Left Termination proof of /tmp/tmpM1m1XK/permutation-bf.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 86 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 19 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 138 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇒, 0 ms)

        ↳27 QDP

        ↳28 MRRProof (⇔, 44 ms)

        ↳29 QDP

        ↳30 PisEmptyProof (⇔, 0 ms)

        ↳31 YES

(0) Obligation:

Clauses:

perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

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: true | perm([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nperm(T1, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: perm([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(X7, .(T7, X8), []), ','(app(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: ','(app(X7, .(T7, X8), []), ','(app(X7, X8, X9), perm(X9, T8))), SCOPE: 2, CLAUSE: 2 | ','(app(X7, .(T7, X8), []), ','(app(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: ','(app(X7, .(T7, X8), []), ','(app(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: ','(app(X25, .(T17, X26), T14), ','(app(X25, X26, X27), perm(X27, T18)))\n\nT14 is ground\nX25 is free\nX26 is free\nX27 is free\nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(app(X25, .(T17, X26), T14), ','(app(X25, X26, X27), perm(X27, T18))), SCOPE: 3, CLAUSE: 2 | ','(app(X25, .(T17, X26), T14), ','(app(X25, X26, X27), perm(X27, T18))), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX25 is free\nX26 is free\nX27 is free\nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(app(X25, .(T17, X26), T14), ','(app(X25, X26, X27), perm(X27, T18))), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nX25 is free\nX26 is free\nX27 is free\nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(app(X25, .(T17, X26), T14), ','(app(X25, X26, X27), perm(X27, T18))), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX25 is free\nX26 is free\nX27 is free\nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(app([], T28, X27), perm(X27, T29))\n\nT28 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app([], T28, X27)\n\nT28 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: perm(T31, T29)\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app([], T28, X27), SCOPE: 4, CLAUSE: 2 | app([], T28, X27), SCOPE: 4, CLAUSE: 3\n\nT28 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app([], T28, X27), SCOPE: 4, CLAUSE: 2\n\nT28 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app([], T28, X27), SCOPE: 4, CLAUSE: 3\n\nT28 is ground\nX27 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: ','(app(X68, .(T47, X69), T46), ','(app(.(T45, X68), X69, X27), perm(X27, T48)))\n\nT45 is ground\nT46 is ground\nX27 is free\nX68 is free\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(X68, .(T47, X69), T46)\n\nT46 is ground\nX68 is free\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(app(.(T45, T53), T54, X27), perm(X27, T55))\n\nT45 is ground\nT53 is ground\nT54 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(X68, .(T47, X69), T46), SCOPE: 5, CLAUSE: 2 | app(X68, .(T47, X69), T46), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nX68 is free\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(X68, .(T47, X69), T46), SCOPE: 5, CLAUSE: 2\n\nT46 is ground\nX68 is free\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(X68, .(T47, X69), T46), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nX68 is free\nX69 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: app(X107, .(T79, X108), T78)\n\nT78 is ground\nX107 is free\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(.(T45, T53), T54, X27)\n\nT45 is ground\nT53 is ground\nT54 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: perm(T84, T55)\n\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(.(T45, T53), T54, X27), SCOPE: 6, CLAUSE: 2 | app(.(T45, T53), T54, X27), SCOPE: 6, CLAUSE: 3\n\nT45 is ground\nT53 is ground\nT54 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(.(T45, T53), T54, X27), SCOPE: 6, CLAUSE: 3\n\nT45 is ground\nT53 is ground\nT54 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app(T94, T95, X134)\n\nT94 is ground\nT95 is ground\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T94, T95, X134), SCOPE: 7, CLAUSE: 2 | app(T94, T95, X134), SCOPE: 7, CLAUSE: 3\n\nT94 is ground\nT95 is ground\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(T94, T95, X134), SCOPE: 7, CLAUSE: 2\n\nT94 is ground\nT95 is ground\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app(T94, T95, X134), SCOPE: 7, CLAUSE: 3\n\nT94 is ground\nT95 is ground\nX134 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: app(T110, T111, X156)\n\nT110 is ground\nT111 is ground\nX156 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([], []).\nand substitutionT1 -> [],\nT2 -> []", 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(X22, .(X23, X24)) :- ','(app(X25, .(X23, X26), X22), ','(app(X25, X26, X27), perm(X27, X24))).\nand substitutionT1 -> T14,\nX22 -> T14,\nX23 -> T17,\nX24 -> T18,\nT2 -> .(T17, T18),\nT15 -> T17,\nT16 -> T18", 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, .(X5, X6)) :- ','(app(X7, .(X5, X8), X4), ','(app(X7, X8, X9), perm(X9, X6))).\nand substitutionX4 -> [],\nX5 -> T7,\nX6 -> T8,\nT2 -> .(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: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)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\napp([], X36, X36).\nand substitutionX25 -> [],\nT17 -> T27,\nX26 -> T28,\nX36 -> .(T27, T28),\nX37 -> T28,\nT14 -> .(T27, T28),\nT18 -> T29", 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\napp(.(X63, X64), X65, .(X63, X66)) :- app(X64, X65, X66).\nand substitutionX63 -> T45,\nX64 -> X68,\nX25 -> .(T45, X68),\nT17 -> T47,\nX26 -> X69,\nX65 -> .(T47, X69),\nX67 -> T45,\nX66 -> T46,\nT14 -> .(T45, T46),\nT44 -> T47,\nT18 -> T48", 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:\nT28 is ground\nT31 is ground\nreplacements:X27 -> T31", 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 -> T31\nT2 -> T29", 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\napp([], X44, X44).\nand substitutionT28 -> T37,\nX44 -> T37,\nX27 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 75 [arrowhead = none ];
75 -> 23;

76 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)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:\nT53 is ground\nT47 is ground\nT54 is ground\nT46 is ground\nreplacements:X68 -> T53,\nX69 -> T54,\nT48 -> T55", 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:\nT45 is ground\nT53 is ground\nT54 is ground\nT84 is ground\nreplacements:X27 -> T84", 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\napp([], X86, X86).\nand substitutionX68 -> [],\nT47 -> T68,\nX69 -> T69,\nX86 -> .(T68, T69),\nX87 -> T69,\nT46 -> .(T68, T69)", 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\napp(.(X102, X103), X104, .(X102, X105)) :- app(X103, X104, X105).\nand substitutionX102 -> T77,\nX103 -> X107,\nX68 -> .(T77, X107),\nT47 -> T79,\nX69 -> X108,\nX104 -> .(T79, X108),\nX106 -> T77,\nX105 -> T78,\nT46 -> .(T77, T78),\nT76 -> T79", 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:\nX68 -> X107\nT47 -> T79\nX69 -> X108\nT46 -> T78", 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 -> T84\nT2 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 92 [arrowhead = none , style=dashed];
92 -> 1[style=dashed];

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

94 [label="ONLY EVAL with clause\napp(.(X130, X131), X132, .(X130, X133)) :- app(X131, X132, X133).\nand substitutionT45 -> T93,\nX130 -> T93,\nT53 -> T94,\nX131 -> T94,\nT54 -> T95,\nX132 -> T95,\nX133 -> X134,\nX27 -> .(T93, X134)", 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\napp([], X141, X141).\nand substitutionT94 -> [],\nT95 -> T102,\nX141 -> T102,\nX134 -> T102", 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\napp(.(X152, X153), X154, .(X152, X155)) :- app(X153, X154, X155).\nand substitutionX152 -> T109,\nX153 -> T110,\nT94 -> .(T109, T110),\nT95 -> T111,\nX154 -> T111,\nX155 -> X156,\nX134 -> .(T109, X156)", 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:\nT94 -> T110\nT95 -> T111\nX134 -> X156", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 103 [arrowhead = none , style=dashed];
103 -> 42[style=dashed];

}

(2) Obligation:

Clauses:

appA([], T68, T69, .(T68, T69)).
appA(.(T77, X107), T79, X108, .(T77, T78)) :- appA(X107, T79, X108, T78).
appB([], T102, T102).
appB(.(T109, T110), T111, .(T109, X156)) :- appB(T110, T111, X156).
appC(T37, T37).
appD(T93, T94, T95, .(T93, X134)) :- appB(T94, T95, X134).
permE([], []).
permE(.(T27, T28), .(T27, T29)) :- appC(T28, X27).
permE(.(T27, T28), .(T27, T29)) :- ','(appC(T28, T31), permE(T31, T29)).
permE(.(T45, T46), .(T47, T48)) :- appA(X68, T47, X69, T46).
permE(.(T45, T46), .(T47, T55)) :- ','(appA(T53, T47, T54, T46), appD(T45, T53, T54, X27)).
permE(.(T45, T46), .(T47, T55)) :- ','(appA(T53, T47, T54, T46), ','(appD(T45, T53, T54, T84), permE(T84, T55))).

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)
appD_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([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

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([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

(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(.(T27, T28), .(T27, T29)) → U4_GA(T27, T28, T29, appC_in_ga(T28, X27))
PERME_IN_GA(.(T27, T28), .(T27, T29)) → APPC_IN_GA(T28, X27)
PERME_IN_GA(.(T27, T28), .(T27, T29)) → U5_GA(T27, T28, T29, appC_in_ga(T28, T31))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → U6_GA(T27, T28, T29, permE_in_ga(T31, T29))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → PERME_IN_GA(T31, T29)
PERME_IN_GA(.(T45, T46), .(T47, T48)) → U7_GA(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
PERME_IN_GA(.(T45, T46), .(T47, T48)) → APPA_IN_AAAG(X68, T47, X69, T46)
APPA_IN_AAAG(.(T77, X107), T79, X108, .(T77, T78)) → U1_AAAG(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
APPA_IN_AAAG(.(T77, X107), T79, X108, .(T77, T78)) → APPA_IN_AAAG(X107, T79, X108, T78)
PERME_IN_GA(.(T45, T46), .(T47, T55)) → U8_GA(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → APPD_IN_GGGA(T45, T53, T54, X27)
APPD_IN_GGGA(T93, T94, T95, .(T93, X134)) → U3_GGGA(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
APPD_IN_GGGA(T93, T94, T95, .(T93, X134)) → APPB_IN_GGA(T94, T95, X134)
APPB_IN_GGA(.(T109, T110), T111, .(T109, X156)) → U2_GGA(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
APPB_IN_GGA(.(T109, T110), T111, .(T109, X156)) → APPB_IN_GGA(T110, T111, X156)
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_GA(T45, T46, T47, T55, permE_in_ga(T84, T55))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → PERME_IN_GA(T84, T55)

The TRS R consists of the following rules:

permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
permE_in_ga(x1, x2) = permE_in_ga(x1)
[] = []
permE_out_ga(x1, x2) = permE_out_ga
.(x1, x2) = .(x1, x2)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
appC_in_ga(x1, x2) = appC_in_ga(x1)
appC_out_ga(x1, x2) = appC_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)
appA_in_aaag(x1, x2, x3, x4) = appA_in_aaag(x4)
appA_out_aaag(x1, x2, x3, x4) = appA_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6) = U1_aaag(x1, x6)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x1, x5)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x5)
appD_in_ggga(x1, x2, x3, x4) = appD_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)
appB_out_gga(x1, x2, x3) = appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
appD_out_ggga(x1, x2, x3, x4) = appD_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)
APPC_IN_GA(x1, x2) = APPC_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)
APPA_IN_AAAG(x1, x2, x3, x4) = APPA_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6) = U1_AAAG(x1, x6)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x1, x5)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x5)
APPD_IN_GGGA(x1, x2, x3, x4) = APPD_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)
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(.(T27, T28), .(T27, T29)) → U4_GA(T27, T28, T29, appC_in_ga(T28, X27))
PERME_IN_GA(.(T27, T28), .(T27, T29)) → APPC_IN_GA(T28, X27)
PERME_IN_GA(.(T27, T28), .(T27, T29)) → U5_GA(T27, T28, T29, appC_in_ga(T28, T31))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → U6_GA(T27, T28, T29, permE_in_ga(T31, T29))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → PERME_IN_GA(T31, T29)
PERME_IN_GA(.(T45, T46), .(T47, T48)) → U7_GA(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
PERME_IN_GA(.(T45, T46), .(T47, T48)) → APPA_IN_AAAG(X68, T47, X69, T46)
APPA_IN_AAAG(.(T77, X107), T79, X108, .(T77, T78)) → U1_AAAG(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
APPA_IN_AAAG(.(T77, X107), T79, X108, .(T77, T78)) → APPA_IN_AAAG(X107, T79, X108, T78)
PERME_IN_GA(.(T45, T46), .(T47, T55)) → U8_GA(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → APPD_IN_GGGA(T45, T53, T54, X27)
APPD_IN_GGGA(T93, T94, T95, .(T93, X134)) → U3_GGGA(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
APPD_IN_GGGA(T93, T94, T95, .(T93, X134)) → APPB_IN_GGA(T94, T95, X134)
APPB_IN_GGA(.(T109, T110), T111, .(T109, X156)) → U2_GGA(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
APPB_IN_GGA(.(T109, T110), T111, .(T109, X156)) → APPB_IN_GGA(T110, T111, X156)
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_GA(T45, T46, T47, T55, permE_in_ga(T84, T55))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → PERME_IN_GA(T84, T55)

The TRS R consists of the following rules:

permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

(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(.(T109, T110), T111, .(T109, X156)) → APPB_IN_GGA(T110, T111, X156)

The TRS R consists of the following rules:

permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

(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(.(T109, T110), T111, .(T109, X156)) → APPB_IN_GGA(T110, T111, X156)

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(.(T109, T110), T111) → APPB_IN_GGA(T110, T111)

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(.(T109, T110), T111) → APPB_IN_GGA(T110, T111)
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(.(T77, X107), T79, X108, .(T77, T78)) → APPA_IN_AAAG(X107, T79, X108, T78)

The TRS R consists of the following rules:

permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

(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(.(T77, X107), T79, X108, .(T77, T78)) → APPA_IN_AAAG(X107, T79, X108, T78)

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(.(T77, T78)) → APPA_IN_AAAG(T78)

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(.(T77, T78)) → APPA_IN_AAAG(T78)
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(.(T27, T28), .(T27, T29)) → U5_GA(T27, T28, T29, appC_in_ga(T28, T31))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → PERME_IN_GA(T31, T29)
PERME_IN_GA(.(T45, T46), .(T47, T55)) → U8_GA(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → PERME_IN_GA(T84, T55)

The TRS R consists of the following rules:

permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T27, T28), .(T27, T29)) → U4_ga(T27, T28, T29, appC_in_ga(T28, X27))
appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
U4_ga(T27, T28, T29, appC_out_ga(T28, X27)) → permE_out_ga(.(T27, T28), .(T27, T29))
permE_in_ga(.(T27, T28), .(T27, T29)) → U5_ga(T27, T28, T29, appC_in_ga(T28, T31))
U5_ga(T27, T28, T29, appC_out_ga(T28, T31)) → U6_ga(T27, T28, T29, permE_in_ga(T31, T29))
permE_in_ga(.(T45, T46), .(T47, T48)) → U7_ga(T45, T46, T47, T48, appA_in_aaag(X68, T47, X69, T46))
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U7_ga(T45, T46, T47, T48, appA_out_aaag(X68, T47, X69, T46)) → permE_out_ga(.(T45, T46), .(T47, T48))
permE_in_ga(.(T45, T46), .(T47, T55)) → U8_ga(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U9_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, X27))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
U9_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, X27)) → permE_out_ga(.(T45, T46), .(T47, T55))
U8_ga(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_ga(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_ga(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → U11_ga(T45, T46, T47, T55, permE_in_ga(T84, T55))
U11_ga(T45, T46, T47, T55, permE_out_ga(T84, T55)) → permE_out_ga(.(T45, T46), .(T47, T55))
U6_ga(T27, T28, T29, permE_out_ga(T31, T29)) → permE_out_ga(.(T27, T28), .(T27, T29))

(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(.(T27, T28), .(T27, T29)) → U5_GA(T27, T28, T29, appC_in_ga(T28, T31))
U5_GA(T27, T28, T29, appC_out_ga(T28, T31)) → PERME_IN_GA(T31, T29)
PERME_IN_GA(.(T45, T46), .(T47, T55)) → U8_GA(T45, T46, T47, T55, appA_in_aaag(T53, T47, T54, T46))
U8_GA(T45, T46, T47, T55, appA_out_aaag(T53, T47, T54, T46)) → U10_GA(T45, T46, T47, T55, appD_in_ggga(T45, T53, T54, T84))
U10_GA(T45, T46, T47, T55, appD_out_ggga(T45, T53, T54, T84)) → PERME_IN_GA(T84, T55)

The TRS R consists of the following rules:

appC_in_ga(T37, T37) → appC_out_ga(T37, T37)
appA_in_aaag([], T68, T69, .(T68, T69)) → appA_out_aaag([], T68, T69, .(T68, T69))
appA_in_aaag(.(T77, X107), T79, X108, .(T77, T78)) → U1_aaag(T77, X107, T79, X108, T78, appA_in_aaag(X107, T79, X108, T78))
appD_in_ggga(T93, T94, T95, .(T93, X134)) → U3_ggga(T93, T94, T95, X134, appB_in_gga(T94, T95, X134))
U1_aaag(T77, X107, T79, X108, T78, appA_out_aaag(X107, T79, X108, T78)) → appA_out_aaag(.(T77, X107), T79, X108, .(T77, T78))
U3_ggga(T93, T94, T95, X134, appB_out_gga(T94, T95, X134)) → appD_out_ggga(T93, T94, T95, .(T93, X134))
appB_in_gga([], T102, T102) → appB_out_gga([], T102, T102)
appB_in_gga(.(T109, T110), T111, .(T109, X156)) → U2_gga(T109, T110, T111, X156, appB_in_gga(T110, T111, X156))
U2_gga(T109, T110, T111, X156, appB_out_gga(T110, T111, X156)) → appB_out_gga(.(T109, T110), T111, .(T109, X156))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
appC_in_ga(x1, x2) = appC_in_ga(x1)
appC_out_ga(x1, x2) = appC_out_ga(x2)
appA_in_aaag(x1, x2, x3, x4) = appA_in_aaag(x4)
appA_out_aaag(x1, x2, x3, x4) = appA_out_aaag(x1, x2, x3)
U1_aaag(x1, x2, x3, x4, x5, x6) = U1_aaag(x1, x6)
appD_in_ggga(x1, x2, x3, x4) = appD_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)
appB_out_gga(x1, x2, x3) = appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
appD_out_ggga(x1, x2, x3, x4) = appD_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(.(T27, T28)) → U5_GA(appC_in_ga(T28))
U5_GA(appC_out_ga(T31)) → PERME_IN_GA(T31)
PERME_IN_GA(.(T45, T46)) → U8_GA(T45, appA_in_aaag(T46))
U8_GA(T45, appA_out_aaag(T53, T47, T54)) → U10_GA(appD_in_ggga(T45, T53, T54))
U10_GA(appD_out_ggga(T84)) → PERME_IN_GA(T84)

The TRS R consists of the following rules:

appC_in_ga(T37) → appC_out_ga(T37)
appA_in_aaag(.(T68, T69)) → appA_out_aaag([], T68, T69)
appA_in_aaag(.(T77, T78)) → U1_aaag(T77, appA_in_aaag(T78))
appD_in_ggga(T93, T94, T95) → U3_ggga(T93, appB_in_gga(T94, T95))
U1_aaag(T77, appA_out_aaag(X107, T79, X108)) → appA_out_aaag(.(T77, X107), T79, X108)
U3_ggga(T93, appB_out_gga(X134)) → appD_out_ggga(.(T93, X134))
appB_in_gga([], T102) → appB_out_gga(T102)
appB_in_gga(.(T109, T110), T111) → U2_gga(T109, appB_in_gga(T110, T111))
U2_gga(T109, appB_out_gga(X156)) → appB_out_gga(.(T109, X156))

The set Q consists of the following terms:

appC_in_ga(x0)
appA_in_aaag(x0)
appD_in_ggga(x0, x1, x2)
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(.(T27, T28)) → U5_GA(appC_in_ga(T28))
U5_GA(appC_out_ga(T31)) → PERME_IN_GA(T31)
PERME_IN_GA(.(T45, T46)) → U8_GA(T45, appA_in_aaag(T46))
U8_GA(T45, appA_out_aaag(T53, T47, T54)) → U10_GA(appD_in_ggga(T45, T53, T54))
U10_GA(appD_out_ggga(T84)) → PERME_IN_GA(T84)

Strictly oriented rules of the TRS R:

appC_in_ga(T37) → appC_out_ga(T37)
appA_in_aaag(.(T68, T69)) → appA_out_aaag([], T68, T69)
appA_in_aaag(.(T77, T78)) → U1_aaag(T77, appA_in_aaag(T78))
appD_in_ggga(T93, T94, T95) → U3_ggga(T93, appB_in_gga(T94, T95))
U1_aaag(T77, appA_out_aaag(X107, T79, X108)) → appA_out_aaag(.(T77, X107), T79, X108)
U3_ggga(T93, appB_out_gga(X134)) → appD_out_ggga(.(T93, X134))
appB_in_gga([], T102) → appB_out_gga(T102)
appB_in_gga(.(T109, T110), T111) → U2_gga(T109, appB_in_gga(T110, T111))
U2_gga(T109, appB_out_gga(X156)) → appB_out_gga(.(T109, X156))

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

U10_GA₁ > appD_{in_ggga₃} > appA_{in_aaag₁} > appC_{out_ga₁} > .₂ > U5_GA₁ > PERME_{IN_GA₁} > U1_aaag₂ > U8_GA₂ > appB_{in_gga₂} > U2_gga₂ > U3_ggga₂ > appD_{out_ggga₁} > appB_{out_gga₁} > [] > appA_{out_aaag₃} > appC_{in_ga₁}

and weight map:

[]=4
appC_in_ga_1=3
appC_out_ga_1=2
appA_in_aaag_1=3
appB_out_gga_1=4
appD_out_ggga_1=3
PERME_IN_GA_1=3
U5_GA_1=1
U10_GA_1=1
._2=1
appA_out_aaag_3=0
U1_aaag_2=1
appD_in_ggga_3=0
U3_ggga_2=0
appB_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:

appC_in_ga(x0)
appA_in_aaag(x0)
appD_in_ggga(x0, x1, x2)
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.

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) MRRProof (EQUIVALENT transformation)

(29) Obligation:

(30) PisEmptyProof (EQUIVALENT transformation)

(31) YES