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

(0) Obligation:

Clauses:

perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
perm([], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, 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: 0 | perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))) | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX4 is free\nX5 is free\nX6 is free\nperm(T1, T2) !=? perm(X4, .(X5, X6))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 2 | ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(X43, .(T26, X44), T25), ','(app(.(T24, X43), X44, X9), perm(X9, T27)))\n\nT24 is ground\nT25 is ground\nX9 is free\nX43 is free\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(X43, .(T26, X44), T25)\n\nT25 is ground\nX43 is free\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(.(T24, T31), T32, X9), perm(X9, T33))\n\nT24 is ground\nT31 is ground\nT32 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(X43, .(T26, X44), T25), SCOPE: 3, CLAUSE: 2 | app(X43, .(T26, X44), T25), SCOPE: 3, CLAUSE: 3\n\nT25 is ground\nX43 is free\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(X43, .(T26, X44), T25), SCOPE: 3, CLAUSE: 2\n\nT25 is ground\nX43 is free\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app(X43, .(T26, X44), T25), SCOPE: 3, CLAUSE: 3\n\nT25 is ground\nX43 is free\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app(X92, .(T52, X93), T51)\n\nT51 is ground\nX92 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(.(T24, T31), T32, X9)\n\nT24 is ground\nT31 is ground\nT32 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: perm(T65, T33)\n\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(.(T24, T31), T32, X9), SCOPE: 4, CLAUSE: 2 | app(.(T24, T31), T32, X9), SCOPE: 4, CLAUSE: 3\n\nT24 is ground\nT31 is ground\nT32 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app(.(T24, T31), T32, X9), SCOPE: 4, CLAUSE: 2\n\nT24 is ground\nT31 is ground\nT32 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(.(T24, T31), T32, X9), SCOPE: 4, CLAUSE: 3\n\nT24 is ground\nT31 is ground\nT32 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(T88, T89, X151)\n\nT88 is ground\nT89 is ground\nX151 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(T88, T89, X151), SCOPE: 5, CLAUSE: 2 | app(T88, T89, X151), SCOPE: 5, CLAUSE: 3\n\nT88 is ground\nT89 is ground\nX151 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T88, T89, X151), SCOPE: 5, CLAUSE: 2\n\nT88 is ground\nT89 is ground\nX151 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T88, T89, X151), SCOPE: 5, CLAUSE: 3\n\nT88 is ground\nT89 is ground\nX151 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T106, T107, X181)\n\nT106 is ground\nT107 is ground\nX181 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(app([], T126, X9), perm(X9, T127))\n\nT126 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app([], T126, X9)\n\nT126 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: perm(T128, T127)\n\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app([], T126, X9), SCOPE: 6, CLAUSE: 2 | app([], T126, X9), SCOPE: 6, CLAUSE: 3\n\nT126 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app([], T126, X9), SCOPE: 6, CLAUSE: 3\n\nT126 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", 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: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="EVAL with clause\nperm(X4, .(X5, X6)) :- ','(app(X7, .(X5, X8), X4), ','(app(X7, X8, X9), perm(X9, X6))).\nand substitutionT1 -> T6,\nX4 -> T6,\nX5 -> T9,\nX6 -> T10,\nT2 -> .(T9, T10),\nT7 -> T9,\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="EVAL with clause\nperm([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 56 [arrowhead = none ];
56 -> 49;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 57 [arrowhead = none ];
57 -> 50;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="EVAL with clause\napp(.(X38, X39), X40, .(X38, X41)) :- app(X39, X40, X41).\nand substitutionX38 -> T24,\nX39 -> X43,\nX7 -> .(T24, X43),\nT9 -> T26,\nX8 -> X44,\nX40 -> .(T26, X44),\nX42 -> T24,\nX41 -> T25,\nT6 -> .(T24, T25),\nT23 -> T26,\nT10 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 61 [arrowhead = none ];
61 -> 9;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 62 [arrowhead = none ];
62 -> 35;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 63 [arrowhead = none ];
63 -> 36;

64 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 64 [arrowhead = none ];
64 -> 10;

65 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT26 is ground\nT32 is ground\nT25 is ground\nreplacements:X43 -> T31,\nX44 -> T32,\nT27 -> T33", fontsize=14, style = filled, fillcolor = violet];
8 -> 65 [arrowhead = none ];
65 -> 11;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 66 [arrowhead = none ];
66 -> 12;

67 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 67 [arrowhead = none ];
67 -> 20;

68 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT31 is ground\nT32 is ground\nT65 is ground\nreplacements:X9 -> T65", fontsize=14, style = filled, fillcolor = violet];
11 -> 68 [arrowhead = none ];
68 -> 21;

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

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 70 [arrowhead = none ];
70 -> 14;

71 [label="EVAL with clause\napp(.(X87, X88), X89, .(X87, X90)) :- app(X88, X89, X90).\nand substitutionX87 -> T50,\nX88 -> X92,\nX43 -> .(T50, X92),\nT26 -> T52,\nX44 -> X93,\nX89 -> .(T52, X93),\nX91 -> T50,\nX90 -> T51,\nT25 -> .(T50, T51),\nT49 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 71 [arrowhead = none ];
71 -> 15;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 72 [arrowhead = none ];
72 -> 16;

73 [label="EVAL with clause\napp([], X105, X105).\nand substitutionX43 -> [],\nT26 -> T60,\nX44 -> T61,\nX105 -> .(T60, T61),\nX106 -> T61,\nT25 -> .(T60, T61)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 73 [arrowhead = none ];
73 -> 17;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 74 [arrowhead = none ];
74 -> 18;

75 [label="INSTANCE with matching:\nX43 -> X92\nT26 -> T52\nX44 -> X93\nT25 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 75 [arrowhead = none , style=dashed];
75 -> 10[style=dashed];

76 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 77 [arrowhead = none ];
77 -> 22;

78 [label="INSTANCE with matching:\nT1 -> T65\nT2 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 78 [arrowhead = none , style=dashed];
78 -> 1[style=dashed];

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 79 [arrowhead = none ];
79 -> 23;

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

81 [label="ONLY EVAL with clause\napp(.(X147, X148), X149, .(X147, X150)) :- app(X148, X149, X150).\nand substitutionT24 -> T87,\nX147 -> T87,\nT31 -> T88,\nX148 -> T88,\nT32 -> T89,\nX149 -> T89,\nX150 -> X151,\nX9 -> .(T87, X151)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 81 [arrowhead = none ];
81 -> 25;

82 [label="BACKTRACK\nfor clause: app([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 82 [arrowhead = none ];
82 -> 34;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 83 [arrowhead = none ];
83 -> 26;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 84 [arrowhead = none ];
84 -> 27;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 85 [arrowhead = none ];
85 -> 28;

86 [label="EVAL with clause\napp(.(X177, X178), X179, .(X177, X180)) :- app(X178, X179, X180).\nand substitutionX177 -> T105,\nX178 -> T106,\nT88 -> .(T105, T106),\nT89 -> T107,\nX179 -> T107,\nX180 -> X181,\nX151 -> .(T105, X181)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 86 [arrowhead = none ];
86 -> 29;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 87 [arrowhead = none ];
87 -> 30;

88 [label="EVAL with clause\napp([], X189, X189).\nand substitutionT88 -> [],\nT89 -> T113,\nX189 -> T113,\nX151 -> T113", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 88 [arrowhead = none ];
88 -> 31;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 89 [arrowhead = none ];
89 -> 32;

90 [label="INSTANCE with matching:\nT88 -> T106\nT89 -> T107\nX151 -> X181", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 90 [arrowhead = none , style=dashed];
90 -> 25[style=dashed];

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 91 [arrowhead = none ];
91 -> 33;

92 [label="EVAL with clause\napp([], X205, X205).\nand substitutionX7 -> [],\nT9 -> T125,\nX8 -> T126,\nX205 -> .(T125, T126),\nX206 -> T126,\nT6 -> .(T125, T126),\nT10 -> T127", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 92 [arrowhead = none ];
92 -> 37;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 93 [arrowhead = none ];
93 -> 38;

94 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 94 [arrowhead = none ];
94 -> 45;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 95 [arrowhead = none ];
95 -> 39;

96 [label="SPLIT 2\nnew knowledge:\nT126 is ground\nT128 is ground\nreplacements:X9 -> T128", fontsize=14, style = filled, fillcolor = violet];
37 -> 96 [arrowhead = none ];
96 -> 40;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 97 [arrowhead = none ];
97 -> 41;

98 [label="INSTANCE with matching:\nT1 -> T128\nT2 -> T127", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 98 [arrowhead = none , style=dashed];
98 -> 1[style=dashed];

99 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 99 [arrowhead = none ];
99 -> 42;

100 [label="ONLY EVAL with clause\napp([], X221, X221).\nand substitutionT126 -> T131,\nX221 -> T131,\nX9 -> T131", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 100 [arrowhead = none ];
100 -> 43;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 101 [arrowhead = none ];
101 -> 44;

102 [label="EVAL with clause\nperm([], []).\nand substitutionT6 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 102 [arrowhead = none ];
102 -> 46;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 104 [arrowhead = none ];
104 -> 48;

105 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 105 [arrowhead = none ];
105 -> 51;

}

(2) Obligation:

Clauses:

appA(.(T50, X92), T52, X93, .(T50, T51)) :- appA(X92, T52, X93, T51).
appA([], T60, T61, .(T60, T61)).
appB(.(T105, T106), T107, .(T105, X181)) :- appB(T106, T107, X181).
appB([], T113, T113).
appC(T87, T88, T89, .(T87, X151)) :- appB(T88, T89, X151).
appD(T131, T131).
permE(.(T24, T25), .(T26, T27)) :- appA(X43, T26, X44, T25).
permE(.(T24, T25), .(T26, T33)) :- ','(appA(T31, T26, T32, T25), appC(T24, T31, T32, X9)).
permE(.(T24, T25), .(T26, T33)) :- ','(appA(T31, T26, T32, T25), ','(appC(T24, T31, T32, T65), permE(T65, T33))).
permE(.(T125, T126), .(T125, T127)) :- appD(T126, X9).
permE(.(T125, T126), .(T125, T127)) :- ','(appD(T126, T128), permE(T128, T127)).
permE([], []).
permE([], []).

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)
appA_in: (f,f,f,b)
appC_in: (b,b,b,f)
appB_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(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)

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(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)

(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(.(T24, T25), .(T26, T27)) → U4_GA(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
PERME_IN_GA(.(T24, T25), .(T26, T27)) → APPA_IN_AAAG(X43, T26, X44, T25)
APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → U1_AAAG(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → APPA_IN_AAAG(X92, T52, X93, T51)
PERME_IN_GA(.(T24, T25), .(T26, T33)) → U5_GA(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → APPC_IN_GGGA(T24, T31, T32, X9)
APPC_IN_GGGA(T87, T88, T89, .(T87, X151)) → U3_GGGA(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
APPC_IN_GGGA(T87, T88, T89, .(T87, X151)) → APPB_IN_GGA(T88, T89, X151)
APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → U2_GGA(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → APPB_IN_GGA(T106, T107, X181)
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_GA(T24, T25, T26, T33, permE_in_ga(T65, T33))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → PERME_IN_GA(T65, T33)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U9_GA(T125, T126, T127, appD_in_ga(T126, X9))
PERME_IN_GA(.(T125, T126), .(T125, T127)) → APPD_IN_GA(T126, X9)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U10_GA(T125, T126, T127, appD_in_ga(T126, T128))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → U11_GA(T125, T126, T127, permE_in_ga(T128, T127))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → PERME_IN_GA(T128, T127)

The TRS R consists of the following rules:

permE_in_ga(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x5)
APPC_IN_GGGA(x1, x2, x3, x4)  =  APPC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
APPD_IN_GA(x1, x2)  =  APPD_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)

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(.(T24, T25), .(T26, T27)) → U4_GA(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
PERME_IN_GA(.(T24, T25), .(T26, T27)) → APPA_IN_AAAG(X43, T26, X44, T25)
APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → U1_AAAG(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → APPA_IN_AAAG(X92, T52, X93, T51)
PERME_IN_GA(.(T24, T25), .(T26, T33)) → U5_GA(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → APPC_IN_GGGA(T24, T31, T32, X9)
APPC_IN_GGGA(T87, T88, T89, .(T87, X151)) → U3_GGGA(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
APPC_IN_GGGA(T87, T88, T89, .(T87, X151)) → APPB_IN_GGA(T88, T89, X151)
APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → U2_GGA(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → APPB_IN_GGA(T106, T107, X181)
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_GA(T24, T25, T26, T33, permE_in_ga(T65, T33))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → PERME_IN_GA(T65, T33)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U9_GA(T125, T126, T127, appD_in_ga(T126, X9))
PERME_IN_GA(.(T125, T126), .(T125, T127)) → APPD_IN_GA(T126, X9)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U10_GA(T125, T126, T127, appD_in_ga(T126, T128))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → U11_GA(T125, T126, T127, permE_in_ga(T128, T127))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → PERME_IN_GA(T128, T127)

The TRS R consists of the following rules:

permE_in_ga(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x5)
APPC_IN_GGGA(x1, x2, x3, x4)  =  APPC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x5)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
APPD_IN_GA(x1, x2)  =  APPD_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)

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:

APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → APPB_IN_GGA(T106, T107, X181)

The TRS R consists of the following rules:

permE_in_ga(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_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:

APPB_IN_GGA(.(T105, T106), T107, .(T105, X181)) → APPB_IN_GGA(T106, T107, X181)

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

APPB_IN_GGA(.(T105, T106), T107) → APPB_IN_GGA(T106, T107)

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:

• APPB_IN_GGA(.(T105, T106), T107) → APPB_IN_GGA(T106, T107)
  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:

APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → APPA_IN_AAAG(X92, T52, X93, T51)

The TRS R consists of the following rules:

permE_in_ga(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)

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:

APPA_IN_AAAG(.(T50, X92), T52, X93, .(T50, T51)) → APPA_IN_AAAG(X92, T52, X93, T51)

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

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:

APPA_IN_AAAG(.(T50, T51)) → APPA_IN_AAAG(T51)

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:

• APPA_IN_AAAG(.(T50, T51)) → APPA_IN_AAAG(T51)
  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(.(T24, T25), .(T26, T33)) → U5_GA(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → PERME_IN_GA(T65, T33)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U10_GA(T125, T126, T127, appD_in_ga(T126, T128))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → PERME_IN_GA(T128, T127)

The TRS R consists of the following rules:

permE_in_ga(.(T24, T25), .(T26, T27)) → U4_ga(T24, T25, T26, T27, appA_in_aaag(X43, T26, X44, T25))
appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U4_ga(T24, T25, T26, T27, appA_out_aaag(X43, T26, X44, T25)) → permE_out_ga(.(T24, T25), .(T26, T27))
permE_in_ga(.(T24, T25), .(T26, T33)) → U5_ga(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U6_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, X9))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
U6_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, X9)) → permE_out_ga(.(T24, T25), .(T26, T33))
U5_ga(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_ga(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_ga(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → U8_ga(T24, T25, T26, T33, permE_in_ga(T65, T33))
permE_in_ga(.(T125, T126), .(T125, T127)) → U9_ga(T125, T126, T127, appD_in_ga(T126, X9))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U9_ga(T125, T126, T127, appD_out_ga(T126, X9)) → permE_out_ga(.(T125, T126), .(T125, T127))
permE_in_ga(.(T125, T126), .(T125, T127)) → U10_ga(T125, T126, T127, appD_in_ga(T126, T128))
U10_ga(T125, T126, T127, appD_out_ga(T126, T128)) → U11_ga(T125, T126, T127, permE_in_ga(T128, T127))
permE_in_ga([], []) → permE_out_ga([], [])
U11_ga(T125, T126, T127, permE_out_ga(T128, T127)) → permE_out_ga(.(T125, T126), .(T125, T127))
U8_ga(T24, T25, T26, T33, permE_out_ga(T65, T33)) → permE_out_ga(.(T24, T25), .(T26, T33))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
permE_out_ga(x1, x2)  =  permE_out_ga
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)

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(.(T24, T25), .(T26, T33)) → U5_GA(T24, T25, T26, T33, appA_in_aaag(T31, T26, T32, T25))
U5_GA(T24, T25, T26, T33, appA_out_aaag(T31, T26, T32, T25)) → U7_GA(T24, T25, T26, T33, appC_in_ggga(T24, T31, T32, T65))
U7_GA(T24, T25, T26, T33, appC_out_ggga(T24, T31, T32, T65)) → PERME_IN_GA(T65, T33)
PERME_IN_GA(.(T125, T126), .(T125, T127)) → U10_GA(T125, T126, T127, appD_in_ga(T126, T128))
U10_GA(T125, T126, T127, appD_out_ga(T126, T128)) → PERME_IN_GA(T128, T127)

The TRS R consists of the following rules:

appA_in_aaag(.(T50, X92), T52, X93, .(T50, T51)) → U1_aaag(T50, X92, T52, X93, T51, appA_in_aaag(X92, T52, X93, T51))
appA_in_aaag([], T60, T61, .(T60, T61)) → appA_out_aaag([], T60, T61, .(T60, T61))
appC_in_ggga(T87, T88, T89, .(T87, X151)) → U3_ggga(T87, T88, T89, X151, appB_in_gga(T88, T89, X151))
appD_in_ga(T131, T131) → appD_out_ga(T131, T131)
U1_aaag(T50, X92, T52, X93, T51, appA_out_aaag(X92, T52, X93, T51)) → appA_out_aaag(.(T50, X92), T52, X93, .(T50, T51))
U3_ggga(T87, T88, T89, X151, appB_out_gga(T88, T89, X151)) → appC_out_ggga(T87, T88, T89, .(T87, X151))
appB_in_gga(.(T105, T106), T107, .(T105, X181)) → U2_gga(T105, T106, T107, X181, appB_in_gga(T106, T107, X181))
appB_in_gga([], T113, T113) → appB_out_gga([], T113, T113)
U2_gga(T105, T106, T107, X181, appB_out_gga(T106, T107, X181)) → appB_out_gga(.(T105, T106), T107, .(T105, X181))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
U1_aaag(x1, x2, x3, x4, x5, x6)  =  U1_aaag(x1, x6)
appA_out_aaag(x1, x2, x3, x4)  =  appA_out_aaag(x1, x2, x3)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
[]  =  []
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
appC_out_ggga(x1, x2, x3, x4)  =  appC_out_ggga(x4)
appD_in_ga(x1, x2)  =  appD_in_ga(x1)
appD_out_ga(x1, x2)  =  appD_out_ga(x2)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)

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(.(T24, T25)) → U5_GA(T24, appA_in_aaag(T25))
U5_GA(T24, appA_out_aaag(T31, T26, T32)) → U7_GA(appC_in_ggga(T24, T31, T32))
U7_GA(appC_out_ggga(T65)) → PERME_IN_GA(T65)
PERME_IN_GA(.(T125, T126)) → U10_GA(appD_in_ga(T126))
U10_GA(appD_out_ga(T128)) → PERME_IN_GA(T128)

The TRS R consists of the following rules:

appA_in_aaag(.(T50, T51)) → U1_aaag(T50, appA_in_aaag(T51))
appA_in_aaag(.(T60, T61)) → appA_out_aaag([], T60, T61)
appC_in_ggga(T87, T88, T89) → U3_ggga(T87, appB_in_gga(T88, T89))
appD_in_ga(T131) → appD_out_ga(T131)
U1_aaag(T50, appA_out_aaag(X92, T52, X93)) → appA_out_aaag(.(T50, X92), T52, X93)
U3_ggga(T87, appB_out_gga(X151)) → appC_out_ggga(.(T87, X151))
appB_in_gga(.(T105, T106), T107) → U2_gga(T105, appB_in_gga(T106, T107))
appB_in_gga([], T113) → appB_out_gga(T113)
U2_gga(T105, appB_out_gga(X181)) → appB_out_gga(.(T105, X181))

The set Q consists of the following terms:

appA_in_aaag(x0)
appC_in_ggga(x0, x1, x2)
appD_in_ga(x0)
U1_aaag(x0, x1)
U3_ggga(x0, x1)
appB_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(.(T24, T25)) → U5_GA(T24, appA_in_aaag(T25))
U5_GA(T24, appA_out_aaag(T31, T26, T32)) → U7_GA(appC_in_ggga(T24, T31, T32))
U7_GA(appC_out_ggga(T65)) → PERME_IN_GA(T65)
PERME_IN_GA(.(T125, T126)) → U10_GA(appD_in_ga(T126))
U10_GA(appD_out_ga(T128)) → PERME_IN_GA(T128)

Strictly oriented rules of the TRS R:

appA_in_aaag(.(T50, T51)) → U1_aaag(T50, appA_in_aaag(T51))
appA_in_aaag(.(T60, T61)) → appA_out_aaag([], T60, T61)
appC_in_ggga(T87, T88, T89) → U3_ggga(T87, appB_in_gga(T88, T89))
appD_in_ga(T131) → appD_out_ga(T131)
U1_aaag(T50, appA_out_aaag(X92, T52, X93)) → appA_out_aaag(.(T50, X92), T52, X93)
U3_ggga(T87, appB_out_gga(X151)) → appC_out_ggga(.(T87, X151))
appB_in_gga(.(T105, T106), T107) → U2_gga(T105, appB_in_gga(T106, T107))
appB_in_gga([], T113) → appB_out_gga(T113)
U2_gga(T105, appB_out_gga(X181)) → appB_out_gga(.(T105, X181))

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

U10_GA1 > appC_inggga3 > U3_ggga2 > U5_GA2 > appB_ingga2 > U7_GA1 > PERME_INGA1 > appA_inaaag1 > U1_aaag2 > U2_gga2 > appC_outggga1 > appB_outgga1 > appA_outaaag3 > appD_inga1 > appD_outga1 > [] > .₂

and weight map:

[]=5
appA_in_aaag_1=3
appD_in_ga_1=2
appD_out_ga_1=1
appB_out_gga_1=5
appC_out_ggga_1=3
PERME_IN_GA_1=4
U7_GA_1=1
U10_GA_1=3
._2=2
U1_aaag_2=2
appA_out_aaag_3=0
appC_in_ggga_3=0
U3_ggga_2=0
appB_in_gga_2=0
U2_gga_2=2
U5_GA_2=0

The variable weight is 1

(29) Obligation:

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

appA_in_aaag(x0)
appC_in_ggga(x0, x1, x2)
appD_in_ga(x0)
U1_aaag(x0, x1)
U3_ggga(x0, x1)
appB_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