Left Termination of the query pattern
perm1(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)).

Query: perm1(g,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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(T43, .(T45, T46)), same_sets(T44, .(T45, T46)))\n\nT43 is ground\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: member(T43, .(T45, T46))\n\nT43 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: same_sets(T44, .(T45, T46))\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: member(T43, .(T45, T46)), SCOPE: 6, CLAUSE: 3 | member(T43, .(T45, T46)), SCOPE: 6, CLAUSE: 4\n\nT43 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: member(T43, .(T45, T46)), SCOPE: 6, CLAUSE: 3\n\nT43 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member(T43, .(T45, T46)), SCOPE: 6, CLAUSE: 4\n\nT43 is ground\nT45 is ground\nT46 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(T75, T77)\n\nT75 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: member(T75, T77), SCOPE: 7, CLAUSE: 3 | member(T75, T77), SCOPE: 7, CLAUSE: 4\n\nT75 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: member(T75, T77), SCOPE: 7, CLAUSE: 3\n\nT75 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: member(T75, T77), SCOPE: 7, CLAUSE: 4\n\nT75 is ground\nT77 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(T98, T100)\n\nT98 is ground\nT100 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: same_sets(T44, .(T45, T46)), SCOPE: 8, CLAUSE: 5 | same_sets(T44, .(T45, T46)), SCOPE: 8, CLAUSE: 6\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: same_sets(T44, .(T45, T46)), SCOPE: 8, CLAUSE: 5\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: same_sets(T44, .(T45, T46)), SCOPE: 8, CLAUSE: 6\n\nT44 is ground\nT45 is ground\nT46 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(T127, .(T129, T130)), same_sets(T128, .(T129, T130)))\n\nT127 is ground\nT128 is ground\nT129 is ground\nT130 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 substitutionT1 -> T5,\nX8 -> T5,\nT2 -> T6,\nX9 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT5 -> [],\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 60 [arrowhead = none ];
60 -> 7;

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

62 [label="EVAL with clause\neq_len1(.(X27, X28), .(X29, X30)) :- eq_len1(X28, X30).\nand substitutionX27 -> 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 = lightgreen];
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 substitutionX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
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\nnew knowledge:\nT16 is ground\nT18 is ground", 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 substitutionT16 -> [],\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 76 [arrowhead = none ];
76 -> 22;

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

78 [label="EVAL with clause\neq_len1(.(X39, X40), .(X41, X42)) :- eq_len1(X40, X42).\nand substitutionX39 -> T27,\nX40 -> T28,\nT16 -> .(T27, T28),\nX41 -> T29,\nX42 -> 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 = lightgreen];
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 = lightgreen];
27 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="ONLY EVAL with clause\nsame_sets(.(X56, X57), X58) :- ','(member(X56, X58), same_sets(X57, X58)).\nand substitutionT15 -> T43,\nX56 -> T43,\nT16 -> T44,\nX57 -> T44,\nT17 -> T45,\nT18 -> T46,\nX58 -> .(T45, T46)", fontsize=14, style = filled, fillcolor = lightblue];
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\nnew knowledge:\nT43 is ground\nT45 is ground\nT46 is ground", 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(X75, .(X75, X76)).\nand substitutionT43 -> T63,\nX75 -> T63,\nT45 -> T63,\nT46 -> T64,\nX76 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 90 [arrowhead = none ];
90 -> 35;

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

92 [label="ONLY EVAL with clause\nmember(X87, .(X88, X89)) :- member(X87, X89).\nand substitutionT43 -> T75,\nX87 -> T75,\nT45 -> T76,\nX88 -> T76,\nT46 -> T77,\nX89 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
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(X102, .(X102, X103)).\nand substitutionT75 -> T90,\nX102 -> T90,\nX103 -> T91,\nT77 -> .(T90, T91)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 97 [arrowhead = none ];
97 -> 42;

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

99 [label="EVAL with clause\nmember(X110, .(X111, X112)) :- member(X110, X112).\nand substitutionT75 -> T98,\nX110 -> T98,\nX111 -> T99,\nX112 -> T100,\nT77 -> .(T99, T100)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT75 -> T98\nT77 -> T100", 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([], X123).\nand substitutionT44 -> [],\nT45 -> T117,\nT46 -> T118,\nX123 -> .(T117, T118)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 105 [arrowhead = none ];
105 -> 50;

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

107 [label="EVAL with clause\nsame_sets(.(X130, X131), X132) :- ','(member(X130, X132), same_sets(X131, X132)).\nand substitutionX130 -> T127,\nX131 -> T128,\nT44 -> .(T127, T128),\nT45 -> T129,\nT46 -> T130,\nX132 -> .(T129, T130)", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 107 [arrowhead = none ];
107 -> 53;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT43 -> T127\nT45 -> T129\nT46 -> T130\nT44 -> T128", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 110 [arrowhead = none , style=dashed];
110 -> 29[style=dashed];

}

(2) Obligation:

Clauses:

eq_len1A([], []).
eq_len1A(.(T27, T28), .(T29, T30)) :- eq_len1A(T28, T30).
memberB(T90, .(T90, T91)).
memberB(T98, .(T99, T100)) :- memberB(T98, T100).
pC(T43, T45, T46, T44) :- memberD(T43, T45, T46).
pC(T43, T117, T118, []) :- memberD(T43, T117, T118).
pC(T43, T129, T130, .(T127, T128)) :- ','(memberD(T43, T129, T130), pC(T127, T129, T130, T128)).
memberD(T63, T63, T64).
memberD(T75, T76, T77) :- memberB(T75, T77).
perm1E([], []).
perm1E(.(T15, T16), .(T17, T18)) :- eq_len1A(T16, T18).
perm1E(.(T43, T44), .(T45, T46)) :- ','(eq_len1A(T44, T46), pC(T43, T45, T46, T44)).

Query: perm1E(g,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:
perm1E_in: (b,b)
eq_len1A_in: (b,b)
pC_in: (b,b,b,b)
memberD_in: (b,b,b)
memberB_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)

(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:

PERM1E_IN_GG(.(T15, T16), .(T17, T18)) → U8_GG(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
PERM1E_IN_GG(.(T15, T16), .(T17, T18)) → EQ_LEN1A_IN_GG(T16, T18)
EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → U1_GG(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)
PERM1E_IN_GG(.(T43, T44), .(T45, T46)) → U9_GG(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_GG(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_GG(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
U9_GG(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → PC_IN_GGGG(T43, T45, T46, T44)
PC_IN_GGGG(T43, T45, T46, T44) → U3_GGGG(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
PC_IN_GGGG(T43, T45, T46, T44) → MEMBERD_IN_GGG(T43, T45, T46)
MEMBERD_IN_GGG(T75, T76, T77) → U7_GGG(T75, T76, T77, memberB_in_gg(T75, T77))
MEMBERD_IN_GGG(T75, T76, T77) → MEMBERB_IN_GG(T75, T77)
MEMBERB_IN_GG(T98, .(T99, T100)) → U2_GG(T98, T99, T100, memberB_in_gg(T98, T100))
MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)
PC_IN_GGGG(T43, T117, T118, []) → U4_GGGG(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
PC_IN_GGGG(T43, T117, T118, []) → MEMBERD_IN_GGG(T43, T117, T118)
PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → MEMBERD_IN_GGG(T43, T129, T130)
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_GGGG(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → PC_IN_GGGG(T127, T129, T130, T128)

The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)
PERM1E_IN_GG(x1, x2)  =  PERM1E_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x5)
EQ_LEN1A_IN_GG(x1, x2)  =  EQ_LEN1A_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U9_GG(x1, x2, x3, x4, x5)  =  U9_GG(x1, x2, x3, x4, x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x5)
PC_IN_GGGG(x1, x2, x3, x4)  =  PC_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5)  =  U3_GGGG(x5)
MEMBERD_IN_GGG(x1, x2, x3)  =  MEMBERD_IN_GGG(x1, x2, x3)
U7_GGG(x1, x2, x3, x4)  =  U7_GGG(x4)
MEMBERB_IN_GG(x1, x2)  =  MEMBERB_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U4_GGGG(x1, x2, x3, x4)  =  U4_GGGG(x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x2, x3, x4, x5, x6)
U6_GGGG(x1, x2, x3, x4, x5, x6)  =  U6_GGGG(x6)

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

(6) Obligation:

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

PERM1E_IN_GG(.(T15, T16), .(T17, T18)) → U8_GG(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
PERM1E_IN_GG(.(T15, T16), .(T17, T18)) → EQ_LEN1A_IN_GG(T16, T18)
EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → U1_GG(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)
PERM1E_IN_GG(.(T43, T44), .(T45, T46)) → U9_GG(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_GG(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_GG(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
U9_GG(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → PC_IN_GGGG(T43, T45, T46, T44)
PC_IN_GGGG(T43, T45, T46, T44) → U3_GGGG(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
PC_IN_GGGG(T43, T45, T46, T44) → MEMBERD_IN_GGG(T43, T45, T46)
MEMBERD_IN_GGG(T75, T76, T77) → U7_GGG(T75, T76, T77, memberB_in_gg(T75, T77))
MEMBERD_IN_GGG(T75, T76, T77) → MEMBERB_IN_GG(T75, T77)
MEMBERB_IN_GG(T98, .(T99, T100)) → U2_GG(T98, T99, T100, memberB_in_gg(T98, T100))
MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)
PC_IN_GGGG(T43, T117, T118, []) → U4_GGGG(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
PC_IN_GGGG(T43, T117, T118, []) → MEMBERD_IN_GGG(T43, T117, T118)
PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → MEMBERD_IN_GGG(T43, T129, T130)
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_GGGG(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → PC_IN_GGGG(T127, T129, T130, T128)

The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)
PERM1E_IN_GG(x1, x2)  =  PERM1E_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x5)
EQ_LEN1A_IN_GG(x1, x2)  =  EQ_LEN1A_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U9_GG(x1, x2, x3, x4, x5)  =  U9_GG(x1, x2, x3, x4, x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x5)
PC_IN_GGGG(x1, x2, x3, x4)  =  PC_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5)  =  U3_GGGG(x5)
MEMBERD_IN_GGG(x1, x2, x3)  =  MEMBERD_IN_GGG(x1, x2, x3)
U7_GGG(x1, x2, x3, x4)  =  U7_GGG(x4)
MEMBERB_IN_GG(x1, x2)  =  MEMBERB_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U4_GGGG(x1, x2, x3, x4)  =  U4_GGGG(x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x2, x3, x4, x5, x6)
U6_GGGG(x1, x2, x3, x4, x5, x6)  =  U6_GGGG(x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)

The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)
MEMBERB_IN_GG(x1, x2)  =  MEMBERB_IN_GG(x1, 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:

MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)

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:

• MEMBERB_IN_GG(T98, .(T99, T100)) → MEMBERB_IN_GG(T98, T100)
  The graph contains the following edges 1 >= 1, 2 > 2

(15) YES

(16) Obligation:

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

PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → PC_IN_GGGG(T127, T129, T130, T128)

The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)
PC_IN_GGGG(x1, x2, x3, x4)  =  PC_IN_GGGG(x1, x2, x3, x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x2, x3, x4, x5, x6)

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:

PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_GGGG(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → PC_IN_GGGG(T127, T129, T130, T128)

The TRS R consists of the following rules:

memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
PC_IN_GGGG(x1, x2, x3, x4)  =  PC_IN_GGGG(x1, x2, x3, x4)
U5_GGGG(x1, x2, x3, x4, x5, x6)  =  U5_GGGG(x2, x3, x4, x5, x6)

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:

PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_GGGG(T129, T130, T127, T128, memberD_out_ggg) → PC_IN_GGGG(T127, T129, T130, T128)

The TRS R consists of the following rules:

memberD_in_ggg(T63, T63, T64) → memberD_out_ggg
memberD_in_ggg(T75, T76, T77) → U7_ggg(memberB_in_gg(T75, T77))
U7_ggg(memberB_out_gg) → memberD_out_ggg
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg
memberB_in_gg(T98, .(T99, T100)) → U2_gg(memberB_in_gg(T98, T100))
U2_gg(memberB_out_gg) → memberB_out_gg

The set Q consists of the following terms:

memberD_in_ggg(x0, x1, x2)
U7_ggg(x0)
memberB_in_gg(x0, x1)
U2_gg(x0)

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:

• U5_GGGG(T129, T130, T127, T128, memberD_out_ggg) → PC_IN_GGGG(T127, T129, T130, T128)
  The graph contains the following edges 3 >= 1, 1 >= 2, 2 >= 3, 4 >= 4

• PC_IN_GGGG(T43, T129, T130, .(T127, T128)) → U5_GGGG(T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
  The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3, 4 > 4

(22) YES

(23) Obligation:

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

EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)

The TRS R consists of the following rules:

perm1E_in_gg([], []) → perm1E_out_gg([], [])
perm1E_in_gg(.(T15, T16), .(T17, T18)) → U8_gg(T15, T16, T17, T18, eq_len1A_in_gg(T16, T18))
eq_len1A_in_gg([], []) → eq_len1A_out_gg([], [])
eq_len1A_in_gg(.(T27, T28), .(T29, T30)) → U1_gg(T27, T28, T29, T30, eq_len1A_in_gg(T28, T30))
U1_gg(T27, T28, T29, T30, eq_len1A_out_gg(T28, T30)) → eq_len1A_out_gg(.(T27, T28), .(T29, T30))
U8_gg(T15, T16, T17, T18, eq_len1A_out_gg(T16, T18)) → perm1E_out_gg(.(T15, T16), .(T17, T18))
perm1E_in_gg(.(T43, T44), .(T45, T46)) → U9_gg(T43, T44, T45, T46, eq_len1A_in_gg(T44, T46))
U9_gg(T43, T44, T45, T46, eq_len1A_out_gg(T44, T46)) → U10_gg(T43, T44, T45, T46, pC_in_gggg(T43, T45, T46, T44))
pC_in_gggg(T43, T45, T46, T44) → U3_gggg(T43, T45, T46, T44, memberD_in_ggg(T43, T45, T46))
memberD_in_ggg(T63, T63, T64) → memberD_out_ggg(T63, T63, T64)
memberD_in_ggg(T75, T76, T77) → U7_ggg(T75, T76, T77, memberB_in_gg(T75, T77))
memberB_in_gg(T90, .(T90, T91)) → memberB_out_gg(T90, .(T90, T91))
memberB_in_gg(T98, .(T99, T100)) → U2_gg(T98, T99, T100, memberB_in_gg(T98, T100))
U2_gg(T98, T99, T100, memberB_out_gg(T98, T100)) → memberB_out_gg(T98, .(T99, T100))
U7_ggg(T75, T76, T77, memberB_out_gg(T75, T77)) → memberD_out_ggg(T75, T76, T77)
U3_gggg(T43, T45, T46, T44, memberD_out_ggg(T43, T45, T46)) → pC_out_gggg(T43, T45, T46, T44)
pC_in_gggg(T43, T117, T118, []) → U4_gggg(T43, T117, T118, memberD_in_ggg(T43, T117, T118))
U4_gggg(T43, T117, T118, memberD_out_ggg(T43, T117, T118)) → pC_out_gggg(T43, T117, T118, [])
pC_in_gggg(T43, T129, T130, .(T127, T128)) → U5_gggg(T43, T129, T130, T127, T128, memberD_in_ggg(T43, T129, T130))
U5_gggg(T43, T129, T130, T127, T128, memberD_out_ggg(T43, T129, T130)) → U6_gggg(T43, T129, T130, T127, T128, pC_in_gggg(T127, T129, T130, T128))
U6_gggg(T43, T129, T130, T127, T128, pC_out_gggg(T127, T129, T130, T128)) → pC_out_gggg(T43, T129, T130, .(T127, T128))
U10_gg(T43, T44, T45, T46, pC_out_gggg(T43, T45, T46, T44)) → perm1E_out_gg(.(T43, T44), .(T45, T46))

The argument filtering Pi contains the following mapping:
perm1E_in_gg(x1, x2)  =  perm1E_in_gg(x1, x2)
[]  =  []
perm1E_out_gg(x1, x2)  =  perm1E_out_gg
.(x1, x2)  =  .(x1, x2)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x5)
eq_len1A_in_gg(x1, x2)  =  eq_len1A_in_gg(x1, x2)
eq_len1A_out_gg(x1, x2)  =  eq_len1A_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U9_gg(x1, x2, x3, x4, x5)  =  U9_gg(x1, x2, x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
pC_in_gggg(x1, x2, x3, x4)  =  pC_in_gggg(x1, x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5)  =  U3_gggg(x5)
memberD_in_ggg(x1, x2, x3)  =  memberD_in_ggg(x1, x2, x3)
memberD_out_ggg(x1, x2, x3)  =  memberD_out_ggg
U7_ggg(x1, x2, x3, x4)  =  U7_ggg(x4)
memberB_in_gg(x1, x2)  =  memberB_in_gg(x1, x2)
memberB_out_gg(x1, x2)  =  memberB_out_gg
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
pC_out_gggg(x1, x2, x3, x4)  =  pC_out_gggg
U4_gggg(x1, x2, x3, x4)  =  U4_gggg(x4)
U5_gggg(x1, x2, x3, x4, x5, x6)  =  U5_gggg(x2, x3, x4, x5, x6)
U6_gggg(x1, x2, x3, x4, x5, x6)  =  U6_gggg(x6)
EQ_LEN1A_IN_GG(x1, x2)  =  EQ_LEN1A_IN_GG(x1, x2)

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:

EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)

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

(26) PiDPToQDPProof (EQUIVALENT 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:

EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)

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:

• EQ_LEN1A_IN_GG(.(T27, T28), .(T29, T30)) → EQ_LEN1A_IN_GG(T28, T30)
  The graph contains the following edges 1 > 1, 2 > 2

(29) YES