Left Termination proof of /tmp/tmpJNWcye/qicksort.pl

Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

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 QDPOrderProof (⇔)

        ↳50 QDP

        ↳51 DependencyGraphProof (⇔)

        ↳52 TRUE

    ↳53 PiDP

        ↳54 UsableRulesProof (⇔)

        ↳55 PiDP

        ↳56 PiDPToQDPProof (⇐)

        ↳57 QDP

        ↳58 QDPOrderProof (⇔)

        ↳59 QDP

        ↳60 DependencyGraphProof (⇔)

        ↳61 TRUE

(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(0)).
le(0, 0).

Queries:

qs(g,a).

(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\nT1 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\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))) | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 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\nT1 is ground\nqs(T1, T2) !=? qs(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(gt(T22, T23), ','(part(T22, T24, X47, X48), ','(qs(.(T23, X47), X9), ','(qs(X48, X10), app(X9, .(T22, X10), T9)))))\n\nT22 is ground\nT23 is ground\nT24 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(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T22, T24, X47, X48), ','(qs(.(T23, X47), X9), ','(qs(X48, X10), app(X9, .(T22, X10), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX9 is free; \nX10 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: gt(T22, T23), SCOPE: 3, CLAUSE: 7 | gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: gt(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: gt(T37, T38)\n\nT37 is ground\nT38 is ground", 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(T22, T24, X47, X48)\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(qs(.(T23, T44), X9), ','(qs(T45, X10), app(X9, .(T22, X10), T9)))\n\nT22 is ground\nT23 is ground\nT44 is ground\nT45 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(gt(T61, T62), part(T61, T63, X119, X120))\n\nT61 is ground\nT62 is ground\nT63 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(T61, T62)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T61, T63, X119, X120)\n\nT61 is ground\nT63 is ground\nX119 is free; \nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: part(T22, T24, X47, X48), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(le(T83, T84), part(T83, T85, X170, X171))\n\nT83 is ground\nT84 is ground\nT85 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(T83, T84)\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T83, T85, X170, X171)\n\nT83 is ground\nT85 is ground\nX170 is free; \nX171 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: le(T83, T84), SCOPE: 5, CLAUSE: 9 | le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T83, T84), SCOPE: 5, CLAUSE: 9\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T98, T99)\n\nT98 is ground\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T83, T84), SCOPE: 5, CLAUSE: 10\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 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(.(T23, T44), X9)\n\nT23 is ground\nT44 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qs(T45, X10), app(T110, .(T22, X10), T9))\n\nT22 is ground\nT45 is ground\nT110 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: qs(T45, X10)\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: app(T110, .(T22, T115), T9)\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: qs(T45, X10), SCOPE: 6, CLAUSE: 0 | qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: qs(T45, X10), SCOPE: 6, CLAUSE: 0\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(part(T126, T127, X265, X266), ','(qs(X265, X267), ','(qs(X266, X268), app(X267, .(T126, X268), X269))))\n\nT126 is ground\nT127 is ground\nX269 is free; \nX265 is free; \nX266 is free; \nX267 is free; \nX268 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: part(T126, T127, X265, X266)\n\nT126 is ground\nT127 is ground\nX265 is free; \nX266 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(qs(T131, X267), ','(qs(T132, X268), app(X267, .(T126, X268), X269)))\n\nT126 is ground\nT131 is ground\nT132 is ground\nX269 is free; \nX267 is free; \nX268 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: qs(T131, X267)\n\nT131 is ground\nX267 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(qs(T132, X268), app(T138, .(T126, X268), X269))\n\nT126 is ground\nT132 is ground\nT138 is ground\nX269 is free; \nX268 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: qs(T132, X268)\n\nT132 is ground\nX268 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: app(T138, .(T126, T143), X269)\n\nT126 is ground\nT138 is ground\nT143 is ground\nX269 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app(T138, .(T126, T143), X269), SCOPE: 7, CLAUSE: 5 | app(T138, .(T126, T143), X269), SCOPE: 7, CLAUSE: 6\n\nT126 is ground\nT138 is ground\nT143 is ground\nX269 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app(T138, .(T126, T143), X269), SCOPE: 7, CLAUSE: 5\n\nT126 is ground\nT138 is ground\nT143 is ground\nX269 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: app(T138, .(T126, T143), X269), SCOPE: 7, CLAUSE: 6\n\nT126 is ground\nT138 is ground\nT143 is ground\nX269 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: app(T167, .(T168, T169), X348)\n\nT167 is ground\nT168 is ground\nT169 is ground\nX348 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 5 | app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 5\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: app(T206, .(T207, T208), T210)\n\nT206 is ground\nT207 is ground\nT208 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(le(T234, T235), ','(part(T234, T236, X431, X432), ','(qs(X431, X9), ','(qs(.(T235, X432), X10), app(X9, .(T234, X10), T9)))))\n\nT234 is ground\nT235 is ground\nT236 is ground\nX9 is free; \nX10 is free; \nX431 is free; \nX432 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: le(T234, T235)\n\nT234 is ground\nT235 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(part(T234, T236, X431, X432), ','(qs(X431, X9), ','(qs(.(T235, X432), X10), app(X9, .(T234, X10), T9))))\n\nT234 is ground\nT235 is ground\nT236 is ground\nX9 is free; \nX10 is free; \nX431 is free; \nX432 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: part(T234, T236, X431, X432)\n\nT234 is ground\nT236 is ground\nX431 is free; \nX432 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(qs(T244, X9), ','(qs(.(T235, T245), X10), app(X9, .(T234, X10), T9)))\n\nT234 is ground\nT235 is ground\nT244 is ground\nT245 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: qs(T244, X9)\n\nT244 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(qs(.(T235, T245), X10), app(T251, .(T234, X10), T9))\n\nT234 is ground\nT235 is ground\nT245 is ground\nT251 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX7 is free; \nX8 is free; \nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(qs([], X9), ','(qs([], X10), app(X9, .(T260, X10), T9)))\n\nT260 is ground\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: qs([], X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(qs([], X10), app(T263, .(T260, X10), T9))\n\nT260 is ground\nT263 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: qs([], X9), SCOPE: 9, CLAUSE: 0 | qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: qs([], X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: app(T263, .(T260, T268), T9)\n\nT260 is ground\nT263 is ground\nT268 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 112 [arrowhead = none ];
112 -> 2;

113 [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 -> T6\nX5 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX6 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 113 [arrowhead = none ];
113 -> 3;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 114 [arrowhead = none ];
114 -> 4;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 115 [arrowhead = none ];
115 -> 5;

116 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 116 [arrowhead = none ];
116 -> 109;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 117 [arrowhead = none ];
117 -> 110;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 118 [arrowhead = none ];
118 -> 6;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 119 [arrowhead = none ];
119 -> 7;

120 [label="EVAL with clause\npart(X42, .(X43, X44), .(X43, X45), X46) :- ','(gt(X42, X43), part(X42, X44, X45, X46)).\nand substitution\nT6 -> T22\nX42 -> T22\nX43 -> T23\nX44 -> T24\nT7 -> .(T23, T24)\nX45 -> X47\nX7 -> .(T23, X47)\nX8 -> X48\nX46 -> X48", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 120 [arrowhead = none ];
120 -> 8;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 121 [arrowhead = none ];
121 -> 9;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 122 [arrowhead = none ];
122 -> 85;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 123 [arrowhead = none ];
123 -> 86;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 124 [arrowhead = none ];
124 -> 10;

125 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
8 -> 125 [arrowhead = none ];
125 -> 11;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 126 [arrowhead = none ];
126 -> 12;

127 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 127 [arrowhead = none ];
127 -> 20;

128 [label="SPLIT 2\nnew knowledge:\nT44 is ground\nT45 is ground\nreplacements:\nX47 -> T44\nX48 -> T45", fontsize=14, style = filled, fillcolor = violet];
11 -> 128 [arrowhead = none ];
128 -> 21;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 129 [arrowhead = none ];
129 -> 13;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 130 [arrowhead = none ];
130 -> 14;

131 [label="EVAL with clause\ngt(s(X66), s(X67)) :- gt(X66, X67).\nand substitution\nX66 -> T37\nT22 -> s(T37)\nX67 -> T38\nT23 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 131 [arrowhead = none ];
131 -> 15;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 132 [arrowhead = none ];
132 -> 16;

133 [label="EVAL with clause\ngt(s(0), 0).\nand substitution\nT22 -> s(0)\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 133 [arrowhead = none ];
133 -> 17;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 134 [arrowhead = none ];
134 -> 18;

135 [label="INSTANCE with matching:\nT22 -> T37\nT23 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 135 [arrowhead = none , style=dashed];
135 -> 10[style=dashed];

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 136 [arrowhead = none ];
136 -> 19;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 137 [arrowhead = none ];
137 -> 22;

138 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:\nX9 -> T110", fontsize=14, style = filled, fillcolor = violet];
21 -> 139 [arrowhead = none ];
139 -> 52;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 140 [arrowhead = none ];
140 -> 23;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 141 [arrowhead = none ];
141 -> 24;

142 [label="EVAL with clause\npart(X114, .(X115, X116), .(X115, X117), X118) :- ','(gt(X114, X115), part(X114, X116, X117, X118)).\nand substitution\nT22 -> T61\nX114 -> T61\nX115 -> T62\nX116 -> T63\nT24 -> .(T62, T63)\nX117 -> X119\nX47 -> .(T62, X119)\nX48 -> X120\nX118 -> X120", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 142 [arrowhead = none ];
142 -> 25;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 143 [arrowhead = none ];
143 -> 26;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 144 [arrowhead = none ];
144 -> 29;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 145 [arrowhead = none ];
145 -> 30;

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 146 [arrowhead = none ];
146 -> 27;

147 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 147 [arrowhead = none ];
147 -> 28;

148 [label="INSTANCE with matching:\nT22 -> T61\nT23 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 148 [arrowhead = none , style=dashed];
148 -> 10[style=dashed];

149 [label="INSTANCE with matching:\nT22 -> T61\nT24 -> T63\nX47 -> X119\nX48 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 149 [arrowhead = none , style=dashed];
149 -> 20[style=dashed];

150 [label="EVAL with clause\npart(X165, .(X166, X167), X168, .(X166, X169)) :- ','(le(X165, X166), part(X165, X167, X168, X169)).\nand substitution\nT22 -> T83\nX165 -> T83\nX166 -> T84\nX167 -> T85\nT24 -> .(T84, T85)\nX47 -> X170\nX168 -> X170\nX169 -> X171\nX48 -> .(T84, X171)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 150 [arrowhead = none ];
150 -> 31;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 151 [arrowhead = none ];
151 -> 32;

152 [label="EVAL with clause\npart(X204, [], [], []).\nand substitution\nT22 -> T107\nX204 -> T107\nT24 -> []\nX47 -> []\nX48 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 152 [arrowhead = none ];
152 -> 48;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 153 [arrowhead = none ];
153 -> 49;

154 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 154 [arrowhead = none ];
154 -> 33;

155 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
31 -> 155 [arrowhead = none ];
155 -> 34;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 156 [arrowhead = none ];
156 -> 35;

157 [label="INSTANCE with matching:\nT22 -> T83\nT24 -> T85\nX47 -> X170\nX48 -> X171", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 157 [arrowhead = none , style=dashed];
157 -> 20[style=dashed];

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 158 [arrowhead = none ];
158 -> 36;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 159 [arrowhead = none ];
159 -> 37;

160 [label="EVAL with clause\nle(s(X189), s(X190)) :- le(X189, X190).\nand substitution\nX189 -> T98\nT83 -> s(T98)\nX190 -> T99\nT84 -> s(T99)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 160 [arrowhead = none ];
160 -> 38;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 161 [arrowhead = none ];
161 -> 39;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 162 [arrowhead = none ];
162 -> 40;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 163 [arrowhead = none ];
163 -> 41;

164 [label="INSTANCE with matching:\nT83 -> T98\nT84 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 164 [arrowhead = none , style=dashed];
164 -> 33[style=dashed];

165 [label="EVAL with clause\nle(0, s(0)).\nand substitution\nT83 -> 0\nT84 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 165 [arrowhead = none ];
165 -> 42;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 166 [arrowhead = none ];
166 -> 43;

167 [label="EVAL with clause\nle(0, 0).\nand substitution\nT83 -> 0\nT84 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 167 [arrowhead = none ];
167 -> 45;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 168 [arrowhead = none ];
168 -> 46;

169 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 169 [arrowhead = none ];
169 -> 44;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 170 [arrowhead = none ];
170 -> 47;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 171 [arrowhead = none ];
171 -> 50;

172 [label="INSTANCE with matching:\nT1 -> .(T23, T44)\nT2 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 172 [arrowhead = none , style=dashed];
172 -> 1[style=dashed];

173 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 173 [arrowhead = none ];
173 -> 53;

174 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nreplacements:\nX10 -> T115", fontsize=14, style = filled, fillcolor = violet];
52 -> 174 [arrowhead = none ];
174 -> 54;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 175 [arrowhead = none ];
175 -> 55;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 176 [arrowhead = none ];
176 -> 77;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 177 [arrowhead = none ];
177 -> 56;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 178 [arrowhead = none ];
178 -> 57;

179 [label="EVAL with clause\nqs(.(X262, X263), X264) :- ','(part(X262, X263, X265, X266), ','(qs(X265, X267), ','(qs(X266, X268), app(X267, .(X262, X268), X264)))).\nand substitution\nX262 -> T126\nX263 -> T127\nT45 -> .(T126, T127)\nX10 -> X269\nX264 -> X269", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 179 [arrowhead = none ];
179 -> 58;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 180 [arrowhead = none ];
180 -> 59;

181 [label="EVAL with clause\nqs([], []).\nand substitution\nT45 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 181 [arrowhead = none ];
181 -> 74;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 182 [arrowhead = none ];
182 -> 75;

183 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 183 [arrowhead = none ];
183 -> 60;

184 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nT132 is ground\nreplacements:\nX265 -> T131\nX266 -> T132", fontsize=14, style = filled, fillcolor = violet];
58 -> 184 [arrowhead = none ];
184 -> 61;

185 [label="INSTANCE with matching:\nT22 -> T126\nT24 -> T127\nX47 -> X265\nX48 -> X266", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 185 [arrowhead = none , style=dashed];
185 -> 20[style=dashed];

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 186 [arrowhead = none ];
186 -> 62;

187 [label="SPLIT 2\nnew knowledge:\nT138 is ground\nreplacements:\nX267 -> T138", fontsize=14, style = filled, fillcolor = violet];
61 -> 187 [arrowhead = none ];
187 -> 63;

188 [label="INSTANCE with matching:\nT45 -> T131\nX10 -> X267", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 188 [arrowhead = none , style=dashed];
188 -> 53[style=dashed];

189 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
63 -> 189 [arrowhead = none ];
189 -> 64;

190 [label="SPLIT 2\nnew knowledge:\nT143 is ground\nreplacements:\nX268 -> T143", fontsize=14, style = filled, fillcolor = violet];
63 -> 190 [arrowhead = none ];
190 -> 65;

191 [label="INSTANCE with matching:\nT45 -> T132\nX10 -> X268", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 191 [arrowhead = none , style=dashed];
191 -> 53[style=dashed];

192 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 192 [arrowhead = none ];
192 -> 66;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 193 [arrowhead = none ];
193 -> 67;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 194 [arrowhead = none ];
194 -> 68;

195 [label="EVAL with clause\napp(.(X344, X345), X346, .(X344, X347)) :- app(X345, X346, X347).\nand substitution\nX344 -> T166\nX345 -> T167\nT138 -> .(T166, T167)\nT126 -> T168\nT143 -> T169\nX346 -> .(T168, T169)\nX347 -> X348\nX269 -> .(T166, X348)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 195 [arrowhead = none ];
195 -> 69;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 196 [arrowhead = none ];
196 -> 70;

197 [label="EVAL with clause\napp([], X360, X360).\nand substitution\nT138 -> []\nT126 -> T178\nT143 -> T179\nX360 -> .(T178, T179)\nX269 -> .(T178, T179)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 197 [arrowhead = none ];
197 -> 71;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 198 [arrowhead = none ];
198 -> 72;

199 [label="INSTANCE with matching:\nT138 -> T167\nT126 -> T168\nT143 -> T169\nX269 -> X348", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 199 [arrowhead = none , style=dashed];
199 -> 65[style=dashed];

200 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 200 [arrowhead = none ];
200 -> 73;

201 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 201 [arrowhead = none ];
201 -> 76;

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 202 [arrowhead = none ];
202 -> 78;

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 203 [arrowhead = none ];
203 -> 79;

204 [label="EVAL with clause\napp(.(X382, X383), X384, .(X382, X385)) :- app(X383, X384, X385).\nand substitution\nX382 -> T205\nX383 -> T206\nT110 -> .(T205, T206)\nT22 -> T207\nT115 -> T208\nX384 -> .(T207, T208)\nX385 -> T210\nT9 -> .(T205, T210)\nT209 -> T210", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 204 [arrowhead = none ];
204 -> 80;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 205 [arrowhead = none ];
205 -> 81;

206 [label="EVAL with clause\napp([], X396, X396).\nand substitution\nT110 -> []\nT22 -> T220\nT115 -> T221\nX396 -> .(T220, T221)\nT9 -> .(T220, T221)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 206 [arrowhead = none ];
206 -> 82;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 207 [arrowhead = none ];
207 -> 83;

208 [label="INSTANCE with matching:\nT110 -> T206\nT22 -> T207\nT115 -> T208\nT9 -> T210", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 208 [arrowhead = none , style=dashed];
208 -> 54[style=dashed];

209 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 209 [arrowhead = none ];
209 -> 84;

210 [label="EVAL with clause\npart(X426, .(X427, X428), X429, .(X427, X430)) :- ','(le(X426, X427), part(X426, X428, X429, X430)).\nand substitution\nT6 -> T234\nX426 -> T234\nX427 -> T235\nX428 -> T236\nT7 -> .(T235, T236)\nX7 -> X431\nX429 -> X431\nX430 -> X432\nX8 -> .(T235, X432)", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 210 [arrowhead = none ];
210 -> 87;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 211 [arrowhead = none ];
211 -> 88;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 212 [arrowhead = none ];
212 -> 95;

213 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 213 [arrowhead = none ];
213 -> 96;

214 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 214 [arrowhead = none ];
214 -> 89;

215 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
87 -> 215 [arrowhead = none ];
215 -> 90;

216 [label="INSTANCE with matching:\nT83 -> T234\nT84 -> T235", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 216 [arrowhead = none , style=dashed];
216 -> 33[style=dashed];

217 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 217 [arrowhead = none ];
217 -> 91;

218 [label="SPLIT 2\nnew knowledge:\nT244 is ground\nT245 is ground\nreplacements:\nX431 -> T244\nX432 -> T245", fontsize=14, style = filled, fillcolor = violet];
90 -> 218 [arrowhead = none ];
218 -> 92;

219 [label="INSTANCE with matching:\nT22 -> T234\nT24 -> T236\nX47 -> X431\nX48 -> X432", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 219 [arrowhead = none , style=dashed];
219 -> 20[style=dashed];

220 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 220 [arrowhead = none ];
220 -> 93;

221 [label="SPLIT 2\nnew knowledge:\nT251 is ground\nreplacements:\nX9 -> T251", fontsize=14, style = filled, fillcolor = violet];
92 -> 221 [arrowhead = none ];
221 -> 94;

222 [label="INSTANCE with matching:\nT45 -> T244\nX10 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 222 [arrowhead = none , style=dashed];
222 -> 53[style=dashed];

223 [label="INSTANCE with matching:\nT45 -> .(T235, T245)\nT110 -> T251\nT22 -> T234", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 223 [arrowhead = none , style=dashed];
223 -> 52[style=dashed];

224 [label="EVAL with clause\npart(X484, [], [], []).\nand substitution\nT6 -> T260\nX484 -> T260\nT7 -> []\nX7 -> []\nX8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 224 [arrowhead = none ];
224 -> 97;

225 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 225 [arrowhead = none ];
225 -> 98;

226 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 226 [arrowhead = none ];
226 -> 107;

227 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
97 -> 227 [arrowhead = none ];
227 -> 99;

228 [label="SPLIT 2\nnew knowledge:\nT263 is ground\nreplacements:\nX9 -> T263", fontsize=14, style = filled, fillcolor = violet];
97 -> 228 [arrowhead = none ];
228 -> 100;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 229 [arrowhead = none ];
229 -> 101;

230 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
100 -> 230 [arrowhead = none ];
230 -> 105;

231 [label="SPLIT 2\nnew knowledge:\nT268 is ground\nreplacements:\nX10 -> T268", fontsize=14, style = filled, fillcolor = violet];
100 -> 231 [arrowhead = none ];
231 -> 106;

232 [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];
101 -> 232 [arrowhead = none ];
232 -> 102;

233 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX9 -> []", fontsize=14, style = filled, fillcolor = white];
102 -> 233 [arrowhead = none ];
233 -> 103;

234 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 234 [arrowhead = none ];
234 -> 104;

235 [label="INSTANCE with matching:\nX9 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 235 [arrowhead = none , style=dashed];
235 -> 99[style=dashed];

236 [label="INSTANCE with matching:\nT110 -> T263\nT22 -> T260\nT115 -> T268", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 236 [arrowhead = none , style=dashed];
236 -> 54[style=dashed];

237 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
107 -> 237 [arrowhead = none ];
237 -> 108;

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 238 [arrowhead = none ];
238 -> 111;

}

(2) Obligation:

Triples:

gt10(s(T37), s(T38)) :- gt10(T37, T38).
part20(T61, .(T62, T63), .(T62, X119), X120) :- gt10(T61, T62).
part20(T61, .(T62, T63), .(T62, X119), X120) :- ','(gtc10(T61, T62), part20(T61, T63, X119, X120)).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- le33(T83, T84).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- ','(lec33(T83, T84), part20(T83, T85, X170, X171)).
le33(s(T98), s(T99)) :- le33(T98, T99).
qs53(.(T126, T127), X269) :- part20(T126, T127, X265, X266).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), qs53(T131, X267)).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), qs53(T132, X268))).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), ','(qsc53(T132, T143), app65(T138, T126, T143, X269)))).
app65(.(T166, T167), T168, T169, .(T166, X348)) :- app65(T167, T168, T169, X348).
app54(.(T205, T206), T207, T208, .(T205, T210)) :- app54(T206, T207, T208, T210).
p52(T45, X10, T110, T22, T9) :- qs53(T45, X10).
p52(T45, T115, T110, T22, T9) :- ','(qsc53(T45, T115), app54(T110, T22, T115, T9)).
qs1(.(T22, .(T23, T24)), T9) :- gt10(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), part20(T22, T24, X47, X48)).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), qs1(.(T23, T44), X9))).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), ','(qsc1(.(T23, T44), T110), p52(T45, X10, T110, T22, T9)))).
qs1(.(T234, .(T235, T236)), T9) :- le33(T234, T235).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), part20(T234, T236, X431, X432)).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), qs53(T244, X9))).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), ','(qsc53(T244, T251), p52(.(T235, T245), X10, T251, T234, T9)))).
qs1(.(T260, []), T9) :- qs99(X9).
qs1(.(T260, []), T9) :- ','(qsc99(T263), qs99(X10)).
qs1(.(T260, []), T9) :- ','(qsc99(T263), ','(qsc99(T268), app54(T263, T260, T268, T9))).

Clauses:

gtc10(s(T37), s(T38)) :- gtc10(T37, T38).
gtc10(s(0), 0).
partc20(T61, .(T62, T63), .(T62, X119), X120) :- ','(gtc10(T61, T62), partc20(T61, T63, X119, X120)).
partc20(T83, .(T84, T85), X170, .(T84, X171)) :- ','(lec33(T83, T84), partc20(T83, T85, X170, X171)).
partc20(T107, [], [], []).
lec33(s(T98), s(T99)) :- lec33(T98, T99).
lec33(0, s(0)).
lec33(0, 0).
qsc1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), ','(qsc1(.(T23, T44), T110), qc52(T45, X10, T110, T22, T9)))).
qsc1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), ','(qsc53(T244, T251), qc52(.(T235, T245), X10, T251, T234, T9)))).
qsc1(.(T260, []), T9) :- ','(qsc99(T263), ','(qsc99(T268), appc54(T263, T260, T268, T9))).
qsc1([], []).
qsc53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), ','(qsc53(T132, T143), appc65(T138, T126, T143, X269)))).
qsc53([], []).
appc65(.(T166, T167), T168, T169, .(T166, X348)) :- appc65(T167, T168, T169, X348).
appc65([], T178, T179, .(T178, T179)).
appc54(.(T205, T206), T207, T208, .(T205, T210)) :- appc54(T206, T207, T208, T210).
appc54([], T220, T221, .(T220, T221)).
qc52(T45, T115, T110, T22, T9) :- ','(qsc53(T45, T115), appc54(T110, T22, T115, T9)).
qsc99([]).

Afs:

qs1(x1, x2) = qs1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

gt10(s(T37), s(T38)) :- gt10(T37, T38).
part20(T61, .(T62, T63), .(T62, X119), X120) :- gt10(T61, T62).
part20(T61, .(T62, T63), .(T62, X119), X120) :- ','(gtc10(T61, T62), part20(T61, T63, X119, X120)).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- le33(T83, T84).
part20(T83, .(T84, T85), X170, .(T84, X171)) :- ','(lec33(T83, T84), part20(T83, T85, X170, X171)).
le33(s(T98), s(T99)) :- le33(T98, T99).
qs53(.(T126, T127), X269) :- part20(T126, T127, X265, X266).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), qs53(T131, X267)).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), qs53(T132, X268))).
qs53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), ','(qsc53(T132, T143), app65(T138, T126, T143, X269)))).
app65(.(T166, T167), T168, T169, .(T166, X348)) :- app65(T167, T168, T169, X348).
app54(.(T205, T206), T207, T208, .(T205, T210)) :- app54(T206, T207, T208, T210).
p52(T45, X10, T110, T22, T9) :- qs53(T45, X10).
p52(T45, T115, T110, T22, T9) :- ','(qsc53(T45, T115), app54(T110, T22, T115, T9)).
qs1(.(T22, .(T23, T24)), T9) :- gt10(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), part20(T22, T24, X47, X48)).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), qs1(.(T23, T44), X9))).
qs1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), ','(qsc1(.(T23, T44), T110), p52(T45, X10, T110, T22, T9)))).
qs1(.(T234, .(T235, T236)), T9) :- le33(T234, T235).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), part20(T234, T236, X431, X432)).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), qs53(T244, X9))).
qs1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), ','(qsc53(T244, T251), p52(.(T235, T245), X10, T251, T234, T9)))).
qs1(.(T260, []), T9) :- ','(qsc99(T263), ','(qsc99(T268), app54(T263, T260, T268, T9))).

Clauses:

gtc10(s(T37), s(T38)) :- gtc10(T37, T38).
gtc10(s(0), 0).
partc20(T61, .(T62, T63), .(T62, X119), X120) :- ','(gtc10(T61, T62), partc20(T61, T63, X119, X120)).
partc20(T83, .(T84, T85), X170, .(T84, X171)) :- ','(lec33(T83, T84), partc20(T83, T85, X170, X171)).
partc20(T107, [], [], []).
lec33(s(T98), s(T99)) :- lec33(T98, T99).
lec33(0, s(0)).
lec33(0, 0).
qsc1(.(T22, .(T23, T24)), T9) :- ','(gtc10(T22, T23), ','(partc20(T22, T24, T44, T45), ','(qsc1(.(T23, T44), T110), qc52(T45, X10, T110, T22, T9)))).
qsc1(.(T234, .(T235, T236)), T9) :- ','(lec33(T234, T235), ','(partc20(T234, T236, T244, T245), ','(qsc53(T244, T251), qc52(.(T235, T245), X10, T251, T234, T9)))).
qsc1(.(T260, []), T9) :- ','(qsc99(T263), ','(qsc99(T268), appc54(T263, T260, T268, T9))).
qsc1([], []).
qsc53(.(T126, T127), X269) :- ','(partc20(T126, T127, T131, T132), ','(qsc53(T131, T138), ','(qsc53(T132, T143), appc65(T138, T126, T143, X269)))).
qsc53([], []).
appc65(.(T166, T167), T168, T169, .(T166, X348)) :- appc65(T167, T168, T169, X348).
appc65([], T178, T179, .(T178, T179)).
appc54(.(T205, T206), T207, T208, .(T205, T210)) :- appc54(T206, T207, T208, T210).
appc54([], T220, T221, .(T220, T221)).
qc52(T45, T115, T110, T22, T9) :- ','(qsc53(T45, T115), appc54(T110, T22, T115, T9)).
qsc99([]).

Afs:

qs1(x1, x2) = qs1(x1)

(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: (b,f)
gt10_in: (b,b)
gtc10_in: (b,b)
part20_in: (b,b,f,f)
le33_in: (b,b)
lec33_in: (b,b)
partc20_in: (b,b,f,f)
qs53_in: (b,f)
qsc53_in: (b,f)
appc65_in: (b,b,b,f)
app65_in: (b,b,b,f)
p52_in: (b,f,b,b,f)
app54_in: (b,b,b,f)
qsc1_in: (b,f)
qc52_in: (b,f,b,b,f)
appc54_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U21_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
QS1_IN_GA(.(T22, .(T23, T24)), T9) → U22_GA(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U23_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U3_GGAA(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U4_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U5_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U8_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U6_GGAA(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U7_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U25_GA(T22, T23, T24, T9, qs1_in_ga(.(T23, T44), X9))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44), X9)
QS1_IN_GA(.(T234, .(T235, T236)), T9) → U28_GA(T234, T235, T236, T9, le33_in_gg(T234, T235))
QS1_IN_GA(.(T234, .(T235, T236)), T9) → LE33_IN_GG(T234, T235)
QS1_IN_GA(.(T234, .(T235, T236)), T9) → U29_GA(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U30_GA(T234, T235, T236, T9, part20_in_ggaa(T234, T236, X431, X432))
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → PART20_IN_GGAA(T234, T236, X431, X432)
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U31_GA(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U32_GA(T234, T235, T236, T9, qs53_in_ga(T244, X9))
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → QS53_IN_GA(T244, X9)
QS53_IN_GA(.(T126, T127), X269) → U9_GA(T126, T127, X269, part20_in_ggaa(T126, T127, X265, X266))
QS53_IN_GA(.(T126, T127), X269) → PART20_IN_GGAA(T126, T127, X265, X266)
QS53_IN_GA(.(T126, T127), X269) → U10_GA(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U11_GA(T126, T127, X269, qs53_in_ga(T131, X267))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131, X267)
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U13_GA(T126, T127, X269, qs53_in_ga(T132, X268))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132, X268)
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U14_GA(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U14_GA(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U15_GA(T126, T127, X269, app65_in_ggga(T138, T126, T143, X269))
U14_GA(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → APP65_IN_GGGA(T138, T126, T143, X269)
APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → U16_GGGA(T166, T167, T168, T169, X348, app65_in_ggga(T167, T168, T169, X348))
APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → APP65_IN_GGGA(T167, T168, T169, X348)
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U33_GA(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U33_GA(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U34_GA(T234, T235, T236, T9, p52_in_gagga(.(T235, T245), X10, T251, T234, T9))
U33_GA(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → P52_IN_GAGGA(.(T235, T245), X10, T251, T234, T9)
P52_IN_GAGGA(T45, X10, T110, T22, T9) → U18_GAGGA(T45, X10, T110, T22, T9, qs53_in_ga(T45, X10))
P52_IN_GAGGA(T45, X10, T110, T22, T9) → QS53_IN_GA(T45, X10)
P52_IN_GAGGA(T45, T115, T110, T22, T9) → U19_GAGGA(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U19_GAGGA(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U20_GAGGA(T45, T115, T110, T22, T9, app54_in_ggga(T110, T22, T115, T9))
U19_GAGGA(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → APP54_IN_GGGA(T110, T22, T115, T9)
APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → U17_GGGA(T205, T206, T207, T208, T210, app54_in_ggga(T206, T207, T208, T210))
APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APP54_IN_GGGA(T206, T207, T208, T210)
QS1_IN_GA(.(T260, []), T9) → U35_GA(T260, T9, qsc99_in_a(T263))
U35_GA(T260, T9, qsc99_out_a(T263)) → U36_GA(T260, T9, T263, qsc99_in_a(T268))
U36_GA(T260, T9, T263, qsc99_out_a(T268)) → U37_GA(T260, T9, app54_in_ggga(T263, T260, T268, T9))
U36_GA(T260, T9, T263, qsc99_out_a(T268)) → APP54_IN_GGGA(T263, T260, T268, T9)
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U26_GA(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
U26_GA(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U27_GA(T22, T23, T24, T9, p52_in_gagga(T45, X10, T110, T22, T9))
U26_GA(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → P52_IN_GAGGA(T45, X10, T110, T22, T9)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2) = qs1_in_ga(x1)
.(x1, x2) = .(x1, x2)
gt10_in_gg(x1, x2) = gt10_in_gg(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_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)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qs53_in_ga(x1, x2) = qs53_in_ga(x1)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
app65_in_ggga(x1, x2, x3, x4) = app65_in_ggga(x1, x2, x3)
p52_in_gagga(x1, x2, x3, x4, x5) = p52_in_gagga(x1, x3, x4)
app54_in_ggga(x1, x2, x3, x4) = app54_in_ggga(x1, x2, x3)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2) = QS1_IN_GA(x1)
U21_GA(x1, x2, x3, x4, x5) = U21_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)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)
U23_GA(x1, x2, x3, x4, x5) = U23_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)
U24_GA(x1, x2, x3, x4, x5) = U24_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5) = U25_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5) = U28_GA(x1, x2, x3, x5)
U29_GA(x1, x2, x3, x4, x5) = U29_GA(x1, x2, x3, x5)
U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3, x4, x5) = U31_GA(x1, x2, x3, x5)
U32_GA(x1, x2, x3, x4, x5) = U32_GA(x1, x2, x3, x5)
QS53_IN_GA(x1, x2) = QS53_IN_GA(x1)
U9_GA(x1, x2, x3, x4) = U9_GA(x1, x2, x4)
U10_GA(x1, x2, x3, x4) = U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4) = U11_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x4, x5)
U13_GA(x1, x2, x3, x4) = U13_GA(x1, x2, x4)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x4, x5)
U15_GA(x1, x2, x3, x4) = U15_GA(x1, x2, x4)
APP65_IN_GGGA(x1, x2, x3, x4) = APP65_IN_GGGA(x1, x2, x3)
U16_GGGA(x1, x2, x3, x4, x5, x6) = U16_GGGA(x1, x2, x3, x4, x6)
U33_GA(x1, x2, x3, x4, x5, x6) = U33_GA(x1, x2, x3, x5, x6)
U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x2, x3, x5)
P52_IN_GAGGA(x1, x2, x3, x4, x5) = P52_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6) = U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6) = U19_GAGGA(x1, x3, x4, x6)
U20_GAGGA(x1, x2, x3, x4, x5, x6) = U20_GAGGA(x1, x3, x4, x6)
APP54_IN_GGGA(x1, x2, x3, x4) = APP54_IN_GGGA(x1, x2, x3)
U17_GGGA(x1, x2, x3, x4, x5, x6) = U17_GGGA(x1, x2, x3, x4, x6)
U35_GA(x1, x2, x3) = U35_GA(x1, x3)
U36_GA(x1, x2, x3, x4) = U36_GA(x1, x3, x4)
U37_GA(x1, x2, x3) = U37_GA(x1, x3)
U26_GA(x1, x2, x3, x4, x5, x6) = U26_GA(x1, x2, x3, x5, x6)
U27_GA(x1, x2, x3, x4, x5) = U27_GA(x1, x2, x3, x5)

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_GA(.(T22, .(T23, T24)), T9) → U21_GA(T22, T23, T24, T9, gt10_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → GT10_IN_GG(T22, T23)
GT10_IN_GG(s(T37), s(T38)) → U1_GG(T37, T38, gt10_in_gg(T37, T38))
GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
QS1_IN_GA(.(T22, .(T23, T24)), T9) → U22_GA(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U23_GA(T22, T23, T24, T9, part20_in_ggaa(T22, T24, X47, X48))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → PART20_IN_GGAA(T22, T24, X47, X48)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U2_GGAA(T61, T62, T63, X119, X120, gt10_in_gg(T61, T62))
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → GT10_IN_GG(T61, T62)
PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U3_GGAA(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U4_GGAA(T61, T62, T63, X119, X120, part20_in_ggaa(T61, T63, X119, X120))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U5_GGAA(T83, T84, T85, X170, X171, le33_in_gg(T83, T84))
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → LE33_IN_GG(T83, T84)
LE33_IN_GG(s(T98), s(T99)) → U8_GG(T98, T99, le33_in_gg(T98, T99))
LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U6_GGAA(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U7_GGAA(T83, T84, T85, X170, X171, part20_in_ggaa(T83, T85, X170, X171))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U25_GA(T22, T23, T24, T9, qs1_in_ga(.(T23, T44), X9))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44), X9)
QS1_IN_GA(.(T234, .(T235, T236)), T9) → U28_GA(T234, T235, T236, T9, le33_in_gg(T234, T235))
QS1_IN_GA(.(T234, .(T235, T236)), T9) → LE33_IN_GG(T234, T235)
QS1_IN_GA(.(T234, .(T235, T236)), T9) → U29_GA(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U30_GA(T234, T235, T236, T9, part20_in_ggaa(T234, T236, X431, X432))
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → PART20_IN_GGAA(T234, T236, X431, X432)
U29_GA(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U31_GA(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U32_GA(T234, T235, T236, T9, qs53_in_ga(T244, X9))
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → QS53_IN_GA(T244, X9)
QS53_IN_GA(.(T126, T127), X269) → U9_GA(T126, T127, X269, part20_in_ggaa(T126, T127, X265, X266))
QS53_IN_GA(.(T126, T127), X269) → PART20_IN_GGAA(T126, T127, X265, X266)
QS53_IN_GA(.(T126, T127), X269) → U10_GA(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U11_GA(T126, T127, X269, qs53_in_ga(T131, X267))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131, X267)
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U13_GA(T126, T127, X269, qs53_in_ga(T132, X268))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132, X268)
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U14_GA(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U14_GA(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U15_GA(T126, T127, X269, app65_in_ggga(T138, T126, T143, X269))
U14_GA(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → APP65_IN_GGGA(T138, T126, T143, X269)
APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → U16_GGGA(T166, T167, T168, T169, X348, app65_in_ggga(T167, T168, T169, X348))
APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → APP65_IN_GGGA(T167, T168, T169, X348)
U31_GA(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U33_GA(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U33_GA(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U34_GA(T234, T235, T236, T9, p52_in_gagga(.(T235, T245), X10, T251, T234, T9))
U33_GA(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → P52_IN_GAGGA(.(T235, T245), X10, T251, T234, T9)
P52_IN_GAGGA(T45, X10, T110, T22, T9) → U18_GAGGA(T45, X10, T110, T22, T9, qs53_in_ga(T45, X10))
P52_IN_GAGGA(T45, X10, T110, T22, T9) → QS53_IN_GA(T45, X10)
P52_IN_GAGGA(T45, T115, T110, T22, T9) → U19_GAGGA(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U19_GAGGA(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U20_GAGGA(T45, T115, T110, T22, T9, app54_in_ggga(T110, T22, T115, T9))
U19_GAGGA(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → APP54_IN_GGGA(T110, T22, T115, T9)
APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → U17_GGGA(T205, T206, T207, T208, T210, app54_in_ggga(T206, T207, T208, T210))
APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APP54_IN_GGGA(T206, T207, T208, T210)
QS1_IN_GA(.(T260, []), T9) → U35_GA(T260, T9, qsc99_in_a(T263))
U35_GA(T260, T9, qsc99_out_a(T263)) → U36_GA(T260, T9, T263, qsc99_in_a(T268))
U36_GA(T260, T9, T263, qsc99_out_a(T268)) → U37_GA(T260, T9, app54_in_ggga(T263, T260, T268, T9))
U36_GA(T260, T9, T263, qsc99_out_a(T268)) → APP54_IN_GGGA(T263, T260, T268, T9)
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U26_GA(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
U26_GA(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U27_GA(T22, T23, T24, T9, p52_in_gagga(T45, X10, T110, T22, T9))
U26_GA(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → P52_IN_GAGGA(T45, X10, T110, T22, T9)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 45 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APP54_IN_GGGA(T206, T207, T208, T210)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
APP54_IN_GGGA(x1, x2, x3, x4) = APP54_IN_GGGA(x1, x2, x3)

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:

APP54_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APP54_IN_GGGA(T206, T207, T208, T210)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP54_IN_GGGA(x1, x2, x3, x4) = APP54_IN_GGGA(x1, x2, x3)

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:

APP54_IN_GGGA(.(T205, T206), T207, T208) → APP54_IN_GGGA(T206, T207, T208)

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:

APP54_IN_GGGA(.(T205, T206), T207, T208) → APP54_IN_GGGA(T206, T207, T208)
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:

APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → APP65_IN_GGGA(T167, T168, T169, X348)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
APP65_IN_GGGA(x1, x2, x3, x4) = APP65_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:

APP65_IN_GGGA(.(T166, T167), T168, T169, .(T166, X348)) → APP65_IN_GGGA(T167, T168, T169, X348)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP65_IN_GGGA(x1, x2, x3, x4) = APP65_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:

APP65_IN_GGGA(.(T166, T167), T168, T169) → APP65_IN_GGGA(T167, T168, T169)

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:

APP65_IN_GGGA(.(T166, T167), T168, T169) → APP65_IN_GGGA(T167, T168, T169)
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:

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
LE33_IN_GG(x1, x2) = LE33_IN_GG(x1, x2)

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:

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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

(26) PiDPToQDPProof (EQUIVALENT 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:

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)

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:

LE33_IN_GG(s(T98), s(T99)) → LE33_IN_GG(T98, T99)
The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
GT10_IN_GG(x1, x2) = GT10_IN_GG(x1, x2)

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:

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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

(33) PiDPToQDPProof (EQUIVALENT 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:

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)

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:

GT10_IN_GG(s(T37), s(T38)) → GT10_IN_GG(T37, T38)
The graph contains the following edges 1 > 1, 2 > 2

(36) YES

(37) Obligation:

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

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U3_GGAA(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U6_GGAA(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
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

(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:

PART20_IN_GGAA(T61, .(T62, T63), .(T62, X119), X120) → U3_GGAA(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U3_GGAA(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63, X119, X120)
PART20_IN_GGAA(T83, .(T84, T85), X170, .(T84, X171)) → U6_GGAA(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U6_GGAA(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85, X170, X171)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))

(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:

PART20_IN_GGAA(T61, .(T62, T63)) → U3_GGAA(T61, T62, T63, gtc10_in_gg(T61, T62))
U3_GGAA(T61, T62, T63, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63)
PART20_IN_GGAA(T83, .(T84, T85)) → U6_GGAA(T83, T84, T85, lec33_in_gg(T83, T84))
U6_GGAA(T83, T84, T85, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtc10_in_gg(x0, x1)
lec33_in_gg(x0, x1)
U39_gg(x0, x1, x2)
U44_gg(x0, x1, x2)

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:

U3_GGAA(T61, T62, T63, gtc10_out_gg(T61, T62)) → PART20_IN_GGAA(T61, T63)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
U6_GGAA(T83, T84, T85, lec33_out_gg(T83, T84)) → PART20_IN_GGAA(T83, T85)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
PART20_IN_GGAA(T61, .(T62, T63)) → U3_GGAA(T61, T62, T63, gtc10_in_gg(T61, T62))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
PART20_IN_GGAA(T83, .(T84, T85)) → U6_GGAA(T83, T84, T85, lec33_in_gg(T83, T84))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(43) YES

(44) Obligation:

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

QS53_IN_GA(.(T126, T127), X269) → U10_GA(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131, X267)
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132, X268)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
QS53_IN_GA(x1, x2) = QS53_IN_GA(x1)
U10_GA(x1, x2, x3, x4) = U10_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x4, x5)

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:

QS53_IN_GA(.(T126, T127), X269) → U10_GA(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131, X267)
U10_GA(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
U12_GA(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132, X268)

The TRS R consists of the following rules:

partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))

(47) PiDPToQDPProof (SOUND 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:

QS53_IN_GA(.(T126, T127)) → U10_GA(T126, T127, partc20_in_ggaa(T126, T127))
U10_GA(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131)
U10_GA(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, T132, qsc53_in_ga(T131))
U12_GA(T126, T127, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132)

The TRS R consists of the following rules:

partc20_in_ggaa(T61, .(T62, T63)) → U40_ggaa(T61, T62, T63, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85)) → U42_ggaa(T83, T84, T85, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, []) → partc20_out_ggaa(T107, [], [], [])
qsc53_in_ga(.(T126, T127)) → U56_ga(T126, T127, partc20_in_ggaa(T126, T127))
qsc53_in_ga([]) → qsc53_out_ga([], [])
U40_ggaa(T61, T62, T63, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, partc20_in_ggaa(T61, T63))
U42_ggaa(T83, T84, T85, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, partc20_in_ggaa(T83, T85))
U56_ga(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, T132, qsc53_in_ga(T131))
gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U41_ggaa(T61, T62, T63, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U57_ga(T126, T127, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, T138, qsc53_in_ga(T132))
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
U58_ga(T126, T127, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, appc65_in_ggga(T138, T126, T143))
U59_ga(T126, T127, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
appc65_in_ggga(.(T166, T167), T168, T169) → U60_ggga(T166, T167, T168, T169, appc65_in_ggga(T167, T168, T169))
appc65_in_ggga([], T178, T179) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))

The set Q consists of the following terms:

partc20_in_ggaa(x0, x1)
qsc53_in_ga(x0)
U40_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
U56_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U41_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2, x3)
U39_gg(x0, x1, x2)
U44_gg(x0, x1, x2)
U58_ga(x0, x1, x2, x3)
U59_ga(x0, x1, x2)
appc65_in_ggga(x0, x1, x2)
U60_ggga(x0, x1, x2, x3, x4)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QS53_IN_GA(.(T126, T127)) → U10_GA(T126, T127, partc20_in_ggaa(T126, T127))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QS53_IN_GA(x₁)) = x₁
POL(U10_GA(x₁, x₂, x₃)) = x₃
POL(U12_GA(x₁, x₂, x₃, x₄)) = x₃
POL(U39_gg(x₁, x₂, x₃)) = 0
POL(U40_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U41_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U42_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U43_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U44_gg(x₁, x₂, x₃)) = 0
POL(U56_ga(x₁, x₂, x₃)) = 0
POL(U57_ga(x₁, x₂, x₃, x₄)) = 0
POL(U58_ga(x₁, x₂, x₃, x₄)) = 0
POL(U59_ga(x₁, x₂, x₃)) = 0
POL(U60_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL([]) = 0
POL(appc65_in_ggga(x₁, x₂, x₃)) = 0
POL(appc65_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)) = 1
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(qsc53_in_ga(x₁)) = 0
POL(qsc53_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

partc20_in_ggaa(T61, .(T62, T63)) → U40_ggaa(T61, T62, T63, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85)) → U42_ggaa(T83, T84, T85, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, []) → partc20_out_ggaa(T107, [], [], [])
U40_ggaa(T61, T62, T63, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, partc20_in_ggaa(T61, T63))
U41_ggaa(T61, T62, T63, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
U42_ggaa(T83, T84, T85, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, partc20_in_ggaa(T83, T85))
U43_ggaa(T83, T84, T85, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))

(50) Obligation:

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

U10_GA(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → QS53_IN_GA(T131)
U10_GA(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → U12_GA(T126, T127, T132, qsc53_in_ga(T131))
U12_GA(T126, T127, T132, qsc53_out_ga(T131, T138)) → QS53_IN_GA(T132)

The TRS R consists of the following rules:

partc20_in_ggaa(T61, .(T62, T63)) → U40_ggaa(T61, T62, T63, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85)) → U42_ggaa(T83, T84, T85, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, []) → partc20_out_ggaa(T107, [], [], [])
qsc53_in_ga(.(T126, T127)) → U56_ga(T126, T127, partc20_in_ggaa(T126, T127))
qsc53_in_ga([]) → qsc53_out_ga([], [])
U40_ggaa(T61, T62, T63, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, partc20_in_ggaa(T61, T63))
U42_ggaa(T83, T84, T85, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, partc20_in_ggaa(T83, T85))
U56_ga(T126, T127, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, T132, qsc53_in_ga(T131))
gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U41_ggaa(T61, T62, T63, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U57_ga(T126, T127, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, T138, qsc53_in_ga(T132))
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
U58_ga(T126, T127, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, appc65_in_ggga(T138, T126, T143))
U59_ga(T126, T127, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
appc65_in_ggga(.(T166, T167), T168, T169) → U60_ggga(T166, T167, T168, T169, appc65_in_ggga(T167, T168, T169))
appc65_in_ggga([], T178, T179) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))

The set Q consists of the following terms:

partc20_in_ggaa(x0, x1)
qsc53_in_ga(x0)
U40_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
U56_ga(x0, x1, x2)
gtc10_in_gg(x0, x1)
U41_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2, x3)
U39_gg(x0, x1, x2)
U44_gg(x0, x1, x2)
U58_ga(x0, x1, x2, x3)
U59_ga(x0, x1, x2)
appc65_in_ggga(x0, x1, x2)
U60_ggga(x0, x1, x2, x3, x4)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

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

(52) TRUE

(53) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U22_GA(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44), X9)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
qsc53_in_ga(.(T126, T127), X269) → U56_ga(T126, T127, X269, partc20_in_ggaa(T126, T127, T131, T132))
U56_ga(T126, T127, X269, partc20_out_ggaa(T126, T127, T131, T132)) → U57_ga(T126, T127, X269, T132, qsc53_in_ga(T131, T138))
qsc53_in_ga([], []) → qsc53_out_ga([], [])
U57_ga(T126, T127, X269, T132, qsc53_out_ga(T131, T138)) → U58_ga(T126, T127, X269, T138, qsc53_in_ga(T132, T143))
U58_ga(T126, T127, X269, T138, qsc53_out_ga(T132, T143)) → U59_ga(T126, T127, X269, appc65_in_ggga(T138, T126, T143, X269))
appc65_in_ggga(.(T166, T167), T168, T169, .(T166, X348)) → U60_ggga(T166, T167, T168, T169, X348, appc65_in_ggga(T167, T168, T169, X348))
appc65_in_ggga([], T178, T179, .(T178, T179)) → appc65_out_ggga([], T178, T179, .(T178, T179))
U60_ggga(T166, T167, T168, T169, X348, appc65_out_ggga(T167, T168, T169, X348)) → appc65_out_ggga(.(T166, T167), T168, T169, .(T166, X348))
U59_ga(T126, T127, X269, appc65_out_ggga(T138, T126, T143, X269)) → qsc53_out_ga(.(T126, T127), X269)
qsc99_in_a([]) → qsc99_out_a([])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U45_ga(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U45_ga(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U46_ga(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U46_ga(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → U47_ga(T22, T23, T24, T9, T45, qsc1_in_ga(.(T23, T44), T110))
qsc1_in_ga(.(T234, .(T235, T236)), T9) → U49_ga(T234, T235, T236, T9, lec33_in_gg(T234, T235))
U49_ga(T234, T235, T236, T9, lec33_out_gg(T234, T235)) → U50_ga(T234, T235, T236, T9, partc20_in_ggaa(T234, T236, T244, T245))
U50_ga(T234, T235, T236, T9, partc20_out_ggaa(T234, T236, T244, T245)) → U51_ga(T234, T235, T236, T9, T245, qsc53_in_ga(T244, T251))
U51_ga(T234, T235, T236, T9, T245, qsc53_out_ga(T244, T251)) → U52_ga(T234, T235, T236, T9, qc52_in_gagga(.(T235, T245), X10, T251, T234, T9))
qc52_in_gagga(T45, T115, T110, T22, T9) → U62_gagga(T45, T115, T110, T22, T9, qsc53_in_ga(T45, T115))
U62_gagga(T45, T115, T110, T22, T9, qsc53_out_ga(T45, T115)) → U63_gagga(T45, T115, T110, T22, T9, appc54_in_ggga(T110, T22, T115, T9))
appc54_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U61_ggga(T205, T206, T207, T208, T210, appc54_in_ggga(T206, T207, T208, T210))
appc54_in_ggga([], T220, T221, .(T220, T221)) → appc54_out_ggga([], T220, T221, .(T220, T221))
U61_ggga(T205, T206, T207, T208, T210, appc54_out_ggga(T206, T207, T208, T210)) → appc54_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U63_gagga(T45, T115, T110, T22, T9, appc54_out_ggga(T110, T22, T115, T9)) → qc52_out_gagga(T45, T115, T110, T22, T9)
U52_ga(T234, T235, T236, T9, qc52_out_gagga(.(T235, T245), X10, T251, T234, T9)) → qsc1_out_ga(.(T234, .(T235, T236)), T9)
qsc1_in_ga(.(T260, []), T9) → U53_ga(T260, T9, qsc99_in_a(T263))
U53_ga(T260, T9, qsc99_out_a(T263)) → U54_ga(T260, T9, T263, qsc99_in_a(T268))
U54_ga(T260, T9, T263, qsc99_out_a(T268)) → U55_ga(T260, T9, appc54_in_ggga(T263, T260, T268, T9))
U55_ga(T260, T9, appc54_out_ggga(T263, T260, T268, T9)) → qsc1_out_ga(.(T260, []), T9)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
U47_ga(T22, T23, T24, T9, T45, qsc1_out_ga(.(T23, T44), T110)) → U48_ga(T22, T23, T24, T9, qc52_in_gagga(T45, X10, T110, T22, T9))
U48_ga(T22, T23, T24, T9, qc52_out_gagga(T45, X10, T110, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
gtc10_in_gg(x1, x2) = gtc10_in_gg(x1, x2)
U39_gg(x1, x2, x3) = U39_gg(x1, x2, x3)
0 = 0
gtc10_out_gg(x1, x2) = gtc10_out_gg(x1, x2)
lec33_in_gg(x1, x2) = lec33_in_gg(x1, x2)
U44_gg(x1, x2, x3) = U44_gg(x1, x2, x3)
lec33_out_gg(x1, x2) = lec33_out_gg(x1, x2)
partc20_in_ggaa(x1, x2, x3, x4) = partc20_in_ggaa(x1, x2)
U40_ggaa(x1, x2, x3, x4, x5, x6) = U40_ggaa(x1, x2, x3, x6)
U41_ggaa(x1, x2, x3, x4, x5, x6) = U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6) = U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6) = U43_ggaa(x1, x2, x3, x6)
[] = []
partc20_out_ggaa(x1, x2, x3, x4) = partc20_out_ggaa(x1, x2, x3, x4)
qsc53_in_ga(x1, x2) = qsc53_in_ga(x1)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4, x5) = U57_ga(x1, x2, x4, x5)
qsc53_out_ga(x1, x2) = qsc53_out_ga(x1, x2)
U58_ga(x1, x2, x3, x4, x5) = U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4) = U59_ga(x1, x2, x4)
appc65_in_ggga(x1, x2, x3, x4) = appc65_in_ggga(x1, x2, x3)
U60_ggga(x1, x2, x3, x4, x5, x6) = U60_ggga(x1, x2, x3, x4, x6)
appc65_out_ggga(x1, x2, x3, x4) = appc65_out_ggga(x1, x2, x3, x4)
qsc99_in_a(x1) = qsc99_in_a
qsc99_out_a(x1) = qsc99_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
U45_ga(x1, x2, x3, x4, x5) = U45_ga(x1, x2, x3, x5)
U46_ga(x1, x2, x3, x4, x5) = U46_ga(x1, x2, x3, x5)
U47_ga(x1, x2, x3, x4, x5, x6) = U47_ga(x1, x2, x3, x5, x6)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6) = U51_ga(x1, x2, x3, x5, x6)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
qc52_in_gagga(x1, x2, x3, x4, x5) = qc52_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6) = U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6) = U63_gagga(x1, x2, x3, x4, x6)
appc54_in_ggga(x1, x2, x3, x4) = appc54_in_ggga(x1, x2, x3)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appc54_out_ggga(x1, x2, x3, x4) = appc54_out_ggga(x1, x2, x3, x4)
qc52_out_gagga(x1, x2, x3, x4, x5) = qc52_out_gagga(x1, x2, x3, x4, x5)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x3, x4)
U55_ga(x1, x2, x3) = U55_ga(x1, x3)
U48_ga(x1, x2, x3, x4, x5) = U48_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2) = QS1_IN_GA(x1)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)
U24_GA(x1, x2, x3, x4, x5) = U24_GA(x1, x2, x3, x5)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U22_GA(T22, T23, T24, T9, gtc10_in_gg(T22, T23))
U22_GA(T22, T23, T24, T9, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, T9, partc20_in_ggaa(T22, T24, T44, T45))
U24_GA(T22, T23, T24, T9, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44), X9)

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
partc20_in_ggaa(T61, .(T62, T63), .(T62, X119), X120) → U40_ggaa(T61, T62, T63, X119, X120, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85), X170, .(T84, X171)) → U42_ggaa(T83, T84, T85, X170, X171, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, [], [], []) → partc20_out_ggaa(T107, [], [], [])
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U40_ggaa(T61, T62, T63, X119, X120, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, X119, X120, partc20_in_ggaa(T61, T63, X119, X120))
U42_ggaa(T83, T84, T85, X170, X171, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, X170, X171, partc20_in_ggaa(T83, T85, X170, X171))
U41_ggaa(T61, T62, T63, X119, X120, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, X170, X171, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24))) → U22_GA(T22, T23, T24, gtc10_in_gg(T22, T23))
U22_GA(T22, T23, T24, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, partc20_in_ggaa(T22, T24))
U24_GA(T22, T23, T24, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44))

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
partc20_in_ggaa(T61, .(T62, T63)) → U40_ggaa(T61, T62, T63, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85)) → U42_ggaa(T83, T84, T85, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, []) → partc20_out_ggaa(T107, [], [], [])
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U40_ggaa(T61, T62, T63, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, partc20_in_ggaa(T61, T63))
U42_ggaa(T83, T84, T85, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, partc20_in_ggaa(T83, T85))
U41_ggaa(T61, T62, T63, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtc10_in_gg(x0, x1)
partc20_in_ggaa(x0, x1)
U39_gg(x0, x1, x2)
U40_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
U41_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U43_ggaa(x0, x1, x2, x3)
U44_gg(x0, x1, x2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QS1_IN_GA(.(T22, .(T23, T24))) → U22_GA(T22, T23, T24, gtc10_in_gg(T22, T23))

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

POL(.(x₁, x₂)) = x₁
POL(0) = 0
POL(QS1_IN_GA(x₁)) = 1 + x₁
POL(U22_GA(x₁, x₂, x₃, x₄)) = x₄
POL(U24_GA(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(U39_gg(x₁, x₂, x₃)) = 1 + x₃
POL(U40_ggaa(x₁, x₂, x₃, x₄)) = 0
POL(U41_ggaa(x₁, x₂, x₃, x₄)) = 0
POL(U42_ggaa(x₁, x₂, x₃, x₄)) = 0
POL(U43_ggaa(x₁, x₂, x₃, x₄)) = 0
POL(U44_gg(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(gtc10_in_gg(x₁, x₂)) = x₁
POL(gtc10_out_gg(x₁, x₂)) = 1 + x₂
POL(lec33_in_gg(x₁, x₂)) = 0
POL(lec33_out_gg(x₁, x₂)) = 0
POL(partc20_in_ggaa(x₁, x₂)) = 0
POL(partc20_out_ggaa(x₁, x₂, x₃, x₄)) = 0
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))

(59) Obligation:

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

U22_GA(T22, T23, T24, gtc10_out_gg(T22, T23)) → U24_GA(T22, T23, T24, partc20_in_ggaa(T22, T24))
U24_GA(T22, T23, T24, partc20_out_ggaa(T22, T24, T44, T45)) → QS1_IN_GA(.(T23, T44))

The TRS R consists of the following rules:

gtc10_in_gg(s(T37), s(T38)) → U39_gg(T37, T38, gtc10_in_gg(T37, T38))
gtc10_in_gg(s(0), 0) → gtc10_out_gg(s(0), 0)
partc20_in_ggaa(T61, .(T62, T63)) → U40_ggaa(T61, T62, T63, gtc10_in_gg(T61, T62))
partc20_in_ggaa(T83, .(T84, T85)) → U42_ggaa(T83, T84, T85, lec33_in_gg(T83, T84))
partc20_in_ggaa(T107, []) → partc20_out_ggaa(T107, [], [], [])
U39_gg(T37, T38, gtc10_out_gg(T37, T38)) → gtc10_out_gg(s(T37), s(T38))
U40_ggaa(T61, T62, T63, gtc10_out_gg(T61, T62)) → U41_ggaa(T61, T62, T63, partc20_in_ggaa(T61, T63))
U42_ggaa(T83, T84, T85, lec33_out_gg(T83, T84)) → U43_ggaa(T83, T84, T85, partc20_in_ggaa(T83, T85))
U41_ggaa(T61, T62, T63, partc20_out_ggaa(T61, T63, X119, X120)) → partc20_out_ggaa(T61, .(T62, T63), .(T62, X119), X120)
lec33_in_gg(s(T98), s(T99)) → U44_gg(T98, T99, lec33_in_gg(T98, T99))
lec33_in_gg(0, s(0)) → lec33_out_gg(0, s(0))
lec33_in_gg(0, 0) → lec33_out_gg(0, 0)
U43_ggaa(T83, T84, T85, partc20_out_ggaa(T83, T85, X170, X171)) → partc20_out_ggaa(T83, .(T84, T85), X170, .(T84, X171))
U44_gg(T98, T99, lec33_out_gg(T98, T99)) → lec33_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtc10_in_gg(x0, x1)
partc20_in_ggaa(x0, x1)
U39_gg(x0, x1, x2)
U40_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
U41_ggaa(x0, x1, x2, x3)
lec33_in_gg(x0, x1)
U43_ggaa(x0, x1, x2, x3)
U44_gg(x0, x1, x2)

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

(60) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

(4) Obligation:

(5) TriplesToPiDPProof (SOUND transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPSizeChangeProof (EQUIVALENT transformation)

(36) YES

(37) Obligation:

(38) UsableRulesProof (EQUIVALENT transformation)

(39) Obligation:

(40) PiDPToQDPProof (SOUND transformation)

(41) Obligation:

(42) QDPSizeChangeProof (EQUIVALENT transformation)

(43) YES

(44) Obligation:

(45) UsableRulesProof (EQUIVALENT transformation)

(46) Obligation:

(47) PiDPToQDPProof (SOUND transformation)

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) DependencyGraphProof (EQUIVALENT transformation)

(52) TRUE

(53) Obligation:

(54) UsableRulesProof (EQUIVALENT transformation)

(55) Obligation:

(56) PiDPToQDPProof (SOUND transformation)

(57) Obligation:

(58) QDPOrderProof (EQUIVALENT transformation)

(59) Obligation:

(60) DependencyGraphProof (EQUIVALENT transformation)

(61) TRUE