Left Termination proof of /tmp/tmp7qQtW8/rev.pl

Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 Narrowing (⇐)

                ↳29 QDP

                ↳30 Narrowing (⇐)

                ↳31 QDP

                ↳32 Narrowing (⇐)

                ↳33 QDP

                ↳34 Narrowing (⇐)

                ↳35 QDP

                ↳36 Narrowing (⇐)

                ↳37 QDP

                ↳38 Narrowing (⇐)

                ↳39 QDP

                ↳40 Instantiation (⇔)

                ↳41 QDP

                ↳42 Instantiation (⇔)

                ↳43 QDP

                ↳44 Instantiation (⇔)

                ↳45 QDP

                ↳46 Instantiation (⇔)

                ↳47 QDP

                ↳48 Instantiation (⇔)

                ↳49 QDP

                ↳50 ForwardInstantiation (⇔)

                ↳51 QDP

                ↳52 DependencyGraphProof (⇔)

                ↳53 QDP

                ↳54 ForwardInstantiation (⇔)

                ↳55 QDP

                ↳56 ForwardInstantiation (⇔)

                ↳57 QDP

                ↳58 NonTerminationProof (⇔)

                ↳59 NO

            ↳60 PiDP

                ↳61 UsableRulesProof (⇔)

                ↳62 PiDP

                ↳63 PiDPToQDPProof (⇐)

                ↳64 QDP

                ↳65 QDPSizeChangeProof (⇔)

                ↳66 YES

    ↳67 PrologToPiTRSProof (⇐)

        ↳68 PiTRS

        ↳69 DependencyPairsProof (⇔)

        ↳70 PiDP

        ↳71 DependencyGraphProof (⇔)

        ↳72 AND

            ↳73 PiDP

                ↳74 UsableRulesProof (⇔)

                ↳75 PiDP

                ↳76 PiDPToQDPProof (⇐)

                ↳77 QDP

                ↳78 NonTerminationProof (⇔)

                ↳79 NO

            ↳80 PiDP

                ↳81 UsableRulesProof (⇔)

                ↳82 PiDP

                ↳83 PiDPToQDPProof (⇐)

                ↳84 QDP

                ↳85 NonTerminationProof (⇔)

                ↳86 NO

            ↳87 PiDP

                ↳88 UsableRulesProof (⇔)

                ↳89 PiDP

                ↳90 PiDPToQDPProof (⇐)

                ↳91 QDP

                ↳92 Narrowing (⇐)

                ↳93 QDP

                ↳94 Narrowing (⇐)

                ↳95 QDP

                ↳96 Narrowing (⇐)

                ↳97 QDP

                ↳98 Narrowing (⇐)

                ↳99 QDP

                ↳100 Narrowing (⇐)

                ↳101 QDP

                ↳102 Narrowing (⇐)

                ↳103 QDP

                ↳104 Instantiation (⇔)

                ↳105 QDP

                ↳106 Instantiation (⇔)

                ↳107 QDP

                ↳108 Instantiation (⇔)

                ↳109 QDP

                ↳110 Instantiation (⇔)

                ↳111 QDP

                ↳112 Instantiation (⇔)

                ↳113 QDP

                ↳114 Instantiation (⇔)

                ↳115 QDP

                ↳116 Instantiation (⇔)

                ↳117 QDP

                ↳118 Instantiation (⇔)

                ↳119 QDP

                ↳120 Instantiation (⇔)

                ↳121 QDP

                ↳122 Instantiation (⇔)

                ↳123 QDP

                ↳124 ForwardInstantiation (⇔)

                ↳125 QDP

                ↳126 DependencyGraphProof (⇔)

                ↳127 QDP

                ↳128 ForwardInstantiation (⇔)

                ↳129 QDP

                ↳130 ForwardInstantiation (⇔)

                ↳131 QDP

            ↳132 PiDP

                ↳133 UsableRulesProof (⇔)

                ↳134 PiDP

(0) Obligation:

Clauses:

rev([], []).
rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS)).
rev1(X, [], X).
rev1(X, .(Y, YS), Z) :- rev1(Y, YS, Z).
rev2(X, [], []).
rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS))).

Queries:

rev(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(rev1(T7, T8, T11), rev2(T7, T8, T12))\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 2 | ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: rev2(T17, [], T18)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 4 | rev2(T17, [], T18), SCOPE: 3, CLAUSE: 5\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 4\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 5\n\nT17 is ground", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(rev1(T36, T37, T39), rev2(T35, .(T36, T37), T40))\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: rev1(T36, T37, T39)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: rev2(T35, .(T36, T37), T43)\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 2 | rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 3\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 3\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: rev1(T60, T61, T63)\n\nT60 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 4 | rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 5\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 5\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(rev2(T85, T86, X104), ','(rev(X104, X105), rev(.(T84, X105), T88)))\n\nT84 is ground\nT85 is ground\nT86 is ground\nX104 is free; \nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: rev2(T85, T86, X104)\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(rev(T91, X105), rev(.(T84, X105), T88))\n\nT84 is ground\nT91 is ground\nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 4 | rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 5\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 4\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 5\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(rev2(T106, T107, X132), ','(rev(X132, X133), rev(.(T105, X133), X134)))\n\nT105 is ground\nT106 is ground\nT107 is ground\nX134 is free; \nX132 is free; \nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: rev2(T106, T107, X132)\n\nT106 is ground\nT107 is ground\nX132 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(rev(T110, X133), rev(.(T105, X133), X134))\n\nT105 is ground\nT110 is ground\nX134 is free; \nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: rev(T110, X133)\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: rev(.(T105, T113), X134)\n\nT105 is ground\nT113 is ground\nX134 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: rev(T110, X133), SCOPE: 7, CLAUSE: 0 | rev(T110, X133), SCOPE: 7, CLAUSE: 1\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: rev(T110, X133), SCOPE: 7, CLAUSE: 0\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: rev(T110, X133), SCOPE: 7, CLAUSE: 1\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(rev1(T118, T119, X159), rev2(T118, T119, X160))\n\nT118 is ground\nT119 is ground\nX159 is free; \nX160 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: rev1(T118, T119, X159)\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: rev2(T118, T119, X160)\n\nT118 is ground\nT119 is ground\nX160 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 2 | rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 3\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 2\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 3\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: rev1(T137, T138, X189)\n\nT137 is ground\nT138 is ground\nX189 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: rev(T91, X105)\n\nT91 is ground\nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: rev(.(T84, T145), T88)\n\nT84 is ground\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 70 [arrowhead = none ];
70 -> 2;

71 [label="EVAL with clause\nrev([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 71 [arrowhead = none ];
71 -> 3;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 72 [arrowhead = none ];
72 -> 4;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 73 [arrowhead = none ];
73 -> 5;

74 [label="EVAL with clause\nrev(.(X9, X10), .(X11, X12)) :- ','(rev1(X9, X10, X11), rev2(X9, X10, X12)).\nand substitution\nX9 -> T7\nX10 -> T8\nT1 -> .(T7, T8)\nX11 -> T11\nX12 -> T12\nT2 -> .(T11, T12)\nT9 -> T11\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="BACKTRACK\nfor clause: rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 76 [arrowhead = none ];
76 -> 6;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 77 [arrowhead = none ];
77 -> 9;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 78 [arrowhead = none ];
78 -> 10;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 79 [arrowhead = none ];
79 -> 11;

80 [label="EVAL with clause\nrev1(X21, [], X21).\nand substitution\nT7 -> T17\nX21 -> T17\nT8 -> []\nT11 -> T17\nT12 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="EVAL with clause\nrev1(X43, .(X44, X45), X46) :- rev1(X44, X45, X46).\nand substitution\nT7 -> T35\nX43 -> T35\nX44 -> T36\nX45 -> T37\nT8 -> .(T36, T37)\nT11 -> T39\nX46 -> T39\nT38 -> T39\nT12 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL with clause\nrev2(X29, [], []).\nand substitution\nT17 -> T25\nX29 -> T25\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 88 [arrowhead = none ];
88 -> 18;

89 [label="BACKTRACK\nfor clause: rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 89 [arrowhead = none ];
89 -> 20;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 91 [arrowhead = none ];
91 -> 23;

92 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nreplacements:\nT40 -> T43", fontsize=14, style = filled, fillcolor = violet];
21 -> 92 [arrowhead = none ];
92 -> 24;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 93 [arrowhead = none ];
93 -> 25;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 94 [arrowhead = none ];
94 -> 33;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 95 [arrowhead = none ];
95 -> 26;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 96 [arrowhead = none ];
96 -> 27;

97 [label="EVAL with clause\nrev1(X59, [], X59).\nand substitution\nT36 -> T50\nX59 -> T50\nT37 -> []\nT39 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="EVAL with clause\nrev1(X69, .(X70, X71), X72) :- rev1(X70, X71, X72).\nand substitution\nT36 -> T59\nX69 -> T59\nX70 -> T60\nX71 -> T61\nT37 -> .(T60, T61)\nT39 -> T63\nX72 -> T63\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 99 [arrowhead = none ];
99 -> 31;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 100 [arrowhead = none ];
100 -> 32;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 101 [arrowhead = none ];
101 -> 30;

102 [label="INSTANCE with matching:\nT36 -> T60\nT37 -> T61\nT39 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 102 [arrowhead = none , style=dashed];
102 -> 23[style=dashed];

103 [label="BACKTRACK\nfor clause: rev2(X, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
33 -> 103 [arrowhead = none ];
103 -> 34;

104 [label="ONLY EVAL with clause\nrev2(X100, .(X101, X102), X103) :- ','(rev2(X101, X102, X104), ','(rev(X104, X105), rev(.(X100, X105), X103))).\nand substitution\nT35 -> T84\nX100 -> T84\nT36 -> T85\nX101 -> T85\nT37 -> T86\nX102 -> T86\nT43 -> T88\nX103 -> T88\nT87 -> T88", fontsize=14, style = filled, fillcolor = white];
34 -> 104 [arrowhead = none ];
104 -> 35;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 105 [arrowhead = none ];
105 -> 36;

106 [label="SPLIT 2\nnew knowledge:\nT91 is ground\nreplacements:\nX104 -> T91", fontsize=14, style = filled, fillcolor = violet];
35 -> 106 [arrowhead = none ];
106 -> 37;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 107 [arrowhead = none ];
107 -> 38;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 108 [arrowhead = none ];
108 -> 68;

109 [label="SPLIT 2\nnew knowledge:\nT145 is ground\nreplacements:\nX105 -> T145", fontsize=14, style = filled, fillcolor = violet];
37 -> 109 [arrowhead = none ];
109 -> 69;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 110 [arrowhead = none ];
110 -> 39;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 111 [arrowhead = none ];
111 -> 40;

112 [label="EVAL with clause\nrev2(X114, [], []).\nand substitution\nT85 -> T98\nX114 -> T98\nT86 -> []\nX104 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 112 [arrowhead = none ];
112 -> 41;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 113 [arrowhead = none ];
113 -> 42;

114 [label="EVAL with clause\nrev2(X128, .(X129, X130), X131) :- ','(rev2(X129, X130, X132), ','(rev(X132, X133), rev(.(X128, X133), X131))).\nand substitution\nT85 -> T105\nX128 -> T105\nX129 -> T106\nX130 -> T107\nT86 -> .(T106, T107)\nX104 -> X134\nX131 -> X134", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 115 [arrowhead = none ];
115 -> 45;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 116 [arrowhead = none ];
116 -> 43;

117 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 117 [arrowhead = none ];
117 -> 46;

118 [label="SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:\nX132 -> T110", fontsize=14, style = filled, fillcolor = violet];
44 -> 118 [arrowhead = none ];
118 -> 47;

119 [label="INSTANCE with matching:\nT85 -> T106\nT86 -> T107\nX104 -> X132", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 119 [arrowhead = none , style=dashed];
119 -> 36[style=dashed];

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nreplacements:\nX133 -> T113", fontsize=14, style = filled, fillcolor = violet];
47 -> 121 [arrowhead = none ];
121 -> 49;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 122 [arrowhead = none ];
122 -> 50;

123 [label="INSTANCE with matching:\nT1 -> .(T105, T113)\nT2 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 123 [arrowhead = none , style=dashed];
123 -> 1[style=dashed];

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 124 [arrowhead = none ];
124 -> 51;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="EVAL with clause\nrev([], []).\nand substitution\nT110 -> []\nX133 -> []", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 126 [arrowhead = none ];
126 -> 53;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 127 [arrowhead = none ];
127 -> 54;

128 [label="EVAL with clause\nrev(.(X155, X156), .(X157, X158)) :- ','(rev1(X155, X156, X157), rev2(X155, X156, X158)).\nand substitution\nX155 -> T118\nX156 -> T119\nT110 -> .(T118, T119)\nX157 -> X159\nX158 -> X160\nX133 -> .(X159, X160)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 128 [arrowhead = none ];
128 -> 56;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 129 [arrowhead = none ];
129 -> 57;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 131 [arrowhead = none ];
131 -> 58;

132 [label="SPLIT 2\nnew knowledge:\nT122 is ground\nreplacements:\nX159 -> T122", fontsize=14, style = filled, fillcolor = violet];
56 -> 132 [arrowhead = none ];
132 -> 59;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 133 [arrowhead = none ];
133 -> 60;

134 [label="INSTANCE with matching:\nT85 -> T118\nT86 -> T119\nX104 -> X160", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 134 [arrowhead = none , style=dashed];
134 -> 36[style=dashed];

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 135 [arrowhead = none ];
135 -> 61;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 136 [arrowhead = none ];
136 -> 62;

137 [label="EVAL with clause\nrev1(X173, [], X173).\nand substitution\nT118 -> T129\nX173 -> T129\nT119 -> []\nX159 -> T129", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 137 [arrowhead = none ];
137 -> 63;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 138 [arrowhead = none ];
138 -> 64;

139 [label="EVAL with clause\nrev1(X185, .(X186, X187), X188) :- rev1(X186, X187, X188).\nand substitution\nT118 -> T136\nX185 -> T136\nX186 -> T137\nX187 -> T138\nT119 -> .(T137, T138)\nX159 -> X189\nX188 -> X189", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 139 [arrowhead = none ];
139 -> 66;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 67;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 141 [arrowhead = none ];
141 -> 65;

142 [label="INSTANCE with matching:\nT118 -> T137\nT119 -> T138\nX159 -> X189", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 142 [arrowhead = none , style=dashed];
142 -> 58[style=dashed];

143 [label="INSTANCE with matching:\nT110 -> T91\nX133 -> X105", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 143 [arrowhead = none , style=dashed];
143 -> 48[style=dashed];

144 [label="INSTANCE with matching:\nT1 -> .(T84, T145)\nT2 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

rev123(T50, [], T50).
rev123(T59, .(T60, T61), T63) :- rev123(T60, T61, T63).
rev236(T98, [], []).
rev236(T105, .(T106, T107), X134) :- rev236(T106, T107, X132).
rev236(T105, .(T106, T107), X134) :- ','(rev236(T106, T107, T110), rev48(T110, X133)).
rev236(T105, .(T106, T107), X134) :- ','(rev236(T106, T107, T110), ','(rev48(T110, T113), rev11(.(T105, T113), X134))).
rev158(T129, [], T129).
rev158(T136, .(T137, T138), X189) :- rev158(T137, T138, X189).
rev48([], []).
rev48(.(T118, T119), .(X159, X160)) :- rev158(T118, T119, X159).
rev48(.(T118, T119), .(T122, X160)) :- ','(rev158(T118, T119, T122), rev236(T118, T119, X160)).
rev11([], []).
rev11(.(T25, []), .(T25, [])).
rev11(.(T35, .(T36, T37)), .(T39, T40)) :- rev123(T36, T37, T39).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev123(T85, T86, T39), rev236(T85, T86, X104)).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev123(T85, T86, T39), ','(rev236(T85, T86, T91), rev48(T91, X105))).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev123(T85, T86, T39), ','(rev236(T85, T86, T91), ','(rev48(T91, T145), rev11(.(T84, T145), T88)))).

Queries:

rev11(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev11_in: (b,f) (f,f)
rev123_in: (f,b,f) (f,f,f)
rev236_in: (f,b,f)
rev48_in: (f,f)
rev158_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(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:

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AGA(T36, T37, T39)
REV123_IN_AGA(T59, .(T60, T61), T63) → U1_AGA(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
REV236_IN_AGA(T105, .(T106, T107), X134) → U2_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_AGA(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(X159, X160)) → U8_AA(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
REV48_IN_AA(.(T118, T119), .(X159, X160)) → REV158_IN_AAA(T118, T119, X159)
REV158_IN_AAA(T136, .(T137, T138), X189) → U7_AAA(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_AA(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_AGA(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → U11_AA(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AAA(T36, T37, T39)
REV123_IN_AAA(T59, .(T60, T61), T63) → U1_AAA(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_AA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

The argument filtering Pi contains the following mapping:
rev11_in_ga(x1, x2) = rev11_in_ga(x1)
[] = []
rev11_out_ga(x1, x2) = rev11_out_ga
.(x1, x2) = .(x2)
U11_ga(x1, x2, x3, x4, x5, x6) = U11_ga(x6)
U12_ga(x1, x2, x3, x4, x5, x6) = U12_ga(x3, x6)
U13_ga(x1, x2, x3, x4, x5, x6) = U13_ga(x6)
rev48_in_aa(x1, x2) = rev48_in_aa
rev48_out_aa(x1, x2) = rev48_out_aa(x1)
U8_aa(x1, x2, x3, x4, x5) = U8_aa(x5)
rev158_in_aaa(x1, x2, x3) = rev158_in_aaa
rev158_out_aaa(x1, x2, x3) = rev158_out_aaa(x2)
U7_aaa(x1, x2, x3, x4, x5) = U7_aaa(x5)
rev123_in_aga(x1, x2, x3) = rev123_in_aga(x2)
rev123_out_aga(x1, x2, x3) = rev123_out_aga
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5)
rev236_in_aga(x1, x2, x3) = rev236_in_aga(x2)
rev236_out_aga(x1, x2, x3) = rev236_out_aga
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5)
U4_aga(x1, x2, x3, x4, x5) = U4_aga(x5)
U9_aa(x1, x2, x3, x4, x5) = U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5) = U10_aa(x2, x5)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x5)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
rev11_in_aa(x1, x2) = rev11_in_aa
rev11_out_aa(x1, x2) = rev11_out_aa(x1)
U11_aa(x1, x2, x3, x4, x5, x6) = U11_aa(x6)
rev123_in_aaa(x1, x2, x3) = rev123_in_aaa
rev123_out_aaa(x1, x2, x3) = rev123_out_aaa(x2)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U12_aa(x1, x2, x3, x4, x5, x6) = U12_aa(x6)
U13_aa(x1, x2, x3, x4, x5, x6) = U13_aa(x3, x6)
U14_aa(x1, x2, x3, x4, x5, x6) = U14_aa(x3, x6)
U15_aa(x1, x2, x3, x4, x5, x6) = U15_aa(x3, x6)
U16_aa(x1, x2, x3, x4, x5, x6) = U16_aa(x3, x6)
U17_aa(x1, x2, x3, x4, x5, x6) = U17_aa(x3, x6)
U14_ga(x1, x2, x3, x4, x5, x6) = U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6) = U15_ga(x6)
U16_ga(x1, x2, x3, x4, x5, x6) = U16_ga(x6)
U17_ga(x1, x2, x3, x4, x5, x6) = U17_ga(x6)
REV11_IN_GA(x1, x2) = REV11_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6) = U11_GA(x6)
REV123_IN_AGA(x1, x2, x3) = REV123_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x5)
U12_GA(x1, x2, x3, x4, x5, x6) = U12_GA(x3, x6)
U13_GA(x1, x2, x3, x4, x5, x6) = U13_GA(x6)
REV236_IN_AGA(x1, x2, x3) = REV236_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x5)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5)
U4_AGA(x1, x2, x3, x4, x5) = U4_AGA(x5)
REV48_IN_AA(x1, x2) = REV48_IN_AA
U8_AA(x1, x2, x3, x4, x5) = U8_AA(x5)
REV158_IN_AAA(x1, x2, x3) = REV158_IN_AAA
U7_AAA(x1, x2, x3, x4, x5) = U7_AAA(x5)
U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5) = U10_AA(x2, x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5)
REV11_IN_AA(x1, x2) = REV11_IN_AA
U11_AA(x1, x2, x3, x4, x5, x6) = U11_AA(x6)
REV123_IN_AAA(x1, x2, x3) = REV123_IN_AAA
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U12_AA(x1, x2, x3, x4, x5, x6) = U12_AA(x6)
U13_AA(x1, x2, x3, x4, x5, x6) = U13_AA(x3, x6)
U14_AA(x1, x2, x3, x4, x5, x6) = U14_AA(x3, x6)
U15_AA(x1, x2, x3, x4, x5, x6) = U15_AA(x3, x6)
U16_AA(x1, x2, x3, x4, x5, x6) = U16_AA(x3, x6)
U17_AA(x1, x2, x3, x4, x5, x6) = U17_AA(x3, x6)
U14_GA(x1, x2, x3, x4, x5, x6) = U14_GA(x6)
U15_GA(x1, x2, x3, x4, x5, x6) = U15_GA(x6)
U16_GA(x1, x2, x3, x4, x5, x6) = U16_GA(x6)
U17_GA(x1, x2, x3, x4, x5, x6) = U17_GA(x6)

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

(6) Obligation:

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

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AGA(T36, T37, T39)
REV123_IN_AGA(T59, .(T60, T61), T63) → U1_AGA(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
REV236_IN_AGA(T105, .(T106, T107), X134) → U2_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_AGA(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(X159, X160)) → U8_AA(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
REV48_IN_AA(.(T118, T119), .(X159, X160)) → REV158_IN_AAA(T118, T119, X159)
REV158_IN_AAA(T136, .(T137, T138), X189) → U7_AAA(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_AA(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_AGA(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → U11_AA(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AAA(T36, T37, T39)
REV123_IN_AAA(T59, .(T60, T61), T63) → U1_AAA(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_AA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 25 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(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:

REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV123_IN_AAA(x1, x2, x3) = REV123_IN_AAA

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:

REV123_IN_AAA → REV123_IN_AAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = REV123_IN_AAA evaluates to t =REV123_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV123_IN_AAA to REV123_IN_AAA.

(15) NO

(16) Obligation:

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

REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(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:

REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV158_IN_AAA(x1, x2, x3) = REV158_IN_AAA

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:

REV158_IN_AAA → REV158_IN_AAA

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = REV158_IN_AAA evaluates to t =REV158_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV158_IN_AAA to REV158_IN_AAA.

(22) NO

(23) Obligation:

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

REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(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:

REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
rev48_in_aa(x1, x2) = rev48_in_aa
rev48_out_aa(x1, x2) = rev48_out_aa(x1)
U8_aa(x1, x2, x3, x4, x5) = U8_aa(x5)
rev158_in_aaa(x1, x2, x3) = rev158_in_aaa
rev158_out_aaa(x1, x2, x3) = rev158_out_aaa(x2)
U7_aaa(x1, x2, x3, x4, x5) = U7_aaa(x5)
rev236_in_aga(x1, x2, x3) = rev236_in_aga(x2)
rev236_out_aga(x1, x2, x3) = rev236_out_aga
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5)
U4_aga(x1, x2, x3, x4, x5) = U4_aga(x5)
U9_aa(x1, x2, x3, x4, x5) = U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5) = U10_aa(x2, x5)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x5)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
rev11_in_aa(x1, x2) = rev11_in_aa
rev11_out_aa(x1, x2) = rev11_out_aa(x1)
U11_aa(x1, x2, x3, x4, x5, x6) = U11_aa(x6)
rev123_in_aaa(x1, x2, x3) = rev123_in_aaa
rev123_out_aaa(x1, x2, x3) = rev123_out_aaa(x2)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U12_aa(x1, x2, x3, x4, x5, x6) = U12_aa(x6)
U13_aa(x1, x2, x3, x4, x5, x6) = U13_aa(x3, x6)
U14_aa(x1, x2, x3, x4, x5, x6) = U14_aa(x3, x6)
U15_aa(x1, x2, x3, x4, x5, x6) = U15_aa(x3, x6)
U16_aa(x1, x2, x3, x4, x5, x6) = U16_aa(x3, x6)
U17_aa(x1, x2, x3, x4, x5, x6) = U17_aa(x3, x6)
REV236_IN_AGA(x1, x2, x3) = REV236_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5)
REV48_IN_AA(x1, x2) = REV48_IN_AA
U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x5)
REV11_IN_AA(x1, x2) = REV11_IN_AA
U12_AA(x1, x2, x3, x4, x5, x6) = U12_AA(x6)
U14_AA(x1, x2, x3, x4, x5, x6) = U14_AA(x3, x6)
U16_AA(x1, x2, x3, x4, x5, x6) = U16_AA(x3, x6)

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:

REV236_IN_AGA(.(T107)) → U3_AGA(rev236_in_aga(T107))
U3_AGA(rev236_out_aga) → REV48_IN_AA
REV48_IN_AA → U9_AA(rev158_in_aaa)
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(rev236_out_aga) → U5_AGA(rev48_in_aa)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV236_IN_AGA(.(T107)) → U3_AGA(rev236_in_aga(T107)) at position [0] we obtained the following new rules [LPAR04]:

REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))

(29) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
REV48_IN_AA → U9_AA(rev158_in_aaa)
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(rev236_out_aga) → U5_AGA(rev48_in_aa)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV48_IN_AA → U9_AA(rev158_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))

(31) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(rev236_out_aga) → U5_AGA(rev48_in_aa)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(32) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_AGA(rev236_out_aga) → U5_AGA(rev48_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))

(33) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(34) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV11_IN_AA → U12_AA(rev123_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))

(35) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(36) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86)) at position [1] we obtained the following new rules [LPAR04]:

U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))

(37) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(38) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U14_AA(T86, rev236_out_aga) → U16_AA(T86, rev48_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U14_AA(y0, rev236_out_aga) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa))

(39) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U14_AA(T86, rev236_out_aga) → REV48_IN_AA
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(40) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(T86, rev236_out_aga) → REV48_IN_AA we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA

(41) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(42) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga) → U16_AA(y0, rev48_out_aa([])) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))

(43) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(44) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga) → U16_AA(y0, U8_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))

(45) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga) → U16_AA(y0, U9_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))

(47) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(48) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA we obtained the following new rules [LPAR04]:

U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

(49) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(50) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119) we obtained the following new rules [LPAR04]:

U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

(51) Obligation:

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

U3_AGA(rev236_out_aga) → REV48_IN_AA
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(53) Obligation:

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

REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
U3_AGA(rev236_out_aga) → REV48_IN_AA
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(54) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107) we obtained the following new rules [LPAR04]:

REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

(55) Obligation:

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

REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
U3_AGA(rev236_out_aga) → REV48_IN_AA
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(56) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86) we obtained the following new rules [LPAR04]:

U12_AA(rev123_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U12_AA(rev123_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.(.([])))) → REV236_IN_AGA(.(.([])))
U12_AA(rev123_out_aaa(.(.(.(y_0))))) → REV236_IN_AGA(.(.(.(y_0))))

(57) Obligation:

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

REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.([])) → U3_AGA(rev236_out_aga)
U3_AGA(rev236_out_aga) → REV48_IN_AA
U3_AGA(rev236_out_aga) → U5_AGA(rev48_out_aa([]))
U5_AGA(rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U2_aga(rev236_in_aga(x0)))
U3_AGA(rev236_out_aga) → U5_AGA(U8_aa(rev158_in_aaa))
U3_AGA(rev236_out_aga) → U5_AGA(U9_aa(rev158_in_aaa))
REV236_IN_AGA(.(.(x0))) → U3_AGA(U3_aga(rev236_in_aga(x0)))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
U14_AA([], rev236_out_aga) → REV48_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(rev236_in_aga(x0)))
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga) → U16_AA([], U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U12_AA(rev123_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.(.([])))) → REV236_IN_AGA(.(.([])))
U12_AA(rev123_out_aaa(.(.(.(y_0))))) → REV236_IN_AGA(.(.(.(y_0))))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga
rev236_in_aga(.(T107)) → U2_aga(rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(rev236_out_aga) → rev236_out_aga
U3_aga(rev236_out_aga) → U4_aga(rev48_in_aa)
U3_aga(rev236_out_aga) → U5_aga(rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(rev48_out_aa(T110)) → rev236_out_aga
U5_aga(rev48_out_aa(T110)) → U6_aga(rev11_in_aa)
U10_aa(T119, rev236_out_aga) → rev48_out_aa(.(T119))
U6_aga(rev11_out_aa(.(T113))) → rev236_out_aga
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0)
U3_aga(x0)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0)
U5_aga(x0)
U10_aa(x0, x1)
U6_aga(x0)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(58) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U12_AA(rev123_out_aaa([])) evaluates to t =U12_AA(rev123_out_aaa([]))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga)
with rule U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga) at position [] and matcher [ ]

U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([]))
with rule U14_AA([], rev236_out_aga) → U16_AA([], rev48_out_aa([])) at position [] and matcher [ ]

U16_AA([], rev48_out_aa([])) → REV11_IN_AA
with rule U16_AA([], rev48_out_aa([])) → REV11_IN_AA at position [] and matcher [ ]

REV11_IN_AA → U12_AA(rev123_out_aaa([]))
with rule REV11_IN_AA → U12_AA(rev123_out_aaa([]))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(59) NO

(60) Obligation:

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

REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(61) UsableRulesProof (EQUIVALENT transformation)

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

(62) Obligation:

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

REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV123_IN_AGA(x1, x2, x3) = REV123_IN_AGA(x2)

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

(63) PiDPToQDPProof (SOUND transformation)

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

(64) Obligation:

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

REV123_IN_AGA(.(T61)) → REV123_IN_AGA(T61)

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

(65) 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:

REV123_IN_AGA(.(T61)) → REV123_IN_AGA(T61)
The graph contains the following edges 1 > 1

(66) YES

(67) PrologToPiTRSProof (SOUND transformation)

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(68) Obligation:

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

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(69) 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:

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AGA(T36, T37, T39)
REV123_IN_AGA(T59, .(T60, T61), T63) → U1_AGA(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
REV236_IN_AGA(T105, .(T106, T107), X134) → U2_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_AGA(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(X159, X160)) → U8_AA(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
REV48_IN_AA(.(T118, T119), .(X159, X160)) → REV158_IN_AAA(T118, T119, X159)
REV158_IN_AAA(T136, .(T137, T138), X189) → U7_AAA(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_AA(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_AGA(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → U11_AA(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AAA(T36, T37, T39)
REV123_IN_AAA(T59, .(T60, T61), T63) → U1_AAA(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_AA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

The argument filtering Pi contains the following mapping:
rev11_in_ga(x1, x2) = rev11_in_ga(x1)
[] = []
rev11_out_ga(x1, x2) = rev11_out_ga(x1)
.(x1, x2) = .(x2)
U11_ga(x1, x2, x3, x4, x5, x6) = U11_ga(x3, x6)
U12_ga(x1, x2, x3, x4, x5, x6) = U12_ga(x3, x6)
U13_ga(x1, x2, x3, x4, x5, x6) = U13_ga(x3, x6)
rev48_in_aa(x1, x2) = rev48_in_aa
rev48_out_aa(x1, x2) = rev48_out_aa(x1)
U8_aa(x1, x2, x3, x4, x5) = U8_aa(x5)
rev158_in_aaa(x1, x2, x3) = rev158_in_aaa
rev158_out_aaa(x1, x2, x3) = rev158_out_aaa(x2)
U7_aaa(x1, x2, x3, x4, x5) = U7_aaa(x5)
rev123_in_aga(x1, x2, x3) = rev123_in_aga(x2)
rev123_out_aga(x1, x2, x3) = rev123_out_aga(x2)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5)
rev236_in_aga(x1, x2, x3) = rev236_in_aga(x2)
rev236_out_aga(x1, x2, x3) = rev236_out_aga(x2)
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5)
U4_aga(x1, x2, x3, x4, x5) = U4_aga(x3, x5)
U9_aa(x1, x2, x3, x4, x5) = U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5) = U10_aa(x2, x5)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x3, x5)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x3, x5)
rev11_in_aa(x1, x2) = rev11_in_aa
rev11_out_aa(x1, x2) = rev11_out_aa(x1)
U11_aa(x1, x2, x3, x4, x5, x6) = U11_aa(x6)
rev123_in_aaa(x1, x2, x3) = rev123_in_aaa
rev123_out_aaa(x1, x2, x3) = rev123_out_aaa(x2)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U12_aa(x1, x2, x3, x4, x5, x6) = U12_aa(x6)
U13_aa(x1, x2, x3, x4, x5, x6) = U13_aa(x3, x6)
U14_aa(x1, x2, x3, x4, x5, x6) = U14_aa(x3, x6)
U15_aa(x1, x2, x3, x4, x5, x6) = U15_aa(x3, x6)
U16_aa(x1, x2, x3, x4, x5, x6) = U16_aa(x3, x6)
U17_aa(x1, x2, x3, x4, x5, x6) = U17_aa(x3, x6)
U14_ga(x1, x2, x3, x4, x5, x6) = U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6) = U15_ga(x3, x6)
U16_ga(x1, x2, x3, x4, x5, x6) = U16_ga(x3, x6)
U17_ga(x1, x2, x3, x4, x5, x6) = U17_ga(x3, x6)
REV11_IN_GA(x1, x2) = REV11_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6) = U11_GA(x3, x6)
REV123_IN_AGA(x1, x2, x3) = REV123_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6) = U12_GA(x3, x6)
U13_GA(x1, x2, x3, x4, x5, x6) = U13_GA(x3, x6)
REV236_IN_AGA(x1, x2, x3) = REV236_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x3, x5)
U4_AGA(x1, x2, x3, x4, x5) = U4_AGA(x3, x5)
REV48_IN_AA(x1, x2) = REV48_IN_AA
U8_AA(x1, x2, x3, x4, x5) = U8_AA(x5)
REV158_IN_AAA(x1, x2, x3) = REV158_IN_AAA
U7_AAA(x1, x2, x3, x4, x5) = U7_AAA(x5)
U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5) = U10_AA(x2, x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x3, x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x3, x5)
REV11_IN_AA(x1, x2) = REV11_IN_AA
U11_AA(x1, x2, x3, x4, x5, x6) = U11_AA(x6)
REV123_IN_AAA(x1, x2, x3) = REV123_IN_AAA
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U12_AA(x1, x2, x3, x4, x5, x6) = U12_AA(x6)
U13_AA(x1, x2, x3, x4, x5, x6) = U13_AA(x3, x6)
U14_AA(x1, x2, x3, x4, x5, x6) = U14_AA(x3, x6)
U15_AA(x1, x2, x3, x4, x5, x6) = U15_AA(x3, x6)
U16_AA(x1, x2, x3, x4, x5, x6) = U16_AA(x3, x6)
U17_AA(x1, x2, x3, x4, x5, x6) = U17_AA(x3, x6)
U14_GA(x1, x2, x3, x4, x5, x6) = U14_GA(x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6) = U15_GA(x3, x6)
U16_GA(x1, x2, x3, x4, x5, x6) = U16_GA(x3, x6)
U17_GA(x1, x2, x3, x4, x5, x6) = U17_GA(x3, x6)

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

(70) Obligation:

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

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AGA(T36, T37, T39)
REV123_IN_AGA(T59, .(T60, T61), T63) → U1_AGA(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
REV236_IN_AGA(T105, .(T106, T107), X134) → U2_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_AGA(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(X159, X160)) → U8_AA(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
REV48_IN_AA(.(T118, T119), .(X159, X160)) → REV158_IN_AAA(T118, T119, X159)
REV158_IN_AAA(T136, .(T137, T138), X189) → U7_AAA(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_AA(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_AGA(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → U11_AA(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
REV11_IN_AA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_AAA(T36, T37, T39)
REV123_IN_AAA(T59, .(T60, T61), T63) → U1_AAA(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_AA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)
U12_GA(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(71) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 25 less nodes.

(72) Complex Obligation (AND)

(73) Obligation:

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

REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(74) UsableRulesProof (EQUIVALENT transformation)

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

(75) Obligation:

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

REV123_IN_AAA(T59, .(T60, T61), T63) → REV123_IN_AAA(T60, T61, T63)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV123_IN_AAA(x1, x2, x3) = REV123_IN_AAA

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

(76) PiDPToQDPProof (SOUND transformation)

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

(77) Obligation:

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

REV123_IN_AAA → REV123_IN_AAA

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

(78) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV123_IN_AAA to REV123_IN_AAA.

(79) NO

(80) Obligation:

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

REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(81) UsableRulesProof (EQUIVALENT transformation)

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

(82) Obligation:

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

REV158_IN_AAA(T136, .(T137, T138), X189) → REV158_IN_AAA(T137, T138, X189)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV158_IN_AAA(x1, x2, x3) = REV158_IN_AAA

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

(83) PiDPToQDPProof (SOUND transformation)

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

(84) Obligation:

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

REV158_IN_AAA → REV158_IN_AAA

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

(85) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV158_IN_AAA to REV158_IN_AAA.

(86) NO

(87) Obligation:

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

REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(88) UsableRulesProof (EQUIVALENT transformation)

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

(89) Obligation:

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

REV236_IN_AGA(T105, .(T106, T107), X134) → U3_AGA(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → REV48_IN_AA(T110, X133)
REV48_IN_AA(.(T118, T119), .(T122, X160)) → U9_AA(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_AA(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → REV236_IN_AGA(T118, T119, X160)
REV236_IN_AGA(T105, .(T106, T107), X134) → REV236_IN_AGA(T106, T107, X132)
U3_AGA(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_AGA(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_AGA(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → REV11_IN_AA(.(T105, T113), X134)
REV11_IN_AA(.(T84, .(T85, T86)), .(T39, T88)) → U12_AA(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → REV236_IN_AGA(T85, T86, X104)
U12_AA(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_AA(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → REV48_IN_AA(T91, X105)
U14_AA(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_AA(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_AA(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → REV11_IN_AA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
rev48_in_aa(x1, x2) = rev48_in_aa
rev48_out_aa(x1, x2) = rev48_out_aa(x1)
U8_aa(x1, x2, x3, x4, x5) = U8_aa(x5)
rev158_in_aaa(x1, x2, x3) = rev158_in_aaa
rev158_out_aaa(x1, x2, x3) = rev158_out_aaa(x2)
U7_aaa(x1, x2, x3, x4, x5) = U7_aaa(x5)
rev236_in_aga(x1, x2, x3) = rev236_in_aga(x2)
rev236_out_aga(x1, x2, x3) = rev236_out_aga(x2)
U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5)
U4_aga(x1, x2, x3, x4, x5) = U4_aga(x3, x5)
U9_aa(x1, x2, x3, x4, x5) = U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5) = U10_aa(x2, x5)
U5_aga(x1, x2, x3, x4, x5) = U5_aga(x3, x5)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x3, x5)
rev11_in_aa(x1, x2) = rev11_in_aa
rev11_out_aa(x1, x2) = rev11_out_aa(x1)
U11_aa(x1, x2, x3, x4, x5, x6) = U11_aa(x6)
rev123_in_aaa(x1, x2, x3) = rev123_in_aaa
rev123_out_aaa(x1, x2, x3) = rev123_out_aaa(x2)
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
U12_aa(x1, x2, x3, x4, x5, x6) = U12_aa(x6)
U13_aa(x1, x2, x3, x4, x5, x6) = U13_aa(x3, x6)
U14_aa(x1, x2, x3, x4, x5, x6) = U14_aa(x3, x6)
U15_aa(x1, x2, x3, x4, x5, x6) = U15_aa(x3, x6)
U16_aa(x1, x2, x3, x4, x5, x6) = U16_aa(x3, x6)
U17_aa(x1, x2, x3, x4, x5, x6) = U17_aa(x3, x6)
REV236_IN_AGA(x1, x2, x3) = REV236_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x3, x5)
REV48_IN_AA(x1, x2) = REV48_IN_AA
U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5)
U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x3, x5)
REV11_IN_AA(x1, x2) = REV11_IN_AA
U12_AA(x1, x2, x3, x4, x5, x6) = U12_AA(x6)
U14_AA(x1, x2, x3, x4, x5, x6) = U14_AA(x3, x6)
U16_AA(x1, x2, x3, x4, x5, x6) = U16_AA(x3, x6)

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

(90) PiDPToQDPProof (SOUND transformation)

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

(91) Obligation:

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

REV236_IN_AGA(.(T107)) → U3_AGA(T107, rev236_in_aga(T107))
U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
REV48_IN_AA → U9_AA(rev158_in_aaa)
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(T107, rev236_out_aga(T107)) → U5_AGA(T107, rev48_in_aa)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(92) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV236_IN_AGA(.(T107)) → U3_AGA(T107, rev236_in_aga(T107)) at position [1] we obtained the following new rules [LPAR04]:

REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))

(93) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
REV48_IN_AA → U9_AA(rev158_in_aaa)
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(T107, rev236_out_aga(T107)) → U5_AGA(T107, rev48_in_aa)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV48_IN_AA → U9_AA(rev158_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))

(95) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U3_AGA(T107, rev236_out_aga(T107)) → U5_AGA(T107, rev48_in_aa)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(96) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_AGA(T107, rev236_out_aga(T107)) → U5_AGA(T107, rev48_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))

(97) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_in_aaa)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(98) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV11_IN_AA → U12_AA(rev123_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))

(99) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86))
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(100) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U12_AA(rev123_out_aaa(T86)) → U14_AA(T86, rev236_in_aga(T86)) at position [1] we obtained the following new rules [LPAR04]:

U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))

(101) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(102) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U14_AA(T86, rev236_out_aga(T86)) → U16_AA(T86, rev48_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))

(103) Obligation:

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

U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA
U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(104) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGA(T107, rev236_out_aga(T107)) → REV48_IN_AA we obtained the following new rules [LPAR04]:

U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA

(105) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(106) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(T86, rev236_out_aga(T86)) → REV48_IN_AA we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA

(107) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([]))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(108) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, rev48_out_aa([])) we obtained the following new rules [LPAR04]:

U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))

(109) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(110) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U8_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))

(111) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(112) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGA(y0, rev236_out_aga(y0)) → U5_AGA(y0, U9_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))

(113) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(114) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_AGA(T107, rev48_out_aa(T110)) → REV11_IN_AA we obtained the following new rules [LPAR04]:

U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

(115) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([]))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(116) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, rev48_out_aa([])) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))

(117) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(118) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U8_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))

(119) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(120) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_AA(y0, rev236_out_aga(y0)) → U16_AA(y0, U9_aa(rev158_in_aaa)) we obtained the following new rules [LPAR04]:

U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))

(121) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(122) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_AA(T86, rev48_out_aa(T91)) → REV11_IN_AA we obtained the following new rules [LPAR04]:

U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

(123) Obligation:

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

U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119)
REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(124) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_AA(rev158_out_aaa(T119)) → REV236_IN_AGA(T119) we obtained the following new rules [LPAR04]:

U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

(125) Obligation:

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

REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
REV48_IN_AA → U9_AA(rev158_out_aaa([]))
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(126) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(127) Obligation:

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

REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107)
REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(128) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REV236_IN_AGA(.(T107)) → REV236_IN_AGA(T107) we obtained the following new rules [LPAR04]:

REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

(129) Obligation:

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

REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86)
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(130) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U12_AA(rev123_out_aaa(T86)) → REV236_IN_AGA(T86) we obtained the following new rules [LPAR04]:

U12_AA(rev123_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U12_AA(rev123_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.(.([])))) → REV236_IN_AGA(.(.([])))
U12_AA(rev123_out_aaa(.(.(.(y_0))))) → REV236_IN_AGA(.(.(.(y_0))))

(131) Obligation:

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

REV236_IN_AGA(.([])) → U3_AGA([], rev236_out_aga([]))
U3_AGA([], rev236_out_aga([])) → REV48_IN_AA
REV48_IN_AA → U9_AA(U7_aaa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), rev48_out_aa([]))
U5_AGA(.(z0), rev48_out_aa([])) → REV11_IN_AA
REV11_IN_AA → U12_AA(rev123_out_aaa([]))
REV236_IN_AGA(.(.(x0))) → U3_AGA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U8_aa(rev158_in_aaa))
U5_AGA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
REV11_IN_AA → U12_AA(U1_aaa(rev123_in_aaa))
U12_AA(rev123_out_aaa([])) → U14_AA([], rev236_out_aga([]))
U14_AA([], rev236_out_aga([])) → REV48_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], rev48_out_aa([]))
U16_AA([], rev48_out_aa([])) → REV11_IN_AA
U16_AA([], rev48_out_aa(x1)) → REV11_IN_AA
U14_AA([], rev236_out_aga([])) → U16_AA([], U8_aa(rev158_in_aaa))
U14_AA([], rev236_out_aga([])) → U16_AA([], U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U2_aga(x0, rev236_in_aga(x0)))
U14_AA(.(z0), rev236_out_aga(.(z0))) → REV48_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), rev48_out_aa([]))
U16_AA(.(z0), rev48_out_aa([])) → REV11_IN_AA
U16_AA(.(z0), rev48_out_aa(x1)) → REV11_IN_AA
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U8_aa(rev158_in_aaa))
U14_AA(.(z0), rev236_out_aga(.(z0))) → U16_AA(.(z0), U9_aa(rev158_in_aaa))
U12_AA(rev123_out_aaa(.(x0))) → U14_AA(.(x0), U3_aga(x0, rev236_in_aga(x0)))
U3_AGA(.(z0), rev236_out_aga(.(z0))) → U5_AGA(.(z0), U9_aa(rev158_in_aaa))
U9_AA(rev158_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U9_AA(rev158_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], rev48_out_aa([]))
U5_AGA([], rev48_out_aa([])) → REV11_IN_AA
U5_AGA([], rev48_out_aa(x1)) → REV11_IN_AA
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U8_aa(rev158_in_aaa))
U3_AGA([], rev236_out_aga([])) → U5_AGA([], U9_aa(rev158_in_aaa))
REV236_IN_AGA(.(.(y_0))) → REV236_IN_AGA(.(y_0))
REV236_IN_AGA(.(.([]))) → REV236_IN_AGA(.([]))
REV236_IN_AGA(.(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.([]))) → REV236_IN_AGA(.([]))
U12_AA(rev123_out_aaa(.(.(y_0)))) → REV236_IN_AGA(.(.(y_0)))
U12_AA(rev123_out_aaa(.(.([])))) → REV236_IN_AGA(.(.([])))
U12_AA(rev123_out_aaa(.(.(.(y_0))))) → REV236_IN_AGA(.(.(.(y_0))))

The TRS R consists of the following rules:

rev236_in_aga([]) → rev236_out_aga([])
rev236_in_aga(.(T107)) → U2_aga(T107, rev236_in_aga(T107))
rev236_in_aga(.(T107)) → U3_aga(T107, rev236_in_aga(T107))
rev158_in_aaa → rev158_out_aaa([])
rev158_in_aaa → U7_aaa(rev158_in_aaa)
rev48_in_aa → rev48_out_aa([])
rev48_in_aa → U8_aa(rev158_in_aaa)
rev48_in_aa → U9_aa(rev158_in_aaa)
rev123_in_aaa → rev123_out_aaa([])
rev123_in_aaa → U1_aaa(rev123_in_aaa)
U2_aga(T107, rev236_out_aga(T107)) → rev236_out_aga(.(T107))
U3_aga(T107, rev236_out_aga(T107)) → U4_aga(T107, rev48_in_aa)
U3_aga(T107, rev236_out_aga(T107)) → U5_aga(T107, rev48_in_aa)
U7_aaa(rev158_out_aaa(T138)) → rev158_out_aaa(.(T138))
U8_aa(rev158_out_aaa(T119)) → rev48_out_aa(.(T119))
U9_aa(rev158_out_aaa(T119)) → U10_aa(T119, rev236_in_aga(T119))
U1_aaa(rev123_out_aaa(T61)) → rev123_out_aaa(.(T61))
U4_aga(T107, rev48_out_aa(T110)) → rev236_out_aga(.(T107))
U5_aga(T107, rev48_out_aa(T110)) → U6_aga(T107, rev11_in_aa)
U10_aa(T119, rev236_out_aga(T119)) → rev48_out_aa(.(T119))
U6_aga(T107, rev11_out_aa(.(T113))) → rev236_out_aga(.(T107))
rev11_in_aa → rev11_out_aa(.([]))
rev11_in_aa → U11_aa(rev123_in_aaa)
rev11_in_aa → U12_aa(rev123_in_aaa)
U11_aa(rev123_out_aaa(T37)) → rev11_out_aa(.(.(T37)))
U12_aa(rev123_out_aaa(T86)) → U13_aa(T86, rev236_in_aga(T86))
U12_aa(rev123_out_aaa(T86)) → U14_aa(T86, rev236_in_aga(T86))
U13_aa(T86, rev236_out_aga(T86)) → rev11_out_aa(.(.(T86)))
U14_aa(T86, rev236_out_aga(T86)) → U15_aa(T86, rev48_in_aa)
U14_aa(T86, rev236_out_aga(T86)) → U16_aa(T86, rev48_in_aa)
U15_aa(T86, rev48_out_aa(T91)) → rev11_out_aa(.(.(T86)))
U16_aa(T86, rev48_out_aa(T91)) → U17_aa(T86, rev11_in_aa)
U17_aa(T86, rev11_out_aa(.(T145))) → rev11_out_aa(.(.(T86)))

The set Q consists of the following terms:

rev236_in_aga(x0)
rev158_in_aaa
rev48_in_aa
rev123_in_aaa
U2_aga(x0, x1)
U3_aga(x0, x1)
U7_aaa(x0)
U8_aa(x0)
U9_aa(x0)
U1_aaa(x0)
U4_aga(x0, x1)
U5_aga(x0, x1)
U10_aa(x0, x1)
U6_aga(x0, x1)
rev11_in_aa
U11_aa(x0)
U12_aa(x0)
U13_aa(x0, x1)
U14_aa(x0, x1)
U15_aa(x0, x1)
U16_aa(x0, x1)
U17_aa(x0, x1)

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

(132) Obligation:

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

REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)

The TRS R consists of the following rules:

rev11_in_ga([], []) → rev11_out_ga([], [])
rev11_in_ga(.(T25, []), .(T25, [])) → rev11_out_ga(.(T25, []), .(T25, []))
rev11_in_ga(.(T35, .(T36, T37)), .(T39, T40)) → U11_ga(T35, T36, T37, T39, T40, rev123_in_aga(T36, T37, T39))
rev123_in_aga(T50, [], T50) → rev123_out_aga(T50, [], T50)
rev123_in_aga(T59, .(T60, T61), T63) → U1_aga(T59, T60, T61, T63, rev123_in_aga(T60, T61, T63))
U1_aga(T59, T60, T61, T63, rev123_out_aga(T60, T61, T63)) → rev123_out_aga(T59, .(T60, T61), T63)
U11_ga(T35, T36, T37, T39, T40, rev123_out_aga(T36, T37, T39)) → rev11_out_ga(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U12_ga(T84, T85, T86, T39, T88, rev123_in_aga(T85, T86, T39))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U13_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
rev236_in_aga(T98, [], []) → rev236_out_aga(T98, [], [])
rev236_in_aga(T105, .(T106, T107), X134) → U2_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, X132))
rev236_in_aga(T105, .(T106, T107), X134) → U3_aga(T105, T106, T107, X134, rev236_in_aga(T106, T107, T110))
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U4_aga(T105, T106, T107, X134, rev48_in_aa(T110, X133))
rev48_in_aa([], []) → rev48_out_aa([], [])
rev48_in_aa(.(T118, T119), .(X159, X160)) → U8_aa(T118, T119, X159, X160, rev158_in_aaa(T118, T119, X159))
rev158_in_aaa(T129, [], T129) → rev158_out_aaa(T129, [], T129)
rev158_in_aaa(T136, .(T137, T138), X189) → U7_aaa(T136, T137, T138, X189, rev158_in_aaa(T137, T138, X189))
U7_aaa(T136, T137, T138, X189, rev158_out_aaa(T137, T138, X189)) → rev158_out_aaa(T136, .(T137, T138), X189)
U8_aa(T118, T119, X159, X160, rev158_out_aaa(T118, T119, X159)) → rev48_out_aa(.(T118, T119), .(X159, X160))
rev48_in_aa(.(T118, T119), .(T122, X160)) → U9_aa(T118, T119, T122, X160, rev158_in_aaa(T118, T119, T122))
U9_aa(T118, T119, T122, X160, rev158_out_aaa(T118, T119, T122)) → U10_aa(T118, T119, T122, X160, rev236_in_aga(T118, T119, X160))
U10_aa(T118, T119, T122, X160, rev236_out_aga(T118, T119, X160)) → rev48_out_aa(.(T118, T119), .(T122, X160))
U4_aga(T105, T106, T107, X134, rev48_out_aa(T110, X133)) → rev236_out_aga(T105, .(T106, T107), X134)
U3_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, T110)) → U5_aga(T105, T106, T107, X134, rev48_in_aa(T110, T113))
U5_aga(T105, T106, T107, X134, rev48_out_aa(T110, T113)) → U6_aga(T105, T106, T107, X134, rev11_in_aa(.(T105, T113), X134))
rev11_in_aa([], []) → rev11_out_aa([], [])
rev11_in_aa(.(T25, []), .(T25, [])) → rev11_out_aa(.(T25, []), .(T25, []))
rev11_in_aa(.(T35, .(T36, T37)), .(T39, T40)) → U11_aa(T35, T36, T37, T39, T40, rev123_in_aaa(T36, T37, T39))
rev123_in_aaa(T50, [], T50) → rev123_out_aaa(T50, [], T50)
rev123_in_aaa(T59, .(T60, T61), T63) → U1_aaa(T59, T60, T61, T63, rev123_in_aaa(T60, T61, T63))
U1_aaa(T59, T60, T61, T63, rev123_out_aaa(T60, T61, T63)) → rev123_out_aaa(T59, .(T60, T61), T63)
U11_aa(T35, T36, T37, T39, T40, rev123_out_aaa(T36, T37, T39)) → rev11_out_aa(.(T35, .(T36, T37)), .(T39, T40))
rev11_in_aa(.(T84, .(T85, T86)), .(T39, T88)) → U12_aa(T84, T85, T86, T39, T88, rev123_in_aaa(T85, T86, T39))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U13_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, X104))
U13_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U12_aa(T84, T85, T86, T39, T88, rev123_out_aaa(T85, T86, T39)) → U14_aa(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U14_aa(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_aa(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_aa(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_aa(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_aa(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_aa(.(T84, .(T85, T86)), .(T39, T88))
U6_aga(T105, T106, T107, X134, rev11_out_aa(.(T105, T113), X134)) → rev236_out_aga(T105, .(T106, T107), X134)
U2_aga(T105, T106, T107, X134, rev236_out_aga(T106, T107, X132)) → rev236_out_aga(T105, .(T106, T107), X134)
U13_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, X104)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U12_ga(T84, T85, T86, T39, T88, rev123_out_aga(T85, T86, T39)) → U14_ga(T84, T85, T86, T39, T88, rev236_in_aga(T85, T86, T91))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U15_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, X105))
U15_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, X105)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U14_ga(T84, T85, T86, T39, T88, rev236_out_aga(T85, T86, T91)) → U16_ga(T84, T85, T86, T39, T88, rev48_in_aa(T91, T145))
U16_ga(T84, T85, T86, T39, T88, rev48_out_aa(T91, T145)) → U17_ga(T84, T85, T86, T39, T88, rev11_in_aa(.(T84, T145), T88))
U17_ga(T84, T85, T86, T39, T88, rev11_out_aa(.(T84, T145), T88)) → rev11_out_ga(.(T84, .(T85, T86)), .(T39, T88))

(133) UsableRulesProof (EQUIVALENT transformation)

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

(134) Obligation:

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

REV123_IN_AGA(T59, .(T60, T61), T63) → REV123_IN_AGA(T60, T61, T63)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REV123_IN_AGA(x1, x2, x3) = REV123_IN_AGA(x2)

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