
llc35(s(X233), .(T69, T71)) :- llc35(X233, T71).
llc35(0, []).
qc20(T58, T21) :- ','(llc35(T58, T21), tc1(T58)).
selectc19(X137, X138, T34, .(X138, X139)) :- selectc24(X137, T34, X139).
selectc19(X205, X205, T55, T55).

Afs:

t1(x1)  =  t1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
t1_in: (b)
ll9_in: (b,f)
llc9_in: (b,f)
select24_in: (f,b,f)
selectc19_in: (f,f,b,f)
selectc24_in: (f,b,f)
p20_in: (f,b)
ll35_in: (f,b)
llc35_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

T1_IN_G(s(T8)) → U7_G(T8, ll9_in_ga(T8, X32))
T1_IN_G(s(T8)) → LL9_IN_GA(T8, X32)
LL9_IN_GA(s(T16), .(X69, X70)) → U1_GA(T16, X69, X70, ll9_in_ga(T16, X70))
LL9_IN_GA(s(T16), .(X69, X70)) → LL9_IN_GA(T16, X70)
T1_IN_G(s(T8)) → U8_G(T8, llc9_in_ga(T8, T34))
U8_G(T8, llc9_out_ga(T8, T34)) → U9_G(T8, select24_in_aga(X137, T34, X139))
U8_G(T8, llc9_out_ga(T8, T34)) → SELECT24_IN_AGA(X137, T34, X139)
SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → U2_AGA(X174, T45, T47, X175, select24_in_aga(X174, T47, X175))
SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → SELECT24_IN_AGA(X174, T47, X175)
T1_IN_G(s(T8)) → U10_G(T8, llc9_in_ga(T8, T10))
U10_G(T8, llc9_out_ga(T8, T10)) → U11_G(T8, selectc19_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectc19_out_aaga(T19, T20, T10, T21)) → U12_G(T8, p20_in_ag(X7, T21))
U11_G(T8, selectc19_out_aaga(T19, T20, T10, T21)) → P20_IN_AG(X7, T21)
P20_IN_AG(X7, T21) → U4_AG(X7, T21, ll35_in_ag(X7, T21))
P20_IN_AG(X7, T21) → LL35_IN_AG(X7, T21)
LL35_IN_AG(s(X233), .(T69, T71)) → U3_AG(X233, T69, T71, ll35_in_ag(X233, T71))
LL35_IN_AG(s(X233), .(T69, T71)) → LL35_IN_AG(X233, T71)
P20_IN_AG(T58, T21) → U5_AG(T58, T21, llc35_in_ag(T58, T21))
U5_AG(T58, T21, llc35_out_ag(T58, T21)) → U6_AG(T58, T21, t1_in_g(T58))
U5_AG(T58, T21, llc35_out_ag(T58, T21)) → T1_IN_G(T58)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
t1_in_g(x1)  =  t1_in_g(x1)
s(x1)  =  s(x1)
ll9_in_ga(x1, x2)  =  ll9_in_ga(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
select24_in_aga(x1, x2, x3)  =  select24_in_aga(x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
p20_in_ag(x1, x2)  =  p20_in_ag(x2)
ll35_in_ag(x1, x2)  =  ll35_in_ag(x2)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
T1_IN_G(x1)  =  T1_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
LL9_IN_GA(x1, x2)  =  LL9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
SELECT24_IN_AGA(x1, x2, x3)  =  SELECT24_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
P20_IN_AG(x1, x2)  =  P20_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
LL35_IN_AG(x1, x2)  =  LL35_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
U6_AG(x1, x2, x3)  =  U6_AG(x1, x2, x3)

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:

T1_IN_G(s(T8)) → U7_G(T8, ll9_in_ga(T8, X32))
T1_IN_G(s(T8)) → LL9_IN_GA(T8, X32)
LL9_IN_GA(s(T16), .(X69, X70)) → U1_GA(T16, X69, X70, ll9_in_ga(T16, X70))
LL9_IN_GA(s(T16), .(X69, X70)) → LL9_IN_GA(T16, X70)
T1_IN_G(s(T8)) → U8_G(T8, llc9_in_ga(T8, T34))
U8_G(T8, llc9_out_ga(T8, T34)) → U9_G(T8, select24_in_aga(X137, T34, X139))
U8_G(T8, llc9_out_ga(T8, T34)) → SELECT24_IN_AGA(X137, T34, X139)
SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → U2_AGA(X174, T45, T47, X175, select24_in_aga(X174, T47, X175))
SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → SELECT24_IN_AGA(X174, T47, X175)
T1_IN_G(s(T8)) → U10_G(T8, llc9_in_ga(T8, T10))
U10_G(T8, llc9_out_ga(T8, T10)) → U11_G(T8, selectc19_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectc19_out_aaga(T19, T20, T10, T21)) → U12_G(T8, p20_in_ag(X7, T21))
U11_G(T8, selectc19_out_aaga(T19, T20, T10, T21)) → P20_IN_AG(X7, T21)
P20_IN_AG(X7, T21) → U4_AG(X7, T21, ll35_in_ag(X7, T21))
P20_IN_AG(X7, T21) → LL35_IN_AG(X7, T21)
LL35_IN_AG(s(X233), .(T69, T71)) → U3_AG(X233, T69, T71, ll35_in_ag(X233, T71))
LL35_IN_AG(s(X233), .(T69, T71)) → LL35_IN_AG(X233, T71)
P20_IN_AG(T58, T21) → U5_AG(T58, T21, llc35_in_ag(T58, T21))
U5_AG(T58, T21, llc35_out_ag(T58, T21)) → U6_AG(T58, T21, t1_in_g(T58))
U5_AG(T58, T21, llc35_out_ag(T58, T21)) → T1_IN_G(T58)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
t1_in_g(x1)  =  t1_in_g(x1)
s(x1)  =  s(x1)
ll9_in_ga(x1, x2)  =  ll9_in_ga(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
select24_in_aga(x1, x2, x3)  =  select24_in_aga(x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
p20_in_ag(x1, x2)  =  p20_in_ag(x2)
ll35_in_ag(x1, x2)  =  ll35_in_ag(x2)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
T1_IN_G(x1)  =  T1_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
LL9_IN_GA(x1, x2)  =  LL9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
SELECT24_IN_AGA(x1, x2, x3)  =  SELECT24_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
P20_IN_AG(x1, x2)  =  P20_IN_AG(x2)
U4_AG(x1, x2, x3)  =  U4_AG(x2, x3)
LL35_IN_AG(x1, x2)  =  LL35_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
U6_AG(x1, x2, x3)  =  U6_AG(x1, x2, 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 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LL35_IN_AG(s(X233), .(T69, T71)) → LL35_IN_AG(X233, T71)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
LL35_IN_AG(x1, x2)  =  LL35_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:

LL35_IN_AG(s(X233), .(T69, T71)) → LL35_IN_AG(X233, T71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
LL35_IN_AG(x1, x2)  =  LL35_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:

LL35_IN_AG(.(T71)) → LL35_IN_AG(T71)

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:

• LL35_IN_AG(.(T71)) → LL35_IN_AG(T71)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → SELECT24_IN_AGA(X174, T47, X175)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
SELECT24_IN_AGA(x1, x2, x3)  =  SELECT24_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:

SELECT24_IN_AGA(X174, .(T45, T47), .(T45, X175)) → SELECT24_IN_AGA(X174, T47, X175)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SELECT24_IN_AGA(x1, x2, x3)  =  SELECT24_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:

SELECT24_IN_AGA(.(T47)) → SELECT24_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:

• SELECT24_IN_AGA(.(T47)) → SELECT24_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:

LL9_IN_GA(s(T16), .(X69, X70)) → LL9_IN_GA(T16, X70)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
LL9_IN_GA(x1, x2)  =  LL9_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:

LL9_IN_GA(s(T16), .(X69, X70)) → LL9_IN_GA(T16, X70)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
LL9_IN_GA(x1, x2)  =  LL9_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:

LL9_IN_GA(s(T16)) → LL9_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:

• LL9_IN_GA(s(T16)) → LL9_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:

T1_IN_G(s(T8)) → U10_G(T8, llc9_in_ga(T8, T10))
U10_G(T8, llc9_out_ga(T8, T10)) → U11_G(T8, selectc19_in_aaga(T19, T20, T10, T21))
U11_G(T8, selectc19_out_aaga(T19, T20, T10, T21)) → P20_IN_AG(X7, T21)
P20_IN_AG(T58, T21) → U5_AG(T58, T21, llc35_in_ag(T58, T21))
U5_AG(T58, T21, llc35_out_ag(T58, T21)) → T1_IN_G(T58)

The TRS R consists of the following rules:

llc9_in_ga(s(T16), .(X69, X70)) → U14_ga(T16, X69, X70, llc9_in_ga(T16, X70))
llc9_in_ga(0, []) → llc9_out_ga(0, [])
U14_ga(T16, X69, X70, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X69, X70))
selectc19_in_aaga(X137, X138, T34, .(X138, X139)) → U22_aaga(X137, X138, T34, X139, selectc24_in_aga(X137, T34, X139))
selectc24_in_aga(X174, .(T45, T47), .(T45, X175)) → U15_aga(X174, T45, T47, X175, selectc24_in_aga(X174, T47, X175))
selectc24_in_aga(X194, .(X194, T52), T52) → selectc24_out_aga(X194, .(X194, T52), T52)
U15_aga(X174, T45, T47, X175, selectc24_out_aga(X174, T47, X175)) → selectc24_out_aga(X174, .(T45, T47), .(T45, X175))
U22_aaga(X137, X138, T34, X139, selectc24_out_aga(X137, T34, X139)) → selectc19_out_aaga(X137, X138, T34, .(X138, X139))
selectc19_in_aaga(X205, X205, T55, T55) → selectc19_out_aaga(X205, X205, T55, T55)
llc35_in_ag(s(X233), .(T69, T71)) → U19_ag(X233, T69, T71, llc35_in_ag(X233, T71))
llc35_in_ag(0, []) → llc35_out_ag(0, [])
U19_ag(X233, T69, T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T69, T71))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
llc9_in_ga(x1, x2)  =  llc9_in_ga(x1)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
0  =  0
llc9_out_ga(x1, x2)  =  llc9_out_ga(x1, x2)
selectc19_in_aaga(x1, x2, x3, x4)  =  selectc19_in_aaga(x3)
U22_aaga(x1, x2, x3, x4, x5)  =  U22_aaga(x3, x5)
selectc24_in_aga(x1, x2, x3)  =  selectc24_in_aga(x2)
U15_aga(x1, x2, x3, x4, x5)  =  U15_aga(x3, x5)
selectc24_out_aga(x1, x2, x3)  =  selectc24_out_aga(x2, x3)
selectc19_out_aaga(x1, x2, x3, x4)  =  selectc19_out_aaga(x3, x4)
llc35_in_ag(x1, x2)  =  llc35_in_ag(x2)
U19_ag(x1, x2, x3, x4)  =  U19_ag(x3, x4)
[]  =  []
llc35_out_ag(x1, x2)  =  llc35_out_ag(x1, x2)
T1_IN_G(x1)  =  T1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
U11_G(x1, x2)  =  U11_G(x1, x2)
P20_IN_AG(x1, x2)  =  P20_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)

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:

T1_IN_G(s(T8)) → U10_G(T8, llc9_in_ga(T8))
U10_G(T8, llc9_out_ga(T8, T10)) → U11_G(T8, selectc19_in_aaga(T10))
U11_G(T8, selectc19_out_aaga(T10, T21)) → P20_IN_AG(T21)
P20_IN_AG(T21) → U5_AG(T21, llc35_in_ag(T21))
U5_AG(T21, llc35_out_ag(T58, T21)) → T1_IN_G(T58)

The TRS R consists of the following rules:

llc9_in_ga(s(T16)) → U14_ga(T16, llc9_in_ga(T16))
llc9_in_ga(0) → llc9_out_ga(0, [])
U14_ga(T16, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X70))
selectc19_in_aaga(T34) → U22_aaga(T34, selectc24_in_aga(T34))
selectc24_in_aga(.(T47)) → U15_aga(T47, selectc24_in_aga(T47))
selectc24_in_aga(.(T52)) → selectc24_out_aga(.(T52), T52)
U15_aga(T47, selectc24_out_aga(T47, X175)) → selectc24_out_aga(.(T47), .(X175))
U22_aaga(T34, selectc24_out_aga(T34, X139)) → selectc19_out_aaga(T34, .(X139))
selectc19_in_aaga(T55) → selectc19_out_aaga(T55, T55)
llc35_in_ag(.(T71)) → U19_ag(T71, llc35_in_ag(T71))
llc35_in_ag([]) → llc35_out_ag(0, [])
U19_ag(T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T71))

The set Q consists of the following terms:

llc9_in_ga(x0)
U14_ga(x0, x1)
selectc19_in_aaga(x0)
selectc24_in_aga(x0)
U15_aga(x0, x1)
U22_aaga(x0, x1)
llc35_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].

The following pairs can be oriented strictly and are deleted.

T1_IN_G(s(T8)) → U10_G(T8, llc9_in_ga(T8))

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

POL(.(x₁)) = 1 + x₁   
POL(0) = 0   
POL(P20_IN_AG(x₁)) = x₁   
POL(T1_IN_G(x₁)) = x₁   
POL(U10_G(x₁, x₂)) = x₂   
POL(U11_G(x₁, x₂)) = x₂   
POL(U14_ga(x₁, x₂)) = 1 + x₂   
POL(U15_aga(x₁, x₂)) = 1 + x₂   
POL(U19_ag(x₁, x₂)) = 1 + x₂   
POL(U22_aaga(x₁, x₂)) = x₂   
POL(U5_AG(x₁, x₂)) = x₂   
POL([]) = 0   
POL(llc35_in_ag(x₁)) = x₁   
POL(llc35_out_ag(x₁, x₂)) = x₁   
POL(llc9_in_ga(x₁)) = x₁   
POL(llc9_out_ga(x₁, x₂)) = x₂   
POL(s(x₁)) = 1 + x₁   
POL(selectc19_in_aaga(x₁)) = x₁   
POL(selectc19_out_aaga(x₁, x₂)) = x₂   
POL(selectc24_in_aga(x₁)) = x₁   
POL(selectc24_out_aga(x₁, x₂)) = 1 + x₂   

The following usable rules [FROCOS05] were oriented:

llc9_in_ga(s(T16)) → U14_ga(T16, llc9_in_ga(T16))
llc9_in_ga(0) → llc9_out_ga(0, [])
selectc19_in_aaga(T34) → U22_aaga(T34, selectc24_in_aga(T34))
selectc19_in_aaga(T55) → selectc19_out_aaga(T55, T55)
llc35_in_ag(.(T71)) → U19_ag(T71, llc35_in_ag(T71))
llc35_in_ag([]) → llc35_out_ag(0, [])
U14_ga(T16, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X70))
selectc24_in_aga(.(T47)) → U15_aga(T47, selectc24_in_aga(T47))
selectc24_in_aga(.(T52)) → selectc24_out_aga(.(T52), T52)
U22_aaga(T34, selectc24_out_aga(T34, X139)) → selectc19_out_aaga(T34, .(X139))
U15_aga(T47, selectc24_out_aga(T47, X175)) → selectc24_out_aga(.(T47), .(X175))
U19_ag(T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T71))

(34) Obligation:

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

U10_G(T8, llc9_out_ga(T8, T10)) → U11_G(T8, selectc19_in_aaga(T10))
U11_G(T8, selectc19_out_aaga(T10, T21)) → P20_IN_AG(T21)
P20_IN_AG(T21) → U5_AG(T21, llc35_in_ag(T21))
U5_AG(T21, llc35_out_ag(T58, T21)) → T1_IN_G(T58)

The TRS R consists of the following rules:

llc9_in_ga(s(T16)) → U14_ga(T16, llc9_in_ga(T16))
llc9_in_ga(0) → llc9_out_ga(0, [])
U14_ga(T16, llc9_out_ga(T16, X70)) → llc9_out_ga(s(T16), .(X70))
selectc19_in_aaga(T34) → U22_aaga(T34, selectc24_in_aga(T34))
selectc24_in_aga(.(T47)) → U15_aga(T47, selectc24_in_aga(T47))
selectc24_in_aga(.(T52)) → selectc24_out_aga(.(T52), T52)
U15_aga(T47, selectc24_out_aga(T47, X175)) → selectc24_out_aga(.(T47), .(X175))
U22_aaga(T34, selectc24_out_aga(T34, X139)) → selectc19_out_aaga(T34, .(X139))
selectc19_in_aaga(T55) → selectc19_out_aaga(T55, T55)
llc35_in_ag(.(T71)) → U19_ag(T71, llc35_in_ag(T71))
llc35_in_ag([]) → llc35_out_ag(0, [])
U19_ag(T71, llc35_out_ag(X233, T71)) → llc35_out_ag(s(X233), .(T71))

The set Q consists of the following terms:

llc9_in_ga(x0)
U14_ga(x0, x1)
selectc19_in_aaga(x0)
selectc24_in_aga(x0)
U15_aga(x0, x1)
U22_aaga(x0, x1)
llc35_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 4 less nodes.

(36) TRUE