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

(0) Obligation:

Clauses:

append(nil, XS, XS).
append(cons(X, XS1), XS2, cons(X, YS)) :- append(XS1, XS2, YS).
split(XS, nil, XS).
split(cons(X, XS), cons(X, YS1), YS2) :- split(XS, YS1, YS2).
perm(nil, nil).
perm(XS, cons(Y, YS)) :- ','(split(XS, YS1, cons(Y, YS2)), ','(append(YS1, YS2, ZS), perm(ZS, YS))).

Query: perm(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: perm(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: perm(T1, T2), SCOPE: 1, CLAUSE: 4 | perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | perm(nil, T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nperm(T1, T2) !=? perm(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: perm(nil, T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(split(nil, X7, cons(T7, X8)), ','(append(X7, X8, X9), perm(X9, T8)))\n\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(split(nil, X7, cons(T7, X8)), ','(append(X7, X8, X9), perm(X9, T8))), SCOPE: 2, CLAUSE: 2 | ','(split(nil, X7, cons(T7, X8)), ','(append(X7, X8, X9), perm(X9, T8))), SCOPE: 2, CLAUSE: 3\n\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(split(nil, X7, cons(T7, X8)), ','(append(X7, X8, X9), perm(X9, T8))), SCOPE: 2, CLAUSE: 3\n\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(split(T12, X21, cons(T15, X22)), ','(append(X21, X22, X23), perm(X23, T16)))\n\nT12 is ground\nX21 is free\nX22 is free\nX23 is free\nperm(T12, T2) !=? perm(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(split(T12, X21, cons(T15, X22)), ','(append(X21, X22, X23), perm(X23, T16))), SCOPE: 3, CLAUSE: 2 | ','(split(T12, X21, cons(T15, X22)), ','(append(X21, X22, X23), perm(X23, T16))), SCOPE: 3, CLAUSE: 3\n\nT12 is ground\nX21 is free\nX22 is free\nX23 is free\nperm(T12, T2) !=? perm(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split(T12, X21, cons(T15, X22)), ','(append(X21, X22, X23), perm(X23, T16))), SCOPE: 3, CLAUSE: 2\n\nT12 is ground\nX21 is free\nX22 is free\nX23 is free\nperm(T12, T2) !=? perm(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(T12, X21, cons(T15, X22)), ','(append(X21, X22, X23), perm(X23, T16))), SCOPE: 3, CLAUSE: 3\n\nT12 is ground\nX21 is free\nX22 is free\nX23 is free\nperm(T12, T2) !=? perm(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(append(nil, T31, X23), perm(X23, T32))\n\nT31 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(nil, T31, X23)\n\nT31 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: perm(T34, T32)\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(nil, T31, X23), SCOPE: 4, CLAUSE: 0 | append(nil, T31, X23), SCOPE: 4, CLAUSE: 1\n\nT31 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(nil, T31, X23), SCOPE: 4, CLAUSE: 0\n\nT31 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append(nil, T31, X23), SCOPE: 4, CLAUSE: 1\n\nT31 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(split(T48, X56, cons(T50, X57)), ','(append(cons(T47, X56), X57, X23), perm(X23, T51)))\n\nT47 is ground\nT48 is ground\nX23 is free\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: split(T48, X56, cons(T50, X57))\n\nT48 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(append(cons(T47, T58), T59, X23), perm(X23, T60))\n\nT47 is ground\nT58 is ground\nT59 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: split(T48, X56, cons(T50, X57)), SCOPE: 5, CLAUSE: 2 | split(T48, X56, cons(T50, X57)), SCOPE: 5, CLAUSE: 3\n\nT48 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: split(T48, X56, cons(T50, X57)), SCOPE: 5, CLAUSE: 2\n\nT48 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: split(T48, X56, cons(T50, X57)), SCOPE: 5, CLAUSE: 3\n\nT48 is ground\nX56 is free\nX57 is free", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: split(T89, X83, cons(T91, X84))\n\nT89 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: append(cons(T47, T58), T59, X23)\n\nT47 is ground\nT58 is ground\nT59 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: perm(T98, T60)\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: append(cons(T47, T58), T59, X23), SCOPE: 6, CLAUSE: 0 | append(cons(T47, T58), T59, X23), SCOPE: 6, CLAUSE: 1\n\nT47 is ground\nT58 is ground\nT59 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: append(cons(T47, T58), T59, X23), SCOPE: 6, CLAUSE: 1\n\nT47 is ground\nT58 is ground\nT59 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: append(T108, T109, X108)\n\nT108 is ground\nT109 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append(T108, T109, X108), SCOPE: 7, CLAUSE: 0 | append(T108, T109, X108), SCOPE: 7, CLAUSE: 1\n\nT108 is ground\nT109 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append(T108, T109, X108), SCOPE: 7, CLAUSE: 0\n\nT108 is ground\nT109 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: append(T108, T109, X108), SCOPE: 7, CLAUSE: 1\n\nT108 is ground\nT109 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: append(T124, T125, X130)\n\nT124 is ground\nT125 is ground\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 2;

52 [label="EVAL with clause\nperm(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="EVAL with clause\nperm(X18, cons(X19, X20)) :- ','(split(X18, X21, cons(X19, X22)), ','(append(X21, X22, X23), perm(X23, X20))).\nand substitutionT1 -> T12,\nX18 -> T12,\nX19 -> T15,\nX20 -> T16,\nT2 -> cons(T15, T16),\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="EVAL with clause\nperm(X4, cons(X5, X6)) :- ','(split(X4, X7, cons(X5, X8)), ','(append(X7, X8, X9), perm(X9, X6))).\nand substitutionX4 -> nil,\nX5 -> T7,\nX6 -> T8,\nT2 -> cons(T7, T8),\nT5 -> T7,\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 57 [arrowhead = none ];
57 -> 6;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="BACKTRACK\nfor clause: split(XS, nil, XS)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="BACKTRACK\nfor clause: split(cons(X, XS), cons(X, YS1), YS2) :- split(XS, YS1, YS2)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 64 [arrowhead = none ];
64 -> 15;

65 [label="EVAL with clause\nsplit(X28, nil, X28).\nand substitutionT12 -> cons(T30, T31),\nX28 -> cons(T30, T31),\nX21 -> nil,\nT15 -> T30,\nX22 -> T31,\nT29 -> cons(T30, T31),\nT16 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 65 [arrowhead = none ];
65 -> 16;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 66 [arrowhead = none ];
66 -> 17;

67 [label="EVAL with clause\nsplit(cons(X52, X53), cons(X52, X54), X55) :- split(X53, X54, X55).\nand substitutionX52 -> T47,\nX53 -> T48,\nT12 -> cons(T47, T48),\nX54 -> X56,\nX21 -> cons(T47, X56),\nT15 -> T50,\nX22 -> X57,\nX55 -> cons(T50, X57),\nT49 -> T50,\nT16 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 67 [arrowhead = none ];
67 -> 26;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 68 [arrowhead = none ];
68 -> 27;

69 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 69 [arrowhead = none ];
69 -> 18;

70 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT34 is ground\nreplacements:X23 -> T34", fontsize=14, style = filled, fillcolor = violet];
16 -> 70 [arrowhead = none ];
70 -> 19;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 71 [arrowhead = none ];
71 -> 20;

72 [label="INSTANCE with matching:\nT1 -> T34\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 73 [arrowhead = none ];
73 -> 21;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 74 [arrowhead = none ];
74 -> 22;

75 [label="ONLY EVAL with clause\nappend(nil, X35, X35).\nand substitutionT31 -> T40,\nX35 -> T40,\nX23 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 75 [arrowhead = none ];
75 -> 23;

76 [label="BACKTRACK\nfor clause: append(cons(X, XS1), XS2, cons(X, YS)) :- append(XS1, XS2, YS)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 78 [arrowhead = none ];
78 -> 28;

79 [label="SPLIT 2\nnew knowledge:\nT48 is ground\nT58 is ground\nT50 is ground\nT59 is ground\nreplacements:X56 -> T58,\nX57 -> T59,\nT51 -> T60", fontsize=14, style = filled, fillcolor = violet];
26 -> 79 [arrowhead = none ];
79 -> 29;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 80 [arrowhead = none ];
80 -> 30;

81 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 81 [arrowhead = none ];
81 -> 38;

82 [label="SPLIT 2\nnew knowledge:\nT47 is ground\nT58 is ground\nT59 is ground\nT98 is ground\nreplacements:X23 -> T98", fontsize=14, style = filled, fillcolor = violet];
29 -> 82 [arrowhead = none ];
82 -> 39;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 83 [arrowhead = none ];
83 -> 31;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="EVAL with clause\nsplit(X66, nil, X66).\nand substitutionT48 -> cons(T80, T81),\nX66 -> cons(T80, T81),\nX56 -> nil,\nT50 -> T80,\nX57 -> T81,\nT79 -> cons(T80, T81)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 85 [arrowhead = none ];
85 -> 33;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 86 [arrowhead = none ];
86 -> 34;

87 [label="EVAL with clause\nsplit(cons(X79, X80), cons(X79, X81), X82) :- split(X80, X81, X82).\nand substitutionX79 -> T88,\nX80 -> T89,\nT48 -> cons(T88, T89),\nX81 -> X83,\nX56 -> cons(T88, X83),\nT50 -> T91,\nX57 -> X84,\nX82 -> cons(T91, X84),\nT90 -> T91", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 87 [arrowhead = none ];
87 -> 36;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 88 [arrowhead = none ];
88 -> 37;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="INSTANCE with matching:\nT48 -> T89\nX56 -> X83\nT50 -> T91\nX57 -> X84", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 90 [arrowhead = none , style=dashed];
90 -> 28[style=dashed];

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="INSTANCE with matching:\nT1 -> T98\nT2 -> T60", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 92 [arrowhead = none , style=dashed];
92 -> 1[style=dashed];

93 [label="BACKTRACK\nfor clause: append(nil, XS, XS)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 93 [arrowhead = none ];
93 -> 41;

94 [label="ONLY EVAL with clause\nappend(cons(X104, X105), X106, cons(X104, X107)) :- append(X105, X106, X107).\nand substitutionT47 -> T107,\nX104 -> T107,\nT58 -> T108,\nX105 -> T108,\nT59 -> T109,\nX106 -> T109,\nX107 -> X108,\nX23 -> cons(T107, X108)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 94 [arrowhead = none ];
94 -> 42;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 95 [arrowhead = none ];
95 -> 43;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 96 [arrowhead = none ];
96 -> 44;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 97 [arrowhead = none ];
97 -> 45;

98 [label="EVAL with clause\nappend(nil, X115, X115).\nand substitutionT108 -> nil,\nT109 -> T116,\nX115 -> T116,\nX108 -> T116", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 98 [arrowhead = none ];
98 -> 46;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 99 [arrowhead = none ];
99 -> 47;

100 [label="EVAL with clause\nappend(cons(X126, X127), X128, cons(X126, X129)) :- append(X127, X128, X129).\nand substitutionX126 -> T123,\nX127 -> T124,\nT108 -> cons(T123, T124),\nT109 -> T125,\nX128 -> T125,\nX129 -> X130,\nX108 -> cons(T123, X130)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 100 [arrowhead = none ];
100 -> 49;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="INSTANCE with matching:\nT108 -> T124\nT109 -> T125\nX108 -> X130", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 103 [arrowhead = none , style=dashed];
103 -> 42[style=dashed];

}

(2) Obligation:

Clauses:

splitA(cons(T80, T81), nil, T80, T81).
splitA(cons(T88, T89), cons(T88, X83), T91, X84) :- splitA(T89, X83, T91, X84).
appendB(nil, T116, T116).
appendB(cons(T123, T124), T125, cons(T123, X130)) :- appendB(T124, T125, X130).
appendC(T40, T40).
appendD(T107, T108, T109, cons(T107, X108)) :- appendB(T108, T109, X108).
permE(nil, nil).
permE(cons(T30, T31), cons(T30, T32)) :- appendC(T31, X23).
permE(cons(T30, T31), cons(T30, T32)) :- ','(appendC(T31, T34), permE(T34, T32)).
permE(cons(T47, T48), cons(T50, T51)) :- splitA(T48, X56, T50, X57).
permE(cons(T47, T48), cons(T50, T60)) :- ','(splitA(T48, T58, T50, T59), appendD(T47, T58, T59, X23)).
permE(cons(T47, T48), cons(T50, T60)) :- ','(splitA(T48, T58, T50, T59), ','(appendD(T47, T58, T59, T98), permE(T98, T60))).

Query: permE(g,a)

(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:
permE_in: (b,f)
splitA_in: (b,f,f,f)
appendD_in: (b,b,b,f)
appendB_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

(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:

PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U4_GA(T30, T31, T32, appendC_in_ga(T31, X23))
PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → APPENDC_IN_GA(T31, X23)
PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, appendC_in_ga(T31, T34))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_GA(T30, T31, T32, permE_in_ga(T34, T32))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → PERME_IN_GA(T34, T32)
PERME_IN_GA(cons(T47, T48), cons(T50, T51)) → U7_GA(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
PERME_IN_GA(cons(T47, T48), cons(T50, T51)) → SPLITA_IN_GAAA(T48, X56, T50, X57)
SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → U1_GAAA(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → SPLITA_IN_GAAA(T89, X83, T91, X84)
PERME_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → APPENDD_IN_GGGA(T47, T58, T59, X23)
APPENDD_IN_GGGA(T107, T108, T109, cons(T107, X108)) → U3_GGGA(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
APPENDD_IN_GGGA(T107, T108, T109, cons(T107, X108)) → APPENDB_IN_GGA(T108, T109, X108)
APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → U2_GGA(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → APPENDB_IN_GGA(T124, T125, X130)
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_GA(T47, T48, T50, T60, permE_in_ga(T98, T60))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → PERME_IN_GA(T98, T60)

The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPENDC_IN_GA(x1, x2)  =  APPENDC_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
APPENDD_IN_GGGA(x1, x2, x3, x4)  =  APPENDD_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPENDB_IN_GGA(x1, x2, x3)  =  APPENDB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(6) Obligation:

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

PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U4_GA(T30, T31, T32, appendC_in_ga(T31, X23))
PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → APPENDC_IN_GA(T31, X23)
PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, appendC_in_ga(T31, T34))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_GA(T30, T31, T32, permE_in_ga(T34, T32))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → PERME_IN_GA(T34, T32)
PERME_IN_GA(cons(T47, T48), cons(T50, T51)) → U7_GA(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
PERME_IN_GA(cons(T47, T48), cons(T50, T51)) → SPLITA_IN_GAAA(T48, X56, T50, X57)
SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → U1_GAAA(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → SPLITA_IN_GAAA(T89, X83, T91, X84)
PERME_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → APPENDD_IN_GGGA(T47, T58, T59, X23)
APPENDD_IN_GGGA(T107, T108, T109, cons(T107, X108)) → U3_GGGA(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
APPENDD_IN_GGGA(T107, T108, T109, cons(T107, X108)) → APPENDB_IN_GGA(T108, T109, X108)
APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → U2_GGA(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → APPENDB_IN_GGA(T124, T125, X130)
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_GA(T47, T48, T50, T60, permE_in_ga(T98, T60))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → PERME_IN_GA(T98, T60)

The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPENDC_IN_GA(x1, x2)  =  APPENDC_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
APPENDD_IN_GGGA(x1, x2, x3, x4)  =  APPENDD_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPENDB_IN_GGA(x1, x2, x3)  =  APPENDB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → APPENDB_IN_GGA(T124, T125, X130)

The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
APPENDB_IN_GGA(x1, x2, x3)  =  APPENDB_IN_GGA(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:

APPENDB_IN_GGA(cons(T123, T124), T125, cons(T123, X130)) → APPENDB_IN_GGA(T124, T125, X130)

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

APPENDB_IN_GGA(cons(T123, T124), T125) → APPENDB_IN_GGA(T124, T125)

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:

• APPENDB_IN_GGA(cons(T123, T124), T125) → APPENDB_IN_GGA(T124, T125)
  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:

SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → SPLITA_IN_GAAA(T89, X83, T91, X84)

The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)

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:

SPLITA_IN_GAAA(cons(T88, T89), cons(T88, X83), T91, X84) → SPLITA_IN_GAAA(T89, X83, T91, X84)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)

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:

SPLITA_IN_GAAA(cons(T88, T89)) → SPLITA_IN_GAAA(T89)

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:

• SPLITA_IN_GAAA(cons(T88, T89)) → SPLITA_IN_GAAA(T89)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, appendC_in_ga(T31, T34))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → PERME_IN_GA(T34, T32)
PERME_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → PERME_IN_GA(T98, T60)

The TRS R consists of the following rules:

permE_in_ga(nil, nil) → permE_out_ga(nil, nil)
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U4_ga(T30, T31, T32, appendC_in_ga(T31, X23))
appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
U4_ga(T30, T31, T32, appendC_out_ga(T31, X23)) → permE_out_ga(cons(T30, T31), cons(T30, T32))
permE_in_ga(cons(T30, T31), cons(T30, T32)) → U5_ga(T30, T31, T32, appendC_in_ga(T31, T34))
U5_ga(T30, T31, T32, appendC_out_ga(T31, T34)) → U6_ga(T30, T31, T32, permE_in_ga(T34, T32))
permE_in_ga(cons(T47, T48), cons(T50, T51)) → U7_ga(T47, T48, T50, T51, splitA_in_gaaa(T48, X56, T50, X57))
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U7_ga(T47, T48, T50, T51, splitA_out_gaaa(T48, X56, T50, X57)) → permE_out_ga(cons(T47, T48), cons(T50, T51))
permE_in_ga(cons(T47, T48), cons(T50, T60)) → U8_ga(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U9_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, X23))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
U9_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, X23)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U8_ga(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_ga(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_ga(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → U11_ga(T47, T48, T50, T60, permE_in_ga(T98, T60))
U11_ga(T47, T48, T50, T60, permE_out_ga(T98, T60)) → permE_out_ga(cons(T47, T48), cons(T50, T60))
U6_ga(T30, T31, T32, permE_out_ga(T34, T32)) → permE_out_ga(cons(T30, T31), cons(T30, T32))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
nil  =  nil
permE_out_ga(x1, x2)  =  permE_out_ga
cons(x1, x2)  =  cons(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)

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:

PERME_IN_GA(cons(T30, T31), cons(T30, T32)) → U5_GA(T30, T31, T32, appendC_in_ga(T31, T34))
U5_GA(T30, T31, T32, appendC_out_ga(T31, T34)) → PERME_IN_GA(T34, T32)
PERME_IN_GA(cons(T47, T48), cons(T50, T60)) → U8_GA(T47, T48, T50, T60, splitA_in_gaaa(T48, T58, T50, T59))
U8_GA(T47, T48, T50, T60, splitA_out_gaaa(T48, T58, T50, T59)) → U10_GA(T47, T48, T50, T60, appendD_in_ggga(T47, T58, T59, T98))
U10_GA(T47, T48, T50, T60, appendD_out_ggga(T47, T58, T59, T98)) → PERME_IN_GA(T98, T60)

The TRS R consists of the following rules:

appendC_in_ga(T40, T40) → appendC_out_ga(T40, T40)
splitA_in_gaaa(cons(T80, T81), nil, T80, T81) → splitA_out_gaaa(cons(T80, T81), nil, T80, T81)
splitA_in_gaaa(cons(T88, T89), cons(T88, X83), T91, X84) → U1_gaaa(T88, T89, X83, T91, X84, splitA_in_gaaa(T89, X83, T91, X84))
appendD_in_ggga(T107, T108, T109, cons(T107, X108)) → U3_ggga(T107, T108, T109, X108, appendB_in_gga(T108, T109, X108))
U1_gaaa(T88, T89, X83, T91, X84, splitA_out_gaaa(T89, X83, T91, X84)) → splitA_out_gaaa(cons(T88, T89), cons(T88, X83), T91, X84)
U3_ggga(T107, T108, T109, X108, appendB_out_gga(T108, T109, X108)) → appendD_out_ggga(T107, T108, T109, cons(T107, X108))
appendB_in_gga(nil, T116, T116) → appendB_out_gga(nil, T116, T116)
appendB_in_gga(cons(T123, T124), T125, cons(T123, X130)) → U2_gga(T123, T124, T125, X130, appendB_in_gga(T124, T125, X130))
U2_gga(T123, T124, T125, X130, appendB_out_gga(T124, T125, X130)) → appendB_out_gga(cons(T123, T124), T125, cons(T123, X130))

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
appendC_in_ga(x1, x2)  =  appendC_in_ga(x1)
appendC_out_ga(x1, x2)  =  appendC_out_ga(x2)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitA_out_gaaa(x1, x2, x3, x4)  =  splitA_out_gaaa(x2, x3, x4)
U1_gaaa(x1, x2, x3, x4, x5, x6)  =  U1_gaaa(x1, x6)
appendD_in_ggga(x1, x2, x3, x4)  =  appendD_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appendB_in_gga(x1, x2, x3)  =  appendB_in_gga(x1, x2)
appendB_out_gga(x1, x2, x3)  =  appendB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
appendD_out_ggga(x1, x2, x3, x4)  =  appendD_out_ggga(x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)

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:

PERME_IN_GA(cons(T30, T31)) → U5_GA(appendC_in_ga(T31))
U5_GA(appendC_out_ga(T34)) → PERME_IN_GA(T34)
PERME_IN_GA(cons(T47, T48)) → U8_GA(T47, splitA_in_gaaa(T48))
U8_GA(T47, splitA_out_gaaa(T58, T50, T59)) → U10_GA(appendD_in_ggga(T47, T58, T59))
U10_GA(appendD_out_ggga(T98)) → PERME_IN_GA(T98)

The TRS R consists of the following rules:

appendC_in_ga(T40) → appendC_out_ga(T40)
splitA_in_gaaa(cons(T80, T81)) → splitA_out_gaaa(nil, T80, T81)
splitA_in_gaaa(cons(T88, T89)) → U1_gaaa(T88, splitA_in_gaaa(T89))
appendD_in_ggga(T107, T108, T109) → U3_ggga(T107, appendB_in_gga(T108, T109))
U1_gaaa(T88, splitA_out_gaaa(X83, T91, X84)) → splitA_out_gaaa(cons(T88, X83), T91, X84)
U3_ggga(T107, appendB_out_gga(X108)) → appendD_out_ggga(cons(T107, X108))
appendB_in_gga(nil, T116) → appendB_out_gga(T116)
appendB_in_gga(cons(T123, T124), T125) → U2_gga(T123, appendB_in_gga(T124, T125))
U2_gga(T123, appendB_out_gga(X130)) → appendB_out_gga(cons(T123, X130))

The set Q consists of the following terms:

appendC_in_ga(x0)
splitA_in_gaaa(x0)
appendD_in_ggga(x0, x1, x2)
U1_gaaa(x0, x1)
U3_ggga(x0, x1)
appendB_in_gga(x0, x1)
U2_gga(x0, x1)

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

(28) MRRProof (EQUIVALENT transformation)

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

PERME_IN_GA(cons(T30, T31)) → U5_GA(appendC_in_ga(T31))
U5_GA(appendC_out_ga(T34)) → PERME_IN_GA(T34)
PERME_IN_GA(cons(T47, T48)) → U8_GA(T47, splitA_in_gaaa(T48))
U8_GA(T47, splitA_out_gaaa(T58, T50, T59)) → U10_GA(appendD_in_ggga(T47, T58, T59))
U10_GA(appendD_out_ggga(T98)) → PERME_IN_GA(T98)

Strictly oriented rules of the TRS R:

appendC_in_ga(T40) → appendC_out_ga(T40)
splitA_in_gaaa(cons(T80, T81)) → splitA_out_gaaa(nil, T80, T81)
splitA_in_gaaa(cons(T88, T89)) → U1_gaaa(T88, splitA_in_gaaa(T89))
appendD_in_ggga(T107, T108, T109) → U3_ggga(T107, appendB_in_gga(T108, T109))
U1_gaaa(T88, splitA_out_gaaa(X83, T91, X84)) → splitA_out_gaaa(cons(T88, X83), T91, X84)
U3_ggga(T107, appendB_out_gga(X108)) → appendD_out_ggga(cons(T107, X108))
appendB_in_gga(nil, T116) → appendB_out_gga(T116)
appendB_in_gga(cons(T123, T124), T125) → U2_gga(T123, appendB_in_gga(T124, T125))
U2_gga(T123, appendB_out_gga(X130)) → appendB_out_gga(cons(T123, X130))

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

U10_GA1 > appendD_inggga3 > splitA_ingaaa1 > appendC_outga1 > cons₂ > U5_GA1 > PERME_INGA1 > U1_gaaa2 > U8_GA2 > appendB_ingga2 > U2_gga2 > U3_ggga2 > appendD_outggga1 > appendB_outgga1 > nil > splitA_outgaaa3 > appendC_inga1

and weight map:

nil=4
appendC_in_ga_1=3
appendC_out_ga_1=2
splitA_in_gaaa_1=3
appendB_out_gga_1=4
appendD_out_ggga_1=3
PERME_IN_GA_1=3
U5_GA_1=1
U10_GA_1=1
cons_2=1
splitA_out_gaaa_3=0
U1_gaaa_2=1
appendD_in_ggga_3=0
U3_ggga_2=0
appendB_in_gga_2=0
U2_gga_2=1
U8_GA_2=1

The variable weight is 1

(29) Obligation:

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

appendC_in_ga(x0)
splitA_in_gaaa(x0)
appendD_in_ggga(x0, x1, x2)
U1_gaaa(x0, x1)
U3_ggga(x0, x1)
appendB_in_gga(x0, x1)
U2_gga(x0, x1)

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

(30) PisEmptyProof (EQUIVALENT transformation)

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

(31) YES