Left Termination proof of /tmp/tmpRSdiq0/quicksort-bf.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 QDPOrderProof (⇔)

        ↳43 QDP

        ↳44 DependencyGraphProof (⇔)

        ↳45 TRUE

    ↳46 PiDP

        ↳47 UsableRulesProof (⇔)

        ↳48 PiDP

        ↳49 PiDPToQDPProof (⇐)

        ↳50 QDP

        ↳51 QDPOrderProof (⇔)

        ↳52 QDP

        ↳53 DependencyGraphProof (⇔)

        ↳54 TRUE

(0) Obligation:

Clauses:

qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X1, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

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: true | qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqs(T1, T2) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9))))\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(less(T22, T23), ','(part(T22, T24, X52, X53), ','(qs(.(T23, X52), X14), ','(qs(X53, X15), app(X14, .(T22, X15), T9)))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free; \nX15 is free; \nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(part(T22, T24, X52, X53), ','(qs(.(T23, X52), X14), ','(qs(X53, X15), app(X14, .(T22, X15), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free; \nX15 is free; \nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(T22, T23), SCOPE: 3, CLAUSE: 7 | less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: less(T38, T39)\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: part(T22, T24, X52, X53)\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(qs(.(T23, T45), X14), ','(qs(T46, X15), app(X14, .(T22, X15), T9)))\n\nT22 is ground\nT23 is ground\nT45 is ground\nT46 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(less(T62, T63), part(T62, T64, X126, X127))\n\nT62 is ground\nT63 is ground\nT64 is ground\nX126 is free; \nX127 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: less(T62, T63)\n\nT62 is ground\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: part(T62, T64, X126, X127)\n\nT62 is ground\nT64 is ground\nX126 is free; \nX127 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T22, T24, X52, X53), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: part(T84, T86, X177, X178)\n\nT84 is ground\nT86 is ground\nX177 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: qs(.(T23, T45), X14)\n\nT23 is ground\nT45 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(qs(T46, X15), app(T95, .(T22, X15), T9))\n\nT22 is ground\nT46 is ground\nT95 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: qs(T46, X15)\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T95, .(T22, T98), T9)\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: qs(T46, X15), SCOPE: 5, CLAUSE: 0 | qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: qs(T46, X15), SCOPE: 5, CLAUSE: 0\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(part(T103, T104, X226, X227), ','(qs(X226, X228), ','(qs(X227, X229), app(X228, .(T103, X229), X230))))\n\nT103 is ground\nT104 is ground\nX230 is free; \nX226 is free; \nX227 is free; \nX228 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: part(T103, T104, X226, X227)\n\nT103 is ground\nT104 is ground\nX226 is free; \nX227 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(qs(T108, X228), ','(qs(T109, X229), app(X228, .(T103, X229), X230)))\n\nT103 is ground\nT108 is ground\nT109 is ground\nX230 is free; \nX228 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: qs(T108, X228)\n\nT108 is ground\nX228 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(qs(T109, X229), app(T113, .(T103, X229), X230))\n\nT103 is ground\nT109 is ground\nT113 is ground\nX230 is free; \nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: qs(T109, X229)\n\nT109 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: app(T113, .(T103, T114), X230)\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 5 | app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 6\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 5\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T113, .(T103, T114), X230), SCOPE: 6, CLAUSE: 6\n\nT103 is ground\nT113 is ground\nT114 is ground\nX230 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app(T138, .(T139, T140), X270)\n\nT138 is ground\nT139 is ground\nT140 is ground\nX270 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 5 | app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 5\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: app(T95, .(T22, T98), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT95 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T170, .(T171, T172), T174)\n\nT170 is ground\nT171 is ground\nT172 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(part(T191, T193, X336, X337), ','(qs(X336, X14), ','(qs(.(T192, X337), X15), app(X14, .(T191, X15), T9))))\n\nT191 is ground\nT192 is ground\nT193 is ground\nX14 is free; \nX15 is free; \nX336 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: part(T191, T193, X336, X337)\n\nT191 is ground\nT193 is ground\nX336 is free; \nX337 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(qs(T197, X14), ','(qs(.(T192, T198), X15), app(X14, .(T191, X15), T9)))\n\nT191 is ground\nT192 is ground\nT197 is ground\nT198 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: qs(T197, X14)\n\nT197 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(qs(.(T192, T198), X15), app(T202, .(T191, X15), T9))\n\nT191 is ground\nT192 is ground\nT198 is ground\nT202 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(qs([], X14), ','(qs([], X15), app(X14, .(T207, X15), T9)))\n\nT207 is ground\nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: qs([], X14)\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(qs([], X15), app(T210, .(T207, X15), T9))\n\nT207 is ground\nT210 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: qs([], X14), SCOPE: 8, CLAUSE: 0 | qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: qs([], X14), SCOPE: 8, CLAUSE: 0\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: qs([], X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: app(T210, .(T207, T213), T9)\n\nT207 is ground\nT210 is ground\nT213 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 94 [arrowhead = none ];
94 -> 2;

95 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 95 [arrowhead = none ];
95 -> 3;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 96 [arrowhead = none ];
96 -> 4;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 97 [arrowhead = none ];
97 -> 5;

98 [label="EVAL with clause\nqs(.(X9, X10), X11) :- ','(part(X9, X10, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(X9, X15), X11)))).\nand substitution\nX9 -> T6\nX10 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX11 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 99 [arrowhead = none ];
99 -> 8;

100 [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];
5 -> 100 [arrowhead = none ];
100 -> 6;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 101 [arrowhead = none ];
101 -> 9;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 102 [arrowhead = none ];
102 -> 10;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 103 [arrowhead = none ];
103 -> 11;

104 [label="EVAL with clause\npart(X47, .(X48, X49), .(X48, X50), X51) :- ','(less(X47, X48), part(X47, X49, X50, X51)).\nand substitution\nT6 -> T22\nX47 -> T22\nX48 -> T23\nX49 -> T24\nT7 -> .(T23, T24)\nX50 -> X52\nX12 -> .(T23, X52)\nX13 -> X53\nX51 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 104 [arrowhead = none ];
104 -> 12;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 105 [arrowhead = none ];
105 -> 13;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 106 [arrowhead = none ];
106 -> 74;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 107 [arrowhead = none ];
107 -> 75;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 108 [arrowhead = none ];
108 -> 14;

109 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
12 -> 109 [arrowhead = none ];
109 -> 15;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 110 [arrowhead = none ];
110 -> 16;

111 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 111 [arrowhead = none ];
111 -> 24;

112 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nreplacements:\nX52 -> T45\nX53 -> T46", fontsize=14, style = filled, fillcolor = violet];
15 -> 112 [arrowhead = none ];
112 -> 25;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 113 [arrowhead = none ];
113 -> 17;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 114 [arrowhead = none ];
114 -> 18;

115 [label="EVAL with clause\nless(0, s(X67)).\nand substitution\nT22 -> 0\nX67 -> T33\nT23 -> s(T33)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 115 [arrowhead = none ];
115 -> 19;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 116 [arrowhead = none ];
116 -> 20;

117 [label="EVAL with clause\nless(s(X73), s(X74)) :- less(X73, X74).\nand substitution\nX73 -> T38\nT22 -> s(T38)\nX74 -> T39\nT23 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 118 [arrowhead = none ];
118 -> 23;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 119 [arrowhead = none ];
119 -> 21;

120 [label="INSTANCE with matching:\nT22 -> T38\nT23 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 120 [arrowhead = none , style=dashed];
120 -> 14[style=dashed];

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 121 [arrowhead = none ];
121 -> 26;

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 122 [arrowhead = none ];
122 -> 40;

123 [label="SPLIT 2\nnew knowledge:\nT95 is ground\nreplacements:\nX14 -> T95", fontsize=14, style = filled, fillcolor = violet];
25 -> 123 [arrowhead = none ];
123 -> 41;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 124 [arrowhead = none ];
124 -> 27;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 125 [arrowhead = none ];
125 -> 28;

126 [label="EVAL with clause\npart(X121, .(X122, X123), .(X122, X124), X125) :- ','(less(X121, X122), part(X121, X123, X124, X125)).\nand substitution\nT22 -> T62\nX121 -> T62\nX122 -> T63\nX123 -> T64\nT24 -> .(T63, T64)\nX124 -> X126\nX52 -> .(T63, X126)\nX53 -> X127\nX125 -> X127", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 126 [arrowhead = none ];
126 -> 29;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 127 [arrowhead = none ];
127 -> 30;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 128 [arrowhead = none ];
128 -> 33;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 129 [arrowhead = none ];
129 -> 34;

130 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 130 [arrowhead = none ];
130 -> 31;

131 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 131 [arrowhead = none ];
131 -> 32;

132 [label="INSTANCE with matching:\nT22 -> T62\nT23 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 132 [arrowhead = none , style=dashed];
132 -> 14[style=dashed];

133 [label="INSTANCE with matching:\nT22 -> T62\nT24 -> T64\nX52 -> X126\nX53 -> X127", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 133 [arrowhead = none , style=dashed];
133 -> 24[style=dashed];

134 [label="EVAL with clause\npart(X172, .(X173, X174), X175, .(X173, X176)) :- part(X172, X174, X175, X176).\nand substitution\nT22 -> T84\nX172 -> T84\nX173 -> T85\nX174 -> T86\nT24 -> .(T85, T86)\nX52 -> X177\nX175 -> X177\nX176 -> X178\nX53 -> .(T85, X178)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 134 [arrowhead = none ];
134 -> 35;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 135 [arrowhead = none ];
135 -> 36;

136 [label="EVAL with clause\npart(X193, [], [], []).\nand substitution\nT22 -> T92\nX193 -> T92\nT24 -> []\nX52 -> []\nX53 -> []", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 136 [arrowhead = none ];
136 -> 37;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 137 [arrowhead = none ];
137 -> 38;

138 [label="INSTANCE with matching:\nT22 -> T84\nT24 -> T86\nX52 -> X177\nX53 -> X178", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 138 [arrowhead = none , style=dashed];
138 -> 24[style=dashed];

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 139 [arrowhead = none ];
139 -> 39;

140 [label="INSTANCE with matching:\nT1 -> .(T23, T45)\nT2 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 140 [arrowhead = none , style=dashed];
140 -> 1[style=dashed];

141 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 141 [arrowhead = none ];
141 -> 42;

142 [label="SPLIT 2\nnew knowledge:\nT98 is ground\nreplacements:\nX15 -> T98", fontsize=14, style = filled, fillcolor = violet];
41 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 143 [arrowhead = none ];
143 -> 44;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 145 [arrowhead = none ];
145 -> 45;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 146 [arrowhead = none ];
146 -> 46;

147 [label="EVAL with clause\nqs([], []).\nand substitution\nT46 -> []\nX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 147 [arrowhead = none ];
147 -> 47;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 148 [arrowhead = none ];
148 -> 48;

149 [label="EVAL with clause\nqs(.(X223, X224), X225) :- ','(part(X223, X224, X226, X227), ','(qs(X226, X228), ','(qs(X227, X229), app(X228, .(X223, X229), X225)))).\nand substitution\nX223 -> T103\nX224 -> T104\nT46 -> .(T103, T104)\nX15 -> X230\nX225 -> X230", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 149 [arrowhead = none ];
149 -> 50;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 150 [arrowhead = none ];
150 -> 51;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 151 [arrowhead = none ];
151 -> 49;

152 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 152 [arrowhead = none ];
152 -> 52;

153 [label="SPLIT 2\nnew knowledge:\nT108 is ground\nT109 is ground\nreplacements:\nX226 -> T108\nX227 -> T109", fontsize=14, style = filled, fillcolor = violet];
50 -> 153 [arrowhead = none ];
153 -> 53;

154 [label="INSTANCE with matching:\nT22 -> T103\nT24 -> T104\nX52 -> X226\nX53 -> X227", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 154 [arrowhead = none , style=dashed];
154 -> 24[style=dashed];

155 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 155 [arrowhead = none ];
155 -> 54;

156 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nreplacements:\nX228 -> T113", fontsize=14, style = filled, fillcolor = violet];
53 -> 156 [arrowhead = none ];
156 -> 55;

157 [label="INSTANCE with matching:\nT46 -> T108\nX15 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 157 [arrowhead = none , style=dashed];
157 -> 42[style=dashed];

158 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 158 [arrowhead = none ];
158 -> 56;

159 [label="SPLIT 2\nnew knowledge:\nT114 is ground\nreplacements:\nX229 -> T114", fontsize=14, style = filled, fillcolor = violet];
55 -> 159 [arrowhead = none ];
159 -> 57;

160 [label="INSTANCE with matching:\nT46 -> T109\nX15 -> X229", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 160 [arrowhead = none , style=dashed];
160 -> 42[style=dashed];

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 161 [arrowhead = none ];
161 -> 58;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 162 [arrowhead = none ];
162 -> 59;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 163 [arrowhead = none ];
163 -> 60;

164 [label="EVAL with clause\napp([], X254, X254).\nand substitution\nT113 -> []\nT103 -> T127\nT114 -> T128\nX254 -> .(T127, T128)\nX230 -> .(T127, T128)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 164 [arrowhead = none ];
164 -> 61;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 165 [arrowhead = none ];
165 -> 62;

166 [label="EVAL with clause\napp(.(X266, X267), X268, .(X266, X269)) :- app(X267, X268, X269).\nand substitution\nX266 -> T137\nX267 -> T138\nT113 -> .(T137, T138)\nT103 -> T139\nT114 -> T140\nX268 -> .(T139, T140)\nX269 -> X270\nX230 -> .(T137, X270)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 166 [arrowhead = none ];
166 -> 64;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 167 [arrowhead = none ];
167 -> 65;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 168 [arrowhead = none ];
168 -> 63;

169 [label="INSTANCE with matching:\nT113 -> T138\nT103 -> T139\nT114 -> T140\nX230 -> X270", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 169 [arrowhead = none , style=dashed];
169 -> 57[style=dashed];

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 170 [arrowhead = none ];
170 -> 67;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 171 [arrowhead = none ];
171 -> 68;

172 [label="EVAL with clause\napp([], X283, X283).\nand substitution\nT95 -> []\nT22 -> T157\nT98 -> T158\nX283 -> .(T157, T158)\nT9 -> .(T157, T158)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 172 [arrowhead = none ];
172 -> 69;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 173 [arrowhead = none ];
173 -> 70;

174 [label="EVAL with clause\napp(.(X293, X294), X295, .(X293, X296)) :- app(X294, X295, X296).\nand substitution\nX293 -> T169\nX294 -> T170\nT95 -> .(T169, T170)\nT22 -> T171\nT98 -> T172\nX295 -> .(T171, T172)\nX296 -> T174\nT9 -> .(T169, T174)\nT173 -> T174", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 174 [arrowhead = none ];
174 -> 72;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 175 [arrowhead = none ];
175 -> 73;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 176 [arrowhead = none ];
176 -> 71;

177 [label="INSTANCE with matching:\nT95 -> T170\nT22 -> T171\nT98 -> T172\nT9 -> T174", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 177 [arrowhead = none , style=dashed];
177 -> 43[style=dashed];

178 [label="EVAL with clause\npart(X331, .(X332, X333), X334, .(X332, X335)) :- part(X331, X333, X334, X335).\nand substitution\nT6 -> T191\nX331 -> T191\nX332 -> T192\nX333 -> T193\nT7 -> .(T192, T193)\nX12 -> X336\nX334 -> X336\nX335 -> X337\nX13 -> .(T192, X337)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 178 [arrowhead = none ];
178 -> 76;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 179 [arrowhead = none ];
179 -> 77;

180 [label="EVAL with clause\npart(X367, [], [], []).\nand substitution\nT6 -> T207\nX367 -> T207\nT7 -> []\nX12 -> []\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 180 [arrowhead = none ];
180 -> 82;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 181 [arrowhead = none ];
181 -> 83;

182 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
76 -> 182 [arrowhead = none ];
182 -> 78;

183 [label="SPLIT 2\nnew knowledge:\nT197 is ground\nT198 is ground\nreplacements:\nX336 -> T197\nX337 -> T198", fontsize=14, style = filled, fillcolor = violet];
76 -> 183 [arrowhead = none ];
183 -> 79;

184 [label="INSTANCE with matching:\nT22 -> T191\nT24 -> T193\nX52 -> X336\nX53 -> X337", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 184 [arrowhead = none , style=dashed];
184 -> 24[style=dashed];

185 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
79 -> 185 [arrowhead = none ];
185 -> 80;

186 [label="SPLIT 2\nnew knowledge:\nT202 is ground\nreplacements:\nX14 -> T202", fontsize=14, style = filled, fillcolor = violet];
79 -> 186 [arrowhead = none ];
186 -> 81;

187 [label="INSTANCE with matching:\nT46 -> T197\nX15 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 187 [arrowhead = none , style=dashed];
187 -> 42[style=dashed];

188 [label="INSTANCE with matching:\nT46 -> .(T192, T198)\nT95 -> T202\nT22 -> T191", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 188 [arrowhead = none , style=dashed];
188 -> 41[style=dashed];

189 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 189 [arrowhead = none ];
189 -> 84;

190 [label="SPLIT 2\nnew knowledge:\nT210 is ground\nreplacements:\nX14 -> T210", fontsize=14, style = filled, fillcolor = violet];
82 -> 190 [arrowhead = none ];
190 -> 85;

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 191 [arrowhead = none ];
191 -> 86;

192 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 192 [arrowhead = none ];
192 -> 92;

193 [label="SPLIT 2\nnew knowledge:\nT213 is ground\nreplacements:\nX15 -> T213", fontsize=14, style = filled, fillcolor = violet];
85 -> 193 [arrowhead = none ];
193 -> 93;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 194 [arrowhead = none ];
194 -> 87;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 195 [arrowhead = none ];
195 -> 88;

196 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX14 -> []", fontsize=14, style = filled, fillcolor = white];
87 -> 196 [arrowhead = none ];
196 -> 89;

197 [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];
88 -> 197 [arrowhead = none ];
197 -> 91;

198 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 198 [arrowhead = none ];
198 -> 90;

199 [label="INSTANCE with matching:\nX14 -> X15", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 199 [arrowhead = none , style=dashed];
199 -> 84[style=dashed];

200 [label="INSTANCE with matching:\nT95 -> T210\nT22 -> T207\nT98 -> T213", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 200 [arrowhead = none , style=dashed];
200 -> 43[style=dashed];

}

(2) Obligation:

Triples:

less14(s(T38), s(T39)) :- less14(T38, T39).
part24(T62, .(T63, T64), .(T63, X126), X127) :- less14(T62, T63).
part24(T62, .(T63, T64), .(T63, X126), X127) :- ','(lessc14(T62, T63), part24(T62, T64, X126, X127)).
part24(T84, .(T85, T86), X177, .(T85, X178)) :- part24(T84, T86, X177, X178).
qs42(.(T103, T104), X230) :- part24(T103, T104, X226, X227).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), qs42(T108, X228)).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), qs42(T109, X229))).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), ','(qsc42(T109, T114), app57(T113, T103, T114, X230)))).
app57(.(T137, T138), T139, T140, .(T137, X270)) :- app57(T138, T139, T140, X270).
app43(.(T169, T170), T171, T172, .(T169, T174)) :- app43(T170, T171, T172, T174).
p41(T46, X15, T95, T22, T9) :- qs42(T46, X15).
p41(T46, T98, T95, T22, T9) :- ','(qsc42(T46, T98), app43(T95, T22, T98, T9)).
qs1(.(T22, .(T23, T24)), T9) :- less14(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), part24(T22, T24, X52, X53)).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), qs1(.(T23, T45), X14))).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), ','(qsc1(.(T23, T45), T95), p41(T46, X15, T95, T22, T9)))).
qs1(.(T191, .(T192, T193)), T9) :- part24(T191, T193, X336, X337).
qs1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), qs42(T197, X14)).
qs1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), ','(qsc42(T197, T202), p41(.(T192, T198), X15, T202, T191, T9))).
qs1(.(T207, []), T9) :- qs84(X14).
qs1(.(T207, []), T9) :- ','(qsc84(T210), qs84(X15)).
qs1(.(T207, []), T9) :- ','(qsc84(T210), ','(qsc84(T213), app43(T210, T207, T213, T9))).

Clauses:

lessc14(0, s(T33)).
lessc14(s(T38), s(T39)) :- lessc14(T38, T39).
partc24(T62, .(T63, T64), .(T63, X126), X127) :- ','(lessc14(T62, T63), partc24(T62, T64, X126, X127)).
partc24(T84, .(T85, T86), X177, .(T85, X178)) :- partc24(T84, T86, X177, X178).
partc24(T92, [], [], []).
qsc1([], []).
qsc1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), ','(qsc1(.(T23, T45), T95), qc41(T46, X15, T95, T22, T9)))).
qsc1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), ','(qsc42(T197, T202), qc41(.(T192, T198), X15, T202, T191, T9))).
qsc1(.(T207, []), T9) :- ','(qsc84(T210), ','(qsc84(T213), appc43(T210, T207, T213, T9))).
qsc42([], []).
qsc42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), ','(qsc42(T109, T114), appc57(T113, T103, T114, X230)))).
appc57([], T127, T128, .(T127, T128)).
appc57(.(T137, T138), T139, T140, .(T137, X270)) :- appc57(T138, T139, T140, X270).
appc43([], T157, T158, .(T157, T158)).
appc43(.(T169, T170), T171, T172, .(T169, T174)) :- appc43(T170, T171, T172, T174).
qc41(T46, T98, T95, T22, T9) :- ','(qsc42(T46, T98), appc43(T95, T22, T98, T9)).
qsc84([]).

Afs:

qs1(x1, x2) = qs1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

less14(s(T38), s(T39)) :- less14(T38, T39).
part24(T62, .(T63, T64), .(T63, X126), X127) :- less14(T62, T63).
part24(T62, .(T63, T64), .(T63, X126), X127) :- ','(lessc14(T62, T63), part24(T62, T64, X126, X127)).
part24(T84, .(T85, T86), X177, .(T85, X178)) :- part24(T84, T86, X177, X178).
qs42(.(T103, T104), X230) :- part24(T103, T104, X226, X227).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), qs42(T108, X228)).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), qs42(T109, X229))).
qs42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), ','(qsc42(T109, T114), app57(T113, T103, T114, X230)))).
app57(.(T137, T138), T139, T140, .(T137, X270)) :- app57(T138, T139, T140, X270).
app43(.(T169, T170), T171, T172, .(T169, T174)) :- app43(T170, T171, T172, T174).
p41(T46, X15, T95, T22, T9) :- qs42(T46, X15).
p41(T46, T98, T95, T22, T9) :- ','(qsc42(T46, T98), app43(T95, T22, T98, T9)).
qs1(.(T22, .(T23, T24)), T9) :- less14(T22, T23).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), part24(T22, T24, X52, X53)).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), qs1(.(T23, T45), X14))).
qs1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), ','(qsc1(.(T23, T45), T95), p41(T46, X15, T95, T22, T9)))).
qs1(.(T191, .(T192, T193)), T9) :- part24(T191, T193, X336, X337).
qs1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), qs42(T197, X14)).
qs1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), ','(qsc42(T197, T202), p41(.(T192, T198), X15, T202, T191, T9))).
qs1(.(T207, []), T9) :- ','(qsc84(T210), ','(qsc84(T213), app43(T210, T207, T213, T9))).

Clauses:

lessc14(0, s(T33)).
lessc14(s(T38), s(T39)) :- lessc14(T38, T39).
partc24(T62, .(T63, T64), .(T63, X126), X127) :- ','(lessc14(T62, T63), partc24(T62, T64, X126, X127)).
partc24(T84, .(T85, T86), X177, .(T85, X178)) :- partc24(T84, T86, X177, X178).
partc24(T92, [], [], []).
qsc1([], []).
qsc1(.(T22, .(T23, T24)), T9) :- ','(lessc14(T22, T23), ','(partc24(T22, T24, T45, T46), ','(qsc1(.(T23, T45), T95), qc41(T46, X15, T95, T22, T9)))).
qsc1(.(T191, .(T192, T193)), T9) :- ','(partc24(T191, T193, T197, T198), ','(qsc42(T197, T202), qc41(.(T192, T198), X15, T202, T191, T9))).
qsc1(.(T207, []), T9) :- ','(qsc84(T210), ','(qsc84(T213), appc43(T210, T207, T213, T9))).
qsc42([], []).
qsc42(.(T103, T104), X230) :- ','(partc24(T103, T104, T108, T109), ','(qsc42(T108, T113), ','(qsc42(T109, T114), appc57(T113, T103, T114, X230)))).
appc57([], T127, T128, .(T127, T128)).
appc57(.(T137, T138), T139, T140, .(T137, X270)) :- appc57(T138, T139, T140, X270).
appc43([], T157, T158, .(T157, T158)).
appc43(.(T169, T170), T171, T172, .(T169, T174)) :- appc43(T170, T171, T172, T174).
qc41(T46, T98, T95, T22, T9) :- ','(qsc42(T46, T98), appc43(T95, T22, T98, T9)).
qsc84([]).

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)
less14_in: (b,b)
lessc14_in: (b,b)
part24_in: (b,b,f,f)
partc24_in: (b,b,f,f)
qs42_in: (b,f)
qsc42_in: (b,f)
appc57_in: (b,b,b,f)
app57_in: (b,b,b,f)
p41_in: (b,f,b,b,f)
app43_in: (b,b,b,f)
qsc1_in: (b,f)
qc41_in: (b,f,b,b,f)
appc43_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) → U18_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U3_GGAA(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U4_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U5_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U22_GA(T22, T23, T24, T9, qs1_in_ga(.(T23, T45), X14))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45), X14)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U25_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U26_GA(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U27_GA(T191, T192, T193, T9, qs42_in_ga(T197, X14))
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_GA(T197, X14)
QS42_IN_GA(.(T103, T104), X230) → U6_GA(T103, T104, X230, part24_in_ggaa(T103, T104, X226, X227))
QS42_IN_GA(.(T103, T104), X230) → PART24_IN_GGAA(T103, T104, X226, X227)
QS42_IN_GA(.(T103, T104), X230) → U7_GA(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U8_GA(T103, T104, X230, qs42_in_ga(T108, X228))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108, X228)
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U10_GA(T103, T104, X230, qs42_in_ga(T109, X229))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109, X229)
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U11_GA(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U11_GA(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U12_GA(T103, T104, X230, app57_in_ggga(T113, T103, T114, X230))
U11_GA(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → APP57_IN_GGGA(T113, T103, T114, X230)
APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → U13_GGGA(T137, T138, T139, T140, X270, app57_in_ggga(T138, T139, T140, X270))
APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_GGGA(T138, T139, T140, X270)
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U28_GA(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U28_GA(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U29_GA(T191, T192, T193, T9, p41_in_gagga(.(T192, T198), X15, T202, T191, T9))
U28_GA(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → P41_IN_GAGGA(.(T192, T198), X15, T202, T191, T9)
P41_IN_GAGGA(T46, X15, T95, T22, T9) → U15_GAGGA(T46, X15, T95, T22, T9, qs42_in_ga(T46, X15))
P41_IN_GAGGA(T46, X15, T95, T22, T9) → QS42_IN_GA(T46, X15)
P41_IN_GAGGA(T46, T98, T95, T22, T9) → U16_GAGGA(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U16_GAGGA(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U17_GAGGA(T46, T98, T95, T22, T9, app43_in_ggga(T95, T22, T98, T9))
U16_GAGGA(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → APP43_IN_GGGA(T95, T22, T98, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U14_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)
QS1_IN_GA(.(T207, []), T9) → U30_GA(T207, T9, qsc84_in_a(T210))
U30_GA(T207, T9, qsc84_out_a(T210)) → U31_GA(T207, T9, T210, qsc84_in_a(T213))
U31_GA(T207, T9, T210, qsc84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qsc84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U23_GA(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
U23_GA(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U24_GA(T22, T23, T24, T9, p41_in_gagga(T46, X15, T95, T22, T9))
U23_GA(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → P41_IN_GAGGA(T46, X15, T95, T22, T9)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
less14_in_gg(x1, x2) = less14_in_gg(x1, x2)
s(x1) = s(x1)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
part24_in_ggaa(x1, x2, x3, x4) = part24_in_ggaa(x1, x2)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qs42_in_ga(x1, x2) = qs42_in_ga(x1)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
app57_in_ggga(x1, x2, x3, x4) = app57_in_ggga(x1, x2, x3)
p41_in_gagga(x1, x2, x3, x4, x5) = p41_in_gagga(x1, x3, x4)
app43_in_ggga(x1, x2, x3, x4) = app43_in_ggga(x1, x2, x3)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2) = QS1_IN_GA(x1)
U18_GA(x1, x2, x3, x4, x5) = U18_GA(x1, x2, x3, x5)
LESS14_IN_GG(x1, x2) = LESS14_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3)
U19_GA(x1, x2, x3, x4, x5) = U19_GA(x1, x2, x3, x5)
U20_GA(x1, x2, x3, x4, x5) = U20_GA(x1, x2, x3, x5)
PART24_IN_GGAA(x1, x2, x3, x4) = PART24_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)
U21_GA(x1, x2, x3, x4, x5) = U21_GA(x1, x2, x3, x5)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)
U25_GA(x1, x2, x3, x4, x5) = U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5) = U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5) = U27_GA(x1, x2, x3, x5)
QS42_IN_GA(x1, x2) = QS42_IN_GA(x1)
U6_GA(x1, x2, x3, x4) = U6_GA(x1, x2, x4)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x2, x4)
U8_GA(x1, x2, x3, x4) = U8_GA(x1, x2, x4)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x4, x5)
U10_GA(x1, x2, x3, x4) = U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x4, x5)
U12_GA(x1, x2, x3, x4) = U12_GA(x1, x2, x4)
APP57_IN_GGGA(x1, x2, x3, x4) = APP57_IN_GGGA(x1, x2, x3)
U13_GGGA(x1, x2, x3, x4, x5, x6) = U13_GGGA(x1, x2, x3, x4, x6)
U28_GA(x1, x2, x3, x4, x5, x6) = U28_GA(x1, x2, x3, x5, x6)
U29_GA(x1, x2, x3, x4, x5) = U29_GA(x1, x2, x3, x5)
P41_IN_GAGGA(x1, x2, x3, x4, x5) = P41_IN_GAGGA(x1, x3, x4)
U15_GAGGA(x1, x2, x3, x4, x5, x6) = U15_GAGGA(x1, x3, x4, x6)
U16_GAGGA(x1, x2, x3, x4, x5, x6) = U16_GAGGA(x1, x3, x4, x6)
U17_GAGGA(x1, x2, x3, x4, x5, x6) = U17_GAGGA(x1, x3, x4, x6)
APP43_IN_GGGA(x1, x2, x3, x4) = APP43_IN_GGGA(x1, x2, x3)
U14_GGGA(x1, x2, x3, x4, x5, x6) = U14_GGGA(x1, x2, x3, x4, x6)
U30_GA(x1, x2, x3) = U30_GA(x1, x3)
U31_GA(x1, x2, x3, x4) = U31_GA(x1, x3, x4)
U32_GA(x1, x2, x3) = U32_GA(x1, x3)
U23_GA(x1, x2, x3, x4, x5, x6) = U23_GA(x1, x2, x3, x5, x6)
U24_GA(x1, x2, x3, x4, x5) = U24_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) → U18_GA(T22, T23, T24, T9, less14_in_gg(T22, T23))
QS1_IN_GA(.(T22, .(T23, T24)), T9) → LESS14_IN_GG(T22, T23)
LESS14_IN_GG(s(T38), s(T39)) → U1_GG(T38, T39, less14_in_gg(T38, T39))
LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U20_GA(T22, T23, T24, T9, part24_in_ggaa(T22, T24, X52, X53))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → PART24_IN_GGAA(T22, T24, X52, X53)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U2_GGAA(T62, T63, T64, X126, X127, less14_in_gg(T62, T63))
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → LESS14_IN_GG(T62, T63)
PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U3_GGAA(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U4_GGAA(T62, T63, T64, X126, X127, part24_in_ggaa(T62, T64, X126, X127))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → U5_GGAA(T84, T85, T86, X177, X178, part24_in_ggaa(T84, T86, X177, X178))
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U22_GA(T22, T23, T24, T9, qs1_in_ga(.(T23, T45), X14))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45), X14)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U25_GA(T191, T192, T193, T9, part24_in_ggaa(T191, T193, X336, X337))
QS1_IN_GA(.(T191, .(T192, T193)), T9) → PART24_IN_GGAA(T191, T193, X336, X337)
QS1_IN_GA(.(T191, .(T192, T193)), T9) → U26_GA(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U27_GA(T191, T192, T193, T9, qs42_in_ga(T197, X14))
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → QS42_IN_GA(T197, X14)
QS42_IN_GA(.(T103, T104), X230) → U6_GA(T103, T104, X230, part24_in_ggaa(T103, T104, X226, X227))
QS42_IN_GA(.(T103, T104), X230) → PART24_IN_GGAA(T103, T104, X226, X227)
QS42_IN_GA(.(T103, T104), X230) → U7_GA(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U8_GA(T103, T104, X230, qs42_in_ga(T108, X228))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108, X228)
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U10_GA(T103, T104, X230, qs42_in_ga(T109, X229))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109, X229)
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U11_GA(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U11_GA(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U12_GA(T103, T104, X230, app57_in_ggga(T113, T103, T114, X230))
U11_GA(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → APP57_IN_GGGA(T113, T103, T114, X230)
APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → U13_GGGA(T137, T138, T139, T140, X270, app57_in_ggga(T138, T139, T140, X270))
APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_GGGA(T138, T139, T140, X270)
U26_GA(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U28_GA(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U28_GA(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U29_GA(T191, T192, T193, T9, p41_in_gagga(.(T192, T198), X15, T202, T191, T9))
U28_GA(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → P41_IN_GAGGA(.(T192, T198), X15, T202, T191, T9)
P41_IN_GAGGA(T46, X15, T95, T22, T9) → U15_GAGGA(T46, X15, T95, T22, T9, qs42_in_ga(T46, X15))
P41_IN_GAGGA(T46, X15, T95, T22, T9) → QS42_IN_GA(T46, X15)
P41_IN_GAGGA(T46, T98, T95, T22, T9) → U16_GAGGA(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U16_GAGGA(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U17_GAGGA(T46, T98, T95, T22, T9, app43_in_ggga(T95, T22, T98, T9))
U16_GAGGA(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → APP43_IN_GGGA(T95, T22, T98, T9)
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → U14_GGGA(T169, T170, T171, T172, T174, app43_in_ggga(T170, T171, T172, T174))
APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)
QS1_IN_GA(.(T207, []), T9) → U30_GA(T207, T9, qsc84_in_a(T210))
U30_GA(T207, T9, qsc84_out_a(T210)) → U31_GA(T207, T9, T210, qsc84_in_a(T213))
U31_GA(T207, T9, T210, qsc84_out_a(T213)) → U32_GA(T207, T9, app43_in_ggga(T210, T207, T213, T9))
U31_GA(T207, T9, T210, qsc84_out_a(T213)) → APP43_IN_GGGA(T210, T207, T213, T9)
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U23_GA(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
U23_GA(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U24_GA(T22, T23, T24, T9, p41_in_gagga(T46, X15, T95, T22, T9))
U23_GA(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → P41_IN_GAGGA(T46, X15, T95, T22, T9)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, T22, T9)) → qsc1_out_ga(.(T22, .(T23, T24)), T9)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 39 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
APP43_IN_GGGA(x1, x2, x3, x4) = APP43_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:

APP43_IN_GGGA(.(T169, T170), T171, T172, .(T169, T174)) → APP43_IN_GGGA(T170, T171, T172, T174)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP43_IN_GGGA(x1, x2, x3, x4) = APP43_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:

APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)

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:

APP43_IN_GGGA(.(T169, T170), T171, T172) → APP43_IN_GGGA(T170, T171, T172)
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:

APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_GGGA(T138, T139, T140, X270)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
APP57_IN_GGGA(x1, x2, x3, x4) = APP57_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:

APP57_IN_GGGA(.(T137, T138), T139, T140, .(T137, X270)) → APP57_IN_GGGA(T138, T139, T140, X270)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP57_IN_GGGA(x1, x2, x3, x4) = APP57_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:

APP57_IN_GGGA(.(T137, T138), T139, T140) → APP57_IN_GGGA(T138, T139, T140)

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:

APP57_IN_GGGA(.(T137, T138), T139, T140) → APP57_IN_GGGA(T138, T139, T140)
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:

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
LESS14_IN_GG(x1, x2) = LESS14_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:

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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:

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)

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:

LESS14_IN_GG(s(T38), s(T39)) → LESS14_IN_GG(T38, T39)
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:

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U3_GGAA(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
PART24_IN_GGAA(x1, x2, x3, x4) = PART24_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6)

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:

PART24_IN_GGAA(T62, .(T63, T64), .(T63, X126), X127) → U3_GGAA(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U3_GGAA(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64, X126, X127)
PART24_IN_GGAA(T84, .(T85, T86), X177, .(T85, X178)) → PART24_IN_GGAA(T84, T86, X177, X178)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

PART24_IN_GGAA(T62, .(T63, T64)) → U3_GGAA(T62, T63, T64, lessc14_in_gg(T62, T63))
U3_GGAA(T62, T63, T64, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64)
PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))

The set Q consists of the following terms:

lessc14_in_gg(x0, x1)
U34_gg(x0, x1, x2)

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:

U3_GGAA(T62, T63, T64, lessc14_out_gg(T62, T63)) → PART24_IN_GGAA(T62, T64)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
PART24_IN_GGAA(T84, .(T85, T86)) → PART24_IN_GGAA(T84, T86)
The graph contains the following edges 1 >= 1, 2 > 2
PART24_IN_GGAA(T62, .(T63, T64)) → U3_GGAA(T62, T63, T64, lessc14_in_gg(T62, T63))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(36) YES

(37) Obligation:

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

QS42_IN_GA(.(T103, T104), X230) → U7_GA(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108, X228)
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109, X229)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
QS42_IN_GA(x1, x2) = QS42_IN_GA(x1)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x2, x4)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x4, x5)

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:

QS42_IN_GA(.(T103, T104), X230) → U7_GA(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108, X228)
U7_GA(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U9_GA(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109, X229)

The TRS R consists of the following rules:

partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))

(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:

QS42_IN_GA(.(T103, T104)) → U7_GA(T103, T104, partc24_in_ggaa(T103, T104))
U7_GA(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108)
U7_GA(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, T109, qsc42_in_ga(T108))
U9_GA(T103, T104, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109)

The TRS R consists of the following rules:

partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
qsc42_in_ga([]) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104)) → U48_ga(T103, T104, partc24_in_ggaa(T103, T104))
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U48_ga(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, T109, qsc42_in_ga(T108))
lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U49_ga(T103, T104, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, T113, qsc42_in_ga(T109))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U50_ga(T103, T104, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, appc57_in_ggga(T113, T103, T114))
U51_ga(T103, T104, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
appc57_in_ggga([], T127, T128) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140) → U52_ggga(T137, T138, T139, T140, appc57_in_ggga(T138, T139, T140))
U52_ggga(T137, T138, T139, T140, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))

The set Q consists of the following terms:

partc24_in_ggaa(x0, x1)
qsc42_in_ga(x0)
U35_ggaa(x0, x1, x2, x3)
U37_ggaa(x0, x1, x2, x3)
U48_ga(x0, x1, x2)
lessc14_in_gg(x0, x1)
U36_ggaa(x0, x1, x2, x3)
U49_ga(x0, x1, x2, x3)
U34_gg(x0, x1, x2)
U50_ga(x0, x1, x2, x3)
U51_ga(x0, x1, x2)
appc57_in_ggga(x0, x1, x2)
U52_ggga(x0, x1, x2, x3, x4)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_GA(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → QS42_IN_GA(T108)
U7_GA(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → U9_GA(T103, T104, T109, qsc42_in_ga(T108))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QS42_IN_GA(x₁)) = 1 + x₁
POL(U34_gg(x₁, x₂, x₃)) = 1
POL(U35_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(U36_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U37_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U48_ga(x₁, x₂, x₃)) = 0
POL(U49_ga(x₁, x₂, x₃, x₄)) = 0
POL(U50_ga(x₁, x₂, x₃, x₄)) = 0
POL(U51_ga(x₁, x₂, x₃)) = 0
POL(U52_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U7_GA(x₁, x₂, x₃)) = 1 + x₃
POL(U9_GA(x₁, x₂, x₃, x₄)) = 1 + x₃
POL([]) = 0
POL(appc57_in_ggga(x₁, x₂, x₃)) = 0
POL(appc57_out_ggga(x₁, x₂, x₃, x₄)) = 0
POL(lessc14_in_gg(x₁, x₂)) = 1
POL(lessc14_out_gg(x₁, x₂)) = 1
POL(partc24_in_ggaa(x₁, x₂)) = 1 + x₂
POL(partc24_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(qsc42_in_ga(x₁)) = 0
POL(qsc42_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))

(43) Obligation:

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

QS42_IN_GA(.(T103, T104)) → U7_GA(T103, T104, partc24_in_ggaa(T103, T104))
U9_GA(T103, T104, T109, qsc42_out_ga(T108, T113)) → QS42_IN_GA(T109)

The TRS R consists of the following rules:

partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
qsc42_in_ga([]) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104)) → U48_ga(T103, T104, partc24_in_ggaa(T103, T104))
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U48_ga(T103, T104, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, T109, qsc42_in_ga(T108))
lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U49_ga(T103, T104, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, T113, qsc42_in_ga(T109))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U50_ga(T103, T104, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, appc57_in_ggga(T113, T103, T114))
U51_ga(T103, T104, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
appc57_in_ggga([], T127, T128) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140) → U52_ggga(T137, T138, T139, T140, appc57_in_ggga(T138, T139, T140))
U52_ggga(T137, T138, T139, T140, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))

The set Q consists of the following terms:

partc24_in_ggaa(x0, x1)
qsc42_in_ga(x0)
U35_ggaa(x0, x1, x2, x3)
U37_ggaa(x0, x1, x2, x3)
U48_ga(x0, x1, x2)
lessc14_in_gg(x0, x1)
U36_ggaa(x0, x1, x2, x3)
U49_ga(x0, x1, x2, x3)
U34_gg(x0, x1, x2)
U50_ga(x0, x1, x2, x3)
U51_ga(x0, x1, x2)
appc57_in_ggga(x0, x1, x2)
U52_ggga(x0, x1, x2, x3, x4)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

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

(45) TRUE

(46) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45), X14)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
qsc42_in_ga([], []) → qsc42_out_ga([], [])
qsc42_in_ga(.(T103, T104), X230) → U48_ga(T103, T104, X230, partc24_in_ggaa(T103, T104, T108, T109))
U48_ga(T103, T104, X230, partc24_out_ggaa(T103, T104, T108, T109)) → U49_ga(T103, T104, X230, T109, qsc42_in_ga(T108, T113))
U49_ga(T103, T104, X230, T109, qsc42_out_ga(T108, T113)) → U50_ga(T103, T104, X230, T113, qsc42_in_ga(T109, T114))
U50_ga(T103, T104, X230, T113, qsc42_out_ga(T109, T114)) → U51_ga(T103, T104, X230, appc57_in_ggga(T113, T103, T114, X230))
appc57_in_ggga([], T127, T128, .(T127, T128)) → appc57_out_ggga([], T127, T128, .(T127, T128))
appc57_in_ggga(.(T137, T138), T139, T140, .(T137, X270)) → U52_ggga(T137, T138, T139, T140, X270, appc57_in_ggga(T138, T139, T140, X270))
U52_ggga(T137, T138, T139, T140, X270, appc57_out_ggga(T138, T139, T140, X270)) → appc57_out_ggga(.(T137, T138), T139, T140, .(T137, X270))
U51_ga(T103, T104, X230, appc57_out_ggga(T113, T103, T114, X230)) → qsc42_out_ga(.(T103, T104), X230)
qsc84_in_a([]) → qsc84_out_a([])
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(.(T22, .(T23, T24)), T9) → U38_ga(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U38_ga(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U39_ga(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U39_ga(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → U40_ga(T22, T23, T24, T9, T46, qsc1_in_ga(.(T23, T45), T95))
qsc1_in_ga(.(T191, .(T192, T193)), T9) → U42_ga(T191, T192, T193, T9, partc24_in_ggaa(T191, T193, T197, T198))
U42_ga(T191, T192, T193, T9, partc24_out_ggaa(T191, T193, T197, T198)) → U43_ga(T191, T192, T193, T9, T198, qsc42_in_ga(T197, T202))
U43_ga(T191, T192, T193, T9, T198, qsc42_out_ga(T197, T202)) → U44_ga(T191, T192, T193, T9, qc41_in_gagga(.(T192, T198), X15, T202, T191, T9))
qc41_in_gagga(T46, T98, T95, T22, T9) → U54_gagga(T46, T98, T95, T22, T9, qsc42_in_ga(T46, T98))
U54_gagga(T46, T98, T95, T22, T9, qsc42_out_ga(T46, T98)) → U55_gagga(T46, T98, T95, T22, T9, appc43_in_ggga(T95, T22, T98, T9))
appc43_in_ggga([], T157, T158, .(T157, T158)) → appc43_out_ggga([], T157, T158, .(T157, T158))
appc43_in_ggga(.(T169, T170), T171, T172, .(T169, T174)) → U53_ggga(T169, T170, T171, T172, T174, appc43_in_ggga(T170, T171, T172, T174))
U53_ggga(T169, T170, T171, T172, T174, appc43_out_ggga(T170, T171, T172, T174)) → appc43_out_ggga(.(T169, T170), T171, T172, .(T169, T174))
U55_gagga(T46, T98, T95, T22, T9, appc43_out_ggga(T95, T22, T98, T9)) → qc41_out_gagga(T46, T98, T95, T22, T9)
U44_ga(T191, T192, T193, T9, qc41_out_gagga(.(T192, T198), X15, T202, T191, T9)) → qsc1_out_ga(.(T191, .(T192, T193)), T9)
qsc1_in_ga(.(T207, []), T9) → U45_ga(T207, T9, qsc84_in_a(T210))
U45_ga(T207, T9, qsc84_out_a(T210)) → U46_ga(T207, T9, T210, qsc84_in_a(T213))
U46_ga(T207, T9, T210, qsc84_out_a(T213)) → U47_ga(T207, T9, appc43_in_ggga(T210, T207, T213, T9))
U47_ga(T207, T9, appc43_out_ggga(T210, T207, T213, T9)) → qsc1_out_ga(.(T207, []), T9)
U40_ga(T22, T23, T24, T9, T46, qsc1_out_ga(.(T23, T45), T95)) → U41_ga(T22, T23, T24, T9, qc41_in_gagga(T46, X15, T95, T22, T9))
U41_ga(T22, T23, T24, T9, qc41_out_gagga(T46, X15, T95, 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)
lessc14_in_gg(x1, x2) = lessc14_in_gg(x1, x2)
0 = 0
lessc14_out_gg(x1, x2) = lessc14_out_gg(x1, x2)
U34_gg(x1, x2, x3) = U34_gg(x1, x2, x3)
partc24_in_ggaa(x1, x2, x3, x4) = partc24_in_ggaa(x1, x2)
U35_ggaa(x1, x2, x3, x4, x5, x6) = U35_ggaa(x1, x2, x3, x6)
U36_ggaa(x1, x2, x3, x4, x5, x6) = U36_ggaa(x1, x2, x3, x6)
U37_ggaa(x1, x2, x3, x4, x5, x6) = U37_ggaa(x1, x2, x3, x6)
[] = []
partc24_out_ggaa(x1, x2, x3, x4) = partc24_out_ggaa(x1, x2, x3, x4)
qsc42_in_ga(x1, x2) = qsc42_in_ga(x1)
qsc42_out_ga(x1, x2) = qsc42_out_ga(x1, x2)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
U49_ga(x1, x2, x3, x4, x5) = U49_ga(x1, x2, x4, x5)
U50_ga(x1, x2, x3, x4, x5) = U50_ga(x1, x2, x4, x5)
U51_ga(x1, x2, x3, x4) = U51_ga(x1, x2, x4)
appc57_in_ggga(x1, x2, x3, x4) = appc57_in_ggga(x1, x2, x3)
appc57_out_ggga(x1, x2, x3, x4) = appc57_out_ggga(x1, x2, x3, x4)
U52_ggga(x1, x2, x3, x4, x5, x6) = U52_ggga(x1, x2, x3, x4, x6)
qsc84_in_a(x1) = qsc84_in_a
qsc84_out_a(x1) = qsc84_out_a(x1)
qsc1_in_ga(x1, x2) = qsc1_in_ga(x1)
qsc1_out_ga(x1, x2) = qsc1_out_ga(x1, x2)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5, x6) = U40_ga(x1, x2, x3, x5, x6)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U43_ga(x1, x2, x3, x4, x5, x6) = U43_ga(x1, x2, x3, x5, x6)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
qc41_in_gagga(x1, x2, x3, x4, x5) = qc41_in_gagga(x1, x3, x4)
U54_gagga(x1, x2, x3, x4, x5, x6) = U54_gagga(x1, x3, x4, x6)
U55_gagga(x1, x2, x3, x4, x5, x6) = U55_gagga(x1, x2, x3, x4, x6)
appc43_in_ggga(x1, x2, x3, x4) = appc43_in_ggga(x1, x2, x3)
appc43_out_ggga(x1, x2, x3, x4) = appc43_out_ggga(x1, x2, x3, x4)
U53_ggga(x1, x2, x3, x4, x5, x6) = U53_ggga(x1, x2, x3, x4, x6)
qc41_out_gagga(x1, x2, x3, x4, x5) = qc41_out_gagga(x1, x2, x3, x4, x5)
U45_ga(x1, x2, x3) = U45_ga(x1, x3)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x3, x4)
U47_ga(x1, x2, x3) = U47_ga(x1, x3)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2) = QS1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5) = U19_GA(x1, x2, x3, x5)
U21_GA(x1, x2, x3, x4, x5) = U21_GA(x1, x2, x3, x5)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24)), T9) → U19_GA(T22, T23, T24, T9, lessc14_in_gg(T22, T23))
U19_GA(T22, T23, T24, T9, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, T9, partc24_in_ggaa(T22, T24, T45, T46))
U21_GA(T22, T23, T24, T9, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45), X14)

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
partc24_in_ggaa(T62, .(T63, T64), .(T63, X126), X127) → U35_ggaa(T62, T63, T64, X126, X127, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86), X177, .(T85, X178)) → U37_ggaa(T84, T85, T86, X177, X178, partc24_in_ggaa(T84, T86, X177, X178))
partc24_in_ggaa(T92, [], [], []) → partc24_out_ggaa(T92, [], [], [])
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U35_ggaa(T62, T63, T64, X126, X127, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, X126, X127, partc24_in_ggaa(T62, T64, X126, X127))
U37_ggaa(T84, T85, T86, X177, X178, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, X126, X127, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)

(49) PiDPToQDPProof (SOUND transformation)

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

(50) Obligation:

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

QS1_IN_GA(.(T22, .(T23, T24))) → U19_GA(T22, T23, T24, lessc14_in_gg(T22, T23))
U19_GA(T22, T23, T24, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, partc24_in_ggaa(T22, T24))
U21_GA(T22, T23, T24, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45))

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)

The set Q consists of the following terms:

lessc14_in_gg(x0, x1)
partc24_in_ggaa(x0, x1)
U34_gg(x0, x1, x2)
U35_ggaa(x0, x1, x2, x3)
U37_ggaa(x0, x1, x2, x3)
U36_ggaa(x0, x1, x2, x3)

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

(51) 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))) → U19_GA(T22, T23, T24, lessc14_in_gg(T22, T23))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QS1_IN_GA(x₁)) = x₁
POL(U19_GA(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U21_GA(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U34_gg(x₁, x₂, x₃)) = 0
POL(U35_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U36_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U37_ggaa(x₁, x₂, x₃, x₄)) = x₄
POL([]) = 0
POL(lessc14_in_gg(x₁, x₂)) = 0
POL(lessc14_out_gg(x₁, x₂)) = 0
POL(partc24_in_ggaa(x₁, x₂)) = x₂
POL(partc24_out_ggaa(x₁, x₂, x₃, x₄)) = x₃
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))

(52) Obligation:

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

U19_GA(T22, T23, T24, lessc14_out_gg(T22, T23)) → U21_GA(T22, T23, T24, partc24_in_ggaa(T22, T24))
U21_GA(T22, T23, T24, partc24_out_ggaa(T22, T24, T45, T46)) → QS1_IN_GA(.(T23, T45))

The TRS R consists of the following rules:

lessc14_in_gg(0, s(T33)) → lessc14_out_gg(0, s(T33))
lessc14_in_gg(s(T38), s(T39)) → U34_gg(T38, T39, lessc14_in_gg(T38, T39))
partc24_in_ggaa(T62, .(T63, T64)) → U35_ggaa(T62, T63, T64, lessc14_in_gg(T62, T63))
partc24_in_ggaa(T84, .(T85, T86)) → U37_ggaa(T84, T85, T86, partc24_in_ggaa(T84, T86))
partc24_in_ggaa(T92, []) → partc24_out_ggaa(T92, [], [], [])
U34_gg(T38, T39, lessc14_out_gg(T38, T39)) → lessc14_out_gg(s(T38), s(T39))
U35_ggaa(T62, T63, T64, lessc14_out_gg(T62, T63)) → U36_ggaa(T62, T63, T64, partc24_in_ggaa(T62, T64))
U37_ggaa(T84, T85, T86, partc24_out_ggaa(T84, T86, X177, X178)) → partc24_out_ggaa(T84, .(T85, T86), X177, .(T85, X178))
U36_ggaa(T62, T63, T64, partc24_out_ggaa(T62, T64, X126, X127)) → partc24_out_ggaa(T62, .(T63, T64), .(T63, X126), X127)

The set Q consists of the following terms:

lessc14_in_gg(x0, x1)
partc24_in_ggaa(x0, x1)
U34_gg(x0, x1, x2)
U35_ggaa(x0, x1, x2, x3)
U37_ggaa(x0, x1, x2, x3)
U36_ggaa(x0, x1, x2, x3)

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

(53) 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 (SOUND transformation)

(34) Obligation:

(35) QDPSizeChangeProof (EQUIVALENT transformation)

(36) YES

(37) Obligation:

(38) UsableRulesProof (EQUIVALENT transformation)

(39) Obligation:

(40) PiDPToQDPProof (SOUND transformation)

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) DependencyGraphProof (EQUIVALENT transformation)

(45) TRUE

(46) Obligation:

(47) UsableRulesProof (EQUIVALENT transformation)

(48) Obligation:

(49) PiDPToQDPProof (SOUND transformation)

(50) Obligation:

(51) QDPOrderProof (EQUIVALENT transformation)

(52) Obligation:

(53) DependencyGraphProof (EQUIVALENT transformation)

(54) TRUE