Left Termination proof of /tmp/tmp93MuH3/quicksort-oi.pl

Left Termination of the query pattern qs_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇐)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇐)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇐)

        ↳34 QDP

        ↳35 QDPSizeChangeProof (⇔)

        ↳36 YES

    ↳37 PiDP

        ↳38 UsableRulesProof (⇔)

        ↳39 PiDP

        ↳40 PiDPToQDPProof (⇐)

        ↳41 QDP

        ↳42 QDPSizeChangeProof (⇔)

        ↳43 YES

    ↳44 PiDP

        ↳45 UsableRulesProof (⇔)

        ↳46 PiDP

        ↳47 PiDPToQDPProof (⇔)

        ↳48 QDP

        ↳49 QDPSizeChangeProof (⇔)

        ↳50 YES

    ↳51 PiDP

        ↳52 UsableRulesProof (⇔)

        ↳53 PiDP

        ↳54 PiDPToQDPProof (⇔)

        ↳55 QDP

        ↳56 QDPSizeChangeProof (⇔)

        ↳57 YES

    ↳58 PiDP

        ↳59 UsableRulesProof (⇔)

        ↳60 PiDP

        ↳61 PiDPToQDPProof (⇔)

        ↳62 QDP

        ↳63 QDPSizeChangeProof (⇔)

        ↳64 YES

    ↳65 PiDP

        ↳66 UsableRulesProof (⇔)

        ↳67 PiDP

        ↳68 PiDPToQDPProof (⇐)

        ↳69 QDP

        ↳70 QDPSizeChangeProof (⇔)

        ↳71 YES

    ↳72 PiDP

        ↳73 UsableRulesProof (⇔)

        ↳74 PiDP

        ↳75 PiDPToQDPProof (⇐)

        ↳76 QDP

        ↳77 QDPOrderProof (⇔)

        ↳78 QDP

        ↳79 DependencyGraphProof (⇔)

        ↳80 TRUE

    ↳81 PiDP

        ↳82 UsableRulesProof (⇔)

        ↳83 PiDP

        ↳84 PiDPToQDPProof (⇐)

        ↳85 QDP

        ↳86 QDPOrderProof (⇔)

        ↳87 QDP

        ↳88 DependencyGraphProof (⇔)

        ↳89 TRUE

    ↳90 PiDP

        ↳91 UsableRulesProof (⇔)

        ↳92 PiDP

        ↳93 PiDPToQDPProof (⇐)

        ↳94 QDP

        ↳95 NonTerminationProof (⇔)

        ↳96 NO

    ↳97 PiDP

        ↳98 UsableRulesProof (⇔)

        ↳99 PiDP

        ↳100 PiDPToQDPProof (⇐)

        ↳101 QDP

        ↳102 QDPSizeChangeProof (⇔)

        ↳103 YES

    ↳104 PiDP

        ↳105 UsableRulesProof (⇔)

        ↳106 PiDP

        ↳107 PiDPToQDPProof (⇐)

        ↳108 QDP

        ↳109 QDPSizeChangeProof (⇔)

        ↳110 YES

    ↳111 PiDP

        ↳112 UsableRulesProof (⇔)

        ↳113 PiDP

        ↳114 PiDPToQDPProof (⇐)

        ↳115 QDP

        ↳116 Narrowing (⇐)

        ↳117 QDP

        ↳118 Narrowing (⇐)

        ↳119 QDP

        ↳120 Instantiation (⇔)

        ↳121 QDP

        ↳122 Instantiation (⇔)

        ↳123 QDP

        ↳124 DependencyGraphProof (⇔)

        ↳125 AND

            ↳126 QDP

                ↳127 UsableRulesProof (⇔)

                ↳128 QDP

                ↳129 QReductionProof (⇔)

                ↳130 QDP

                ↳131 NonTerminationProof (⇔)

                ↳132 NO

            ↳133 QDP

                ↳134 NonTerminationProof (⇔)

                ↳135 NO

    ↳136 PiDP

        ↳137 UsableRulesProof (⇔)

        ↳138 PiDP

        ↳139 PiDPToQDPProof (⇐)

        ↳140 QDP

        ↳141 NonTerminationProof (⇔)

        ↳142 NO

    ↳143 PiDP

        ↳144 UsableRulesProof (⇔)

        ↳145 PiDP

        ↳146 PiDPToQDPProof (⇐)

        ↳147 QDP

        ↳148 Narrowing (⇐)

        ↳149 QDP

        ↳150 UsableRulesProof (⇔)

        ↳151 QDP

        ↳152 QReductionProof (⇔)

        ↳153 QDP

        ↳154 Narrowing (⇐)

        ↳155 QDP

        ↳156 NonTerminationProof (⇔)

        ↳157 NO

(0) Obligation:

Clauses:

qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
qs([], []).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(gt(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(le(X, Y), part(X, Xs, Ls, Bs)).
part(X, [], [], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(0), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).

Queries:

qs(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: qs(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))) | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nqs(T1, T2) !=? qs(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 2 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 2\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3 | ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(gt(T26, T27), ','(part(T26, T28, X47, X48), ','(qs(.(T27, X47), X9), ','(qs(X48, X10), app(X9, .(T26, X10), T8)))))\n\nT8 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: gt(T26, T27)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T31, T32, X47, X48), ','(qs(.(T33, X47), X9), ','(qs(X48, X10), app(X9, .(T31, X10), T8))))\n\nT8 is ground\nT31 is ground\nT33 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: gt(T26, T27), SCOPE: 3, CLAUSE: 7 | gt(T26, T27), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: gt(T26, T27), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: gt(T26, T27), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: gt(T46, T47)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: part(T31, T32, X47, X48)\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(qs(.(T33, T53), X9), ','(qs(T54, X10), app(X9, .(T31, X10), T8)))\n\nT8 is ground\nT31 is ground\nT33 is ground\nT53 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 2 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 2\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(gt(T70, T73), part(T70, T74, X119, X120))\n\nT70 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: gt(T70, T73)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T70, T77, X119, X120)\n\nT70 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 3\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: part(T31, T32, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT31 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(le(T95, T98), part(T95, T99, X170, X171))\n\nT95 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T95, T98)\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T95, T102, X170, X171)\n\nT95 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: le(T95, T98), SCOPE: 5, CLAUSE: 9 | le(T95, T98), SCOPE: 5, CLAUSE: 10 | le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T95, T98), SCOPE: 5, CLAUSE: 9\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T95, T98), SCOPE: 5, CLAUSE: 10 | le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T113, T115)\n\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T95, T98), SCOPE: 5, CLAUSE: 10\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T95, T98), SCOPE: 5, CLAUSE: 11\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: qs(.(T33, T53), X9)\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qs(T54, X10), app(T131, .(T31, X10), T8))\n\nT8 is ground\nT31 is ground\nT131 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 0 | qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 1\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 0\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: qs(.(T33, T53), X9), SCOPE: 6, CLAUSE: 1\n\nT33 is ground\nT53 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(part(T148, T149, X269, X270), ','(qs(X269, X271), ','(qs(X270, X272), app(X271, .(T148, X272), X273))))\n\nT148 is ground\nT149 is ground\nX273 is free; \nX269 is free; \nX270 is free; \nX271 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: part(T148, T149, X269, X270)\n\nT148 is ground\nT149 is ground\nX269 is free; \nX270 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(qs(T153, X271), ','(qs(T154, X272), app(X271, .(T148, X272), X273)))\n\nT148 is ground\nT153 is ground\nT154 is ground\nX273 is free; \nX271 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: qs(T153, X271)\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(qs(T154, X272), app(T160, .(T148, X272), X273))\n\nT148 is ground\nT154 is ground\nT160 is ground\nX273 is free; \nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: qs(T153, X271), SCOPE: 7, CLAUSE: 0 | qs(T153, X271), SCOPE: 7, CLAUSE: 1\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: qs(T153, X271), SCOPE: 7, CLAUSE: 0\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: qs(T153, X271), SCOPE: 7, CLAUSE: 1\n\nT153 is ground\nX271 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(part(T171, T172, X331, X332), ','(qs(X331, X333), ','(qs(X332, X334), app(X333, .(T171, X334), X335))))\n\nT171 is ground\nT172 is ground\nX335 is free; \nX331 is free; \nX332 is free; \nX333 is free; \nX334 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: qs(T154, X272)\n\nT154 is ground\nX272 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app(T160, .(T148, T178), X273)\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 5 | app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 6\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 5\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T160, .(T148, T178), X273), SCOPE: 8, CLAUSE: 6\n\nT148 is ground\nT160 is ground\nT178 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(T202, .(T203, T204), X391)\n\nT202 is ground\nT203 is ground\nT204 is ground\nX391 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: qs(T54, X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: app(T131, .(T31, T217), T8)\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: qs(T54, X10), SCOPE: 9, CLAUSE: 0 | qs(T54, X10), SCOPE: 9, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: qs(T54, X10), SCOPE: 9, CLAUSE: 0\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: qs(T54, X10), SCOPE: 9, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(part(T230, T231, X448, X449), ','(qs(X448, X450), ','(qs(X449, X451), app(X450, .(T230, X451), X452))))\n\nX452 is free\nX448 is free; \nX449 is free; \nX450 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: part(T230, T231, X448, X449)\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(qs(T235, X450), ','(qs(T236, X451), app(X450, .(T237, X451), X452)))\n\nT235 is ground\nX452 is free; \nX450 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 2 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 2\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3 | part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(gt(T256, T257), part(T256, T258, X503, X504))\n\nX503 is free\nX504 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: gt(T256, T257)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: part(T261, T262, X503, X504)\n\nT261 is ground\nX503 is free; \nX504 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 3\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: part(T230, T231, X448, X449), SCOPE: 10, CLAUSE: 4\n\nX448 is free\nX449 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(le(T283, T284), part(T283, T285, X554, X555))\n\nX554 is free\nX555 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: le(T283, T284)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: part(T288, T289, X554, X555)\n\nT288 is ground\nX554 is free; \nX555 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: le(T283, T284), SCOPE: 11, CLAUSE: 9 | le(T283, T284), SCOPE: 11, CLAUSE: 10 | le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: le(T283, T284), SCOPE: 11, CLAUSE: 9\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: le(T283, T284), SCOPE: 11, CLAUSE: 10 | le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: le(T302, T303)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: le(T283, T284), SCOPE: 11, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: le(T283, T284), SCOPE: 11, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: qs(T235, X450)\n\nT235 is ground\nX450 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(qs(T236, X451), app(T319, .(T237, X451), X452))\n\nT319 is ground\nX452 is free; \nX451 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: qs(T236, X451)\n\nX451 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: app(T319, .(T325, T324), X452)\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 5 | app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 6\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 5\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: app(T319, .(T325, T324), X452), SCOPE: 12, CLAUSE: 6\n\nT319 is ground\nX452 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: app(T349, .(T352, T353), X657)\n\nT349 is ground\nX657 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 5 | app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 6\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 5\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: app(T131, .(T31, T217), T8), SCOPE: 13, CLAUSE: 6\n\nT8 is ground\nT31 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: app(T390, .(T391, T394), T393)\n\nT390 is ground\nT391 is ground\nT393 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: ','(le(T421, T422), ','(part(T421, T423, X740, X741), ','(qs(X740, X9), ','(qs(.(T422, X741), X10), app(X9, .(T421, X10), T8)))))\n\nT8 is ground\nX9 is free; \nX10 is free; \nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: le(T421, T422)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: ','(part(T426, T427, X740, X741), ','(qs(X740, X9), ','(qs(.(T428, X741), X10), app(X9, .(T426, X10), T8))))\n\nT8 is ground\nT426 is ground\nX9 is free; \nX10 is free; \nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: part(T426, T427, X740, X741)\n\nT426 is ground\nX740 is free; \nX741 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: ','(qs(T434, X9), ','(qs(.(T436, T435), X10), app(X9, .(T426, X10), T8)))\n\nT8 is ground\nT426 is ground\nT434 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: qs(T434, X9)\n\nT434 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: ','(qs(.(T436, T435), X10), app(T442, .(T426, X10), T8))\n\nT8 is ground\nT426 is ground\nT442 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(part(T9, T10, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T9, X10), T8)))), SCOPE: 2, CLAUSE: 4\n\nT8 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: ?_2 | qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: ','(qs([], X9), ','(qs([], X10), app(X9, .(T452, X10), T8)))\n\nT8 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: qs([], X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: ','(qs([], X10), app(T455, .(T452, X10), T8))\n\nT8 is ground\nT455 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: qs([], X9), SCOPE: 14, CLAUSE: 0 | qs([], X9), SCOPE: 14, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: qs([], X9), SCOPE: 14, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: qs([], X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: app(T455, .(T452, T460), T8)\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 5 | app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 6\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 5\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: app(T455, .(T452, T460), T8), SCOPE: 15, CLAUSE: 6\n\nT8 is ground\nT455 is ground\nT460 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: app(T487, .(T491, T489), T490)\n\nT487 is ground\nT489 is ground\nT490 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: qs(T1, T8), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 178 [arrowhead = none ];
178 -> 2;

179 [label="EVAL with clause\nqs(.(X4, X5), X6) :- ','(part(X4, X5, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(X4, X10), X6)))).\nand substitution\nX4 -> T9\nX5 -> T10\nT1 -> .(T9, T10)\nT2 -> T8\nX6 -> T8\nT6 -> T9\nT7 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 179 [arrowhead = none ];
179 -> 3;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 180 [arrowhead = none ];
180 -> 4;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 181 [arrowhead = none ];
181 -> 5;

182 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 182 [arrowhead = none ];
182 -> 175;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 183 [arrowhead = none ];
183 -> 176;

184 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 184 [arrowhead = none ];
184 -> 6;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 185 [arrowhead = none ];
185 -> 7;

186 [label="EVAL with clause\npart(X42, .(X43, X44), .(X43, X45), X46) :- ','(gt(X42, X43), part(X42, X44, X45, X46)).\nand substitution\nT9 -> T26\nX42 -> T26\nX43 -> T27\nX44 -> T28\nT10 -> .(T27, T28)\nX45 -> X47\nX7 -> .(T27, X47)\nX8 -> X48\nX46 -> X48\nT23 -> T26\nT24 -> T27\nT25 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 186 [arrowhead = none ];
186 -> 8;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 187 [arrowhead = none ];
187 -> 9;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 188 [arrowhead = none ];
188 -> 141;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 189 [arrowhead = none ];
189 -> 142;

190 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 190 [arrowhead = none ];
190 -> 10;

191 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT33 is ground\nreplacements:\nT26 -> T31\nT28 -> T32\nT27 -> T33", fontsize=14, style = filled, fillcolor = violet];
8 -> 191 [arrowhead = none ];
191 -> 11;

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 192 [arrowhead = none ];
192 -> 12;

193 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 193 [arrowhead = none ];
193 -> 20;

194 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nreplacements:\nX47 -> T53\nX48 -> T54", fontsize=14, style = filled, fillcolor = violet];
11 -> 194 [arrowhead = none ];
194 -> 21;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 195 [arrowhead = none ];
195 -> 13;

196 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 196 [arrowhead = none ];
196 -> 14;

197 [label="EVAL with clause\ngt(s(X66), s(X67)) :- gt(X66, X67).\nand substitution\nX66 -> T46\nT26 -> s(T46)\nX67 -> T47\nT27 -> s(T47)\nT44 -> T46\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 197 [arrowhead = none ];
197 -> 15;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 198 [arrowhead = none ];
198 -> 16;

199 [label="EVAL with clause\ngt(s(0), 0).\nand substitution\nT26 -> s(0)\nT27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 199 [arrowhead = none ];
199 -> 17;

200 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 200 [arrowhead = none ];
200 -> 18;

201 [label="INSTANCE with matching:\nT26 -> T46\nT27 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 201 [arrowhead = none , style=dashed];
201 -> 10[style=dashed];

202 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 202 [arrowhead = none ];
202 -> 19;

203 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 203 [arrowhead = none ];
203 -> 22;

204 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 204 [arrowhead = none ];
204 -> 51;

205 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nreplacements:\nX9 -> T131", fontsize=14, style = filled, fillcolor = violet];
21 -> 205 [arrowhead = none ];
205 -> 52;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 206 [arrowhead = none ];
206 -> 23;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 207 [arrowhead = none ];
207 -> 24;

208 [label="EVAL with clause\npart(X114, .(X115, X116), .(X115, X117), X118) :- ','(gt(X114, X115), part(X114, X116, X117, X118)).\nand substitution\nT31 -> T70\nX114 -> T70\nX115 -> T73\nX116 -> T74\nT32 -> .(T73, T74)\nX117 -> X119\nX47 -> .(T73, X119)\nX48 -> X120\nX118 -> X120\nT71 -> T73\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 208 [arrowhead = none ];
208 -> 25;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 209 [arrowhead = none ];
209 -> 26;

210 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 210 [arrowhead = none ];
210 -> 29;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 211 [arrowhead = none ];
211 -> 30;

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 212 [arrowhead = none ];
212 -> 27;

213 [label="SPLIT 2\nnew knowledge:\nT73 is ground\nreplacements:\nT74 -> T77", fontsize=14, style = filled, fillcolor = violet];
25 -> 213 [arrowhead = none ];
213 -> 28;

214 [label="INSTANCE with matching:\nT26 -> T70\nT27 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 214 [arrowhead = none , style=dashed];
214 -> 10[style=dashed];

215 [label="INSTANCE with matching:\nT31 -> T70\nT32 -> T77\nX47 -> X119\nX48 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 215 [arrowhead = none , style=dashed];
215 -> 20[style=dashed];

216 [label="EVAL with clause\npart(X165, .(X166, X167), X168, .(X166, X169)) :- ','(le(X165, X166), part(X165, X167, X168, X169)).\nand substitution\nT31 -> T95\nX165 -> T95\nX166 -> T98\nX167 -> T99\nT32 -> .(T98, T99)\nX47 -> X170\nX168 -> X170\nX169 -> X171\nX48 -> .(T98, X171)\nT96 -> T98\nT97 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 216 [arrowhead = none ];
216 -> 31;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 217 [arrowhead = none ];
217 -> 32;

218 [label="EVAL with clause\npart(X210, [], [], []).\nand substitution\nT31 -> T128\nX210 -> T128\nT32 -> []\nX47 -> []\nX48 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 218 [arrowhead = none ];
218 -> 48;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 219 [arrowhead = none ];
219 -> 49;

220 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 220 [arrowhead = none ];
220 -> 33;

221 [label="SPLIT 2\nreplacements:\nT99 -> T102", fontsize=14, style = filled, fillcolor = violet];
31 -> 221 [arrowhead = none ];
221 -> 34;

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 222 [arrowhead = none ];
222 -> 35;

223 [label="INSTANCE with matching:\nT31 -> T95\nT32 -> T102\nX47 -> X170\nX48 -> X171", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 223 [arrowhead = none , style=dashed];
223 -> 20[style=dashed];

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 224 [arrowhead = none ];
224 -> 36;

225 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 225 [arrowhead = none ];
225 -> 37;

226 [label="EVAL with clause\nle(s(X189), s(X190)) :- le(X189, X190).\nand substitution\nX189 -> T113\nT95 -> s(T113)\nX190 -> T115\nT98 -> s(T115)\nT114 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 226 [arrowhead = none ];
226 -> 38;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 227 [arrowhead = none ];
227 -> 39;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 228 [arrowhead = none ];
228 -> 40;

229 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 229 [arrowhead = none ];
229 -> 41;

230 [label="INSTANCE with matching:\nT95 -> T113\nT98 -> T115", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 230 [arrowhead = none , style=dashed];
230 -> 33[style=dashed];

231 [label="EVAL with clause\nle(0, s(X199)).\nand substitution\nT95 -> 0\nX199 -> T122\nT98 -> s(T122)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 231 [arrowhead = none ];
231 -> 42;

232 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 232 [arrowhead = none ];
232 -> 43;

233 [label="EVAL with clause\nle(0, 0).\nand substitution\nT95 -> 0\nT98 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 233 [arrowhead = none ];
233 -> 45;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 234 [arrowhead = none ];
234 -> 46;

235 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 235 [arrowhead = none ];
235 -> 44;

236 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 236 [arrowhead = none ];
236 -> 47;

237 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 237 [arrowhead = none ];
237 -> 50;

238 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 238 [arrowhead = none ];
238 -> 53;

239 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 239 [arrowhead = none ];
239 -> 80;

240 [label="SPLIT 2\nreplacements:\nX10 -> T217", fontsize=14, style = filled, fillcolor = violet];
52 -> 240 [arrowhead = none ];
240 -> 81;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 241 [arrowhead = none ];
241 -> 54;

242 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 242 [arrowhead = none ];
242 -> 55;

243 [label="ONLY EVAL with clause\nqs(.(X266, X267), X268) :- ','(part(X266, X267, X269, X270), ','(qs(X269, X271), ','(qs(X270, X272), app(X271, .(X266, X272), X268)))).\nand substitution\nT33 -> T148\nX266 -> T148\nT53 -> T149\nX267 -> T149\nX9 -> X273\nX268 -> X273", fontsize=14, style = filled, fillcolor = white];
54 -> 243 [arrowhead = none ];
243 -> 56;

244 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 244 [arrowhead = none ];
244 -> 79;

245 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 245 [arrowhead = none ];
245 -> 57;

246 [label="SPLIT 2\nnew knowledge:\nT153 is ground\nT154 is ground\nreplacements:\nX269 -> T153\nX270 -> T154", fontsize=14, style = filled, fillcolor = violet];
56 -> 246 [arrowhead = none ];
246 -> 58;

247 [label="INSTANCE with matching:\nT31 -> T148\nT32 -> T149\nX47 -> X269\nX48 -> X270", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 247 [arrowhead = none , style=dashed];
247 -> 20[style=dashed];

248 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 248 [arrowhead = none ];
248 -> 59;

249 [label="SPLIT 2\nnew knowledge:\nT160 is ground\nreplacements:\nX271 -> T160", fontsize=14, style = filled, fillcolor = violet];
58 -> 249 [arrowhead = none ];
249 -> 60;

250 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 250 [arrowhead = none ];
250 -> 61;

251 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 251 [arrowhead = none ];
251 -> 69;

252 [label="SPLIT 2\nnew knowledge:\nT178 is ground\nreplacements:\nX272 -> T178", fontsize=14, style = filled, fillcolor = violet];
60 -> 252 [arrowhead = none ];
252 -> 70;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 253 [arrowhead = none ];
253 -> 62;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 254 [arrowhead = none ];
254 -> 63;

255 [label="EVAL with clause\nqs(.(X328, X329), X330) :- ','(part(X328, X329, X331, X332), ','(qs(X331, X333), ','(qs(X332, X334), app(X333, .(X328, X334), X330)))).\nand substitution\nX328 -> T171\nX329 -> T172\nT153 -> .(T171, T172)\nX271 -> X335\nX330 -> X335", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 255 [arrowhead = none ];
255 -> 64;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 256 [arrowhead = none ];
256 -> 65;

257 [label="EVAL with clause\nqs([], []).\nand substitution\nT153 -> []\nX271 -> []", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 257 [arrowhead = none ];
257 -> 66;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 258 [arrowhead = none ];
258 -> 67;

259 [label="INSTANCE with matching:\nT148 -> T171\nT149 -> T172\nX269 -> X331\nX270 -> X332\nX271 -> X333\nX272 -> X334\nX273 -> X335", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 259 [arrowhead = none , style=dashed];
259 -> 56[style=dashed];

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 260 [arrowhead = none ];
260 -> 68;

261 [label="INSTANCE with matching:\nT153 -> T154\nX271 -> X272", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 261 [arrowhead = none , style=dashed];
261 -> 59[style=dashed];

262 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 262 [arrowhead = none ];
262 -> 71;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 263 [arrowhead = none ];
263 -> 72;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 264 [arrowhead = none ];
264 -> 73;

265 [label="EVAL with clause\napp(.(X387, X388), X389, .(X387, X390)) :- app(X388, X389, X390).\nand substitution\nX387 -> T201\nX388 -> T202\nT160 -> .(T201, T202)\nT148 -> T203\nT178 -> T204\nX389 -> .(T203, T204)\nX390 -> X391\nX273 -> .(T201, X391)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 265 [arrowhead = none ];
265 -> 74;

266 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 266 [arrowhead = none ];
266 -> 75;

267 [label="EVAL with clause\napp([], X403, X403).\nand substitution\nT160 -> []\nT148 -> T213\nT178 -> T214\nX403 -> .(T213, T214)\nX273 -> .(T213, T214)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 267 [arrowhead = none ];
267 -> 76;

268 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 268 [arrowhead = none ];
268 -> 77;

269 [label="INSTANCE with matching:\nT160 -> T202\nT148 -> T203\nT178 -> T204\nX273 -> X391", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 269 [arrowhead = none , style=dashed];
269 -> 70[style=dashed];

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 270 [arrowhead = none ];
270 -> 78;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 271 [arrowhead = none ];
271 -> 82;

272 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 272 [arrowhead = none ];
272 -> 133;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 273 [arrowhead = none ];
273 -> 83;

274 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 274 [arrowhead = none ];
274 -> 84;

275 [label="EVAL with clause\nqs(.(X445, X446), X447) :- ','(part(X445, X446, X448, X449), ','(qs(X448, X450), ','(qs(X449, X451), app(X450, .(X445, X451), X447)))).\nand substitution\nX445 -> T230\nX446 -> T231\nT54 -> .(T230, T231)\nX10 -> X452\nX447 -> X452\nT228 -> T230\nT229 -> T231", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 275 [arrowhead = none ];
275 -> 85;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 276 [arrowhead = none ];
276 -> 86;

277 [label="EVAL with clause\nqs([], []).\nand substitution\nT54 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 277 [arrowhead = none ];
277 -> 130;

278 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 278 [arrowhead = none ];
278 -> 131;

279 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 279 [arrowhead = none ];
279 -> 87;

280 [label="SPLIT 2\nnew knowledge:\nT235 is ground\nreplacements:\nX448 -> T235\nX449 -> T236\nT230 -> T237", fontsize=14, style = filled, fillcolor = violet];
85 -> 280 [arrowhead = none ];
280 -> 88;

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 281 [arrowhead = none ];
281 -> 89;

282 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
88 -> 282 [arrowhead = none ];
282 -> 118;

283 [label="SPLIT 2\nnew knowledge:\nT319 is ground\nreplacements:\nX450 -> T319", fontsize=14, style = filled, fillcolor = violet];
88 -> 283 [arrowhead = none ];
283 -> 119;

284 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 284 [arrowhead = none ];
284 -> 90;

285 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 285 [arrowhead = none ];
285 -> 91;

286 [label="EVAL with clause\npart(X498, .(X499, X500), .(X499, X501), X502) :- ','(gt(X498, X499), part(X498, X500, X501, X502)).\nand substitution\nT230 -> T256\nX498 -> T256\nX499 -> T257\nX500 -> T258\nT231 -> .(T257, T258)\nX501 -> X503\nX448 -> .(T257, X503)\nX449 -> X504\nX502 -> X504\nT253 -> T256\nT254 -> T257\nT255 -> T258", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 286 [arrowhead = none ];
286 -> 92;

287 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 287 [arrowhead = none ];
287 -> 93;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 288 [arrowhead = none ];
288 -> 96;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 289 [arrowhead = none ];
289 -> 97;

290 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 290 [arrowhead = none ];
290 -> 94;

291 [label="SPLIT 2\nnew knowledge:\nT261 is ground\nT257 is ground\nreplacements:\nT256 -> T261\nT258 -> T262", fontsize=14, style = filled, fillcolor = violet];
92 -> 291 [arrowhead = none ];
291 -> 95;

292 [label="INSTANCE with matching:\nT26 -> T256\nT27 -> T257", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 292 [arrowhead = none , style=dashed];
292 -> 10[style=dashed];

293 [label="INSTANCE with matching:\nT31 -> T261\nT32 -> T262\nX47 -> X503\nX48 -> X504", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 293 [arrowhead = none , style=dashed];
293 -> 20[style=dashed];

294 [label="EVAL with clause\npart(X549, .(X550, X551), X552, .(X550, X553)) :- ','(le(X549, X550), part(X549, X551, X552, X553)).\nand substitution\nT230 -> T283\nX549 -> T283\nX550 -> T284\nX551 -> T285\nT231 -> .(T284, T285)\nX448 -> X554\nX552 -> X554\nX553 -> X555\nX449 -> .(T284, X555)\nT280 -> T283\nT281 -> T284\nT282 -> T285", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 294 [arrowhead = none ];
294 -> 98;

295 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 295 [arrowhead = none ];
295 -> 99;

296 [label="EVAL with clause\npart(X594, [], [], []).\nand substitution\nT230 -> T316\nX594 -> T316\nT231 -> []\nX448 -> []\nX449 -> []", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 296 [arrowhead = none ];
296 -> 115;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 297 [arrowhead = none ];
297 -> 116;

298 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
98 -> 298 [arrowhead = none ];
298 -> 100;

299 [label="SPLIT 2\nnew knowledge:\nT288 is ground\nreplacements:\nT283 -> T288\nT285 -> T289", fontsize=14, style = filled, fillcolor = violet];
98 -> 299 [arrowhead = none ];
299 -> 101;

300 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 300 [arrowhead = none ];
300 -> 102;

301 [label="INSTANCE with matching:\nT31 -> T288\nT32 -> T289\nX47 -> X554\nX48 -> X555", fontsize=14, style = filled, fillcolor = lightgrey];
101 -> 301 [arrowhead = none , style=dashed];
301 -> 20[style=dashed];

302 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 302 [arrowhead = none ];
302 -> 103;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 303 [arrowhead = none ];
303 -> 104;

304 [label="EVAL with clause\nle(s(X573), s(X574)) :- le(X573, X574).\nand substitution\nX573 -> T302\nT283 -> s(T302)\nX574 -> T303\nT284 -> s(T303)\nT300 -> T302\nT301 -> T303", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 304 [arrowhead = none ];
304 -> 105;

305 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 305 [arrowhead = none ];
305 -> 106;

306 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 306 [arrowhead = none ];
306 -> 107;

307 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 307 [arrowhead = none ];
307 -> 108;

308 [label="INSTANCE with matching:\nT283 -> T302\nT284 -> T303", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 308 [arrowhead = none , style=dashed];
308 -> 100[style=dashed];

309 [label="EVAL with clause\nle(0, s(X583)).\nand substitution\nT283 -> 0\nX583 -> T310\nT284 -> s(T310)", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 309 [arrowhead = none ];
309 -> 109;

310 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 310 [arrowhead = none ];
310 -> 110;

311 [label="EVAL with clause\nle(0, 0).\nand substitution\nT283 -> 0\nT284 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 311 [arrowhead = none ];
311 -> 112;

312 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 312 [arrowhead = none ];
312 -> 113;

313 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 313 [arrowhead = none ];
313 -> 111;

314 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 314 [arrowhead = none ];
314 -> 114;

315 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 315 [arrowhead = none ];
315 -> 117;

316 [label="INSTANCE with matching:\nT54 -> T235\nX10 -> X450", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 316 [arrowhead = none , style=dashed];
316 -> 80[style=dashed];

317 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
119 -> 317 [arrowhead = none ];
317 -> 120;

318 [label="SPLIT 2\nreplacements:\nX451 -> T324\nT237 -> T325", fontsize=14, style = filled, fillcolor = violet];
119 -> 318 [arrowhead = none ];
318 -> 121;

319 [label="INSTANCE with matching:\nT54 -> T236\nX10 -> X451", fontsize=14, style = filled, fillcolor = lightgrey];
120 -> 319 [arrowhead = none , style=dashed];
319 -> 80[style=dashed];

320 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
121 -> 320 [arrowhead = none ];
320 -> 122;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 321 [arrowhead = none ];
321 -> 123;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 322 [arrowhead = none ];
322 -> 124;

323 [label="EVAL with clause\napp(.(X653, X654), X655, .(X653, X656)) :- app(X654, X655, X656).\nand substitution\nX653 -> T348\nX654 -> T349\nT319 -> .(T348, T349)\nT325 -> T352\nT324 -> T353\nX655 -> .(T352, T353)\nX656 -> X657\nX452 -> .(T348, X657)\nT350 -> T352\nT351 -> T353", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 323 [arrowhead = none ];
323 -> 125;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 324 [arrowhead = none ];
324 -> 126;

325 [label="EVAL with clause\napp([], X669, X669).\nand substitution\nT319 -> []\nT325 -> T362\nT324 -> T363\nX669 -> .(T362, T363)\nX452 -> .(T362, T363)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 325 [arrowhead = none ];
325 -> 127;

326 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 326 [arrowhead = none ];
326 -> 128;

327 [label="INSTANCE with matching:\nT319 -> T349\nT325 -> T352\nT324 -> T353\nX452 -> X657", fontsize=14, style = filled, fillcolor = lightgrey];
125 -> 327 [arrowhead = none , style=dashed];
327 -> 121[style=dashed];

328 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
127 -> 328 [arrowhead = none ];
328 -> 129;

329 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 329 [arrowhead = none ];
329 -> 132;

330 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 330 [arrowhead = none ];
330 -> 134;

331 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 331 [arrowhead = none ];
331 -> 135;

332 [label="EVAL with clause\napp(.(X691, X692), X693, .(X691, X694)) :- app(X692, X693, X694).\nand substitution\nX691 -> T389\nX692 -> T390\nT131 -> .(T389, T390)\nT31 -> T391\nT217 -> T394\nX693 -> .(T391, T394)\nX694 -> T393\nT8 -> .(T389, T393)\nT392 -> T394", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 332 [arrowhead = none ];
332 -> 136;

333 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 333 [arrowhead = none ];
333 -> 137;

334 [label="EVAL with clause\napp([], X705, X705).\nand substitution\nT131 -> []\nT31 -> T404\nT217 -> T405\nX705 -> .(T404, T405)\nT8 -> .(T404, T405)", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 334 [arrowhead = none ];
334 -> 138;

335 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 335 [arrowhead = none ];
335 -> 139;

336 [label="INSTANCE with matching:\nT131 -> T390\nT31 -> T391\nT217 -> T394\nT8 -> T393", fontsize=14, style = filled, fillcolor = lightgrey];
136 -> 336 [arrowhead = none , style=dashed];
336 -> 81[style=dashed];

337 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 337 [arrowhead = none ];
337 -> 140;

338 [label="EVAL with clause\npart(X735, .(X736, X737), X738, .(X736, X739)) :- ','(le(X735, X736), part(X735, X737, X738, X739)).\nand substitution\nT9 -> T421\nX735 -> T421\nX736 -> T422\nX737 -> T423\nT10 -> .(T422, T423)\nX7 -> X740\nX738 -> X740\nX739 -> X741\nX8 -> .(T422, X741)\nT418 -> T421\nT419 -> T422\nT420 -> T423", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 338 [arrowhead = none ];
338 -> 143;

339 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 339 [arrowhead = none ];
339 -> 144;

340 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 340 [arrowhead = none ];
340 -> 151;

341 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 341 [arrowhead = none ];
341 -> 152;

342 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
143 -> 342 [arrowhead = none ];
342 -> 145;

343 [label="SPLIT 2\nnew knowledge:\nT426 is ground\nreplacements:\nT421 -> T426\nT423 -> T427\nT422 -> T428", fontsize=14, style = filled, fillcolor = violet];
143 -> 343 [arrowhead = none ];
343 -> 146;

344 [label="INSTANCE with matching:\nT283 -> T421\nT284 -> T422", fontsize=14, style = filled, fillcolor = lightgrey];
145 -> 344 [arrowhead = none , style=dashed];
344 -> 100[style=dashed];

345 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
146 -> 345 [arrowhead = none ];
345 -> 147;

346 [label="SPLIT 2\nnew knowledge:\nT434 is ground\nreplacements:\nX740 -> T434\nX741 -> T435\nT428 -> T436", fontsize=14, style = filled, fillcolor = violet];
146 -> 346 [arrowhead = none ];
346 -> 148;

347 [label="INSTANCE with matching:\nT31 -> T426\nT32 -> T427\nX47 -> X740\nX48 -> X741", fontsize=14, style = filled, fillcolor = lightgrey];
147 -> 347 [arrowhead = none , style=dashed];
347 -> 20[style=dashed];

348 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
148 -> 348 [arrowhead = none ];
348 -> 149;

349 [label="SPLIT 2\nnew knowledge:\nT442 is ground\nreplacements:\nX9 -> T442", fontsize=14, style = filled, fillcolor = violet];
148 -> 349 [arrowhead = none ];
349 -> 150;

350 [label="INSTANCE with matching:\nT54 -> T434\nX10 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
149 -> 350 [arrowhead = none , style=dashed];
350 -> 80[style=dashed];

351 [label="INSTANCE with matching:\nT54 -> .(T436, T435)\nT131 -> T442\nT31 -> T426", fontsize=14, style = filled, fillcolor = lightgrey];
150 -> 351 [arrowhead = none , style=dashed];
351 -> 52[style=dashed];

352 [label="EVAL with clause\npart(X793, [], [], []).\nand substitution\nT9 -> T452\nX793 -> T452\nT10 -> []\nX7 -> []\nX8 -> []\nT451 -> T452", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 352 [arrowhead = none ];
352 -> 153;

353 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
151 -> 353 [arrowhead = none ];
353 -> 154;

354 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 354 [arrowhead = none ];
354 -> 171;

355 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
153 -> 355 [arrowhead = none ];
355 -> 155;

356 [label="SPLIT 2\nnew knowledge:\nT455 is ground\nreplacements:\nX9 -> T455", fontsize=14, style = filled, fillcolor = violet];
153 -> 356 [arrowhead = none ];
356 -> 156;

357 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
155 -> 357 [arrowhead = none ];
357 -> 157;

358 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
156 -> 358 [arrowhead = none ];
358 -> 161;

359 [label="SPLIT 2\nnew knowledge:\nT460 is ground\nreplacements:\nX10 -> T460", fontsize=14, style = filled, fillcolor = violet];
156 -> 359 [arrowhead = none ];
359 -> 162;

360 [label="BACKTRACK\nfor clause: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
157 -> 360 [arrowhead = none ];
360 -> 158;

361 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX9 -> []", fontsize=14, style = filled, fillcolor = white];
158 -> 361 [arrowhead = none ];
361 -> 159;

362 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
159 -> 362 [arrowhead = none ];
362 -> 160;

363 [label="INSTANCE with matching:\nX9 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
161 -> 363 [arrowhead = none , style=dashed];
363 -> 155[style=dashed];

364 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
162 -> 364 [arrowhead = none ];
364 -> 163;

365 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
163 -> 365 [arrowhead = none ];
365 -> 164;

366 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
163 -> 366 [arrowhead = none ];
366 -> 165;

367 [label="EVAL with clause\napp(.(X865, X866), X867, .(X865, X868)) :- app(X866, X867, X868).\nand substitution\nX865 -> T486\nX866 -> T487\nT455 -> .(T486, T487)\nT452 -> T491\nT460 -> T489\nX867 -> .(T491, T489)\nX868 -> T490\nT8 -> .(T486, T490)\nT488 -> T491", fontsize=14, style = filled, fillcolor = lightblue];
164 -> 367 [arrowhead = none ];
367 -> 166;

368 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
164 -> 368 [arrowhead = none ];
368 -> 167;

369 [label="EVAL with clause\napp([], X879, X879).\nand substitution\nT455 -> []\nT452 -> T501\nT460 -> T502\nX879 -> .(T501, T502)\nT8 -> .(T501, T502)", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 369 [arrowhead = none ];
369 -> 168;

370 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 370 [arrowhead = none ];
370 -> 169;

371 [label="INSTANCE with matching:\nT455 -> T487\nT452 -> T491\nT460 -> T489\nT8 -> T490", fontsize=14, style = filled, fillcolor = lightgrey];
166 -> 371 [arrowhead = none , style=dashed];
371 -> 162[style=dashed];

372 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
168 -> 372 [arrowhead = none ];
372 -> 170;

373 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 373 [arrowhead = none ];
373 -> 172;

374 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 374 [arrowhead = none ];
374 -> 173;

375 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 375 [arrowhead = none ];
375 -> 174;

376 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
175 -> 376 [arrowhead = none ];
376 -> 177;

}

(2) Obligation:

Triples:

gt10(s(T46), s(T47)) :- gt10(T46, T47).
part20(T70, .(T73, T74), .(T73, X119), X120) :- gt10(T70, T73).
part20(T70, .(T73, T77), .(T73, X119), X120) :- ','(gtc10(T70, T73), part20(T70, T77, X119, X120)).
part20(T95, .(T98, T99), X170, .(T98, X171)) :- le33(T95, T98).
part20(T95, .(T98, T102), X170, .(T98, X171)) :- ','(lec33(T95, T98), part20(T95, T102, X170, X171)).
le33(s(T113), s(T115)) :- le33(T113, T115).
p56(T148, T149, X269, X270, X271, X272, X273) :- part20(T148, T149, X269, X270).
p56(T148, T149, T153, T154, X271, X272, X273) :- ','(partc20(T148, T149, T153, T154), qs59(T153, X271)).
p56(T148, T149, T153, T154, T160, X272, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), qs59(T154, X272))).
p56(T148, T149, T153, T154, T160, T178, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), ','(qsc59(T154, T178), app70(T160, T148, T178, X273)))).
qs59(.(T171, T172), X335) :- p56(T171, T172, X331, X332, X333, X334, X335).
app70(.(T201, T202), T203, T204, .(T201, X391)) :- app70(T202, T203, T204, X391).
le100(s(T302), s(T303)) :- le100(T302, T303).
qs80(.(T256, .(T257, T258)), X452) :- gt10(T256, T257).
qs80(.(T261, .(T257, T262)), X452) :- ','(gtc10(T261, T257), part20(T261, T262, X503, X504)).
qs80(.(T283, .(T284, T285)), X452) :- le100(T283, T284).
qs80(.(T288, .(T284, T289)), X452) :- ','(lec100(T288, T284), part20(T288, T289, X554, X555)).
qs80(.(T237, T231), X452) :- ','(partc87(T237, T231, T235, T236), qs80(T235, X450)).
qs80(.(T237, T231), X452) :- ','(partc87(T237, T231, T235, T236), ','(qsc80(T235, T319), qs80(T236, X451))).
qs80(.(T325, T231), X452) :- ','(partc87(T325, T231, T235, T236), ','(qsc80(T235, T319), ','(qsc80(T236, T324), app121(T319, T325, T324, X452)))).
app121(.(T348, T349), T352, T353, .(T348, X657)) :- app121(T349, T352, T353, X657).
app81(.(T389, T390), T391, T394, .(T389, T393)) :- app81(T390, T391, T394, T393).
p52(T54, X10, T131, T31, T8) :- qs80(T54, X10).
p52(T54, T217, T131, T31, T8) :- ','(qsc80(T54, T217), app81(T131, T31, T217, T8)).
app162(.(T486, T487), T491, T489, .(T486, T490)) :- app162(T487, T491, T489, T490).
qs1(.(T26, .(T27, T28)), T8) :- gt10(T26, T27).
qs1(.(T31, .(T33, T32)), T8) :- ','(gtc10(T31, T33), part20(T31, T32, X47, X48)).
qs1(.(T31, .(T148, T32)), T8) :- ','(gtc10(T31, T148), ','(partc20(T31, T32, T149, T54), p56(T148, T149, X269, X270, X271, X272, X273))).
qs1(.(T31, .(T33, T32)), T8) :- ','(gtc10(T31, T33), ','(partc20(T31, T32, T53, T54), ','(qsc51(T33, T53, T131), p52(T54, X10, T131, T31, T8)))).
qs1(.(T421, .(T422, T423)), T8) :- le100(T421, T422).
qs1(.(T426, .(T428, T427)), T8) :- ','(lec100(T426, T428), part20(T426, T427, X740, X741)).
qs1(.(T426, .(T436, T427)), T8) :- ','(lec100(T426, T436), ','(partc20(T426, T427, T434, T435), qs80(T434, X9))).
qs1(.(T426, .(T436, T427)), T8) :- ','(lec100(T426, T436), ','(partc20(T426, T427, T434, T435), ','(qsc80(T434, T442), p52(.(T436, T435), X10, T442, T426, T8)))).
qs1(.(T452, []), T8) :- qs155(X9).
qs1(.(T452, []), T8) :- ','(qsc155(T455), qs155(X10)).
qs1(.(T452, []), T8) :- ','(qsc155(T455), ','(qsc155(T460), app162(T455, T452, T460, T8))).

Clauses:

gtc10(s(T46), s(T47)) :- gtc10(T46, T47).
gtc10(s(0), 0).
partc20(T70, .(T73, T77), .(T73, X119), X120) :- ','(gtc10(T70, T73), partc20(T70, T77, X119, X120)).
partc20(T95, .(T98, T102), X170, .(T98, X171)) :- ','(lec33(T95, T98), partc20(T95, T102, X170, X171)).
partc20(T128, [], [], []).
lec33(s(T113), s(T115)) :- lec33(T113, T115).
lec33(0, s(T122)).
lec33(0, 0).
qc56(T148, T149, T153, T154, T160, T178, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), ','(qsc59(T154, T178), appc70(T160, T148, T178, X273)))).
qsc59(.(T171, T172), X335) :- qc56(T171, T172, X331, X332, X333, X334, X335).
qsc59([], []).
appc70(.(T201, T202), T203, T204, .(T201, X391)) :- appc70(T202, T203, T204, X391).
appc70([], T213, T214, .(T213, T214)).
lec100(s(T302), s(T303)) :- lec100(T302, T303).
lec100(0, s(T310)).
lec100(0, 0).
qsc80(.(T325, T231), X452) :- ','(partc87(T325, T231, T235, T236), ','(qsc80(T235, T319), ','(qsc80(T236, T324), appc121(T319, T325, T324, X452)))).
qsc80([], []).
appc121(.(T348, T349), T352, T353, .(T348, X657)) :- appc121(T349, T352, T353, X657).
appc121([], T362, T363, .(T362, T363)).
appc81(.(T389, T390), T391, T394, .(T389, T393)) :- appc81(T390, T391, T394, T393).
appc81([], T404, T405, .(T404, T405)).
qc52(T54, T217, T131, T31, T8) :- ','(qsc80(T54, T217), appc81(T131, T31, T217, T8)).
qsc155([]).
appc162(.(T486, T487), T491, T489, .(T486, T490)) :- appc162(T487, T491, T489, T490).
appc162([], T501, T502, .(T501, T502)).
qsc51(T148, T149, X273) :- qc56(T148, T149, X269, X270, X271, X272, X273).
partc87(T261, .(T257, T262), .(T257, X503), X504) :- ','(gtc10(T261, T257), partc20(T261, T262, X503, X504)).
partc87(T288, .(T284, T289), X554, .(T284, X555)) :- ','(lec100(T288, T284), partc20(T288, T289, X554, X555)).
partc87(T316, [], [], []).

Afs:

qs1(x1, x2) = qs1(x2)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

gt10(s(T46), s(T47)) :- gt10(T46, T47).
part20(T70, .(T73, T74), .(T73, X119), X120) :- gt10(T70, T73).
part20(T70, .(T73, T77), .(T73, X119), X120) :- ','(gtc10(T70, T73), part20(T70, T77, X119, X120)).
part20(T95, .(T98, T99), X170, .(T98, X171)) :- le33(T95, T98).
part20(T95, .(T98, T102), X170, .(T98, X171)) :- ','(lec33(T95, T98), part20(T95, T102, X170, X171)).
le33(s(T113), s(T115)) :- le33(T113, T115).
p56(T148, T149, X269, X270, X271, X272, X273) :- part20(T148, T149, X269, X270).
p56(T148, T149, T153, T154, X271, X272, X273) :- ','(partc20(T148, T149, T153, T154), qs59(T153, X271)).
p56(T148, T149, T153, T154, T160, X272, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), qs59(T154, X272))).
p56(T148, T149, T153, T154, T160, T178, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), ','(qsc59(T154, T178), app70(T160, T148, T178, X273)))).
qs59(.(T171, T172), X335) :- p56(T171, T172, X331, X332, X333, X334, X335).
app70(.(T201, T202), T203, T204, .(T201, X391)) :- app70(T202, T203, T204, X391).
le100(s(T302), s(T303)) :- le100(T302, T303).
qs80(.(T256, .(T257, T258)), X452) :- gt10(T256, T257).
qs80(.(T261, .(T257, T262)), X452) :- ','(gtc10(T261, T257), part20(T261, T262, X503, X504)).
qs80(.(T283, .(T284, T285)), X452) :- le100(T283, T284).
qs80(.(T288, .(T284, T289)), X452) :- ','(lec100(T288, T284), part20(T288, T289, X554, X555)).
qs80(.(T237, T231), X452) :- ','(partc87(T237, T231, T235, T236), qs80(T235, X450)).
qs80(.(T237, T231), X452) :- ','(partc87(T237, T231, T235, T236), ','(qsc80(T235, T319), qs80(T236, X451))).
qs80(.(T325, T231), X452) :- ','(partc87(T325, T231, T235, T236), ','(qsc80(T235, T319), ','(qsc80(T236, T324), app121(T319, T325, T324, X452)))).
app121(.(T348, T349), T352, T353, .(T348, X657)) :- app121(T349, T352, T353, X657).
app81(.(T389, T390), T391, T394, .(T389, T393)) :- app81(T390, T391, T394, T393).
p52(T54, X10, T131, T31, T8) :- qs80(T54, X10).
p52(T54, T217, T131, T31, T8) :- ','(qsc80(T54, T217), app81(T131, T31, T217, T8)).
app162(.(T486, T487), T491, T489, .(T486, T490)) :- app162(T487, T491, T489, T490).
qs1(.(T26, .(T27, T28)), T8) :- gt10(T26, T27).
qs1(.(T31, .(T33, T32)), T8) :- ','(gtc10(T31, T33), part20(T31, T32, X47, X48)).
qs1(.(T31, .(T148, T32)), T8) :- ','(gtc10(T31, T148), ','(partc20(T31, T32, T149, T54), p56(T148, T149, X269, X270, X271, X272, X273))).
qs1(.(T31, .(T33, T32)), T8) :- ','(gtc10(T31, T33), ','(partc20(T31, T32, T53, T54), ','(qsc51(T33, T53, T131), p52(T54, X10, T131, T31, T8)))).
qs1(.(T421, .(T422, T423)), T8) :- le100(T421, T422).
qs1(.(T426, .(T428, T427)), T8) :- ','(lec100(T426, T428), part20(T426, T427, X740, X741)).
qs1(.(T426, .(T436, T427)), T8) :- ','(lec100(T426, T436), ','(partc20(T426, T427, T434, T435), qs80(T434, X9))).
qs1(.(T426, .(T436, T427)), T8) :- ','(lec100(T426, T436), ','(partc20(T426, T427, T434, T435), ','(qsc80(T434, T442), p52(.(T436, T435), X10, T442, T426, T8)))).
qs1(.(T452, []), T8) :- ','(qsc155(T455), ','(qsc155(T460), app162(T455, T452, T460, T8))).

Clauses:

gtc10(s(T46), s(T47)) :- gtc10(T46, T47).
gtc10(s(0), 0).
partc20(T70, .(T73, T77), .(T73, X119), X120) :- ','(gtc10(T70, T73), partc20(T70, T77, X119, X120)).
partc20(T95, .(T98, T102), X170, .(T98, X171)) :- ','(lec33(T95, T98), partc20(T95, T102, X170, X171)).
partc20(T128, [], [], []).
lec33(s(T113), s(T115)) :- lec33(T113, T115).
lec33(0, s(T122)).
lec33(0, 0).
qc56(T148, T149, T153, T154, T160, T178, X273) :- ','(partc20(T148, T149, T153, T154), ','(qsc59(T153, T160), ','(qsc59(T154, T178), appc70(T160, T148, T178, X273)))).
qsc59(.(T171, T172), X335) :- qc56(T171, T172, X331, X332, X333, X334, X335).
qsc59([], []).
appc70(.(T201, T202), T203, T204, .(T201, X391)) :- appc70(T202, T203, T204, X391).
appc70([], T213, T214, .(T213, T214)).
lec100(s(T302), s(T303)) :- lec100(T302, T303).
lec100(0, s(T310)).
lec100(0, 0).
qsc80(.(T325, T231), X452) :- ','(partc87(T325, T231, T235, T236), ','(qsc80(T235, T319), ','(qsc80(T236, T324), appc121(T319, T325, T324, X452)))).
qsc80([], []).
appc121(.(T348, T349), T352, T353, .(T348, X657)) :- appc121(T349, T352, T353, X657).
appc121([], T362, T363, .(T362, T363)).
appc81(.(T389, T390), T391, T394, .(T389, T393)) :- appc81(T390, T391, T394, T393).
appc81([], T404, T405, .(T404, T405)).
qc52(T54, T217, T131, T31, T8) :- ','(qsc80(T54, T217), appc81(T131, T31, T217, T8)).
qsc155([]).
appc162(.(T486, T487), T491, T489, .(T486, T490)) :- appc162(T487, T491, T489, T490).
appc162([], T501, T502, .(T501, T502)).
qsc51(T148, T149, X273) :- qc56(T148, T149, X269, X270, X271, X272, X273).
partc87(T261, .(T257, T262), .(T257, X503), X504) :- ','(gtc10(T261, T257), partc20(T261, T262, X503, X504)).
partc87(T288, .(T284, T289), X554, .(T284, X555)) :- ','(lec100(T288, T284), partc20(T288, T289, X554, X555)).
partc87(T316, [], [], []).

Afs:

qs1(x1, x2) = qs1(x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (f,b)
gt10_in: (f,f) (b,f) (b,b)
gtc10_in: (f,f) (b,f) (b,b)
part20_in: (b,f,f,f) (b,b,f,f)
le33_in: (b,f) (b,b)
lec33_in: (b,f) (b,b)
partc20_in: (b,f,f,f) (b,b,f,f)
qsc51_in: (b,b,f)
qc56_in: (b,b,f,f,f,f,f)
qsc59_in: (b,f)
appc70_in: (b,b,b,f)
p52_in: (f,f,b,b,b)
qs80_in: (f,f) (b,f)
le100_in: (f,f) (b,b)
lec100_in: (f,f) (b,b)
partc87_in: (f,f,f,f) (b,b,f,f)
qsc80_in: (b,f) (f,f)
appc121_in: (b,b,b,f) (b,f,f,f)
app121_in: (b,b,b,f) (b,f,f,f)
app81_in: (b,b,f,b)
p56_in: (b,b,f,f,f,f,f)
qs59_in: (b,f)
app70_in: (b,b,b,f)
app162_in: (b,f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_AG(.(T26, .(T27, T28)), T8) → U42_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U43_AG(T31, T33, T32, T8, gtc10_in_aa(T31, T33))
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → U44_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → U45_AG(T31, T33, T32, T8, partc20_in_gaaa(T31, T32, T53, T54))
U45_AG(T31, T33, T32, T8, partc20_out_gaaa(T31, T32, T53, T54)) → U46_AG(T31, T33, T32, T8, T54, qsc51_in_gga(T33, T53, T131))
U46_AG(T31, T33, T32, T8, T54, qsc51_out_gga(T33, T53, T131)) → U47_AG(T31, T33, T32, T8, p52_in_aaggg(T54, X10, T131, T31, T8))
U46_AG(T31, T33, T32, T8, T54, qsc51_out_gga(T33, T53, T131)) → P52_IN_AAGGG(T54, X10, T131, T31, T8)
P52_IN_AAGGG(T54, X10, T131, T31, T8) → U38_AAGGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAGGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T256, .(T257, T258)), X452) → U22_AA(T256, T257, T258, X452, gt10_in_aa(T256, T257))
QS80_IN_AA(.(T256, .(T257, T258)), X452) → GT10_IN_AA(T256, T257)
QS80_IN_AA(.(T261, .(T257, T262)), X452) → U23_AA(T261, T257, T262, X452, gtc10_in_aa(T261, T257))
U23_AA(T261, T257, T262, X452, gtc10_out_aa(T261, T257)) → U24_AA(T261, T257, T262, X452, part20_in_gaaa(T261, T262, X503, X504))
U23_AA(T261, T257, T262, X452, gtc10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
QS80_IN_AA(.(T283, .(T284, T285)), X452) → U25_AA(T283, T284, T285, X452, le100_in_aa(T283, T284))
QS80_IN_AA(.(T283, .(T284, T285)), X452) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
QS80_IN_AA(.(T288, .(T284, T289)), X452) → U26_AA(T288, T284, T289, X452, lec100_in_aa(T288, T284))
U26_AA(T288, T284, T289, X452, lec100_out_aa(T288, T284)) → U27_AA(T288, T284, T289, X452, part20_in_gaaa(T288, T289, X554, X555))
U26_AA(T288, T284, T289, X452, lec100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U28_AA(T237, T231, X452, partc87_in_aaaa(T237, T231, T235, T236))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U29_AA(T237, T231, X452, qs80_in_ga(T235, X450))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
QS80_IN_GA(.(T256, .(T257, T258)), X452) → U22_GA(T256, T257, T258, X452, gt10_in_gg(T256, T257))
QS80_IN_GA(.(T256, .(T257, T258)), X452) → GT10_IN_GG(T256, T257)
GT10_IN_GG(s(T46), s(T47)) → U1_GG(T46, T47, gt10_in_gg(T46, T47))
GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)
QS80_IN_GA(.(T261, .(T257, T262)), X452) → U23_GA(T261, T257, T262, X452, gtc10_in_gg(T261, T257))
U23_GA(T261, T257, T262, X452, gtc10_out_gg(T261, T257)) → U24_GA(T261, T257, T262, X452, part20_in_ggaa(T261, T262, X503, X504))
U23_GA(T261, T257, T262, X452, gtc10_out_gg(T261, T257)) → PART20_IN_GGAA(T261, T262, X503, X504)
PART20_IN_GGAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GGAA(T70, T73, T74, X119, X120, gt10_in_gg(T70, T73))
PART20_IN_GGAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GG(T70, T73)
PART20_IN_GGAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GGAA(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U4_GGAA(T70, T73, T77, X119, X120, part20_in_ggaa(T70, T77, X119, X120))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77, X119, X120)
PART20_IN_GGAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GGAA(T95, T98, T99, X170, X171, le33_in_gg(T95, T98))
PART20_IN_GGAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GG(T95, T98)
LE33_IN_GG(s(T113), s(T115)) → U8_GG(T113, T115, le33_in_gg(T113, T115))
LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)
PART20_IN_GGAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GGAA(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U7_GGAA(T95, T98, T102, X170, X171, part20_in_ggaa(T95, T102, X170, X171))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102, X170, X171)
QS80_IN_GA(.(T283, .(T284, T285)), X452) → U25_GA(T283, T284, T285, X452, le100_in_gg(T283, T284))
QS80_IN_GA(.(T283, .(T284, T285)), X452) → LE100_IN_GG(T283, T284)
LE100_IN_GG(s(T302), s(T303)) → U21_GG(T302, T303, le100_in_gg(T302, T303))
LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)
QS80_IN_GA(.(T288, .(T284, T289)), X452) → U26_GA(T288, T284, T289, X452, lec100_in_gg(T288, T284))
U26_GA(T288, T284, T289, X452, lec100_out_gg(T288, T284)) → U27_GA(T288, T284, T289, X452, part20_in_ggaa(T288, T289, X554, X555))
U26_GA(T288, T284, T289, X452, lec100_out_gg(T288, T284)) → PART20_IN_GGAA(T288, T289, X554, X555)
QS80_IN_GA(.(T237, T231), X452) → U28_GA(T237, T231, X452, partc87_in_ggaa(T237, T231, T235, T236))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U29_GA(T237, T231, X452, qs80_in_ga(T235, X450))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
QS80_IN_GA(.(T325, T231), X452) → U32_GA(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U32_GA(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U33_GA(T325, T231, X452, T236, qsc80_in_ga(T235, T319))
U33_GA(T325, T231, X452, T236, qsc80_out_ga(T235, T319)) → U34_GA(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U34_GA(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U35_GA(T325, T231, X452, app121_in_ggga(T319, T325, T324, X452))
U34_GA(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → APP121_IN_GGGA(T319, T325, T324, X452)
APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → U36_GGGA(T348, T349, T352, T353, X657, app121_in_ggga(T349, T352, T353, X657))
APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GGGA(T349, T352, T353, X657)
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → U31_GA(T237, T231, X452, qs80_in_ga(T236, X451))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236, X451)
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U30_AA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → U31_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_AA(T236, X451)
QS80_IN_AA(.(T325, T231), X452) → U32_AA(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U32_AA(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U33_AA(T325, T231, X452, T236, qsc80_in_ga(T235, T319))
U33_AA(T325, T231, X452, T236, qsc80_out_ga(T235, T319)) → U34_AA(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
U34_AA(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U35_AA(T325, T231, X452, app121_in_gaaa(T319, T325, T324, X452))
U34_AA(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → APP121_IN_GAAA(T319, T325, T324, X452)
APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → U36_GAAA(T348, T349, T352, T353, X657, app121_in_gaaa(T349, T352, T353, X657))
APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GAAA(T349, T352, T353, X657)
P52_IN_AAGGG(T54, T217, T131, T31, T8) → U39_AAGGG(T54, T217, T131, T31, T8, qsc80_in_aa(T54, T217))
U39_AAGGG(T54, T217, T131, T31, T8, qsc80_out_aa(T54, T217)) → U40_AAGGG(T54, T217, T131, T31, T8, app81_in_ggag(T131, T31, T217, T8))
U39_AAGGG(T54, T217, T131, T31, T8, qsc80_out_aa(T54, T217)) → APP81_IN_GGAG(T131, T31, T217, T8)
APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → U37_GGAG(T389, T390, T391, T394, T393, app81_in_ggag(T390, T391, T394, T393))
APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T31, .(T148, T32)), T8) → U48_AG(T31, T148, T32, T8, gtc10_in_aa(T31, T148))
U48_AG(T31, T148, T32, T8, gtc10_out_aa(T31, T148)) → U49_AG(T31, T148, T32, T8, partc20_in_gaaa(T31, T32, T149, T54))
U49_AG(T31, T148, T32, T8, partc20_out_gaaa(T31, T32, T149, T54)) → U50_AG(T31, T148, T32, T8, p56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
U49_AG(T31, T148, T32, T8, partc20_out_gaaa(T31, T32, T149, T54)) → P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GGAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_ggaa(T148, T149, X269, X270))
P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GGAA(T148, T149, X269, X270)
P56_IN_GGAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U11_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_ga(T153, X271))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153, X271)
QS59_IN_GA(.(T171, T172), X335) → U19_GA(T171, T172, X335, p56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_GA(.(T171, T172), X335) → P56_IN_GGAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_in_ga(T153, T160))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → U14_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_ga(T154, X272))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154, X272)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
U15_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U16_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
U16_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U18_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_ggga(T160, T148, T178, X273))
U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → APP70_IN_GGGA(T160, T148, T178, X273)
APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → U20_GGGA(T201, T202, T203, T204, X391, app70_in_ggga(T202, T203, T204, X391))
APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_GGGA(T202, T203, T204, X391)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, lec100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, lec100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, lec100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, lec100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, lec100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, partc20_in_gaaa(T426, T427, T434, T435))
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_ga(T434, X9))
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_GA(T434, X9)
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qsc80_in_ga(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qsc80_out_ga(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaggg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qsc80_out_ga(T434, T442)) → P52_IN_AAGGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qsc155_in_a(T455))
U59_AG(T452, T8, qsc155_out_a(T455)) → U60_AG(T452, T8, T455, qsc155_in_a(T460))
U60_AG(T452, T8, T455, qsc155_out_a(T460)) → U61_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U60_AG(T452, T8, T455, qsc155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U41_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gt10_in_aa(x1, x2) = gt10_in_aa
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
part20_in_gaaa(x1, x2, x3, x4) = part20_in_gaaa(x1)
gt10_in_ga(x1, x2) = gt10_in_ga(x1)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
le33_in_ga(x1, x2) = le33_in_ga(x1)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
p52_in_aaggg(x1, x2, x3, x4, x5) = p52_in_aaggg(x3, x4, x5)
qs80_in_aa(x1, x2) = qs80_in_aa
le100_in_aa(x1, x2) = le100_in_aa
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
qs80_in_ga(x1, x2) = qs80_in_ga(x1)
gt10_in_gg(x1, x2) = gt10_in_gg(x1, x2)
part20_in_ggaa(x1, x2, x3, x4) = part20_in_ggaa(x1, x2)
le33_in_gg(x1, x2) = le33_in_gg(x1, x2)
le100_in_gg(x1, x2) = le100_in_gg(x1, x2)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
app121_in_ggga(x1, x2, x3, x4) = app121_in_ggga(x1, x2, x3)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
app121_in_gaaa(x1, x2, x3, x4) = app121_in_gaaa(x1)
app81_in_ggag(x1, x2, x3, x4) = app81_in_ggag(x1, x2, x4)
p56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = p56_in_ggaaaaa(x1, x2)
qs59_in_ga(x1, x2) = qs59_in_ga(x1)
app70_in_ggga(x1, x2, x3, x4) = app70_in_ggga(x1, x2, x3)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
app162_in_gagg(x1, x2, x3, x4) = app162_in_gagg(x1, x3, x4)
QS1_IN_AG(x1, x2) = QS1_IN_AG(x2)
U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5)
GT10_IN_AA(x1, x2) = GT10_IN_AA
U1_AA(x1, x2, x3) = U1_AA(x3)
U43_AG(x1, x2, x3, x4, x5) = U43_AG(x4, x5)
U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5)
PART20_IN_GAAA(x1, x2, x3, x4) = PART20_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6)
GT10_IN_GA(x1, x2) = GT10_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
LE33_IN_GA(x1, x2) = LE33_IN_GA(x1)
U8_GA(x1, x2, x3) = U8_GA(x1, x3)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6)
U45_AG(x1, x2, x3, x4, x5) = U45_AG(x1, x2, x4, x5)
U46_AG(x1, x2, x3, x4, x5, x6) = U46_AG(x1, x4, x6)
U47_AG(x1, x2, x3, x4, x5) = U47_AG(x4, x5)
P52_IN_AAGGG(x1, x2, x3, x4, x5) = P52_IN_AAGGG(x3, x4, x5)
U38_AAGGG(x1, x2, x3, x4, x5, x6) = U38_AAGGG(x3, x4, x5, x6)
QS80_IN_AA(x1, x2) = QS80_IN_AA
U22_AA(x1, x2, x3, x4, x5) = U22_AA(x5)
U23_AA(x1, x2, x3, x4, x5) = U23_AA(x5)
U24_AA(x1, x2, x3, x4, x5) = U24_AA(x5)
U25_AA(x1, x2, x3, x4, x5) = U25_AA(x5)
LE100_IN_AA(x1, x2) = LE100_IN_AA
U21_AA(x1, x2, x3) = U21_AA(x3)
U26_AA(x1, x2, x3, x4, x5) = U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5) = U27_AA(x5)
U28_AA(x1, x2, x3, x4) = U28_AA(x4)
U29_AA(x1, x2, x3, x4) = U29_AA(x4)
QS80_IN_GA(x1, x2) = QS80_IN_GA(x1)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)
GT10_IN_GG(x1, x2) = GT10_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3)
U23_GA(x1, x2, x3, x4, x5) = U23_GA(x1, x2, x3, x5)
U24_GA(x1, x2, x3, x4, x5) = U24_GA(x1, x2, x3, x5)
PART20_IN_GGAA(x1, x2, x3, x4) = PART20_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6)
U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
LE33_IN_GG(x1, x2) = LE33_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
U25_GA(x1, x2, x3, x4, x5) = U25_GA(x1, x2, x3, x5)
LE100_IN_GG(x1, x2) = LE100_IN_GG(x1, x2)
U21_GG(x1, x2, x3) = U21_GG(x1, x2, x3)
U26_GA(x1, x2, x3, x4, x5) = U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5) = U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4) = U28_GA(x1, x2, x4)
U29_GA(x1, x2, x3, x4) = U29_GA(x1, x2, x4)
U32_GA(x1, x2, x3, x4) = U32_GA(x1, x2, x4)
U33_GA(x1, x2, x3, x4, x5) = U33_GA(x1, x2, x4, x5)
U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x2, x4, x5)
U35_GA(x1, x2, x3, x4) = U35_GA(x1, x2, x4)
APP121_IN_GGGA(x1, x2, x3, x4) = APP121_IN_GGGA(x1, x2, x3)
U36_GGGA(x1, x2, x3, x4, x5, x6) = U36_GGGA(x1, x2, x3, x4, x6)
U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x4, x5)
U31_GA(x1, x2, x3, x4) = U31_GA(x1, x2, x4)
U30_AA(x1, x2, x3, x4, x5) = U30_AA(x5)
U31_AA(x1, x2, x3, x4) = U31_AA(x4)
U32_AA(x1, x2, x3, x4) = U32_AA(x4)
U33_AA(x1, x2, x3, x4, x5) = U33_AA(x5)
U34_AA(x1, x2, x3, x4, x5) = U34_AA(x4, x5)
U35_AA(x1, x2, x3, x4) = U35_AA(x4)
APP121_IN_GAAA(x1, x2, x3, x4) = APP121_IN_GAAA(x1)
U36_GAAA(x1, x2, x3, x4, x5, x6) = U36_GAAA(x1, x2, x6)
U39_AAGGG(x1, x2, x3, x4, x5, x6) = U39_AAGGG(x3, x4, x5, x6)
U40_AAGGG(x1, x2, x3, x4, x5, x6) = U40_AAGGG(x3, x4, x5, x6)
APP81_IN_GGAG(x1, x2, x3, x4) = APP81_IN_GGAG(x1, x2, x4)
U37_GGAG(x1, x2, x3, x4, x5, x6) = U37_GGAG(x1, x2, x3, x5, x6)
U48_AG(x1, x2, x3, x4, x5) = U48_AG(x4, x5)
U49_AG(x1, x2, x3, x4, x5) = U49_AG(x2, x4, x5)
U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5)
P56_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = P56_IN_GGAAAAA(x1, x2)
U9_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U9_GGAAAAA(x1, x2, x8)
U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8)
U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8)
QS59_IN_GA(x1, x2) = QS59_IN_GA(x1)
U19_GA(x1, x2, x3, x4) = U19_GA(x1, x2, x4)
U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8)
U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8)
U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GGAAAAA(x1, x2, x8)
U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GGAAAAA(x1, x2, x8)
U16_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_GGAAAAA(x1, x2, x4, x8)
U17_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U17_GGAAAAA(x1, x2, x5, x8)
U18_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U18_GGAAAAA(x1, x2, x8)
APP70_IN_GGGA(x1, x2, x3, x4) = APP70_IN_GGGA(x1, x2, x3)
U20_GGGA(x1, x2, x3, x4, x5, x6) = U20_GGGA(x1, x2, x3, x4, x6)
U51_AG(x1, x2, x3, x4, x5) = U51_AG(x4, x5)
U52_AG(x1, x2, x3, x4, x5) = U52_AG(x4, x5)
U53_AG(x1, x2, x3, x4, x5) = U53_AG(x4, x5)
U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5) = U55_AG(x4, x5)
U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5)
U57_AG(x1, x2, x3, x4, x5, x6) = U57_AG(x1, x4, x6)
U58_AG(x1, x2, x3, x4, x5) = U58_AG(x4, x5)
U59_AG(x1, x2, x3) = U59_AG(x2, x3)
U60_AG(x1, x2, x3, x4) = U60_AG(x2, x3, x4)
U61_AG(x1, x2, x3) = U61_AG(x2, x3)
APP162_IN_GAGG(x1, x2, x3, x4) = APP162_IN_GAGG(x1, x3, x4)
U41_GAGG(x1, x2, x3, x4, x5, x6) = U41_GAGG(x1, x2, x4, x5, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

QS1_IN_AG(.(T26, .(T27, T28)), T8) → U42_AG(T26, T27, T28, T8, gt10_in_aa(T26, T27))
QS1_IN_AG(.(T26, .(T27, T28)), T8) → GT10_IN_AA(T26, T27)
GT10_IN_AA(s(T46), s(T47)) → U1_AA(T46, T47, gt10_in_aa(T46, T47))
GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)
QS1_IN_AG(.(T31, .(T33, T32)), T8) → U43_AG(T31, T33, T32, T8, gtc10_in_aa(T31, T33))
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → U44_AG(T31, T33, T32, T8, part20_in_gaaa(T31, T32, X47, X48))
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → PART20_IN_GAAA(T31, T32, X47, X48)
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GAAA(T70, T73, T74, X119, X120, gt10_in_ga(T70, T73))
PART20_IN_GAAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GA(T70, T73)
GT10_IN_GA(s(T46), s(T47)) → U1_GA(T46, T47, gt10_in_ga(T46, T47))
GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)
PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X119, X120, part20_in_gaaa(T70, T77, X119, X120))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GAAA(T95, T98, T99, X170, X171, le33_in_ga(T95, T98))
PART20_IN_GAAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GA(T95, T98)
LE33_IN_GA(s(T113), s(T115)) → U8_GA(T113, T115, le33_in_ga(T113, T115))
LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U7_GAAA(T95, T98, T102, X170, X171, part20_in_gaaa(T95, T102, X170, X171))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)
U43_AG(T31, T33, T32, T8, gtc10_out_aa(T31, T33)) → U45_AG(T31, T33, T32, T8, partc20_in_gaaa(T31, T32, T53, T54))
U45_AG(T31, T33, T32, T8, partc20_out_gaaa(T31, T32, T53, T54)) → U46_AG(T31, T33, T32, T8, T54, qsc51_in_gga(T33, T53, T131))
U46_AG(T31, T33, T32, T8, T54, qsc51_out_gga(T33, T53, T131)) → U47_AG(T31, T33, T32, T8, p52_in_aaggg(T54, X10, T131, T31, T8))
U46_AG(T31, T33, T32, T8, T54, qsc51_out_gga(T33, T53, T131)) → P52_IN_AAGGG(T54, X10, T131, T31, T8)
P52_IN_AAGGG(T54, X10, T131, T31, T8) → U38_AAGGG(T54, X10, T131, T31, T8, qs80_in_aa(T54, X10))
P52_IN_AAGGG(T54, X10, T131, T31, T8) → QS80_IN_AA(T54, X10)
QS80_IN_AA(.(T256, .(T257, T258)), X452) → U22_AA(T256, T257, T258, X452, gt10_in_aa(T256, T257))
QS80_IN_AA(.(T256, .(T257, T258)), X452) → GT10_IN_AA(T256, T257)
QS80_IN_AA(.(T261, .(T257, T262)), X452) → U23_AA(T261, T257, T262, X452, gtc10_in_aa(T261, T257))
U23_AA(T261, T257, T262, X452, gtc10_out_aa(T261, T257)) → U24_AA(T261, T257, T262, X452, part20_in_gaaa(T261, T262, X503, X504))
U23_AA(T261, T257, T262, X452, gtc10_out_aa(T261, T257)) → PART20_IN_GAAA(T261, T262, X503, X504)
QS80_IN_AA(.(T283, .(T284, T285)), X452) → U25_AA(T283, T284, T285, X452, le100_in_aa(T283, T284))
QS80_IN_AA(.(T283, .(T284, T285)), X452) → LE100_IN_AA(T283, T284)
LE100_IN_AA(s(T302), s(T303)) → U21_AA(T302, T303, le100_in_aa(T302, T303))
LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)
QS80_IN_AA(.(T288, .(T284, T289)), X452) → U26_AA(T288, T284, T289, X452, lec100_in_aa(T288, T284))
U26_AA(T288, T284, T289, X452, lec100_out_aa(T288, T284)) → U27_AA(T288, T284, T289, X452, part20_in_gaaa(T288, T289, X554, X555))
U26_AA(T288, T284, T289, X452, lec100_out_aa(T288, T284)) → PART20_IN_GAAA(T288, T289, X554, X555)
QS80_IN_AA(.(T237, T231), X452) → U28_AA(T237, T231, X452, partc87_in_aaaa(T237, T231, T235, T236))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U29_AA(T237, T231, X452, qs80_in_ga(T235, X450))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
QS80_IN_GA(.(T256, .(T257, T258)), X452) → U22_GA(T256, T257, T258, X452, gt10_in_gg(T256, T257))
QS80_IN_GA(.(T256, .(T257, T258)), X452) → GT10_IN_GG(T256, T257)
GT10_IN_GG(s(T46), s(T47)) → U1_GG(T46, T47, gt10_in_gg(T46, T47))
GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)
QS80_IN_GA(.(T261, .(T257, T262)), X452) → U23_GA(T261, T257, T262, X452, gtc10_in_gg(T261, T257))
U23_GA(T261, T257, T262, X452, gtc10_out_gg(T261, T257)) → U24_GA(T261, T257, T262, X452, part20_in_ggaa(T261, T262, X503, X504))
U23_GA(T261, T257, T262, X452, gtc10_out_gg(T261, T257)) → PART20_IN_GGAA(T261, T262, X503, X504)
PART20_IN_GGAA(T70, .(T73, T74), .(T73, X119), X120) → U2_GGAA(T70, T73, T74, X119, X120, gt10_in_gg(T70, T73))
PART20_IN_GGAA(T70, .(T73, T74), .(T73, X119), X120) → GT10_IN_GG(T70, T73)
PART20_IN_GGAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GGAA(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U4_GGAA(T70, T73, T77, X119, X120, part20_in_ggaa(T70, T77, X119, X120))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77, X119, X120)
PART20_IN_GGAA(T95, .(T98, T99), X170, .(T98, X171)) → U5_GGAA(T95, T98, T99, X170, X171, le33_in_gg(T95, T98))
PART20_IN_GGAA(T95, .(T98, T99), X170, .(T98, X171)) → LE33_IN_GG(T95, T98)
LE33_IN_GG(s(T113), s(T115)) → U8_GG(T113, T115, le33_in_gg(T113, T115))
LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)
PART20_IN_GGAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GGAA(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U7_GGAA(T95, T98, T102, X170, X171, part20_in_ggaa(T95, T102, X170, X171))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102, X170, X171)
QS80_IN_GA(.(T283, .(T284, T285)), X452) → U25_GA(T283, T284, T285, X452, le100_in_gg(T283, T284))
QS80_IN_GA(.(T283, .(T284, T285)), X452) → LE100_IN_GG(T283, T284)
LE100_IN_GG(s(T302), s(T303)) → U21_GG(T302, T303, le100_in_gg(T302, T303))
LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)
QS80_IN_GA(.(T288, .(T284, T289)), X452) → U26_GA(T288, T284, T289, X452, lec100_in_gg(T288, T284))
U26_GA(T288, T284, T289, X452, lec100_out_gg(T288, T284)) → U27_GA(T288, T284, T289, X452, part20_in_ggaa(T288, T289, X554, X555))
U26_GA(T288, T284, T289, X452, lec100_out_gg(T288, T284)) → PART20_IN_GGAA(T288, T289, X554, X555)
QS80_IN_GA(.(T237, T231), X452) → U28_GA(T237, T231, X452, partc87_in_ggaa(T237, T231, T235, T236))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U29_GA(T237, T231, X452, qs80_in_ga(T235, X450))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
QS80_IN_GA(.(T325, T231), X452) → U32_GA(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U32_GA(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U33_GA(T325, T231, X452, T236, qsc80_in_ga(T235, T319))
U33_GA(T325, T231, X452, T236, qsc80_out_ga(T235, T319)) → U34_GA(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U34_GA(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U35_GA(T325, T231, X452, app121_in_ggga(T319, T325, T324, X452))
U34_GA(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → APP121_IN_GGGA(T319, T325, T324, X452)
APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → U36_GGGA(T348, T349, T352, T353, X657, app121_in_ggga(T349, T352, T353, X657))
APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GGGA(T349, T352, T353, X657)
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → U31_GA(T237, T231, X452, qs80_in_ga(T236, X451))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236, X451)
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U30_AA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → U31_AA(T237, T231, X452, qs80_in_aa(T236, X451))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_AA(T236, X451)
QS80_IN_AA(.(T325, T231), X452) → U32_AA(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U32_AA(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U33_AA(T325, T231, X452, T236, qsc80_in_ga(T235, T319))
U33_AA(T325, T231, X452, T236, qsc80_out_ga(T235, T319)) → U34_AA(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
U34_AA(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U35_AA(T325, T231, X452, app121_in_gaaa(T319, T325, T324, X452))
U34_AA(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → APP121_IN_GAAA(T319, T325, T324, X452)
APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → U36_GAAA(T348, T349, T352, T353, X657, app121_in_gaaa(T349, T352, T353, X657))
APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GAAA(T349, T352, T353, X657)
P52_IN_AAGGG(T54, T217, T131, T31, T8) → U39_AAGGG(T54, T217, T131, T31, T8, qsc80_in_aa(T54, T217))
U39_AAGGG(T54, T217, T131, T31, T8, qsc80_out_aa(T54, T217)) → U40_AAGGG(T54, T217, T131, T31, T8, app81_in_ggag(T131, T31, T217, T8))
U39_AAGGG(T54, T217, T131, T31, T8, qsc80_out_aa(T54, T217)) → APP81_IN_GGAG(T131, T31, T217, T8)
APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → U37_GGAG(T389, T390, T391, T394, T393, app81_in_ggag(T390, T391, T394, T393))
APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T394, T393)
QS1_IN_AG(.(T31, .(T148, T32)), T8) → U48_AG(T31, T148, T32, T8, gtc10_in_aa(T31, T148))
U48_AG(T31, T148, T32, T8, gtc10_out_aa(T31, T148)) → U49_AG(T31, T148, T32, T8, partc20_in_gaaa(T31, T32, T149, T54))
U49_AG(T31, T148, T32, T8, partc20_out_gaaa(T31, T32, T149, T54)) → U50_AG(T31, T148, T32, T8, p56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
U49_AG(T31, T148, T32, T8, partc20_out_gaaa(T31, T32, T149, T54)) → P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273)
P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273) → U9_GGAAAAA(T148, T149, X269, X270, X271, X272, X273, part20_in_ggaa(T148, T149, X269, X270))
P56_IN_GGAAAAA(T148, T149, X269, X270, X271, X272, X273) → PART20_IN_GGAA(T148, T149, X269, X270)
P56_IN_GGAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U11_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, qs59_in_ga(T153, X271))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153, X271)
QS59_IN_GA(.(T171, T172), X335) → U19_GA(T171, T172, X335, p56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
QS59_IN_GA(.(T171, T172), X335) → P56_IN_GGAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_in_ga(T153, T160))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → U14_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qs59_in_ga(T154, X272))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154, X272)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, T178, X273) → U15_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
U15_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U16_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
U16_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U18_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, app70_in_ggga(T160, T148, T178, X273))
U17_GGAAAAA(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → APP70_IN_GGGA(T160, T148, T178, X273)
APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → U20_GGGA(T201, T202, T203, T204, X391, app70_in_ggga(T202, T203, T204, X391))
APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_GGGA(T202, T203, T204, X391)
QS1_IN_AG(.(T421, .(T422, T423)), T8) → U51_AG(T421, T422, T423, T8, le100_in_aa(T421, T422))
QS1_IN_AG(.(T421, .(T422, T423)), T8) → LE100_IN_AA(T421, T422)
QS1_IN_AG(.(T426, .(T428, T427)), T8) → U52_AG(T426, T428, T427, T8, lec100_in_aa(T426, T428))
U52_AG(T426, T428, T427, T8, lec100_out_aa(T426, T428)) → U53_AG(T426, T428, T427, T8, part20_in_gaaa(T426, T427, X740, X741))
U52_AG(T426, T428, T427, T8, lec100_out_aa(T426, T428)) → PART20_IN_GAAA(T426, T427, X740, X741)
QS1_IN_AG(.(T426, .(T436, T427)), T8) → U54_AG(T426, T436, T427, T8, lec100_in_aa(T426, T436))
U54_AG(T426, T436, T427, T8, lec100_out_aa(T426, T436)) → U55_AG(T426, T436, T427, T8, partc20_in_gaaa(T426, T427, T434, T435))
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → U56_AG(T426, T436, T427, T8, qs80_in_ga(T434, X9))
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → QS80_IN_GA(T434, X9)
U55_AG(T426, T436, T427, T8, partc20_out_gaaa(T426, T427, T434, T435)) → U57_AG(T426, T436, T427, T8, T435, qsc80_in_ga(T434, T442))
U57_AG(T426, T436, T427, T8, T435, qsc80_out_ga(T434, T442)) → U58_AG(T426, T436, T427, T8, p52_in_aaggg(.(T436, T435), X10, T442, T426, T8))
U57_AG(T426, T436, T427, T8, T435, qsc80_out_ga(T434, T442)) → P52_IN_AAGGG(.(T436, T435), X10, T442, T426, T8)
QS1_IN_AG(.(T452, []), T8) → U59_AG(T452, T8, qsc155_in_a(T455))
U59_AG(T452, T8, qsc155_out_a(T455)) → U60_AG(T452, T8, T455, qsc155_in_a(T460))
U60_AG(T452, T8, T455, qsc155_out_a(T460)) → U61_AG(T452, T8, app162_in_gagg(T455, T452, T460, T8))
U60_AG(T452, T8, T455, qsc155_out_a(T460)) → APP162_IN_GAGG(T455, T452, T460, T8)
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → U41_GAGG(T486, T487, T491, T489, T490, app162_in_gagg(T487, T491, T489, T490))
APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 17 SCCs with 103 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
APP162_IN_GAGG(x1, x2, x3, x4) = APP162_IN_GAGG(x1, x3, x4)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP162_IN_GAGG(.(T486, T487), T491, T489, .(T486, T490)) → APP162_IN_GAGG(T487, T491, T489, T490)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP162_IN_GAGG(x1, x2, x3, x4) = APP162_IN_GAGG(x1, x3, x4)

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:

APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP162_IN_GAGG(.(T486, T487), T489, .(T486, T490)) → APP162_IN_GAGG(T487, T489, T490)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(15) YES

(16) Obligation:

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

APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_GGGA(T202, T203, T204, X391)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
APP70_IN_GGGA(x1, x2, x3, x4) = APP70_IN_GGGA(x1, x2, x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP70_IN_GGGA(.(T201, T202), T203, T204, .(T201, X391)) → APP70_IN_GGGA(T202, T203, T204, X391)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP70_IN_GGGA(x1, x2, x3, x4) = APP70_IN_GGGA(x1, x2, x3)

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:

APP70_IN_GGGA(.(T201, T202), T203, T204) → APP70_IN_GGGA(T202, T203, T204)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP70_IN_GGGA(.(T201, T202), T203, T204) → APP70_IN_GGGA(T202, T203, T204)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(22) YES

(23) Obligation:

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

APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T394, T393)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
APP81_IN_GGAG(x1, x2, x3, x4) = APP81_IN_GGAG(x1, x2, x4)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APP81_IN_GGAG(.(T389, T390), T391, T394, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T394, T393)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP81_IN_GGAG(x1, x2, x3, x4) = APP81_IN_GGAG(x1, x2, x4)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APP81_IN_GGAG(.(T389, T390), T391, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T393)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP81_IN_GGAG(.(T389, T390), T391, .(T389, T393)) → APP81_IN_GGAG(T390, T391, T393)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

(29) YES

(30) Obligation:

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

APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GAAA(T349, T352, T353, X657)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
APP121_IN_GAAA(x1, x2, x3, x4) = APP121_IN_GAAA(x1)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APP121_IN_GAAA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GAAA(T349, T352, T353, X657)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP121_IN_GAAA(x1, x2, x3, x4) = APP121_IN_GAAA(x1)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP121_IN_GAAA(.(T348, T349)) → APP121_IN_GAAA(T349)

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

(35) 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:

APP121_IN_GAAA(.(T348, T349)) → APP121_IN_GAAA(T349)
The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GGGA(T349, T352, T353, X657)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
APP121_IN_GGGA(x1, x2, x3, x4) = APP121_IN_GGGA(x1, x2, x3)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

APP121_IN_GGGA(.(T348, T349), T352, T353, .(T348, X657)) → APP121_IN_GGGA(T349, T352, T353, X657)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP121_IN_GGGA(x1, x2, x3, x4) = APP121_IN_GGGA(x1, x2, x3)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

APP121_IN_GGGA(.(T348, T349), T352, T353) → APP121_IN_GGGA(T349, T352, T353)

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

(42) 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:

APP121_IN_GGGA(.(T348, T349), T352, T353) → APP121_IN_GGGA(T349, T352, T353)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(43) YES

(44) Obligation:

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

LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
LE100_IN_GG(x1, x2) = LE100_IN_GG(x1, x2)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)

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

(47) PiDPToQDPProof (EQUIVALENT transformation)

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

(48) Obligation:

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

LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)

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

(49) 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:

LE100_IN_GG(s(T302), s(T303)) → LE100_IN_GG(T302, T303)
The graph contains the following edges 1 > 1, 2 > 2

(50) YES

(51) Obligation:

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

LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
LE33_IN_GG(x1, x2) = LE33_IN_GG(x1, x2)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)

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

(54) PiDPToQDPProof (EQUIVALENT transformation)

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

(55) Obligation:

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

LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)

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

(56) 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:

LE33_IN_GG(s(T113), s(T115)) → LE33_IN_GG(T113, T115)
The graph contains the following edges 1 > 1, 2 > 2

(57) YES

(58) Obligation:

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

GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
GT10_IN_GG(x1, x2) = GT10_IN_GG(x1, x2)

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

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)

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

(61) PiDPToQDPProof (EQUIVALENT transformation)

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

(62) Obligation:

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

GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)

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

(63) 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:

GT10_IN_GG(s(T46), s(T47)) → GT10_IN_GG(T46, T47)
The graph contains the following edges 1 > 1, 2 > 2

(64) YES

(65) Obligation:

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

PART20_IN_GGAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GGAA(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77, X119, X120)
PART20_IN_GGAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GGAA(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
PART20_IN_GGAA(x1, x2, x3, x4) = PART20_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

PART20_IN_GGAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GGAA(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
U3_GGAA(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77, X119, X120)
PART20_IN_GGAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GGAA(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
U6_GGAA(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
.(x1, x2) = .(x1, x2)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
PART20_IN_GGAA(x1, x2, x3, x4) = PART20_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

PART20_IN_GGAA(T70, .(T73, T77)) → U3_GGAA(T70, T73, T77, gtc10_in_gg(T70, T73))
U3_GGAA(T70, T73, T77, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77)
PART20_IN_GGAA(T95, .(T98, T102)) → U6_GGAA(T95, T98, T102, lec33_in_gg(T95, T98))
U6_GGAA(T95, T98, T102, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102)

The TRS R consists of the following rules:

gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))

The set Q consists of the following terms:

gtc10_in_gg(x0, x1)
lec33_in_gg(x0, x1)
U63_gg(x0, x1, x2)
U68_gg(x0, x1, x2)

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

(70) 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:

U3_GGAA(T70, T73, T77, gtc10_out_gg(T70, T73)) → PART20_IN_GGAA(T70, T77)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
U6_GGAA(T95, T98, T102, lec33_out_gg(T95, T98)) → PART20_IN_GGAA(T95, T102)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
PART20_IN_GGAA(T70, .(T73, T77)) → U3_GGAA(T70, T73, T77, gtc10_in_gg(T70, T73))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
PART20_IN_GGAA(T95, .(T98, T102)) → U6_GGAA(T95, T98, T102, lec33_in_gg(T95, T98))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(71) YES

(72) Obligation:

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

P56_IN_GGAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153, X271)
QS59_IN_GA(.(T171, T172), X335) → P56_IN_GGAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_in_ga(T153, T160))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154, X272)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
P56_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = P56_IN_GGAAAAA(x1, x2)
U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8)
QS59_IN_GA(x1, x2) = QS59_IN_GA(x1)
U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8)
U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

P56_IN_GGAAAAA(T148, T149, T153, T154, X271, X272, X273) → U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U10_GGAAAAA(T148, T149, T153, T154, X271, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153, X271)
QS59_IN_GA(.(T171, T172), X335) → P56_IN_GGAAAAA(T171, T172, X331, X332, X333, X334, X335)
P56_IN_GGAAAAA(T148, T149, T153, T154, T160, X272, X273) → U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_in_ggaa(T148, T149, T153, T154))
U12_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_in_ga(T153, T160))
U13_GGAAAAA(T148, T149, T153, T154, T160, X272, X273, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154, X272)

The TRS R consists of the following rules:

partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
P56_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = P56_IN_GGAAAAA(x1, x2)
U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8)
QS59_IN_GA(x1, x2) = QS59_IN_GA(x1)
U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8)
U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

P56_IN_GGAAAAA(T148, T149) → U10_GGAAAAA(T148, T149, partc20_in_ggaa(T148, T149))
U10_GGAAAAA(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153)
QS59_IN_GA(.(T171, T172)) → P56_IN_GGAAAAA(T171, T172)
P56_IN_GGAAAAA(T148, T149) → U12_GGAAAAA(T148, T149, partc20_in_ggaa(T148, T149))
U12_GGAAAAA(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T154, qsc59_in_ga(T153))
U13_GGAAAAA(T148, T149, T154, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154)

The TRS R consists of the following rules:

partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
qsc59_in_ga(.(T171, T172)) → U73_ga(T171, T172, qc56_in_ggaaaaa(T171, T172))
qsc59_in_ga([]) → qsc59_out_ga([], [])
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U73_ga(T171, T172, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
qc56_in_ggaaaaa(T148, T149) → U69_ggaaaaa(T148, T149, partc20_in_ggaa(T148, T149))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U69_ggaaaaa(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, qsc59_in_ga(T153))
U70_ggaaaaa(T148, T149, T153, T154, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, qsc59_in_ga(T154))
U71_ggaaaaa(T148, T149, T153, T154, T160, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, appc70_in_ggga(T160, T148, T178))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
appc70_in_ggga(.(T201, T202), T203, T204) → U74_ggga(T201, T202, T203, T204, appc70_in_ggga(T202, T203, T204))
appc70_in_ggga([], T213, T214) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))

The set Q consists of the following terms:

partc20_in_ggaa(x0, x1)
qsc59_in_ga(x0)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U73_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
qc56_in_ggaaaaa(x0, x1)
U63_gg(x0, x1, x2)
U68_gg(x0, x1, x2)
U69_ggaaaaa(x0, x1, x2)
U70_ggaaaaa(x0, x1, x2, x3, x4)
U71_ggaaaaa(x0, x1, x2, x3, x4, x5)
U72_ggaaaaa(x0, x1, x2, x3, x4, x5, x6)
appc70_in_ggga(x0, x1, x2)
U74_ggga(x0, x1, x2, x3, x4)

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P56_IN_GGAAAAA(T148, T149) → U10_GGAAAAA(T148, T149, partc20_in_ggaa(T148, T149))
P56_IN_GGAAAAA(T148, T149) → U12_GGAAAAA(T148, T149, partc20_in_ggaa(T148, T149))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(P56_IN_GGAAAAA(x₁, x₂)) = 1 + x₂
POL(QS59_IN_GA(x₁)) = x₁
POL(U10_GGAAAAA(x₁, x₂, x₃)) = x₃
POL(U12_GGAAAAA(x₁, x₂, x₃)) = x₃
POL(U13_GGAAAAA(x₁, x₂, x₃, x₄)) = x₃
POL(U63_gg(x₁, x₂, x₃)) = 0
POL(U64_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U65_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U66_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U67_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U68_gg(x₁, x₂, x₃)) = 0
POL(U69_ggaaaaa(x₁, x₂, x₃)) = 0
POL(U70_ggaaaaa(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U71_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆)) = 0
POL(U72_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 0
POL(U73_ga(x₁, x₂, x₃)) = 0
POL(U74_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL([]) = 0
POL(appc70_in_ggga(x₁, x₂, x₃)) = 0
POL(appc70_out_ggga(x₁, x₂, x₃, x₄)) = 0
POL(gtc10_in_gg(x₁, x₂)) = 0
POL(gtc10_out_gg(x₁, x₂)) = 0
POL(lec33_in_gg(x₁, x₂)) = 0
POL(lec33_out_gg(x₁, x₂)) = 0
POL(partc20_in_ggaa(x₁, x₂)) = x₂
POL(partc20_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(qc56_in_ggaaaaa(x₁, x₂)) = 0
POL(qc56_out_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 0
POL(qsc59_in_ga(x₁)) = 0
POL(qsc59_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))

(78) Obligation:

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

U10_GGAAAAA(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → QS59_IN_GA(T153)
QS59_IN_GA(.(T171, T172)) → P56_IN_GGAAAAA(T171, T172)
U12_GGAAAAA(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → U13_GGAAAAA(T148, T149, T154, qsc59_in_ga(T153))
U13_GGAAAAA(T148, T149, T154, qsc59_out_ga(T153, T160)) → QS59_IN_GA(T154)

The TRS R consists of the following rules:

partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
qsc59_in_ga(.(T171, T172)) → U73_ga(T171, T172, qc56_in_ggaaaaa(T171, T172))
qsc59_in_ga([]) → qsc59_out_ga([], [])
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U73_ga(T171, T172, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
qc56_in_ggaaaaa(T148, T149) → U69_ggaaaaa(T148, T149, partc20_in_ggaa(T148, T149))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U69_ggaaaaa(T148, T149, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, qsc59_in_ga(T153))
U70_ggaaaaa(T148, T149, T153, T154, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, qsc59_in_ga(T154))
U71_ggaaaaa(T148, T149, T153, T154, T160, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, appc70_in_ggga(T160, T148, T178))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
appc70_in_ggga(.(T201, T202), T203, T204) → U74_ggga(T201, T202, T203, T204, appc70_in_ggga(T202, T203, T204))
appc70_in_ggga([], T213, T214) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))

The set Q consists of the following terms:

partc20_in_ggaa(x0, x1)
qsc59_in_ga(x0)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U73_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
qc56_in_ggaaaaa(x0, x1)
U63_gg(x0, x1, x2)
U68_gg(x0, x1, x2)
U69_ggaaaaa(x0, x1, x2)
U70_ggaaaaa(x0, x1, x2, x3, x4)
U71_ggaaaaa(x0, x1, x2, x3, x4, x5)
U72_ggaaaaa(x0, x1, x2, x3, x4, x5, x6)
appc70_in_ggga(x0, x1, x2)
U74_ggga(x0, x1, x2, x3, x4)

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

(79) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(80) TRUE

(81) Obligation:

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

QS80_IN_GA(.(T237, T231), X452) → U28_GA(T237, T231, X452, partc87_in_ggaa(T237, T231, T235, T236))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236, X451)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
QS80_IN_GA(x1, x2) = QS80_IN_GA(x1)
U28_GA(x1, x2, x3, x4) = U28_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x4, x5)

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

(82) UsableRulesProof (EQUIVALENT transformation)

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

(83) Obligation:

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

QS80_IN_GA(.(T237, T231), X452) → U28_GA(T237, T231, X452, partc87_in_ggaa(T237, T231, T235, T236))
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235, X450)
U28_GA(T237, T231, X452, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_GA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236, X451)

The TRS R consists of the following rules:

partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
QS80_IN_GA(x1, x2) = QS80_IN_GA(x1)
U28_GA(x1, x2, x3, x4) = U28_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x4, x5)

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

(84) PiDPToQDPProof (SOUND transformation)

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

(85) Obligation:

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

QS80_IN_GA(.(T237, T231)) → U28_GA(T237, T231, partc87_in_ggaa(T237, T231))
U28_GA(T237, T231, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235)
U28_GA(T237, T231, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, T236, qsc80_in_ga(T235))
U30_GA(T237, T231, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236)

The TRS R consists of the following rules:

partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))

The set Q consists of the following terms:

partc87_in_ggaa(x0, x1)
qsc80_in_ga(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U76_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U77_ga(x0, x1, x2, x3)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
U78_ga(x0, x1, x2, x3)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
appc121_in_ggga(x0, x1, x2)
U68_gg(x0, x1, x2)
U80_ggga(x0, x1, x2, x3, x4)

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QS80_IN_GA(.(T237, T231)) → U28_GA(T237, T231, partc87_in_ggaa(T237, T231))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QS80_IN_GA(x₁)) = x₁
POL(U28_GA(x₁, x₂, x₃)) = x₃
POL(U30_GA(x₁, x₂, x₃, x₄)) = x₃
POL(U63_gg(x₁, x₂, x₃)) = 0
POL(U64_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U65_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U66_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U67_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U68_gg(x₁, x₂, x₃)) = 0
POL(U75_gg(x₁, x₂, x₃)) = 0
POL(U76_ga(x₁, x₂, x₃)) = 0
POL(U77_ga(x₁, x₂, x₃, x₄)) = 0
POL(U78_ga(x₁, x₂, x₃, x₄)) = 0
POL(U79_ga(x₁, x₂, x₃)) = 0
POL(U80_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U86_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U87_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U88_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U89_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL([]) = 0
POL(appc121_in_ggga(x₁, x₂, x₃)) = 0
POL(appc121_out_ggga(x₁, x₂, x₃, x₄)) = 0
POL(gtc10_in_gg(x₁, x₂)) = 0
POL(gtc10_out_gg(x₁, x₂)) = 0
POL(lec100_in_gg(x₁, x₂)) = 0
POL(lec100_out_gg(x₁, x₂)) = 0
POL(lec33_in_gg(x₁, x₂)) = 0
POL(lec33_out_gg(x₁, x₂)) = 0
POL(partc20_in_ggaa(x₁, x₂)) = x₂
POL(partc20_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(partc87_in_ggaa(x₁, x₂)) = x₂
POL(partc87_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(qsc80_in_ga(x₁)) = 0
POL(qsc80_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))

(87) Obligation:

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

U28_GA(T237, T231, partc87_out_ggaa(T237, T231, T235, T236)) → QS80_IN_GA(T235)
U28_GA(T237, T231, partc87_out_ggaa(T237, T231, T235, T236)) → U30_GA(T237, T231, T236, qsc80_in_ga(T235))
U30_GA(T237, T231, T236, qsc80_out_ga(T235, T319)) → QS80_IN_GA(T236)

The TRS R consists of the following rules:

partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))

The set Q consists of the following terms:

partc87_in_ggaa(x0, x1)
qsc80_in_ga(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U76_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U77_ga(x0, x1, x2, x3)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
U78_ga(x0, x1, x2, x3)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
appc121_in_ggga(x0, x1, x2)
U68_gg(x0, x1, x2)
U80_ggga(x0, x1, x2, x3, x4)

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

(88) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(89) TRUE

(90) Obligation:

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

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
LE100_IN_AA(x1, x2) = LE100_IN_AA

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

(91) UsableRulesProof (EQUIVALENT transformation)

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

(92) Obligation:

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

LE100_IN_AA(s(T302), s(T303)) → LE100_IN_AA(T302, T303)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE100_IN_AA(x1, x2) = LE100_IN_AA

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

(93) PiDPToQDPProof (SOUND transformation)

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

(94) Obligation:

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

LE100_IN_AA → LE100_IN_AA

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

(95) 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 = LE100_IN_AA evaluates to t =LE100_IN_AA

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 LE100_IN_AA to LE100_IN_AA.

(96) NO

(97) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
LE33_IN_GA(x1, x2) = LE33_IN_GA(x1)

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

(98) UsableRulesProof (EQUIVALENT transformation)

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

(99) Obligation:

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

LE33_IN_GA(s(T113), s(T115)) → LE33_IN_GA(T113, T115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE33_IN_GA(x1, x2) = LE33_IN_GA(x1)

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

(100) PiDPToQDPProof (SOUND transformation)

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

(101) Obligation:

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

LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)

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

(102) 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:

LE33_IN_GA(s(T113)) → LE33_IN_GA(T113)
The graph contains the following edges 1 > 1

(103) YES

(104) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
GT10_IN_GA(x1, x2) = GT10_IN_GA(x1)

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

(105) UsableRulesProof (EQUIVALENT transformation)

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

(106) Obligation:

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

GT10_IN_GA(s(T46), s(T47)) → GT10_IN_GA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT10_IN_GA(x1, x2) = GT10_IN_GA(x1)

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

(107) PiDPToQDPProof (SOUND transformation)

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

(108) Obligation:

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

GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)

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

(109) 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:

GT10_IN_GA(s(T46)) → GT10_IN_GA(T46)
The graph contains the following edges 1 > 1

(110) YES

(111) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
PART20_IN_GAAA(x1, x2, x3, x4) = PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)

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

(112) UsableRulesProof (EQUIVALENT transformation)

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

(113) Obligation:

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

PART20_IN_GAAA(T70, .(T73, T77), .(T73, X119), X120) → U3_GAAA(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70, T77, X119, X120)
PART20_IN_GAAA(T95, .(T98, T102), X170, .(T98, X171)) → U6_GAAA(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U6_GAAA(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → PART20_IN_GAAA(T95, T102, X170, X171)

The TRS R consists of the following rules:

gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
.(x1, x2) = .(x1, x2)
PART20_IN_GAAA(x1, x2, x3, x4) = PART20_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)

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

(114) PiDPToQDPProof (SOUND transformation)

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

(115) Obligation:

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

PART20_IN_GAAA(T70) → U3_GAAA(T70, gtc10_in_ga(T70))
U3_GAAA(T70, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, lec33_in_ga(T95))
U6_GAAA(T95, lec33_out_ga(T95)) → PART20_IN_GAAA(T95)

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(116) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T70) → U3_GAAA(T70, gtc10_in_ga(T70)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))

(117) Obligation:

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

U3_GAAA(T70, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
PART20_IN_GAAA(T95) → U6_GAAA(T95, lec33_in_ga(T95))
U6_GAAA(T95, lec33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(118) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART20_IN_GAAA(T95) → U6_GAAA(T95, lec33_in_ga(T95)) at position [1] we obtained the following new rules [LPAR04]:

PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U68_ga(x0, lec33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))

(119) Obligation:

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

U3_GAAA(T70, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70)
U6_GAAA(T95, lec33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U68_ga(x0, lec33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(120) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAAA(T70, gtc10_out_ga(T70, T73)) → PART20_IN_GAAA(T70) we obtained the following new rules [LPAR04]:

U3_GAAA(s(z0), gtc10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))

(121) Obligation:

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

U6_GAAA(T95, lec33_out_ga(T95)) → PART20_IN_GAAA(T95)
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U68_ga(x0, lec33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))
U3_GAAA(s(z0), gtc10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(122) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GAAA(T95, lec33_out_ga(T95)) → PART20_IN_GAAA(T95) we obtained the following new rules [LPAR04]:

U6_GAAA(s(z0), lec33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)

(123) Obligation:

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

PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U68_ga(x0, lec33_in_ga(x0)))
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))
U3_GAAA(s(z0), gtc10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
U6_GAAA(s(z0), lec33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))
U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(124) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(125) Complex Obligation (AND)

(126) Obligation:

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

U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(127) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(128) Obligation:

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

U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))

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

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(129) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

(130) Obligation:

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

U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)
PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))

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

(131) 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 = PART20_IN_GAAA(0) evaluates to t =PART20_IN_GAAA(0)

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0))
with rule PART20_IN_GAAA(0) → U6_GAAA(0, lec33_out_ga(0)) at position [] and matcher [ ]

U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)
with rule U6_GAAA(0, lec33_out_ga(0)) → PART20_IN_GAAA(0)

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.

(132) NO

(133) Obligation:

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

U3_GAAA(s(z0), gtc10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))
PART20_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U63_ga(x0, gtc10_in_ga(x0)))
U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))
PART20_IN_GAAA(s(x0)) → U6_GAAA(s(x0), U68_ga(x0, lec33_in_ga(x0)))
U6_GAAA(s(z0), lec33_out_ga(s(z0))) → PART20_IN_GAAA(s(z0))

The TRS R consists of the following rules:

gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))

The set Q consists of the following terms:

gtc10_in_ga(x0)
lec33_in_ga(x0)
U63_ga(x0, x1)
U68_ga(x0, x1)

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

(134) 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 = PART20_IN_GAAA(s(0)) evaluates to t =PART20_IN_GAAA(s(0))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0))
with rule PART20_IN_GAAA(s(0)) → U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) at position [] and matcher [ ]

U3_GAAA(s(0), gtc10_out_ga(s(0), 0)) → PART20_IN_GAAA(s(0))
with rule U3_GAAA(s(z0), gtc10_out_ga(s(z0), x1)) → PART20_IN_GAAA(s(z0))

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.

(135) NO

(136) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
GT10_IN_AA(x1, x2) = GT10_IN_AA

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

(137) UsableRulesProof (EQUIVALENT transformation)

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

(138) Obligation:

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

GT10_IN_AA(s(T46), s(T47)) → GT10_IN_AA(T46, T47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT10_IN_AA(x1, x2) = GT10_IN_AA

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

(139) PiDPToQDPProof (SOUND transformation)

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

(140) Obligation:

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

GT10_IN_AA → GT10_IN_AA

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

(141) 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 = GT10_IN_AA evaluates to t =GT10_IN_AA

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 GT10_IN_AA to GT10_IN_AA.

(142) NO

(143) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U28_AA(T237, T231, X452, partc87_in_aaaa(T237, T231, T235, T236))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U30_AA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
qsc51_in_gga(T148, T149, X273) → U85_gga(T148, T149, X273, qc56_in_ggaaaaa(T148, T149, X269, X270, X271, X272, X273))
qc56_in_ggaaaaa(T148, T149, T153, T154, T160, T178, X273) → U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_in_ggaa(T148, T149, T153, T154))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U69_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, partc20_out_ggaa(T148, T149, T153, T154)) → U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T153, T160))
qsc59_in_ga(.(T171, T172), X335) → U73_ga(T171, T172, X335, qc56_in_ggaaaaa(T171, T172, X331, X332, X333, X334, X335))
U73_ga(T171, T172, X335, qc56_out_ggaaaaa(T171, T172, X331, X332, X333, X334, X335)) → qsc59_out_ga(.(T171, T172), X335)
qsc59_in_ga([], []) → qsc59_out_ga([], [])
U70_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T153, T160)) → U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_in_ga(T154, T178))
U71_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, qsc59_out_ga(T154, T178)) → U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_in_ggga(T160, T148, T178, X273))
appc70_in_ggga(.(T201, T202), T203, T204, .(T201, X391)) → U74_ggga(T201, T202, T203, T204, X391, appc70_in_ggga(T202, T203, T204, X391))
appc70_in_ggga([], T213, T214, .(T213, T214)) → appc70_out_ggga([], T213, T214, .(T213, T214))
U74_ggga(T201, T202, T203, T204, X391, appc70_out_ggga(T202, T203, T204, X391)) → appc70_out_ggga(.(T201, T202), T203, T204, .(T201, X391))
U72_ggaaaaa(T148, T149, T153, T154, T160, T178, X273, appc70_out_ggga(T160, T148, T178, X273)) → qc56_out_ggaaaaa(T148, T149, T153, T154, T160, T178, X273)
U85_gga(T148, T149, X273, qc56_out_ggaaaaa(T148, T149, X269, X270, X271, X272, X273)) → qsc51_out_gga(T148, T149, X273)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
qsc80_in_aa(.(T325, T231), X452) → U76_aa(T325, T231, X452, partc87_in_aaaa(T325, T231, T235, T236))
U76_aa(T325, T231, X452, partc87_out_aaaa(T325, T231, T235, T236)) → U77_aa(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
U77_aa(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_aa(T325, T231, X452, T319, qsc80_in_aa(T236, T324))
qsc80_in_aa([], []) → qsc80_out_aa([], [])
U78_aa(T325, T231, X452, T319, qsc80_out_aa(T236, T324)) → U79_aa(T325, T231, X452, appc121_in_gaaa(T319, T325, T324, X452))
appc121_in_gaaa(.(T348, T349), T352, T353, .(T348, X657)) → U80_gaaa(T348, T349, T352, T353, X657, appc121_in_gaaa(T349, T352, T353, X657))
appc121_in_gaaa([], T362, T363, .(T362, T363)) → appc121_out_gaaa([], T362, T363, .(T362, T363))
U80_gaaa(T348, T349, T352, T353, X657, appc121_out_gaaa(T349, T352, T353, X657)) → appc121_out_gaaa(.(T348, T349), T352, T353, .(T348, X657))
U79_aa(T325, T231, X452, appc121_out_gaaa(T319, T325, T324, X452)) → qsc80_out_aa(.(T325, T231), X452)
qsc155_in_a([]) → qsc155_out_a([])

The argument filtering Pi contains the following mapping:
gtc10_in_aa(x1, x2) = gtc10_in_aa
U63_aa(x1, x2, x3) = U63_aa(x3)
gtc10_out_aa(x1, x2) = gtc10_out_aa(x1, x2)
s(x1) = s(x1)
gtc10_in_ga(x1, x2) = gtc10_in_ga(x1)
U63_ga(x1, x2, x3) = U63_ga(x1, x3)
0 = 0
gtc10_out_ga(x1, x2) = gtc10_out_ga(x1, x2)
lec33_in_ga(x1, x2) = lec33_in_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
lec33_out_ga(x1, x2) = lec33_out_ga(x1)
partc20_in_gaaa(x1, x2, x3, x4) = partc20_in_gaaa(x1)
U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6)
U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x2, x6)
U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6)
U67_gaaa(x1, x2, x3, x4, x5, x6) = U67_gaaa(x1, x6)
partc20_out_gaaa(x1, x2, x3, x4) = partc20_out_gaaa(x1, x3)
qsc51_in_gga(x1, x2, x3) = qsc51_in_gga(x1, x2)
U85_gga(x1, x2, x3, x4) = U85_gga(x1, x2, x4)
qc56_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_in_ggaaaaa(x1, x2)
U69_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U69_ggaaaaa(x1, x2, x8)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
.(x1, x2) = .(x1, x2)
U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3)
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6)
U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U68_gg(x1, x2, x3) = U68_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x3, x4, x8)
qsc59_in_ga(x1, x2) = qsc59_in_ga(x1)
U73_ga(x1, x2, x3, x4) = U73_ga(x1, x2, x4)
qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc56_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
qsc59_out_ga(x1, x2) = qsc59_out_ga(x1, x2)
U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x5, x8)
U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)
appc70_in_ggga(x1, x2, x3, x4) = appc70_in_ggga(x1, x2, x3)
U74_ggga(x1, x2, x3, x4, x5, x6) = U74_ggga(x1, x2, x3, x4, x6)
appc70_out_ggga(x1, x2, x3, x4) = appc70_out_ggga(x1, x2, x3, x4)
qsc51_out_gga(x1, x2, x3) = qsc51_out_gga(x1, x2, x3)
lec100_in_aa(x1, x2) = lec100_in_aa
U75_aa(x1, x2, x3) = U75_aa(x3)
lec100_out_aa(x1, x2) = lec100_out_aa(x1)
partc87_in_aaaa(x1, x2, x3, x4) = partc87_in_aaaa
U86_aaaa(x1, x2, x3, x4, x5, x6) = U86_aaaa(x6)
U87_aaaa(x1, x2, x3, x4, x5, x6) = U87_aaaa(x2, x6)
partc87_out_aaaa(x1, x2, x3, x4) = partc87_out_aaaa(x3)
U88_aaaa(x1, x2, x3, x4, x5, x6) = U88_aaaa(x6)
U89_aaaa(x1, x2, x3, x4, x5, x6) = U89_aaaa(x6)
lec100_in_gg(x1, x2) = lec100_in_gg(x1, x2)
U75_gg(x1, x2, x3) = U75_gg(x1, x2, x3)
lec100_out_gg(x1, x2) = lec100_out_gg(x1, x2)
partc87_in_ggaa(x1, x2, x3, x4) = partc87_in_ggaa(x1, x2)
U86_ggaa(x1, x2, x3, x4, x5, x6) = U86_ggaa(x1, x2, x3, x6)
U87_ggaa(x1, x2, x3, x4, x5, x6) = U87_ggaa(x1, x2, x3, x6)
partc87_out_ggaa(x1, x2, x3, x4) = partc87_out_ggaa(x1, x2, x3, x4)
U88_ggaa(x1, x2, x3, x4, x5, x6) = U88_ggaa(x1, x2, x3, x6)
U89_ggaa(x1, x2, x3, x4, x5, x6) = U89_ggaa(x1, x2, x3, x6)
qsc80_in_ga(x1, x2) = qsc80_in_ga(x1)
U76_ga(x1, x2, x3, x4) = U76_ga(x1, x2, x4)
U77_ga(x1, x2, x3, x4, x5, x6) = U77_ga(x1, x2, x5, x6)
qsc80_out_ga(x1, x2) = qsc80_out_ga(x1, x2)
U78_ga(x1, x2, x3, x4, x5) = U78_ga(x1, x2, x4, x5)
U79_ga(x1, x2, x3, x4) = U79_ga(x1, x2, x4)
appc121_in_ggga(x1, x2, x3, x4) = appc121_in_ggga(x1, x2, x3)
U80_ggga(x1, x2, x3, x4, x5, x6) = U80_ggga(x1, x2, x3, x4, x6)
appc121_out_ggga(x1, x2, x3, x4) = appc121_out_ggga(x1, x2, x3, x4)
qsc80_in_aa(x1, x2) = qsc80_in_aa
U76_aa(x1, x2, x3, x4) = U76_aa(x4)
U77_aa(x1, x2, x3, x4, x5, x6) = U77_aa(x6)
U78_aa(x1, x2, x3, x4, x5) = U78_aa(x4, x5)
qsc80_out_aa(x1, x2) = qsc80_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
appc121_in_gaaa(x1, x2, x3, x4) = appc121_in_gaaa(x1)
U80_gaaa(x1, x2, x3, x4, x5, x6) = U80_gaaa(x1, x2, x6)
appc121_out_gaaa(x1, x2, x3, x4) = appc121_out_gaaa(x1)
qsc155_in_a(x1) = qsc155_in_a
qsc155_out_a(x1) = qsc155_out_a(x1)
QS80_IN_AA(x1, x2) = QS80_IN_AA
U28_AA(x1, x2, x3, x4) = U28_AA(x4)
U30_AA(x1, x2, x3, x4, x5) = U30_AA(x5)

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

(144) UsableRulesProof (EQUIVALENT transformation)

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

(145) Obligation:

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

QS80_IN_AA(.(T237, T231), X452) → U28_AA(T237, T231, X452, partc87_in_aaaa(T237, T231, T235, T236))
U28_AA(T237, T231, X452, partc87_out_aaaa(T237, T231, T235, T236)) → U30_AA(T237, T231, X452, T236, qsc80_in_ga(T235, T319))
U30_AA(T237, T231, X452, T236, qsc80_out_ga(T235, T319)) → QS80_IN_AA(T236, X451)

The TRS R consists of the following rules:

partc87_in_aaaa(T261, .(T257, T262), .(T257, X503), X504) → U86_aaaa(T261, T257, T262, X503, X504, gtc10_in_aa(T261, T257))
partc87_in_aaaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_aaaa(T288, T284, T289, X554, X555, lec100_in_aa(T288, T284))
partc87_in_aaaa(T316, [], [], []) → partc87_out_aaaa(T316, [], [], [])
qsc80_in_ga(.(T325, T231), X452) → U76_ga(T325, T231, X452, partc87_in_ggaa(T325, T231, T235, T236))
qsc80_in_ga([], []) → qsc80_out_ga([], [])
U86_aaaa(T261, T257, T262, X503, X504, gtc10_out_aa(T261, T257)) → U87_aaaa(T261, T257, T262, X503, X504, partc20_in_gaaa(T261, T262, X503, X504))
U88_aaaa(T288, T284, T289, X554, X555, lec100_out_aa(T288, T284)) → U89_aaaa(T288, T284, T289, X554, X555, partc20_in_gaaa(T288, T289, X554, X555))
U76_ga(T325, T231, X452, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, X452, T235, T236, qsc80_in_ga(T235, T319))
gtc10_in_aa(s(T46), s(T47)) → U63_aa(T46, T47, gtc10_in_aa(T46, T47))
gtc10_in_aa(s(0), 0) → gtc10_out_aa(s(0), 0)
U87_aaaa(T261, T257, T262, X503, X504, partc20_out_gaaa(T261, T262, X503, X504)) → partc87_out_aaaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_aa(s(T302), s(T303)) → U75_aa(T302, T303, lec100_in_aa(T302, T303))
lec100_in_aa(0, s(T310)) → lec100_out_aa(0, s(T310))
lec100_in_aa(0, 0) → lec100_out_aa(0, 0)
U89_aaaa(T288, T284, T289, X554, X555, partc20_out_gaaa(T288, T289, X554, X555)) → partc87_out_aaaa(T288, .(T284, T289), X554, .(T284, X555))
partc87_in_ggaa(T261, .(T257, T262), .(T257, X503), X504) → U86_ggaa(T261, T257, T262, X503, X504, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289), X554, .(T284, X555)) → U88_ggaa(T288, T284, T289, X554, X555, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, [], [], []) → partc87_out_ggaa(T316, [], [], [])
U77_ga(T325, T231, X452, T235, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, X452, T319, qsc80_in_ga(T236, T324))
U63_aa(T46, T47, gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
partc20_in_gaaa(T70, .(T73, T77), .(T73, X119), X120) → U64_gaaa(T70, T73, T77, X119, X120, gtc10_in_ga(T70, T73))
partc20_in_gaaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_gaaa(T95, T98, T102, X170, X171, lec33_in_ga(T95, T98))
partc20_in_gaaa(T128, [], [], []) → partc20_out_gaaa(T128, [], [], [])
U75_aa(T302, T303, lec100_out_aa(T302, T303)) → lec100_out_aa(s(T302), s(T303))
U86_ggaa(T261, T257, T262, X503, X504, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, X503, X504, partc20_in_ggaa(T261, T262, X503, X504))
U88_ggaa(T288, T284, T289, X554, X555, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, X554, X555, partc20_in_ggaa(T288, T289, X554, X555))
U78_ga(T325, T231, X452, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, X452, appc121_in_ggga(T319, T325, T324, X452))
U64_gaaa(T70, T73, T77, X119, X120, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, T77, X119, X120, partc20_in_gaaa(T70, T77, X119, X120))
U66_gaaa(T95, T98, T102, X170, X171, lec33_out_ga(T95, T98)) → U67_gaaa(T95, T98, T102, X170, X171, partc20_in_gaaa(T95, T102, X170, X171))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, X503, X504, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, X554, X555, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U79_ga(T325, T231, X452, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
gtc10_in_ga(s(T46), s(T47)) → U63_ga(T46, T47, gtc10_in_ga(T46, T47))
gtc10_in_ga(s(0), 0) → gtc10_out_ga(s(0), 0)
U65_gaaa(T70, T73, T77, X119, X120, partc20_out_gaaa(T70, T77, X119, X120)) → partc20_out_gaaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_ga(s(T113), s(T115)) → U68_ga(T113, T115, lec33_in_ga(T113, T115))
lec33_in_ga(0, s(T122)) → lec33_out_ga(0, s(T122))
lec33_in_ga(0, 0) → lec33_out_ga(0, 0)
U67_gaaa(T95, T98, T102, X170, X171, partc20_out_gaaa(T95, T102, X170, X171)) → partc20_out_gaaa(T95, .(T98, T102), X170, .(T98, X171))
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77), .(T73, X119), X120) → U64_ggaa(T70, T73, T77, X119, X120, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102), X170, .(T98, X171)) → U66_ggaa(T95, T98, T102, X170, X171, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, [], [], []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
appc121_in_ggga(.(T348, T349), T352, T353, .(T348, X657)) → U80_ggga(T348, T349, T352, T353, X657, appc121_in_ggga(T349, T352, T353, X657))
appc121_in_ggga([], T362, T363, .(T362, T363)) → appc121_out_ggga([], T362, T363, .(T362, T363))
U63_ga(T46, T47, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, T115, lec33_out_ga(T113, T115)) → lec33_out_ga(s(T113), s(T115))
U64_ggaa(T70, T73, T77, X119, X120, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, X119, X120, partc20_in_ggaa(T70, T77, X119, X120))
U66_ggaa(T95, T98, T102, X170, X171, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, X170, X171, partc20_in_ggaa(T95, T102, X170, X171))
U80_ggga(T348, T349, T352, T353, X657, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U65_ggaa(T70, T73, T77, X119, X120, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, X170, X171, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))

(146) PiDPToQDPProof (SOUND transformation)

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

(147) Obligation:

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

QS80_IN_AA → U28_AA(partc87_in_aaaa)
U28_AA(partc87_out_aaaa(T235)) → U30_AA(qsc80_in_ga(T235))
U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA

The TRS R consists of the following rules:

partc87_in_aaaa → U86_aaaa(gtc10_in_aa)
partc87_in_aaaa → U88_aaaa(lec100_in_aa)
partc87_in_aaaa → partc87_out_aaaa([])
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
U86_aaaa(gtc10_out_aa(T261, T257)) → U87_aaaa(T257, partc20_in_gaaa(T261))
U88_aaaa(lec100_out_aa(T288)) → U89_aaaa(partc20_in_gaaa(T288))
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
gtc10_in_aa → U63_aa(gtc10_in_aa)
gtc10_in_aa → gtc10_out_aa(s(0), 0)
U87_aaaa(T257, partc20_out_gaaa(T261, X503)) → partc87_out_aaaa(.(T257, X503))
lec100_in_aa → U75_aa(lec100_in_aa)
lec100_in_aa → lec100_out_aa(0)
U89_aaaa(partc20_out_gaaa(T288, X554)) → partc87_out_aaaa(X554)
partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U63_aa(gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
partc20_in_gaaa(T70) → U64_gaaa(T70, gtc10_in_ga(T70))
partc20_in_gaaa(T95) → U66_gaaa(T95, lec33_in_ga(T95))
partc20_in_gaaa(T128) → partc20_out_gaaa(T128, [])
U75_aa(lec100_out_aa(T302)) → lec100_out_aa(s(T302))
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
U64_gaaa(T70, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, partc20_in_gaaa(T70))
U66_gaaa(T95, lec33_out_ga(T95)) → U67_gaaa(T95, partc20_in_gaaa(T95))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
U65_gaaa(T70, T73, partc20_out_gaaa(T70, X119)) → partc20_out_gaaa(T70, .(T73, X119))
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U67_gaaa(T95, partc20_out_gaaa(T95, X170)) → partc20_out_gaaa(T95, X170)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))

The set Q consists of the following terms:

partc87_in_aaaa
qsc80_in_ga(x0)
U86_aaaa(x0)
U88_aaaa(x0)
U76_ga(x0, x1, x2)
gtc10_in_aa
U87_aaaa(x0, x1)
lec100_in_aa
U89_aaaa(x0)
partc87_in_ggaa(x0, x1)
U77_ga(x0, x1, x2, x3)
U63_aa(x0)
partc20_in_gaaa(x0)
U75_aa(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U78_ga(x0, x1, x2, x3)
U64_gaaa(x0, x1)
U66_gaaa(x0, x1)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
gtc10_in_ga(x0)
U65_gaaa(x0, x1, x2)
lec33_in_ga(x0)
U67_gaaa(x0, x1)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
appc121_in_ggga(x0, x1, x2)
U63_ga(x0, x1)
U68_ga(x0, x1)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U80_ggga(x0, x1, x2, x3, x4)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
U68_gg(x0, x1, x2)

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

(148) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS80_IN_AA → U28_AA(partc87_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS80_IN_AA → U28_AA(U86_aaaa(gtc10_in_aa))
QS80_IN_AA → U28_AA(U88_aaaa(lec100_in_aa))
QS80_IN_AA → U28_AA(partc87_out_aaaa([]))

(149) Obligation:

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

U28_AA(partc87_out_aaaa(T235)) → U30_AA(qsc80_in_ga(T235))
U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA
QS80_IN_AA → U28_AA(U86_aaaa(gtc10_in_aa))
QS80_IN_AA → U28_AA(U88_aaaa(lec100_in_aa))
QS80_IN_AA → U28_AA(partc87_out_aaaa([]))

The TRS R consists of the following rules:

partc87_in_aaaa → U86_aaaa(gtc10_in_aa)
partc87_in_aaaa → U88_aaaa(lec100_in_aa)
partc87_in_aaaa → partc87_out_aaaa([])
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
U86_aaaa(gtc10_out_aa(T261, T257)) → U87_aaaa(T257, partc20_in_gaaa(T261))
U88_aaaa(lec100_out_aa(T288)) → U89_aaaa(partc20_in_gaaa(T288))
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
gtc10_in_aa → U63_aa(gtc10_in_aa)
gtc10_in_aa → gtc10_out_aa(s(0), 0)
U87_aaaa(T257, partc20_out_gaaa(T261, X503)) → partc87_out_aaaa(.(T257, X503))
lec100_in_aa → U75_aa(lec100_in_aa)
lec100_in_aa → lec100_out_aa(0)
U89_aaaa(partc20_out_gaaa(T288, X554)) → partc87_out_aaaa(X554)
partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U63_aa(gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
partc20_in_gaaa(T70) → U64_gaaa(T70, gtc10_in_ga(T70))
partc20_in_gaaa(T95) → U66_gaaa(T95, lec33_in_ga(T95))
partc20_in_gaaa(T128) → partc20_out_gaaa(T128, [])
U75_aa(lec100_out_aa(T302)) → lec100_out_aa(s(T302))
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
U64_gaaa(T70, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, partc20_in_gaaa(T70))
U66_gaaa(T95, lec33_out_ga(T95)) → U67_gaaa(T95, partc20_in_gaaa(T95))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
U65_gaaa(T70, T73, partc20_out_gaaa(T70, X119)) → partc20_out_gaaa(T70, .(T73, X119))
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U67_gaaa(T95, partc20_out_gaaa(T95, X170)) → partc20_out_gaaa(T95, X170)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))

The set Q consists of the following terms:

partc87_in_aaaa
qsc80_in_ga(x0)
U86_aaaa(x0)
U88_aaaa(x0)
U76_ga(x0, x1, x2)
gtc10_in_aa
U87_aaaa(x0, x1)
lec100_in_aa
U89_aaaa(x0)
partc87_in_ggaa(x0, x1)
U77_ga(x0, x1, x2, x3)
U63_aa(x0)
partc20_in_gaaa(x0)
U75_aa(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U78_ga(x0, x1, x2, x3)
U64_gaaa(x0, x1)
U66_gaaa(x0, x1)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
gtc10_in_ga(x0)
U65_gaaa(x0, x1, x2)
lec33_in_ga(x0)
U67_gaaa(x0, x1)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
appc121_in_ggga(x0, x1, x2)
U63_ga(x0, x1)
U68_ga(x0, x1)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U80_ggga(x0, x1, x2, x3, x4)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
U68_gg(x0, x1, x2)

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

(150) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(151) Obligation:

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

U28_AA(partc87_out_aaaa(T235)) → U30_AA(qsc80_in_ga(T235))
U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA
QS80_IN_AA → U28_AA(U86_aaaa(gtc10_in_aa))
QS80_IN_AA → U28_AA(U88_aaaa(lec100_in_aa))
QS80_IN_AA → U28_AA(partc87_out_aaaa([]))

The TRS R consists of the following rules:

lec100_in_aa → U75_aa(lec100_in_aa)
lec100_in_aa → lec100_out_aa(0)
U88_aaaa(lec100_out_aa(T288)) → U89_aaaa(partc20_in_gaaa(T288))
partc20_in_gaaa(T70) → U64_gaaa(T70, gtc10_in_ga(T70))
partc20_in_gaaa(T95) → U66_gaaa(T95, lec33_in_ga(T95))
partc20_in_gaaa(T128) → partc20_out_gaaa(T128, [])
U89_aaaa(partc20_out_gaaa(T288, X554)) → partc87_out_aaaa(X554)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U66_gaaa(T95, lec33_out_ga(T95)) → U67_gaaa(T95, partc20_in_gaaa(T95))
U67_gaaa(T95, partc20_out_gaaa(T95, X170)) → partc20_out_gaaa(T95, X170)
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))
gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
U64_gaaa(T70, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, partc20_in_gaaa(T70))
U65_gaaa(T70, T73, partc20_out_gaaa(T70, X119)) → partc20_out_gaaa(T70, .(T73, X119))
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U75_aa(lec100_out_aa(T302)) → lec100_out_aa(s(T302))
gtc10_in_aa → U63_aa(gtc10_in_aa)
gtc10_in_aa → gtc10_out_aa(s(0), 0)
U86_aaaa(gtc10_out_aa(T261, T257)) → U87_aaaa(T257, partc20_in_gaaa(T261))
U87_aaaa(T257, partc20_out_gaaa(T261, X503)) → partc87_out_aaaa(.(T257, X503))
U63_aa(gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)

The set Q consists of the following terms:

partc87_in_aaaa
qsc80_in_ga(x0)
U86_aaaa(x0)
U88_aaaa(x0)
U76_ga(x0, x1, x2)
gtc10_in_aa
U87_aaaa(x0, x1)
lec100_in_aa
U89_aaaa(x0)
partc87_in_ggaa(x0, x1)
U77_ga(x0, x1, x2, x3)
U63_aa(x0)
partc20_in_gaaa(x0)
U75_aa(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U78_ga(x0, x1, x2, x3)
U64_gaaa(x0, x1)
U66_gaaa(x0, x1)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
gtc10_in_ga(x0)
U65_gaaa(x0, x1, x2)
lec33_in_ga(x0)
U67_gaaa(x0, x1)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
appc121_in_ggga(x0, x1, x2)
U63_ga(x0, x1)
U68_ga(x0, x1)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U80_ggga(x0, x1, x2, x3, x4)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
U68_gg(x0, x1, x2)

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

(152) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

partc87_in_aaaa

(153) Obligation:

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

U28_AA(partc87_out_aaaa(T235)) → U30_AA(qsc80_in_ga(T235))
U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA
QS80_IN_AA → U28_AA(U86_aaaa(gtc10_in_aa))
QS80_IN_AA → U28_AA(U88_aaaa(lec100_in_aa))
QS80_IN_AA → U28_AA(partc87_out_aaaa([]))

The TRS R consists of the following rules:

lec100_in_aa → U75_aa(lec100_in_aa)
lec100_in_aa → lec100_out_aa(0)
U88_aaaa(lec100_out_aa(T288)) → U89_aaaa(partc20_in_gaaa(T288))
partc20_in_gaaa(T70) → U64_gaaa(T70, gtc10_in_ga(T70))
partc20_in_gaaa(T95) → U66_gaaa(T95, lec33_in_ga(T95))
partc20_in_gaaa(T128) → partc20_out_gaaa(T128, [])
U89_aaaa(partc20_out_gaaa(T288, X554)) → partc87_out_aaaa(X554)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U66_gaaa(T95, lec33_out_ga(T95)) → U67_gaaa(T95, partc20_in_gaaa(T95))
U67_gaaa(T95, partc20_out_gaaa(T95, X170)) → partc20_out_gaaa(T95, X170)
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))
gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
U64_gaaa(T70, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, partc20_in_gaaa(T70))
U65_gaaa(T70, T73, partc20_out_gaaa(T70, X119)) → partc20_out_gaaa(T70, .(T73, X119))
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U75_aa(lec100_out_aa(T302)) → lec100_out_aa(s(T302))
gtc10_in_aa → U63_aa(gtc10_in_aa)
gtc10_in_aa → gtc10_out_aa(s(0), 0)
U86_aaaa(gtc10_out_aa(T261, T257)) → U87_aaaa(T257, partc20_in_gaaa(T261))
U87_aaaa(T257, partc20_out_gaaa(T261, X503)) → partc87_out_aaaa(.(T257, X503))
U63_aa(gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)

The set Q consists of the following terms:

qsc80_in_ga(x0)
U86_aaaa(x0)
U88_aaaa(x0)
U76_ga(x0, x1, x2)
gtc10_in_aa
U87_aaaa(x0, x1)
lec100_in_aa
U89_aaaa(x0)
partc87_in_ggaa(x0, x1)
U77_ga(x0, x1, x2, x3)
U63_aa(x0)
partc20_in_gaaa(x0)
U75_aa(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U78_ga(x0, x1, x2, x3)
U64_gaaa(x0, x1)
U66_gaaa(x0, x1)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
gtc10_in_ga(x0)
U65_gaaa(x0, x1, x2)
lec33_in_ga(x0)
U67_gaaa(x0, x1)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
appc121_in_ggga(x0, x1, x2)
U63_ga(x0, x1)
U68_ga(x0, x1)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U80_ggga(x0, x1, x2, x3, x4)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
U68_gg(x0, x1, x2)

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

(154) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U28_AA(partc87_out_aaaa(T235)) → U30_AA(qsc80_in_ga(T235)) at position [0] we obtained the following new rules [LPAR04]:

U28_AA(partc87_out_aaaa(.(x0, x1))) → U30_AA(U76_ga(x0, x1, partc87_in_ggaa(x0, x1)))
U28_AA(partc87_out_aaaa([])) → U30_AA(qsc80_out_ga([], []))

(155) Obligation:

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

U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA
QS80_IN_AA → U28_AA(U86_aaaa(gtc10_in_aa))
QS80_IN_AA → U28_AA(U88_aaaa(lec100_in_aa))
QS80_IN_AA → U28_AA(partc87_out_aaaa([]))
U28_AA(partc87_out_aaaa(.(x0, x1))) → U30_AA(U76_ga(x0, x1, partc87_in_ggaa(x0, x1)))
U28_AA(partc87_out_aaaa([])) → U30_AA(qsc80_out_ga([], []))

The TRS R consists of the following rules:

lec100_in_aa → U75_aa(lec100_in_aa)
lec100_in_aa → lec100_out_aa(0)
U88_aaaa(lec100_out_aa(T288)) → U89_aaaa(partc20_in_gaaa(T288))
partc20_in_gaaa(T70) → U64_gaaa(T70, gtc10_in_ga(T70))
partc20_in_gaaa(T95) → U66_gaaa(T95, lec33_in_ga(T95))
partc20_in_gaaa(T128) → partc20_out_gaaa(T128, [])
U89_aaaa(partc20_out_gaaa(T288, X554)) → partc87_out_aaaa(X554)
lec33_in_ga(s(T113)) → U68_ga(T113, lec33_in_ga(T113))
lec33_in_ga(0) → lec33_out_ga(0)
U66_gaaa(T95, lec33_out_ga(T95)) → U67_gaaa(T95, partc20_in_gaaa(T95))
U67_gaaa(T95, partc20_out_gaaa(T95, X170)) → partc20_out_gaaa(T95, X170)
U68_ga(T113, lec33_out_ga(T113)) → lec33_out_ga(s(T113))
gtc10_in_ga(s(T46)) → U63_ga(T46, gtc10_in_ga(T46))
gtc10_in_ga(s(0)) → gtc10_out_ga(s(0), 0)
U64_gaaa(T70, gtc10_out_ga(T70, T73)) → U65_gaaa(T70, T73, partc20_in_gaaa(T70))
U65_gaaa(T70, T73, partc20_out_gaaa(T70, X119)) → partc20_out_gaaa(T70, .(T73, X119))
U63_ga(T46, gtc10_out_ga(T46, T47)) → gtc10_out_ga(s(T46), s(T47))
U75_aa(lec100_out_aa(T302)) → lec100_out_aa(s(T302))
gtc10_in_aa → U63_aa(gtc10_in_aa)
gtc10_in_aa → gtc10_out_aa(s(0), 0)
U86_aaaa(gtc10_out_aa(T261, T257)) → U87_aaaa(T257, partc20_in_gaaa(T261))
U87_aaaa(T257, partc20_out_gaaa(T261, X503)) → partc87_out_aaaa(.(T257, X503))
U63_aa(gtc10_out_aa(T46, T47)) → gtc10_out_aa(s(T46), s(T47))
qsc80_in_ga(.(T325, T231)) → U76_ga(T325, T231, partc87_in_ggaa(T325, T231))
qsc80_in_ga([]) → qsc80_out_ga([], [])
partc87_in_ggaa(T261, .(T257, T262)) → U86_ggaa(T261, T257, T262, gtc10_in_gg(T261, T257))
partc87_in_ggaa(T288, .(T284, T289)) → U88_ggaa(T288, T284, T289, lec100_in_gg(T288, T284))
partc87_in_ggaa(T316, []) → partc87_out_ggaa(T316, [], [], [])
U76_ga(T325, T231, partc87_out_ggaa(T325, T231, T235, T236)) → U77_ga(T325, T231, T236, qsc80_in_ga(T235))
U77_ga(T325, T231, T236, qsc80_out_ga(T235, T319)) → U78_ga(T325, T231, T319, qsc80_in_ga(T236))
U78_ga(T325, T231, T319, qsc80_out_ga(T236, T324)) → U79_ga(T325, T231, appc121_in_ggga(T319, T325, T324))
appc121_in_ggga(.(T348, T349), T352, T353) → U80_ggga(T348, T349, T352, T353, appc121_in_ggga(T349, T352, T353))
appc121_in_ggga([], T362, T363) → appc121_out_ggga([], T362, T363, .(T362, T363))
U79_ga(T325, T231, appc121_out_ggga(T319, T325, T324, X452)) → qsc80_out_ga(.(T325, T231), X452)
U80_ggga(T348, T349, T352, T353, appc121_out_ggga(T349, T352, T353, X657)) → appc121_out_ggga(.(T348, T349), T352, T353, .(T348, X657))
lec100_in_gg(s(T302), s(T303)) → U75_gg(T302, T303, lec100_in_gg(T302, T303))
lec100_in_gg(0, s(T310)) → lec100_out_gg(0, s(T310))
lec100_in_gg(0, 0) → lec100_out_gg(0, 0)
U88_ggaa(T288, T284, T289, lec100_out_gg(T288, T284)) → U89_ggaa(T288, T284, T289, partc20_in_ggaa(T288, T289))
partc20_in_ggaa(T70, .(T73, T77)) → U64_ggaa(T70, T73, T77, gtc10_in_gg(T70, T73))
partc20_in_ggaa(T95, .(T98, T102)) → U66_ggaa(T95, T98, T102, lec33_in_gg(T95, T98))
partc20_in_ggaa(T128, []) → partc20_out_ggaa(T128, [], [], [])
U89_ggaa(T288, T284, T289, partc20_out_ggaa(T288, T289, X554, X555)) → partc87_out_ggaa(T288, .(T284, T289), X554, .(T284, X555))
lec33_in_gg(s(T113), s(T115)) → U68_gg(T113, T115, lec33_in_gg(T113, T115))
lec33_in_gg(0, s(T122)) → lec33_out_gg(0, s(T122))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U66_ggaa(T95, T98, T102, lec33_out_gg(T95, T98)) → U67_ggaa(T95, T98, T102, partc20_in_ggaa(T95, T102))
U67_ggaa(T95, T98, T102, partc20_out_ggaa(T95, T102, X170, X171)) → partc20_out_ggaa(T95, .(T98, T102), X170, .(T98, X171))
U68_gg(T113, T115, lec33_out_gg(T113, T115)) → lec33_out_gg(s(T113), s(T115))
gtc10_in_gg(s(T46), s(T47)) → U63_gg(T46, T47, gtc10_in_gg(T46, T47))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U64_ggaa(T70, T73, T77, gtc10_out_gg(T70, T73)) → U65_ggaa(T70, T73, T77, partc20_in_ggaa(T70, T77))
U65_ggaa(T70, T73, T77, partc20_out_ggaa(T70, T77, X119, X120)) → partc20_out_ggaa(T70, .(T73, T77), .(T73, X119), X120)
U63_gg(T46, T47, gtc10_out_gg(T46, T47)) → gtc10_out_gg(s(T46), s(T47))
U75_gg(T302, T303, lec100_out_gg(T302, T303)) → lec100_out_gg(s(T302), s(T303))
U86_ggaa(T261, T257, T262, gtc10_out_gg(T261, T257)) → U87_ggaa(T261, T257, T262, partc20_in_ggaa(T261, T262))
U87_ggaa(T261, T257, T262, partc20_out_ggaa(T261, T262, X503, X504)) → partc87_out_ggaa(T261, .(T257, T262), .(T257, X503), X504)

The set Q consists of the following terms:

qsc80_in_ga(x0)
U86_aaaa(x0)
U88_aaaa(x0)
U76_ga(x0, x1, x2)
gtc10_in_aa
U87_aaaa(x0, x1)
lec100_in_aa
U89_aaaa(x0)
partc87_in_ggaa(x0, x1)
U77_ga(x0, x1, x2, x3)
U63_aa(x0)
partc20_in_gaaa(x0)
U75_aa(x0)
U86_ggaa(x0, x1, x2, x3)
U88_ggaa(x0, x1, x2, x3)
U78_ga(x0, x1, x2, x3)
U64_gaaa(x0, x1)
U66_gaaa(x0, x1)
gtc10_in_gg(x0, x1)
U87_ggaa(x0, x1, x2, x3)
lec100_in_gg(x0, x1)
U89_ggaa(x0, x1, x2, x3)
U79_ga(x0, x1, x2)
gtc10_in_ga(x0)
U65_gaaa(x0, x1, x2)
lec33_in_ga(x0)
U67_gaaa(x0, x1)
U63_gg(x0, x1, x2)
partc20_in_ggaa(x0, x1)
U75_gg(x0, x1, x2)
appc121_in_ggga(x0, x1, x2)
U63_ga(x0, x1)
U68_ga(x0, x1)
U64_ggaa(x0, x1, x2, x3)
U66_ggaa(x0, x1, x2, x3)
U80_ggga(x0, x1, x2, x3, x4)
U65_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U67_ggaa(x0, x1, x2, x3)
U68_gg(x0, x1, x2)

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

(156) 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 = QS80_IN_AA evaluates to t =QS80_IN_AA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

QS80_IN_AA → U28_AA(partc87_out_aaaa([]))
with rule QS80_IN_AA → U28_AA(partc87_out_aaaa([])) at position [] and matcher [ ]

U28_AA(partc87_out_aaaa([])) → U30_AA(qsc80_out_ga([], []))
with rule U28_AA(partc87_out_aaaa([])) → U30_AA(qsc80_out_ga([], [])) at position [] and matcher [ ]

U30_AA(qsc80_out_ga([], [])) → QS80_IN_AA
with rule U30_AA(qsc80_out_ga(T235, T319)) → QS80_IN_AA

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.