Left Termination of the query pattern perm1_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

perm1(L, M) :- ','(eq_len1(L, M), same_sets(L, M)).
eq_len1([], []).
eq_len1(.(X1, Xs), .(X2, Ys)) :- eq_len1(Xs, Ys).
member(X, .(X, X3)).
member(X, .(X4, T)) :- member(X, T).
same_sets([], X5).
same_sets(.(X, Xs), L) :- ','(member(X, L), same_sets(Xs, L)).

Queries:

perm1(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: perm1(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: perm1(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(eq_len1(T5, T6), same_sets(T5, T6))\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(eq_len1(T5, T6), same_sets(T5, T6)), SCOPE: 2, CLAUSE: 1 | ','(eq_len1(T5, T6), same_sets(T5, T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(eq_len1(T5, T6), same_sets(T5, T6)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(eq_len1(T5, T6), same_sets(T5, T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: same_sets([], [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: same_sets([], []), SCOPE: 3, CLAUSE: 5 | same_sets([], []), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: same_sets([], []), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: same_sets([], []), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(eq_len1(T16, T18), same_sets(.(T15, T16), .(T17, T18)))\n\nT15 is ground\nT16 is ground\nT17 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: eq_len1(T16, T18)\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: same_sets(.(T15, T16), .(T17, T18))\n\nT15 is ground\nT16 is ground\nT17 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: eq_len1(T16, T18), SCOPE: 4, CLAUSE: 1 | eq_len1(T16, T18), SCOPE: 4, CLAUSE: 2\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq_len1(T16, T18), SCOPE: 4, CLAUSE: 1\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq_len1(T16, T18), SCOPE: 4, CLAUSE: 2\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq_len1(T28, T30)\n\nT28 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: same_sets(.(T15, T16), .(T17, T18)), SCOPE: 5, CLAUSE: 5 | same_sets(.(T15, T16), .(T17, T18)), SCOPE: 5, CLAUSE: 6\n\nT15 is ground\nT16 is ground\nT17 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: same_sets(.(T15, T16), .(T17, T18)), SCOPE: 5, CLAUSE: 6\n\nT15 is ground\nT16 is ground\nT17 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(member(T47, .(T49, T50)), same_sets(T48, .(T49, T50)))\n\nT47 is ground\nT48 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: member(T47, .(T49, T50))\n\nT47 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: same_sets(T48, .(T49, T50))\n\nT48 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: member(T47, .(T49, T50)), SCOPE: 6, CLAUSE: 3 | member(T47, .(T49, T50)), SCOPE: 6, CLAUSE: 4\n\nT47 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: member(T47, .(T49, T50)), SCOPE: 6, CLAUSE: 3\n\nT47 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member(T47, .(T49, T50)), SCOPE: 6, CLAUSE: 4\n\nT47 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: member(T79, T81)\n\nT79 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: member(T79, T81), SCOPE: 7, CLAUSE: 3 | member(T79, T81), SCOPE: 7, CLAUSE: 4\n\nT79 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: member(T79, T81), SCOPE: 7, CLAUSE: 3\n\nT79 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: member(T79, T81), SCOPE: 7, CLAUSE: 4\n\nT79 is ground\nT81 is ground", 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: member(T102, T104)\n\nT102 is ground\nT104 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: same_sets(T48, .(T49, T50)), SCOPE: 8, CLAUSE: 5 | same_sets(T48, .(T49, T50)), SCOPE: 8, CLAUSE: 6\n\nT48 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: same_sets(T48, .(T49, T50)), SCOPE: 8, CLAUSE: 5\n\nT48 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: same_sets(T48, .(T49, T50)), SCOPE: 8, CLAUSE: 6\n\nT48 is ground\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(member(T131, .(T133, T134)), same_sets(T132, .(T133, T134)))\n\nT131 is ground\nT132 is ground\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 55 [arrowhead = none ];
55 -> 2;

56 [label="ONLY EVAL with clause\nperm1(X8, X9) :- ','(eq_len1(X8, X9), same_sets(X8, X9)).\nand substitution\nT1 -> T5\nX8 -> T5\nT2 -> T6\nX9 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 56 [arrowhead = none ];
56 -> 3;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 57 [arrowhead = none ];
57 -> 4;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 58 [arrowhead = none ];
58 -> 5;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="EVAL with clause\neq_len1([], []).\nand substitution\nT5 -> []\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 60 [arrowhead = none ];
60 -> 7;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 61 [arrowhead = none ];
61 -> 8;

62 [label="EVAL with clause\neq_len1(.(X27, X28), .(X29, X30)) :- eq_len1(X28, X30).\nand substitution\nX27 -> T15\nX28 -> T16\nT5 -> .(T15, T16)\nX29 -> T17\nX30 -> T18\nT6 -> .(T17, T18)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 62 [arrowhead = none ];
62 -> 15;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 63 [arrowhead = none ];
63 -> 16;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 64 [arrowhead = none ];
64 -> 9;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 65 [arrowhead = none ];
65 -> 10;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 66 [arrowhead = none ];
66 -> 11;

67 [label="ONLY EVAL with clause\nsame_sets([], X15).\nand substitution\nX15 -> []", fontsize=14, style = filled, fillcolor = white];
10 -> 67 [arrowhead = none ];
67 -> 12;

68 [label="BACKTRACK\nfor clause: same_sets(.(X, Xs), L) :- ','(member(X, L), same_sets(Xs, L))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 68 [arrowhead = none ];
68 -> 14;

69 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 69 [arrowhead = none ];
69 -> 13;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 72 [arrowhead = none ];
72 -> 19;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 73 [arrowhead = none ];
73 -> 27;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 74 [arrowhead = none ];
74 -> 20;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 75 [arrowhead = none ];
75 -> 21;

76 [label="EVAL with clause\neq_len1([], []).\nand substitution\nT16 -> []\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 76 [arrowhead = none ];
76 -> 22;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 77 [arrowhead = none ];
77 -> 23;

78 [label="EVAL with clause\neq_len1(.(X43, X44), .(X45, X46)) :- eq_len1(X44, X46).\nand substitution\nX43 -> T27\nX44 -> T28\nT16 -> .(T27, T28)\nX45 -> T29\nX46 -> T30\nT18 -> .(T29, T30)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 78 [arrowhead = none ];
78 -> 25;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="INSTANCE with matching:\nT16 -> T28\nT18 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 81 [arrowhead = none , style=dashed];
81 -> 17[style=dashed];

82 [label="BACKTRACK\nfor clause: same_sets([], X5)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="ONLY EVAL with clause\nsame_sets(.(X67, X68), X69) :- ','(member(X67, X69), same_sets(X68, X69)).\nand substitution\nT15 -> T47\nX67 -> T47\nT16 -> T48\nX68 -> T48\nT17 -> T49\nT18 -> T50\nX69 -> .(T49, T50)", fontsize=14, style = filled, fillcolor = white];
28 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 85 [arrowhead = none ];
85 -> 31;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 87 [arrowhead = none ];
87 -> 47;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 88 [arrowhead = none ];
88 -> 33;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 89 [arrowhead = none ];
89 -> 34;

90 [label="EVAL with clause\nmember(X86, .(X86, X87)).\nand substitution\nT47 -> T67\nX86 -> T67\nT49 -> T67\nT50 -> T68\nX87 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 90 [arrowhead = none ];
90 -> 35;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 91 [arrowhead = none ];
91 -> 36;

92 [label="ONLY EVAL with clause\nmember(X100, .(X101, X102)) :- member(X100, X102).\nand substitution\nT47 -> T79\nX100 -> T79\nT49 -> T80\nX101 -> T80\nT50 -> T81\nX102 -> T81", fontsize=14, style = filled, fillcolor = white];
34 -> 92 [arrowhead = none ];
92 -> 38;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="EVAL with clause\nmember(X115, .(X115, X116)).\nand substitution\nT79 -> T94\nX115 -> T94\nX116 -> T95\nT81 -> .(T94, T95)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 98 [arrowhead = none ];
98 -> 43;

99 [label="EVAL with clause\nmember(X125, .(X126, X127)) :- member(X125, X127).\nand substitution\nT79 -> T102\nX125 -> T102\nX126 -> T103\nX127 -> T104\nT81 -> .(T103, T104)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 101 [arrowhead = none ];
101 -> 44;

102 [label="INSTANCE with matching:\nT79 -> T102\nT81 -> T104", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 102 [arrowhead = none , style=dashed];
102 -> 38[style=dashed];

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 103 [arrowhead = none ];
103 -> 48;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 104 [arrowhead = none ];
104 -> 49;

105 [label="EVAL with clause\nsame_sets([], X141).\nand substitution\nT48 -> []\nT49 -> T121\nT50 -> T122\nX141 -> .(T121, T122)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 106 [arrowhead = none ];
106 -> 51;

107 [label="EVAL with clause\nsame_sets(.(X149, X150), X151) :- ','(member(X149, X151), same_sets(X150, X151)).\nand substitution\nX149 -> T131\nX150 -> T132\nT48 -> .(T131, T132)\nT49 -> T133\nT50 -> T134\nX151 -> .(T133, T134)", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 107 [arrowhead = none ];
107 -> 53;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 108 [arrowhead = none ];
108 -> 54;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 109 [arrowhead = none ];
109 -> 52;

110 [label="INSTANCE with matching:\nT47 -> T131\nT49 -> T133\nT50 -> T134\nT48 -> T132", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 110 [arrowhead = none , style=dashed];
110 -> 29[style=dashed];

}

(2) Obligation:

Triples:

eq_len117(.(T27, T28), .(T29, T30)) :- eq_len117(T28, T30).
member38(T102, .(T103, T104)) :- member38(T102, T104).
p29(T79, T80, T81, T48) :- member38(T79, T81).
p29(T47, T133, T134, .(T131, T132)) :- ','(memberc30(T47, T133, T134), p29(T131, T133, T134, T132)).
perm11(.(T15, T16), .(T17, T18)) :- eq_len117(T16, T18).
perm11(.(T47, T48), .(T49, T50)) :- ','(eq_len1c17(T48, T50), p29(T47, T49, T50, T48)).

Clauses:

eq_len1c17([], []).
eq_len1c17(.(T27, T28), .(T29, T30)) :- eq_len1c17(T28, T30).
memberc38(T94, .(T94, T95)).
memberc38(T102, .(T103, T104)) :- memberc38(T102, T104).
qc29(T47, T121, T122, []) :- memberc30(T47, T121, T122).
qc29(T47, T133, T134, .(T131, T132)) :- ','(memberc30(T47, T133, T134), qc29(T131, T133, T134, T132)).
memberc30(T67, T67, T68).
memberc30(T79, T80, T81) :- memberc38(T79, T81).

Afs:

perm11(x1, x2)  =  perm11(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm11_in: (b,b)
eq_len117_in: (b,b)
eq_len1c17_in: (b,b)
p29_in: (b,b,b,b)
member38_in: (b,b)
memberc30_in: (b,b,b)
memberc38_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERM11_IN_GG(.(T15, T16), .(T17, T18)) → U6_GG(T15, T16, T17, T18, eq_len117_in_gg(T16, T18))
PERM11_IN_GG(.(T15, T16), .(T17, T18)) → EQ_LEN117_IN_GG(T16, T18)
EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → U1_GG(T27, T28, T29, T30, eq_len117_in_gg(T28, T30))
EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)
PERM11_IN_GG(.(T47, T48), .(T49, T50)) → U7_GG(T47, T48, T49, T50, eq_len1c17_in_gg(T48, T50))
U7_GG(T47, T48, T49, T50, eq_len1c17_out_gg(T48, T50)) → U8_GG(T47, T48, T49, T50, p29_in_gggg(T47, T49, T50, T48))
U7_GG(T47, T48, T49, T50, eq_len1c17_out_gg(T48, T50)) → P29_IN_GGGG(T47, T49, T50, T48)
P29_IN_GGGG(T79, T80, T81, T48) → U3_GGGG(T79, T80, T81, T48, member38_in_gg(T79, T81))
P29_IN_GGGG(T79, T80, T81, T48) → MEMBER38_IN_GG(T79, T81)
MEMBER38_IN_GG(T102, .(T103, T104)) → U2_GG(T102, T103, T104, member38_in_gg(T102, T104))
MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)
P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → U5_GGGG(T47, T133, T134, T131, T132, p29_in_gggg(T131, T133, T134, T132))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)

The TRS R consists of the following rules:

eq_len1c17_in_gg([], []) → eq_len1c17_out_gg([], [])
eq_len1c17_in_gg(.(T27, T28), .(T29, T30)) → U10_gg(T27, T28, T29, T30, eq_len1c17_in_gg(T28, T30))
U10_gg(T27, T28, T29, T30, eq_len1c17_out_gg(T28, T30)) → eq_len1c17_out_gg(.(T27, T28), .(T29, T30))
memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PERM11_IN_GG(.(T15, T16), .(T17, T18)) → U6_GG(T15, T16, T17, T18, eq_len117_in_gg(T16, T18))
PERM11_IN_GG(.(T15, T16), .(T17, T18)) → EQ_LEN117_IN_GG(T16, T18)
EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → U1_GG(T27, T28, T29, T30, eq_len117_in_gg(T28, T30))
EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)
PERM11_IN_GG(.(T47, T48), .(T49, T50)) → U7_GG(T47, T48, T49, T50, eq_len1c17_in_gg(T48, T50))
U7_GG(T47, T48, T49, T50, eq_len1c17_out_gg(T48, T50)) → U8_GG(T47, T48, T49, T50, p29_in_gggg(T47, T49, T50, T48))
U7_GG(T47, T48, T49, T50, eq_len1c17_out_gg(T48, T50)) → P29_IN_GGGG(T47, T49, T50, T48)
P29_IN_GGGG(T79, T80, T81, T48) → U3_GGGG(T79, T80, T81, T48, member38_in_gg(T79, T81))
P29_IN_GGGG(T79, T80, T81, T48) → MEMBER38_IN_GG(T79, T81)
MEMBER38_IN_GG(T102, .(T103, T104)) → U2_GG(T102, T103, T104, member38_in_gg(T102, T104))
MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)
P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → U5_GGGG(T47, T133, T134, T131, T132, p29_in_gggg(T131, T133, T134, T132))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)

The TRS R consists of the following rules:

eq_len1c17_in_gg([], []) → eq_len1c17_out_gg([], [])
eq_len1c17_in_gg(.(T27, T28), .(T29, T30)) → U10_gg(T27, T28, T29, T30, eq_len1c17_in_gg(T28, T30))
U10_gg(T27, T28, T29, T30, eq_len1c17_out_gg(T28, T30)) → eq_len1c17_out_gg(.(T27, T28), .(T29, T30))
memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)

The TRS R consists of the following rules:

eq_len1c17_in_gg([], []) → eq_len1c17_out_gg([], [])
eq_len1c17_in_gg(.(T27, T28), .(T29, T30)) → U10_gg(T27, T28, T29, T30, eq_len1c17_in_gg(T28, T30))
U10_gg(T27, T28, T29, T30, eq_len1c17_out_gg(T28, T30)) → eq_len1c17_out_gg(.(T27, T28), .(T29, T30))
memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)

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

(12) 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:

• MEMBER38_IN_GG(T102, .(T103, T104)) → MEMBER38_IN_GG(T102, T104)
  The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)

The TRS R consists of the following rules:

eq_len1c17_in_gg([], []) → eq_len1c17_out_gg([], [])
eq_len1c17_in_gg(.(T27, T28), .(T29, T30)) → U10_gg(T27, T28, T29, T30, eq_len1c17_in_gg(T28, T30))
U10_gg(T27, T28, T29, T30, eq_len1c17_out_gg(T28, T30)) → eq_len1c17_out_gg(.(T27, T28), .(T29, T30))
memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)

The TRS R consists of the following rules:

memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)

The TRS R consists of the following rules:

memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))

The set Q consists of the following terms:

memberc30_in_ggg(x0, x1, x2)
U15_ggg(x0, x1, x2, x3)
memberc38_in_gg(x0, x1)
U11_gg(x0, x1, x2, x3)

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

(19) 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:

• U4_GGGG(T47, T133, T134, T131, T132, memberc30_out_ggg(T47, T133, T134)) → P29_IN_GGGG(T131, T133, T134, T132)
  The graph contains the following edges 4 >= 1, 2 >= 2, 6 > 2, 3 >= 3, 6 > 3, 5 >= 4

• P29_IN_GGGG(T47, T133, T134, .(T131, T132)) → U4_GGGG(T47, T133, T134, T131, T132, memberc30_in_ggg(T47, T133, T134))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 4 > 5

(20) YES

(21) Obligation:

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

EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)

The TRS R consists of the following rules:

eq_len1c17_in_gg([], []) → eq_len1c17_out_gg([], [])
eq_len1c17_in_gg(.(T27, T28), .(T29, T30)) → U10_gg(T27, T28, T29, T30, eq_len1c17_in_gg(T28, T30))
U10_gg(T27, T28, T29, T30, eq_len1c17_out_gg(T28, T30)) → eq_len1c17_out_gg(.(T27, T28), .(T29, T30))
memberc30_in_ggg(T67, T67, T68) → memberc30_out_ggg(T67, T67, T68)
memberc30_in_ggg(T79, T80, T81) → U15_ggg(T79, T80, T81, memberc38_in_gg(T79, T81))
memberc38_in_gg(T94, .(T94, T95)) → memberc38_out_gg(T94, .(T94, T95))
memberc38_in_gg(T102, .(T103, T104)) → U11_gg(T102, T103, T104, memberc38_in_gg(T102, T104))
U11_gg(T102, T103, T104, memberc38_out_gg(T102, T104)) → memberc38_out_gg(T102, .(T103, T104))
U15_ggg(T79, T80, T81, memberc38_out_gg(T79, T81)) → memberc30_out_ggg(T79, T80, T81)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)

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

(24) PiDPToQDPProof (EQUIVALENT transformation)

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

(25) Obligation:

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

EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)

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

(26) 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:

• EQ_LEN117_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN117_IN_GG(T28, T30)
  The graph contains the following edges 1 > 1, 2 > 2

(27) YES