Left Termination proof of /tmp/tmpxX36J2/quicksort-fb.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇐)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇐)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇐)

        ↳34 QDP

        ↳35 NonTerminationProof (⇔)

        ↳36 NO

    ↳37 PiDP

        ↳38 UsableRulesProof (⇔)

        ↳39 PiDP

        ↳40 PiDPToQDPProof (⇐)

        ↳41 QDP

        ↳42 QDPSizeChangeProof (⇔)

        ↳43 YES

    ↳44 PiDP

        ↳45 UsableRulesProof (⇔)

        ↳46 PiDP

        ↳47 PiDPToQDPProof (⇐)

        ↳48 QDP

        ↳49 Narrowing (⇐)

        ↳50 QDP

        ↳51 Instantiation (⇔)

        ↳52 QDP

        ↳53 NonTerminationProof (⇔)

        ↳54 NO

    ↳55 PiDP

        ↳56 UsableRulesProof (⇔)

        ↳57 PiDP

        ↳58 PiDPToQDPProof (⇐)

        ↳59 QDP

        ↳60 NonTerminationProof (⇔)

        ↳61 NO

    ↳62 PiDP

        ↳63 UsableRulesProof (⇔)

        ↳64 PiDP

        ↳65 PiDPToQDPProof (⇐)

        ↳66 QDP

        ↳67 NonTerminationProof (⇔)

        ↳68 NO

    ↳69 PiDP

        ↳70 UsableRulesProof (⇔)

        ↳71 PiDP

        ↳72 PiDPToQDPProof (⇐)

        ↳73 QDP

        ↳74 Narrowing (⇐)

        ↳75 QDP

        ↳76 Narrowing (⇐)

        ↳77 QDP

        ↳78 Narrowing (⇐)

        ↳79 QDP

        ↳80 NonTerminationProof (⇔)

        ↳81 NO

(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(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: qs(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | qs(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nqs(T1, T2) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: qs(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), []))))\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 2 | ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 3 | ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 4\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 2\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 3 | ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 4\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(less(T24, T25), ','(part(T24, T26, X49, X50), ','(qs(.(T25, X49), X11), ','(qs(X50, X12), app(X11, .(T24, X12), [])))))\n\nX11 is free\nX12 is free; \nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T24, T25)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(part(T29, T30, X49, X50), ','(qs(.(T31, X49), X11), ','(qs(X50, X12), app(X11, .(T29, X12), []))))\n\nT29 is ground\nX11 is free; \nX12 is free; \nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(T24, T25), SCOPE: 3, CLAUSE: 7 | less(T24, T25), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(T24, T25), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T24, T25), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: less(T45, T46)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: part(T29, T30, X49, X50)\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(qs(.(T54, T52), X11), ','(qs(T53, X12), app(X11, .(T29, X12), [])))\n\nT29 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 2 | part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 3 | part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 2\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 3 | part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(less(T70, T73), part(T70, T74, X123, X124))\n\nT70 is ground\nX123 is free; \nX124 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T70, T73)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: part(T70, T77, X123, X124)\n\nT70 is ground\nX123 is free; \nX124 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 3\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: part(T29, T30, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: part(T95, T98, X174, X175)\n\nT95 is ground\nX174 is free; \nX175 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: qs(.(T54, T52), X11)\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(qs(T108, X12), app(T107, .(T29, X12), []))\n\nT29 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: qs(.(T54, T52), X11), SCOPE: 5, CLAUSE: 0 | qs(.(T54, T52), X11), SCOPE: 5, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: qs(.(T54, T52), X11), SCOPE: 5, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(part(T122, T123, X230, X231), ','(qs(X230, X232), ','(qs(X231, X233), app(X232, .(T122, X233), X234))))\n\nX234 is free\nX230 is free; \nX231 is free; \nX232 is free; \nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: part(T122, T123, X230, X231)\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(qs(T127, X232), ','(qs(T128, X233), app(X232, .(T129, X233), X234)))\n\nX234 is free\nX232 is free; \nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 2 | part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 3 | part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 4\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 2\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 3 | part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 4\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(less(T148, T149), part(T148, T150, X282, X283))\n\nX282 is free\nX283 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T148, T149)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: part(T153, T154, X282, X283)\n\nT153 is ground\nX282 is free; \nX283 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 3\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: part(T122, T123, X230, X231), SCOPE: 6, CLAUSE: 4\n\nX230 is free\nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: part(T175, T176, X333, X334)\n\nX333 is free\nX334 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: qs(T127, X232)\n\nX232 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(qs(T184, X233), app(T183, .(T185, X233), X234))\n\nX234 is free\nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: qs(T127, X232), SCOPE: 7, CLAUSE: 0 | qs(T127, X232), SCOPE: 7, CLAUSE: 1\n\nX232 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: qs(T127, X232), SCOPE: 7, CLAUSE: 0\n\nX232 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: qs(T127, X232), SCOPE: 7, CLAUSE: 1\n\nX232 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(part(T192, T193, X366, X367), ','(qs(X366, X368), ','(qs(X367, X369), app(X368, .(T192, X369), X370))))\n\nX370 is free\nX366 is free; \nX367 is free; \nX368 is free; \nX369 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: qs(T184, X233)\n\nX233 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: app(T198, .(T199, T197), X234)\n\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T198, .(T199, T197), X234), SCOPE: 8, CLAUSE: 5 | app(T198, .(T199, T197), X234), SCOPE: 8, CLAUSE: 6\n\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T198, .(T199, T197), X234), SCOPE: 8, CLAUSE: 5\n\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(T198, .(T199, T197), X234), SCOPE: 8, CLAUSE: 6\n\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app(T226, .(T227, T228), X403)\n\nX403 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: qs(T108, X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: app(T234, .(T29, T233), [])\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: app(T234, .(T29, T233), []), SCOPE: 9, CLAUSE: 5 | app(T234, .(T29, T233), []), SCOPE: 9, CLAUSE: 6\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: app(T234, .(T29, T233), []), SCOPE: 9, CLAUSE: 6\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 3\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(part(T7, T8, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(T7, X12), [])))), SCOPE: 2, CLAUSE: 4\n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(part(T264, T265, X454, X455), ','(qs(X454, X11), ','(qs(.(T266, X455), X12), app(X11, .(T264, X12), []))))\n\nX11 is free\nX12 is free; \nX454 is free; \nX455 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: part(T264, T265, X454, X455)\n\nX454 is free\nX455 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(qs(T270, X11), ','(qs(.(T272, T271), X12), app(X11, .(T273, X12), [])))\n\nX11 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: qs(T270, X11)\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(qs(.(T278, T279), X12), app(T277, .(T280, X12), []))\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: qs(.(T278, T279), X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: app(T284, .(T285, T283), [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: app(T284, .(T285, T283), []), SCOPE: 10, CLAUSE: 5 | app(T284, .(T285, T283), []), SCOPE: 10, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: app(T284, .(T285, T283), []), SCOPE: 10, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(qs([], X11), ','(qs([], X12), app(X11, .(T305, X12), [])))\n\nX11 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: qs([], X11)\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(qs([], X12), app(T309, .(T305, X12), []))\n\nT309 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: qs([], X11), SCOPE: 11, CLAUSE: 0 | qs([], X11), SCOPE: 11, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: qs([], X11), SCOPE: 11, CLAUSE: 0\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: qs([], X11), SCOPE: 11, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: qs([], X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: app(T309, .(T305, T315), [])\n\nT309 is ground\nT315 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329))))\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 2 | ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 3 | ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 4\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 2\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 3 | ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 4\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: ','(less(T347, T348), ','(part(T347, T349, X581, X582), ','(qs(.(T348, X581), X543), ','(qs(X582, X544), app(X543, .(T347, X544), T329)))))\n\nT329 is ground\nX543 is free; \nX544 is free; \nX581 is free; \nX582 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: less(T347, T348)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: ','(part(T352, T353, X581, X582), ','(qs(.(T354, X581), X543), ','(qs(X582, X544), app(X543, .(T352, X544), T329))))\n\nT329 is ground\nT352 is ground\nX543 is free; \nX544 is free; \nX581 is free; \nX582 is free; \nqs(T355, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: part(T352, T353, X581, X582)\n\nT352 is ground\nX581 is free; \nX582 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(qs(.(T363, T361), X543), ','(qs(T362, X544), app(X543, .(T352, X544), T329)))\n\nT329 is ground\nT352 is ground\nX543 is free; \nX544 is free; \nqs(T364, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: qs(.(T363, T361), X543)\n\nX543 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(qs(T371, X544), app(T370, .(T352, X544), T329))\n\nT329 is ground\nT352 is ground\nX544 is free; \nqs(T372, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: qs(T371, X544)\n\nX544 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: app(T376, .(T352, T375), T329)\n\nT329 is ground\nT352 is ground\nqs(T377, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: app(T376, .(T352, T375), T329), SCOPE: 13, CLAUSE: 5 | app(T376, .(T352, T375), T329), SCOPE: 13, CLAUSE: 6\n\nT329 is ground\nT352 is ground\nqs(T377, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: app(T376, .(T352, T375), T329), SCOPE: 13, CLAUSE: 5\n\nT329 is ground\nT352 is ground\nqs(T377, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: app(T376, .(T352, T375), T329), SCOPE: 13, CLAUSE: 6\n\nT329 is ground\nT352 is ground\nqs(T377, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: app(T407, .(T404, T408), T406)\n\nT404 is ground\nT406 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: app(T407, .(T404, T408), T406), SCOPE: 14, CLAUSE: 5 | app(T407, .(T404, T408), T406), SCOPE: 14, CLAUSE: 6\n\nT404 is ground\nT406 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: app(T407, .(T404, T408), T406), SCOPE: 14, CLAUSE: 5\n\nT404 is ground\nT406 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: app(T407, .(T404, T408), T406), SCOPE: 14, CLAUSE: 6\n\nT404 is ground\nT406 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: app(T438, .(T435, T439), T437)\n\nT435 is ground\nT437 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 3\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: ','(part(T330, T331, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(T330, X544), T329)))), SCOPE: 12, CLAUSE: 4\n\nT329 is ground\nX541 is free; \nX542 is free; \nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: ','(part(T459, T460, X705, X706), ','(qs(X705, X543), ','(qs(.(T461, X706), X544), app(X543, .(T459, X544), T329))))\n\nT329 is ground\nX543 is free; \nX544 is free; \nX705 is free; \nX706 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: part(T459, T460, X705, X706)\n\nX705 is free\nX706 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: ','(qs(T465, X543), ','(qs(.(T467, T466), X544), app(X543, .(T468, X544), T329)))\n\nT329 is ground\nX543 is free; \nX544 is free; \nqs(T469, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: qs(T465, X543)\n\nX543 is free", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: ','(qs(.(T474, T475), X544), app(T473, .(T476, X544), T329))\n\nT329 is ground\nX544 is free; \nqs(T477, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: qs(.(T474, T475), X544)\n\nX544 is free", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: app(T481, .(T482, T480), T329)\n\nT329 is ground\nqs(T483, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: app(T481, .(T482, T480), T329), SCOPE: 15, CLAUSE: 5 | app(T481, .(T482, T480), T329), SCOPE: 15, CLAUSE: 6\n\nT329 is ground\nqs(T483, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: app(T481, .(T482, T480), T329), SCOPE: 15, CLAUSE: 5\n\nT329 is ground\nqs(T483, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: app(T481, .(T482, T480), T329), SCOPE: 15, CLAUSE: 6\n\nT329 is ground\nqs(T483, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: app(T515, .(T516, T517), T514)\n\nT514 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: app(T515, .(T516, T517), T514), SCOPE: 16, CLAUSE: 5 | app(T515, .(T516, T517), T514), SCOPE: 16, CLAUSE: 6\n\nT514 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: app(T515, .(T516, T517), T514), SCOPE: 16, CLAUSE: 5\n\nT514 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: app(T515, .(T516, T517), T514), SCOPE: 16, CLAUSE: 6\n\nT514 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: app(T547, .(T548, T549), T546)\n\nT546 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: ','(qs([], X543), ','(qs([], X544), app(X543, .(T557, X544), T329)))\n\nT329 is ground\nX543 is free; \nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: qs([], X543)\n\nX543 is free", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: ','(qs([], X544), app(T560, .(T557, X544), T329))\n\nT329 is ground\nT560 is ground\nX544 is free; \nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: qs([], X544)\n\nX544 is free", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: app(T560, .(T557, T563), T329)\n\nT329 is ground\nT560 is ground\nT563 is ground\nqs(T1, T329) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 173 [arrowhead = none ];
173 -> 2;

174 [label="EVAL with clause\nqs([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 174 [arrowhead = none ];
174 -> 3;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 175 [arrowhead = none ];
175 -> 4;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 176 [arrowhead = none ];
176 -> 5;

177 [label="EVAL with clause\nqs(.(X538, X539), X540) :- ','(part(X538, X539, X541, X542), ','(qs(X541, X543), ','(qs(X542, X544), app(X543, .(X538, X544), X540)))).\nand substitution\nX538 -> T330\nX539 -> T331\nT1 -> .(T330, T331)\nT2 -> T329\nX540 -> T329\nT327 -> T330\nT328 -> T331", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 177 [arrowhead = none ];
177 -> 110;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 178 [arrowhead = none ];
178 -> 111;

179 [label="EVAL with clause\nqs(.(X6, X7), X8) :- ','(part(X6, X7, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(X6, X12), X8)))).\nand substitution\nX6 -> T7\nX7 -> T8\nT1 -> .(T7, T8)\nX8 -> []\nT5 -> T7\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 179 [arrowhead = none ];
179 -> 6;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 180 [arrowhead = none ];
180 -> 7;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 181 [arrowhead = none ];
181 -> 8;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 182 [arrowhead = none ];
182 -> 9;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 183 [arrowhead = none ];
183 -> 10;

184 [label="EVAL with clause\npart(X44, .(X45, X46), .(X45, X47), X48) :- ','(less(X44, X45), part(X44, X46, X47, X48)).\nand substitution\nT7 -> T24\nX44 -> T24\nX45 -> T25\nX46 -> T26\nT8 -> .(T25, T26)\nX47 -> X49\nX9 -> .(T25, X49)\nX10 -> X50\nX48 -> X50\nT21 -> T24\nT22 -> T25\nT23 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 184 [arrowhead = none ];
184 -> 11;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 185 [arrowhead = none ];
185 -> 12;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 186 [arrowhead = none ];
186 -> 85;

187 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 187 [arrowhead = none ];
187 -> 86;

188 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 188 [arrowhead = none ];
188 -> 13;

189 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:\nT24 -> T29\nT26 -> T30\nT25 -> T31", fontsize=14, style = filled, fillcolor = violet];
11 -> 189 [arrowhead = none ];
189 -> 14;

190 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 190 [arrowhead = none ];
190 -> 15;

191 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 191 [arrowhead = none ];
191 -> 23;

192 [label="SPLIT 2\nreplacements:\nX49 -> T52\nX50 -> T53\nT31 -> T54", fontsize=14, style = filled, fillcolor = violet];
14 -> 192 [arrowhead = none ];
192 -> 24;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 193 [arrowhead = none ];
193 -> 16;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 194 [arrowhead = none ];
194 -> 17;

195 [label="EVAL with clause\nless(0, s(X64)).\nand substitution\nT24 -> 0\nX64 -> T38\nT25 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 195 [arrowhead = none ];
195 -> 18;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 196 [arrowhead = none ];
196 -> 19;

197 [label="EVAL with clause\nless(s(X70), s(X71)) :- less(X70, X71).\nand substitution\nX70 -> T45\nT24 -> s(T45)\nX71 -> T46\nT25 -> s(T46)\nT43 -> T45\nT44 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 197 [arrowhead = none ];
197 -> 21;

198 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 198 [arrowhead = none ];
198 -> 22;

199 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 199 [arrowhead = none ];
199 -> 20;

200 [label="INSTANCE with matching:\nT24 -> T45\nT25 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 200 [arrowhead = none , style=dashed];
200 -> 13[style=dashed];

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 201 [arrowhead = none ];
201 -> 25;

202 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 202 [arrowhead = none ];
202 -> 39;

203 [label="SPLIT 2\nreplacements:\nX11 -> T107\nT53 -> T108", fontsize=14, style = filled, fillcolor = violet];
24 -> 203 [arrowhead = none ];
203 -> 40;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 204 [arrowhead = none ];
204 -> 26;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 205 [arrowhead = none ];
205 -> 27;

206 [label="EVAL with clause\npart(X118, .(X119, X120), .(X119, X121), X122) :- ','(less(X118, X119), part(X118, X120, X121, X122)).\nand substitution\nT29 -> T70\nX118 -> T70\nX119 -> T73\nX120 -> T74\nT30 -> .(T73, T74)\nX121 -> X123\nX49 -> .(T73, X123)\nX50 -> X124\nX122 -> X124\nT71 -> T73\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 206 [arrowhead = none ];
206 -> 28;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 207 [arrowhead = none ];
207 -> 29;

208 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 208 [arrowhead = none ];
208 -> 32;

209 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 209 [arrowhead = none ];
209 -> 33;

210 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
28 -> 210 [arrowhead = none ];
210 -> 30;

211 [label="SPLIT 2\nreplacements:\nT74 -> T77", fontsize=14, style = filled, fillcolor = violet];
28 -> 211 [arrowhead = none ];
211 -> 31;

212 [label="INSTANCE with matching:\nT24 -> T70\nT25 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 212 [arrowhead = none , style=dashed];
212 -> 13[style=dashed];

213 [label="INSTANCE with matching:\nT29 -> T70\nT30 -> T77\nX49 -> X123\nX50 -> X124", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 213 [arrowhead = none , style=dashed];
213 -> 23[style=dashed];

214 [label="EVAL with clause\npart(X169, .(X170, X171), X172, .(X170, X173)) :- part(X169, X171, X172, X173).\nand substitution\nT29 -> T95\nX169 -> T95\nX170 -> T96\nX171 -> T98\nT30 -> .(T96, T98)\nX49 -> X174\nX172 -> X174\nX173 -> X175\nX50 -> .(T96, X175)\nT97 -> T98", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 214 [arrowhead = none ];
214 -> 34;

215 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 215 [arrowhead = none ];
215 -> 35;

216 [label="EVAL with clause\npart(X190, [], [], []).\nand substitution\nT29 -> T104\nX190 -> T104\nT30 -> []\nX49 -> []\nX50 -> []", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 216 [arrowhead = none ];
216 -> 36;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 217 [arrowhead = none ];
217 -> 37;

218 [label="INSTANCE with matching:\nT29 -> T95\nT30 -> T98\nX49 -> X174\nX50 -> X175", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 218 [arrowhead = none , style=dashed];
218 -> 23[style=dashed];

219 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 219 [arrowhead = none ];
219 -> 38;

220 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 220 [arrowhead = none ];
220 -> 41;

221 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 221 [arrowhead = none ];
221 -> 80;

222 [label="SPLIT 2\nreplacements:\nX12 -> T233\nT107 -> T234", fontsize=14, style = filled, fillcolor = violet];
40 -> 222 [arrowhead = none ];
222 -> 81;

223 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
41 -> 223 [arrowhead = none ];
223 -> 42;

224 [label="ONLY EVAL with clause\nqs(.(X227, X228), X229) :- ','(part(X227, X228, X230, X231), ','(qs(X230, X232), ','(qs(X231, X233), app(X232, .(X227, X233), X229)))).\nand substitution\nT54 -> T122\nX227 -> T122\nT52 -> T123\nX228 -> T123\nX11 -> X234\nX229 -> X234\nT120 -> T122\nT121 -> T123", fontsize=14, style = filled, fillcolor = white];
42 -> 224 [arrowhead = none ];
224 -> 43;

225 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 225 [arrowhead = none ];
225 -> 44;

226 [label="SPLIT 2\nreplacements:\nX230 -> T127\nX231 -> T128\nT122 -> T129", fontsize=14, style = filled, fillcolor = violet];
43 -> 226 [arrowhead = none ];
226 -> 45;

227 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 227 [arrowhead = none ];
227 -> 46;

228 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 228 [arrowhead = none ];
228 -> 60;

229 [label="SPLIT 2\nreplacements:\nX232 -> T183\nT128 -> T184\nT129 -> T185", fontsize=14, style = filled, fillcolor = violet];
45 -> 229 [arrowhead = none ];
229 -> 61;

230 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 230 [arrowhead = none ];
230 -> 47;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 231 [arrowhead = none ];
231 -> 48;

232 [label="EVAL with clause\npart(X277, .(X278, X279), .(X278, X280), X281) :- ','(less(X277, X278), part(X277, X279, X280, X281)).\nand substitution\nT122 -> T148\nX277 -> T148\nX278 -> T149\nX279 -> T150\nT123 -> .(T149, T150)\nX280 -> X282\nX230 -> .(T149, X282)\nX231 -> X283\nX281 -> X283\nT145 -> T148\nT146 -> T149\nT147 -> T150", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 232 [arrowhead = none ];
232 -> 49;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 233 [arrowhead = none ];
233 -> 50;

234 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 234 [arrowhead = none ];
234 -> 53;

235 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 235 [arrowhead = none ];
235 -> 54;

236 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 236 [arrowhead = none ];
236 -> 51;

237 [label="SPLIT 2\nnew knowledge:\nT153 is ground\nreplacements:\nT148 -> T153\nT150 -> T154", fontsize=14, style = filled, fillcolor = violet];
49 -> 237 [arrowhead = none ];
237 -> 52;

238 [label="INSTANCE with matching:\nT24 -> T148\nT25 -> T149", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 238 [arrowhead = none , style=dashed];
238 -> 13[style=dashed];

239 [label="INSTANCE with matching:\nT29 -> T153\nT30 -> T154\nX49 -> X282\nX50 -> X283", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 239 [arrowhead = none , style=dashed];
239 -> 23[style=dashed];

240 [label="EVAL with clause\npart(X328, .(X329, X330), X331, .(X329, X332)) :- part(X328, X330, X331, X332).\nand substitution\nT122 -> T175\nX328 -> T175\nX329 -> T173\nX330 -> T176\nT123 -> .(T173, T176)\nX230 -> X333\nX331 -> X333\nX332 -> X334\nX231 -> .(T173, X334)\nT172 -> T175\nT174 -> T176", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 240 [arrowhead = none ];
240 -> 55;

241 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 241 [arrowhead = none ];
241 -> 56;

242 [label="EVAL with clause\npart(X349, [], [], []).\nand substitution\nT122 -> T182\nX349 -> T182\nT123 -> []\nX230 -> []\nX231 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 242 [arrowhead = none ];
242 -> 57;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 243 [arrowhead = none ];
243 -> 58;

244 [label="INSTANCE with matching:\nT122 -> T175\nT123 -> T176\nX230 -> X333\nX231 -> X334", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 244 [arrowhead = none , style=dashed];
244 -> 44[style=dashed];

245 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 245 [arrowhead = none ];
245 -> 59;

246 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 246 [arrowhead = none ];
246 -> 62;

247 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 247 [arrowhead = none ];
247 -> 70;

248 [label="SPLIT 2\nreplacements:\nX233 -> T197\nT183 -> T198\nT185 -> T199", fontsize=14, style = filled, fillcolor = violet];
61 -> 248 [arrowhead = none ];
248 -> 71;

249 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 249 [arrowhead = none ];
249 -> 63;

250 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 250 [arrowhead = none ];
250 -> 64;

251 [label="EVAL with clause\nqs([], []).\nand substitution\nT127 -> []\nX232 -> []", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 251 [arrowhead = none ];
251 -> 65;

252 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 252 [arrowhead = none ];
252 -> 66;

253 [label="EVAL with clause\nqs(.(X363, X364), X365) :- ','(part(X363, X364, X366, X367), ','(qs(X366, X368), ','(qs(X367, X369), app(X368, .(X363, X369), X365)))).\nand substitution\nX363 -> T192\nX364 -> T193\nT127 -> .(T192, T193)\nX232 -> X370\nX365 -> X370\nT190 -> T192\nT191 -> T193", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 253 [arrowhead = none ];
253 -> 68;

254 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 254 [arrowhead = none ];
254 -> 69;

255 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 255 [arrowhead = none ];
255 -> 67;

256 [label="INSTANCE with matching:\nT122 -> T192\nT123 -> T193\nX230 -> X366\nX231 -> X367\nX232 -> X368\nX233 -> X369\nX234 -> X370", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 256 [arrowhead = none , style=dashed];
256 -> 43[style=dashed];

257 [label="INSTANCE with matching:\nT127 -> T184\nX232 -> X233", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 257 [arrowhead = none , style=dashed];
257 -> 60[style=dashed];

258 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 258 [arrowhead = none ];
258 -> 72;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 259 [arrowhead = none ];
259 -> 73;

260 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 260 [arrowhead = none ];
260 -> 74;

261 [label="EVAL with clause\napp([], X387, X387).\nand substitution\nT198 -> []\nT199 -> T212\nT197 -> T213\nX387 -> .(T212, T213)\nX234 -> .(T212, T213)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 261 [arrowhead = none ];
261 -> 75;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 262 [arrowhead = none ];
262 -> 76;

263 [label="EVAL with clause\napp(.(X399, X400), X401, .(X399, X402)) :- app(X400, X401, X402).\nand substitution\nX399 -> T222\nX400 -> T226\nT198 -> .(T222, T226)\nT199 -> T227\nT197 -> T228\nX401 -> .(T227, T228)\nX402 -> X403\nX234 -> .(T222, X403)\nT223 -> T226\nT224 -> T227\nT225 -> T228", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 263 [arrowhead = none ];
263 -> 78;

264 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 264 [arrowhead = none ];
264 -> 79;

265 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 265 [arrowhead = none ];
265 -> 77;

266 [label="INSTANCE with matching:\nT198 -> T226\nT199 -> T227\nT197 -> T228\nX234 -> X403", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 266 [arrowhead = none , style=dashed];
266 -> 71[style=dashed];

267 [label="INSTANCE with matching:\nT127 -> T108\nX232 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 267 [arrowhead = none , style=dashed];
267 -> 60[style=dashed];

268 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 268 [arrowhead = none ];
268 -> 82;

269 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
82 -> 269 [arrowhead = none ];
269 -> 83;

270 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
83 -> 270 [arrowhead = none ];
270 -> 84;

271 [label="EVAL with clause\npart(X449, .(X450, X451), X452, .(X450, X453)) :- part(X449, X451, X452, X453).\nand substitution\nT7 -> T264\nX449 -> T264\nX450 -> T266\nX451 -> T265\nT8 -> .(T266, T265)\nX9 -> X454\nX452 -> X454\nX453 -> X455\nX10 -> .(T266, X455)\nT261 -> T264\nT263 -> T265\nT262 -> T266", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 271 [arrowhead = none ];
271 -> 87;

272 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 272 [arrowhead = none ];
272 -> 88;

273 [label="EVAL with clause\npart(X504, [], [], []).\nand substitution\nT7 -> T305\nX504 -> T305\nT8 -> []\nX9 -> []\nX10 -> []\nT304 -> T305", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 273 [arrowhead = none ];
273 -> 98;

274 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 274 [arrowhead = none ];
274 -> 99;

275 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 275 [arrowhead = none ];
275 -> 89;

276 [label="SPLIT 2\nreplacements:\nX454 -> T270\nX455 -> T271\nT266 -> T272\nT264 -> T273", fontsize=14, style = filled, fillcolor = violet];
87 -> 276 [arrowhead = none ];
276 -> 90;

277 [label="INSTANCE with matching:\nT122 -> T264\nT123 -> T265\nX230 -> X454\nX231 -> X455", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 277 [arrowhead = none , style=dashed];
277 -> 44[style=dashed];

278 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 278 [arrowhead = none ];
278 -> 91;

279 [label="SPLIT 2\nreplacements:\nX11 -> T277\nT272 -> T278\nT271 -> T279\nT273 -> T280", fontsize=14, style = filled, fillcolor = violet];
90 -> 279 [arrowhead = none ];
279 -> 92;

280 [label="INSTANCE with matching:\nT108 -> T270\nX12 -> X11", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 280 [arrowhead = none , style=dashed];
280 -> 80[style=dashed];

281 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 281 [arrowhead = none ];
281 -> 93;

282 [label="SPLIT 2\nreplacements:\nX12 -> T283\nT277 -> T284\nT280 -> T285", fontsize=14, style = filled, fillcolor = violet];
92 -> 282 [arrowhead = none ];
282 -> 94;

283 [label="INSTANCE with matching:\nT54 -> T278\nT52 -> T279\nX11 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 283 [arrowhead = none , style=dashed];
283 -> 39[style=dashed];

284 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 284 [arrowhead = none ];
284 -> 95;

285 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
95 -> 285 [arrowhead = none ];
285 -> 96;

286 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
96 -> 286 [arrowhead = none ];
286 -> 97;

287 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
98 -> 287 [arrowhead = none ];
287 -> 100;

288 [label="SPLIT 2\nnew knowledge:\nT309 is ground\nreplacements:\nX11 -> T309", fontsize=14, style = filled, fillcolor = violet];
98 -> 288 [arrowhead = none ];
288 -> 101;

289 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 289 [arrowhead = none ];
289 -> 102;

290 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
101 -> 290 [arrowhead = none ];
290 -> 108;

291 [label="SPLIT 2\nnew knowledge:\nT315 is ground\nreplacements:\nX12 -> T315", fontsize=14, style = filled, fillcolor = violet];
101 -> 291 [arrowhead = none ];
291 -> 109;

292 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 292 [arrowhead = none ];
292 -> 103;

293 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 293 [arrowhead = none ];
293 -> 104;

294 [label="ONLY EVAL with clause\nqs([], []).\nand substitution\nX11 -> []", fontsize=14, style = filled, fillcolor = white];
103 -> 294 [arrowhead = none ];
294 -> 105;

295 [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];
104 -> 295 [arrowhead = none ];
295 -> 107;

296 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 296 [arrowhead = none ];
296 -> 106;

297 [label="INSTANCE with matching:\nX11 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
108 -> 297 [arrowhead = none , style=dashed];
297 -> 100[style=dashed];

298 [label="INSTANCE with matching:\nT284 -> T309\nT285 -> T305\nT283 -> T315", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 298 [arrowhead = none , style=dashed];
298 -> 94[style=dashed];

299 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
110 -> 299 [arrowhead = none ];
299 -> 112;

300 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 300 [arrowhead = none ];
300 -> 113;

301 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 301 [arrowhead = none ];
301 -> 114;

302 [label="EVAL with clause\npart(X576, .(X577, X578), .(X577, X579), X580) :- ','(less(X576, X577), part(X576, X578, X579, X580)).\nand substitution\nT330 -> T347\nX576 -> T347\nX577 -> T348\nX578 -> T349\nT331 -> .(T348, T349)\nX579 -> X581\nX541 -> .(T348, X581)\nX542 -> X582\nX580 -> X582\nT344 -> T347\nT345 -> T348\nT346 -> T349", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 302 [arrowhead = none ];
302 -> 115;

303 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 303 [arrowhead = none ];
303 -> 116;

304 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 304 [arrowhead = none ];
304 -> 141;

305 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 305 [arrowhead = none ];
305 -> 142;

306 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
115 -> 306 [arrowhead = none ];
306 -> 117;

307 [label="SPLIT 2\nnew knowledge:\nT352 is ground\nreplacements:\nT347 -> T352\nT349 -> T353\nT348 -> T354\nT1 -> T355", fontsize=14, style = filled, fillcolor = violet];
115 -> 307 [arrowhead = none ];
307 -> 118;

308 [label="INSTANCE with matching:\nT24 -> T347\nT25 -> T348", fontsize=14, style = filled, fillcolor = lightgrey];
117 -> 308 [arrowhead = none , style=dashed];
308 -> 13[style=dashed];

309 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
118 -> 309 [arrowhead = none ];
309 -> 119;

310 [label="SPLIT 2\nreplacements:\nX581 -> T361\nX582 -> T362\nT354 -> T363\nT355 -> T364", fontsize=14, style = filled, fillcolor = violet];
118 -> 310 [arrowhead = none ];
310 -> 120;

311 [label="INSTANCE with matching:\nT29 -> T352\nT30 -> T353\nX49 -> X581\nX50 -> X582", fontsize=14, style = filled, fillcolor = lightgrey];
119 -> 311 [arrowhead = none , style=dashed];
311 -> 23[style=dashed];

312 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
120 -> 312 [arrowhead = none ];
312 -> 121;

313 [label="SPLIT 2\nreplacements:\nX543 -> T370\nT362 -> T371\nT364 -> T372", fontsize=14, style = filled, fillcolor = violet];
120 -> 313 [arrowhead = none ];
313 -> 122;

314 [label="INSTANCE with matching:\nT54 -> T363\nT52 -> T361\nX11 -> X543", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 314 [arrowhead = none , style=dashed];
314 -> 39[style=dashed];

315 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
122 -> 315 [arrowhead = none ];
315 -> 123;

316 [label="SPLIT 2\nreplacements:\nX544 -> T375\nT370 -> T376\nT372 -> T377", fontsize=14, style = filled, fillcolor = violet];
122 -> 316 [arrowhead = none ];
316 -> 124;

317 [label="INSTANCE with matching:\nT270 -> T371\nX11 -> X544", fontsize=14, style = filled, fillcolor = lightgrey];
123 -> 317 [arrowhead = none , style=dashed];
317 -> 91[style=dashed];

318 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
124 -> 318 [arrowhead = none ];
318 -> 125;

319 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
125 -> 319 [arrowhead = none ];
319 -> 126;

320 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
125 -> 320 [arrowhead = none ];
320 -> 127;

321 [label="EVAL with clause\napp([], X628, X628).\nand substitution\nT376 -> []\nT352 -> T390\nT375 -> T391\nX628 -> .(T390, T391)\nT329 -> .(T390, T391)", fontsize=14, style = filled, fillcolor = lightblue];
126 -> 321 [arrowhead = none ];
321 -> 128;

322 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
126 -> 322 [arrowhead = none ];
322 -> 129;

323 [label="EVAL with clause\napp(.(X638, X639), X640, .(X638, X641)) :- app(X639, X640, X641).\nand substitution\nX638 -> T402\nX639 -> T407\nT376 -> .(T402, T407)\nT352 -> T404\nT375 -> T408\nX640 -> .(T404, T408)\nX641 -> T406\nT329 -> .(T402, T406)\nT403 -> T407\nT405 -> T408", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 323 [arrowhead = none ];
323 -> 131;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 324 [arrowhead = none ];
324 -> 132;

325 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 325 [arrowhead = none ];
325 -> 130;

326 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
131 -> 326 [arrowhead = none ];
326 -> 133;

327 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 327 [arrowhead = none ];
327 -> 134;

328 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
133 -> 328 [arrowhead = none ];
328 -> 135;

329 [label="EVAL with clause\napp([], X652, X652).\nand substitution\nT407 -> []\nT404 -> T421\nT408 -> T422\nX652 -> .(T421, T422)\nT406 -> .(T421, T422)", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 329 [arrowhead = none ];
329 -> 136;

330 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 330 [arrowhead = none ];
330 -> 137;

331 [label="EVAL with clause\napp(.(X662, X663), X664, .(X662, X665)) :- app(X663, X664, X665).\nand substitution\nX662 -> T433\nX663 -> T438\nT407 -> .(T433, T438)\nT404 -> T435\nT408 -> T439\nX664 -> .(T435, T439)\nX665 -> T437\nT406 -> .(T433, T437)\nT434 -> T438\nT436 -> T439", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 331 [arrowhead = none ];
331 -> 139;

332 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 332 [arrowhead = none ];
332 -> 140;

333 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 333 [arrowhead = none ];
333 -> 138;

334 [label="INSTANCE with matching:\nT407 -> T438\nT404 -> T435\nT408 -> T439\nT406 -> T437", fontsize=14, style = filled, fillcolor = lightgrey];
139 -> 334 [arrowhead = none , style=dashed];
334 -> 131[style=dashed];

335 [label="EVAL with clause\npart(X700, .(X701, X702), X703, .(X701, X704)) :- part(X700, X702, X703, X704).\nand substitution\nT330 -> T459\nX700 -> T459\nX701 -> T461\nX702 -> T460\nT331 -> .(T461, T460)\nX541 -> X705\nX703 -> X705\nX704 -> X706\nX542 -> .(T461, X706)\nT456 -> T459\nT458 -> T460\nT457 -> T461", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 335 [arrowhead = none ];
335 -> 143;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
141 -> 336 [arrowhead = none ];
336 -> 144;

337 [label="EVAL with clause\npart(X794, [], [], []).\nand substitution\nT330 -> T557\nX794 -> T557\nT331 -> []\nX541 -> []\nX542 -> []\nT556 -> T557", fontsize=14, style = filled, fillcolor = lightblue];
142 -> 337 [arrowhead = none ];
337 -> 167;

338 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
142 -> 338 [arrowhead = none ];
338 -> 168;

339 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
143 -> 339 [arrowhead = none ];
339 -> 145;

340 [label="SPLIT 2\nreplacements:\nX705 -> T465\nX706 -> T466\nT461 -> T467\nT459 -> T468\nT1 -> T469", fontsize=14, style = filled, fillcolor = violet];
143 -> 340 [arrowhead = none ];
340 -> 146;

341 [label="INSTANCE with matching:\nT264 -> T459\nT265 -> T460\nX454 -> X705\nX455 -> X706", fontsize=14, style = filled, fillcolor = lightgrey];
145 -> 341 [arrowhead = none , style=dashed];
341 -> 89[style=dashed];

342 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
146 -> 342 [arrowhead = none ];
342 -> 147;

343 [label="SPLIT 2\nreplacements:\nX543 -> T473\nT467 -> T474\nT466 -> T475\nT468 -> T476\nT469 -> T477", fontsize=14, style = filled, fillcolor = violet];
146 -> 343 [arrowhead = none ];
343 -> 148;

344 [label="INSTANCE with matching:\nT270 -> T465\nX11 -> X543", fontsize=14, style = filled, fillcolor = lightgrey];
147 -> 344 [arrowhead = none , style=dashed];
344 -> 91[style=dashed];

345 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
148 -> 345 [arrowhead = none ];
345 -> 149;

346 [label="SPLIT 2\nreplacements:\nX544 -> T480\nT473 -> T481\nT476 -> T482\nT477 -> T483", fontsize=14, style = filled, fillcolor = violet];
148 -> 346 [arrowhead = none ];
346 -> 150;

347 [label="INSTANCE with matching:\nT363 -> T474\nT361 -> T475\nX543 -> X544", fontsize=14, style = filled, fillcolor = lightgrey];
149 -> 347 [arrowhead = none , style=dashed];
347 -> 121[style=dashed];

348 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
150 -> 348 [arrowhead = none ];
348 -> 151;

349 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
151 -> 349 [arrowhead = none ];
349 -> 152;

350 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
151 -> 350 [arrowhead = none ];
350 -> 153;

351 [label="EVAL with clause\napp([], X748, X748).\nand substitution\nT481 -> []\nT482 -> T498\nT480 -> T499\nX748 -> .(T498, T499)\nT329 -> .(T498, T499)", fontsize=14, style = filled, fillcolor = lightblue];
152 -> 351 [arrowhead = none ];
351 -> 154;

352 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
152 -> 352 [arrowhead = none ];
352 -> 155;

353 [label="EVAL with clause\napp(.(X758, X759), X760, .(X758, X761)) :- app(X759, X760, X761).\nand substitution\nX758 -> T510\nX759 -> T515\nT481 -> .(T510, T515)\nT482 -> T516\nT480 -> T517\nX760 -> .(T516, T517)\nX761 -> T514\nT329 -> .(T510, T514)\nT511 -> T515\nT512 -> T516\nT513 -> T517", fontsize=14, style = filled, fillcolor = lightblue];
153 -> 353 [arrowhead = none ];
353 -> 157;

354 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
153 -> 354 [arrowhead = none ];
354 -> 158;

355 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
154 -> 355 [arrowhead = none ];
355 -> 156;

356 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
157 -> 356 [arrowhead = none ];
356 -> 159;

357 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
159 -> 357 [arrowhead = none ];
357 -> 160;

358 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
159 -> 358 [arrowhead = none ];
358 -> 161;

359 [label="EVAL with clause\napp([], X772, X772).\nand substitution\nT515 -> []\nT516 -> T530\nT517 -> T531\nX772 -> .(T530, T531)\nT514 -> .(T530, T531)", fontsize=14, style = filled, fillcolor = lightblue];
160 -> 359 [arrowhead = none ];
359 -> 162;

360 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
160 -> 360 [arrowhead = none ];
360 -> 163;

361 [label="EVAL with clause\napp(.(X782, X783), X784, .(X782, X785)) :- app(X783, X784, X785).\nand substitution\nX782 -> T542\nX783 -> T547\nT515 -> .(T542, T547)\nT516 -> T548\nT517 -> T549\nX784 -> .(T548, T549)\nX785 -> T546\nT514 -> .(T542, T546)\nT543 -> T547\nT544 -> T548\nT545 -> T549", fontsize=14, style = filled, fillcolor = lightblue];
161 -> 361 [arrowhead = none ];
361 -> 165;

362 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
161 -> 362 [arrowhead = none ];
362 -> 166;

363 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
162 -> 363 [arrowhead = none ];
363 -> 164;

364 [label="INSTANCE with matching:\nT515 -> T547\nT516 -> T548\nT517 -> T549\nT514 -> T546", fontsize=14, style = filled, fillcolor = lightgrey];
165 -> 364 [arrowhead = none , style=dashed];
364 -> 157[style=dashed];

365 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
167 -> 365 [arrowhead = none ];
365 -> 169;

366 [label="SPLIT 2\nnew knowledge:\nT560 is ground\nreplacements:\nX543 -> T560", fontsize=14, style = filled, fillcolor = violet];
167 -> 366 [arrowhead = none ];
366 -> 170;

367 [label="INSTANCE with matching:\nX11 -> X543", fontsize=14, style = filled, fillcolor = lightgrey];
169 -> 367 [arrowhead = none , style=dashed];
367 -> 100[style=dashed];

368 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
170 -> 368 [arrowhead = none ];
368 -> 171;

369 [label="SPLIT 2\nnew knowledge:\nT563 is ground\nreplacements:\nX544 -> T563", fontsize=14, style = filled, fillcolor = violet];
170 -> 369 [arrowhead = none ];
369 -> 172;

370 [label="INSTANCE with matching:\nX543 -> X544", fontsize=14, style = filled, fillcolor = lightgrey];
171 -> 370 [arrowhead = none , style=dashed];
370 -> 169[style=dashed];

371 [label="INSTANCE with matching:\nT515 -> T560\nT516 -> T557\nT517 -> T563\nT514 -> T329", fontsize=14, style = filled, fillcolor = lightgrey];
172 -> 371 [arrowhead = none , style=dashed];
371 -> 157[style=dashed];

}

(2) Obligation:

Triples:

less13(s(T45), s(T46)) :- less13(T45, T46).
part23(T70, .(T73, T74), .(T73, X123), X124) :- less13(T70, T73).
part23(T70, .(T73, T77), .(T73, X123), X124) :- ','(lessc13(T70, T73), part23(T70, T77, X123, X124)).
part23(T95, .(T96, T98), X174, .(T96, X175)) :- part23(T95, T98, X174, X175).
part44(T148, .(T149, T150), .(T149, X282), X283) :- less13(T148, T149).
part44(T153, .(T149, T154), .(T149, X282), X283) :- ','(lessc13(T153, T149), part23(T153, T154, X282, X283)).
part44(T175, .(T173, T176), X333, .(T173, X334)) :- part44(T175, T176, X333, X334).
p43(T122, T123, X230, X231, X232, X233, X234) :- part44(T122, T123, X230, X231).
p43(T129, T123, T127, T128, X232, X233, X234) :- ','(partc44(T129, T123, T127, T128), qs60(T127, X232)).
p43(T185, T123, T127, T184, T183, X233, X234) :- ','(partc44(T185, T123, T127, T184), ','(qsc60(T127, T183), qs60(T184, X233))).
p43(T199, T123, T127, T184, T198, T197, X234) :- ','(partc44(T199, T123, T127, T184), ','(qsc60(T127, T198), ','(qsc60(T184, T197), app71(T198, T199, T197, X234)))).
qs60(.(T192, T193), X370) :- p43(T192, T193, X366, X367, X368, X369, X370).
app71(.(T222, T226), T227, T228, .(T222, X403)) :- app71(T226, T227, T228, X403).
qs39(T122, T123, X234) :- p43(T122, T123, X230, X231, X232, X233, X234).
app131(.(T433, T438), T435, T439, .(T433, T437)) :- app131(T438, T435, T439, T437).
app157(.(T542, T547), T548, T549, .(T542, T546)) :- app157(T547, T548, T549, T546).
qs1(.(T24, .(T25, T26)), []) :- less13(T24, T25).
qs1(.(T29, .(T31, T30)), []) :- ','(lessc13(T29, T31), part23(T29, T30, X49, X50)).
qs1(.(T29, .(T54, T30)), []) :- ','(lessc13(T29, T54), ','(partc23(T29, T30, T52, T53), qs39(T54, T52, X11))).
qs1(.(T29, .(T54, T30)), []) :- ','(lessc13(T29, T54), ','(partc23(T29, T30, T52, T108), ','(qsc39(T54, T52, T107), qs60(T108, X12)))).
qs1(.(T264, .(T266, T265)), []) :- part44(T264, T265, X454, X455).
qs1(.(T273, .(T272, T265)), []) :- ','(partc44(T273, T265, T270, T271), qs60(T270, X11)).
qs1(.(T280, .(T278, T265)), []) :- ','(partc44(T280, T265, T270, T279), ','(qsc60(T270, T277), qs39(T278, T279, X12))).
qs1(.(T285, .(T278, T265)), []) :- ','(partc44(T285, T265, T270, T279), ','(qsc60(T270, T284), ','(qsc39(T278, T279, T283), app94(T284, T285, T283)))).
qs1(.(T305, []), []) :- qs100(X11).
qs1(.(T305, []), []) :- ','(qsc100(T309), qs100(X12)).
qs1(.(T305, []), []) :- ','(qsc100(T309), ','(qsc100(T315), app94(T309, T305, T315))).
qs1(.(T347, .(T348, T349)), T329) :- less13(T347, T348).
qs1(.(T352, .(T354, T353)), T329) :- ','(lessc13(T352, T354), part23(T352, T353, X581, X582)).
qs1(.(T352, .(T363, T353)), T329) :- ','(lessc13(T352, T363), ','(partc23(T352, T353, T361, T362), qs39(T363, T361, X543))).
qs1(.(T352, .(T363, T353)), T329) :- ','(lessc13(T352, T363), ','(partc23(T352, T353, T361, T371), ','(qsc39(T363, T361, T370), qs60(T371, X544)))).
qs1(.(T404, .(T363, T353)), .(T402, T406)) :- ','(lessc13(T404, T363), ','(partc23(T404, T353, T361, T371), ','(qsc39(T363, T361, .(T402, T407)), ','(qsc60(T371, T408), app131(T407, T404, T408, T406))))).
qs1(.(T459, .(T461, T460)), T329) :- part44(T459, T460, X705, X706).
qs1(.(T468, .(T467, T460)), T329) :- ','(partc44(T468, T460, T465, T466), qs60(T465, X543)).
qs1(.(T476, .(T474, T460)), T329) :- ','(partc44(T476, T460, T465, T475), ','(qsc60(T465, T473), qs39(T474, T475, X544))).
qs1(.(T516, .(T474, T460)), .(T510, T514)) :- ','(partc44(T516, T460, T465, T475), ','(qsc60(T465, .(T510, T515)), ','(qsc39(T474, T475, T517), app157(T515, T516, T517, T514)))).
qs1(.(T557, []), T329) :- qs100(X543).
qs1(.(T557, []), T329) :- ','(qsc100(T560), qs100(X544)).
qs1(.(T557, []), T329) :- ','(qsc100(T560), ','(qsc100(T563), app157(T560, T557, T563, T329))).

Clauses:

lessc13(0, s(T38)).
lessc13(s(T45), s(T46)) :- lessc13(T45, T46).
partc23(T70, .(T73, T77), .(T73, X123), X124) :- ','(lessc13(T70, T73), partc23(T70, T77, X123, X124)).
partc23(T95, .(T96, T98), X174, .(T96, X175)) :- partc23(T95, T98, X174, X175).
partc23(T104, [], [], []).
partc44(T153, .(T149, T154), .(T149, X282), X283) :- ','(lessc13(T153, T149), partc23(T153, T154, X282, X283)).
partc44(T175, .(T173, T176), X333, .(T173, X334)) :- partc44(T175, T176, X333, X334).
partc44(T182, [], [], []).
qc43(T199, T123, T127, T184, T198, T197, X234) :- ','(partc44(T199, T123, T127, T184), ','(qsc60(T127, T198), ','(qsc60(T184, T197), appc71(T198, T199, T197, X234)))).
qsc60([], []).
qsc60(.(T192, T193), X370) :- qc43(T192, T193, X366, X367, X368, X369, X370).
appc71([], T212, T213, .(T212, T213)).
appc71(.(T222, T226), T227, T228, .(T222, X403)) :- appc71(T226, T227, T228, X403).
qsc39(T122, T123, X234) :- qc43(T122, T123, X230, X231, X232, X233, X234).
qsc100([]).
appc131([], T421, T422, .(T421, T422)).
appc131(.(T433, T438), T435, T439, .(T433, T437)) :- appc131(T438, T435, T439, T437).
appc157([], T530, T531, .(T530, T531)).
appc157(.(T542, T547), T548, T549, .(T542, T546)) :- appc157(T547, T548, T549, T546).

Afs:

qs1(x1, x2) = qs1(x2)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

less13(s(T45), s(T46)) :- less13(T45, T46).
part23(T70, .(T73, T74), .(T73, X123), X124) :- less13(T70, T73).
part23(T70, .(T73, T77), .(T73, X123), X124) :- ','(lessc13(T70, T73), part23(T70, T77, X123, X124)).
part23(T95, .(T96, T98), X174, .(T96, X175)) :- part23(T95, T98, X174, X175).
part44(T148, .(T149, T150), .(T149, X282), X283) :- less13(T148, T149).
part44(T153, .(T149, T154), .(T149, X282), X283) :- ','(lessc13(T153, T149), part23(T153, T154, X282, X283)).
part44(T175, .(T173, T176), X333, .(T173, X334)) :- part44(T175, T176, X333, X334).
p43(T122, T123, X230, X231, X232, X233, X234) :- part44(T122, T123, X230, X231).
p43(T129, T123, T127, T128, X232, X233, X234) :- ','(partc44(T129, T123, T127, T128), qs60(T127, X232)).
p43(T185, T123, T127, T184, T183, X233, X234) :- ','(partc44(T185, T123, T127, T184), ','(qsc60(T127, T183), qs60(T184, X233))).
p43(T199, T123, T127, T184, T198, T197, X234) :- ','(partc44(T199, T123, T127, T184), ','(qsc60(T127, T198), ','(qsc60(T184, T197), app71(T198, T199, T197, X234)))).
qs60(.(T192, T193), X370) :- p43(T192, T193, X366, X367, X368, X369, X370).
app71(.(T222, T226), T227, T228, .(T222, X403)) :- app71(T226, T227, T228, X403).
qs39(T122, T123, X234) :- p43(T122, T123, X230, X231, X232, X233, X234).
app131(.(T433, T438), T435, T439, .(T433, T437)) :- app131(T438, T435, T439, T437).
app157(.(T542, T547), T548, T549, .(T542, T546)) :- app157(T547, T548, T549, T546).
qs1(.(T24, .(T25, T26)), []) :- less13(T24, T25).
qs1(.(T29, .(T31, T30)), []) :- ','(lessc13(T29, T31), part23(T29, T30, X49, X50)).
qs1(.(T29, .(T54, T30)), []) :- ','(lessc13(T29, T54), ','(partc23(T29, T30, T52, T53), qs39(T54, T52, X11))).
qs1(.(T29, .(T54, T30)), []) :- ','(lessc13(T29, T54), ','(partc23(T29, T30, T52, T108), ','(qsc39(T54, T52, T107), qs60(T108, X12)))).
qs1(.(T264, .(T266, T265)), []) :- part44(T264, T265, X454, X455).
qs1(.(T273, .(T272, T265)), []) :- ','(partc44(T273, T265, T270, T271), qs60(T270, X11)).
qs1(.(T280, .(T278, T265)), []) :- ','(partc44(T280, T265, T270, T279), ','(qsc60(T270, T277), qs39(T278, T279, X12))).
qs1(.(T347, .(T348, T349)), T329) :- less13(T347, T348).
qs1(.(T352, .(T354, T353)), T329) :- ','(lessc13(T352, T354), part23(T352, T353, X581, X582)).
qs1(.(T352, .(T363, T353)), T329) :- ','(lessc13(T352, T363), ','(partc23(T352, T353, T361, T362), qs39(T363, T361, X543))).
qs1(.(T352, .(T363, T353)), T329) :- ','(lessc13(T352, T363), ','(partc23(T352, T353, T361, T371), ','(qsc39(T363, T361, T370), qs60(T371, X544)))).
qs1(.(T404, .(T363, T353)), .(T402, T406)) :- ','(lessc13(T404, T363), ','(partc23(T404, T353, T361, T371), ','(qsc39(T363, T361, .(T402, T407)), ','(qsc60(T371, T408), app131(T407, T404, T408, T406))))).
qs1(.(T459, .(T461, T460)), T329) :- part44(T459, T460, X705, X706).
qs1(.(T468, .(T467, T460)), T329) :- ','(partc44(T468, T460, T465, T466), qs60(T465, X543)).
qs1(.(T476, .(T474, T460)), T329) :- ','(partc44(T476, T460, T465, T475), ','(qsc60(T465, T473), qs39(T474, T475, X544))).
qs1(.(T516, .(T474, T460)), .(T510, T514)) :- ','(partc44(T516, T460, T465, T475), ','(qsc60(T465, .(T510, T515)), ','(qsc39(T474, T475, T517), app157(T515, T516, T517, T514)))).
qs1(.(T557, []), T329) :- ','(qsc100(T560), ','(qsc100(T563), app157(T560, T557, T563, T329))).

Clauses:

lessc13(0, s(T38)).
lessc13(s(T45), s(T46)) :- lessc13(T45, T46).
partc23(T70, .(T73, T77), .(T73, X123), X124) :- ','(lessc13(T70, T73), partc23(T70, T77, X123, X124)).
partc23(T95, .(T96, T98), X174, .(T96, X175)) :- partc23(T95, T98, X174, X175).
partc23(T104, [], [], []).
partc44(T153, .(T149, T154), .(T149, X282), X283) :- ','(lessc13(T153, T149), partc23(T153, T154, X282, X283)).
partc44(T175, .(T173, T176), X333, .(T173, X334)) :- partc44(T175, T176, X333, X334).
partc44(T182, [], [], []).
qc43(T199, T123, T127, T184, T198, T197, X234) :- ','(partc44(T199, T123, T127, T184), ','(qsc60(T127, T198), ','(qsc60(T184, T197), appc71(T198, T199, T197, X234)))).
qsc60([], []).
qsc60(.(T192, T193), X370) :- qc43(T192, T193, X366, X367, X368, X369, X370).
appc71([], T212, T213, .(T212, T213)).
appc71(.(T222, T226), T227, T228, .(T222, X403)) :- appc71(T226, T227, T228, X403).
qsc39(T122, T123, X234) :- qc43(T122, T123, X230, X231, X232, X233, X234).
qsc100([]).
appc131([], T421, T422, .(T421, T422)).
appc131(.(T433, T438), T435, T439, .(T433, T437)) :- appc131(T438, T435, T439, T437).
appc157([], T530, T531, .(T530, T531)).
appc157(.(T542, T547), T548, T549, .(T542, T546)) :- appc157(T547, T548, T549, T546).

Afs:

qs1(x1, x2) = qs1(x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (f,b)
less13_in: (f,f) (b,f)
lessc13_in: (f,f) (b,f)
part23_in: (b,f,f,f)
partc23_in: (b,f,f,f)
qs39_in: (f,f,f)
p43_in: (f,f,f,f,f,f,f)
part44_in: (f,f,f,f)
partc44_in: (f,f,f,f)
qs60_in: (f,f)
qsc60_in: (f,f)
qc43_in: (f,f,f,f,f,f,f)
appc71_in: (f,f,f,f)
app71_in: (f,f,f,f)
qsc39_in: (f,f,f)
app131_in: (f,b,f,b)
app157_in: (f,f,f,b) (b,f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_AG(.(T24, .(T25, T26)), []) → U25_AG(T24, T25, T26, less13_in_aa(T24, T25))
QS1_IN_AG(.(T24, .(T25, T26)), []) → LESS13_IN_AA(T24, T25)
LESS13_IN_AA(s(T45), s(T46)) → U1_AA(T45, T46, less13_in_aa(T45, T46))
LESS13_IN_AA(s(T45), s(T46)) → LESS13_IN_AA(T45, T46)
QS1_IN_AG(.(T29, .(T31, T30)), []) → U26_AG(T29, T31, T30, lessc13_in_aa(T29, T31))
U26_AG(T29, T31, T30, lessc13_out_aa(T29, T31)) → U27_AG(T29, T31, T30, part23_in_gaaa(T29, T30, X49, X50))
U26_AG(T29, T31, T30, lessc13_out_aa(T29, T31)) → PART23_IN_GAAA(T29, T30, X49, X50)
PART23_IN_GAAA(T70, .(T73, T74), .(T73, X123), X124) → U2_GAAA(T70, T73, T74, X123, X124, less13_in_ga(T70, T73))
PART23_IN_GAAA(T70, .(T73, T74), .(T73, X123), X124) → LESS13_IN_GA(T70, T73)
LESS13_IN_GA(s(T45), s(T46)) → U1_GA(T45, T46, less13_in_ga(T45, T46))
LESS13_IN_GA(s(T45), s(T46)) → LESS13_IN_GA(T45, T46)
PART23_IN_GAAA(T70, .(T73, T77), .(T73, X123), X124) → U3_GAAA(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X123, X124, part23_in_gaaa(T70, T77, X123, X124))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → PART23_IN_GAAA(T70, T77, X123, X124)
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → U5_GAAA(T95, T96, T98, X174, X175, part23_in_gaaa(T95, T98, X174, X175))
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → PART23_IN_GAAA(T95, T98, X174, X175)
QS1_IN_AG(.(T29, .(T54, T30)), []) → U28_AG(T29, T54, T30, lessc13_in_aa(T29, T54))
U28_AG(T29, T54, T30, lessc13_out_aa(T29, T54)) → U29_AG(T29, T54, T30, partc23_in_gaaa(T29, T30, T52, T53))
U29_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T53)) → U30_AG(T29, T54, T30, qs39_in_aaa(T54, T52, X11))
U29_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T53)) → QS39_IN_AAA(T54, T52, X11)
QS39_IN_AAA(T122, T123, X234) → U22_AAA(T122, T123, X234, p43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
QS39_IN_AAA(T122, T123, X234) → P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234)
P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234) → U10_AAAAAAA(T122, T123, X230, X231, X232, X233, X234, part44_in_aaaa(T122, T123, X230, X231))
P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234) → PART44_IN_AAAA(T122, T123, X230, X231)
PART44_IN_AAAA(T148, .(T149, T150), .(T149, X282), X283) → U6_AAAA(T148, T149, T150, X282, X283, less13_in_aa(T148, T149))
PART44_IN_AAAA(T148, .(T149, T150), .(T149, X282), X283) → LESS13_IN_AA(T148, T149)
PART44_IN_AAAA(T153, .(T149, T154), .(T149, X282), X283) → U7_AAAA(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U7_AAAA(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U8_AAAA(T153, T149, T154, X282, X283, part23_in_gaaa(T153, T154, X282, X283))
U7_AAAA(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → PART23_IN_GAAA(T153, T154, X282, X283)
PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → U9_AAAA(T175, T173, T176, X333, X334, part44_in_aaaa(T175, T176, X333, X334))
PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → PART44_IN_AAAA(T175, T176, X333, X334)
P43_IN_AAAAAAA(T129, T123, T127, T128, X232, X233, X234) → U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_in_aaaa(T129, T123, T127, T128))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → U12_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, qs60_in_aa(T127, X232))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → QS60_IN_AA(T127, X232)
QS60_IN_AA(.(T192, T193), X370) → U20_AA(T192, T193, X370, p43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
QS60_IN_AA(.(T192, T193), X370) → P43_IN_AAAAAAA(T192, T193, X366, X367, X368, X369, X370)
P43_IN_AAAAAAA(T185, T123, T127, T184, T183, X233, X234) → U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_in_aaaa(T185, T123, T127, T184))
U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_out_aaaa(T185, T123, T127, T184)) → U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_in_aa(T127, T183))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → U15_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qs60_in_aa(T184, X233))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → QS60_IN_AA(T184, X233)
P43_IN_AAAAAAA(T199, T123, T127, T184, T198, T197, X234) → U16_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U16_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U17_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U17_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U19_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, app71_in_aaaa(T198, T199, T197, X234))
U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → APP71_IN_AAAA(T198, T199, T197, X234)
APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → U21_AAAA(T222, T226, T227, T228, X403, app71_in_aaaa(T226, T227, T228, X403))
APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → APP71_IN_AAAA(T226, T227, T228, X403)
U28_AG(T29, T54, T30, lessc13_out_aa(T29, T54)) → U31_AG(T29, T54, T30, partc23_in_gaaa(T29, T30, T52, T108))
U31_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T108)) → U32_AG(T29, T54, T30, T108, qsc39_in_aaa(T54, T52, T107))
U32_AG(T29, T54, T30, T108, qsc39_out_aaa(T54, T52, T107)) → U33_AG(T29, T54, T30, qs60_in_aa(T108, X12))
U32_AG(T29, T54, T30, T108, qsc39_out_aaa(T54, T52, T107)) → QS60_IN_AA(T108, X12)
QS1_IN_AG(.(T264, .(T266, T265)), []) → U34_AG(T264, T266, T265, part44_in_aaaa(T264, T265, X454, X455))
QS1_IN_AG(.(T264, .(T266, T265)), []) → PART44_IN_AAAA(T264, T265, X454, X455)
QS1_IN_AG(.(T273, .(T272, T265)), []) → U35_AG(T273, T272, T265, partc44_in_aaaa(T273, T265, T270, T271))
U35_AG(T273, T272, T265, partc44_out_aaaa(T273, T265, T270, T271)) → U36_AG(T273, T272, T265, qs60_in_aa(T270, X11))
U35_AG(T273, T272, T265, partc44_out_aaaa(T273, T265, T270, T271)) → QS60_IN_AA(T270, X11)
QS1_IN_AG(.(T280, .(T278, T265)), []) → U37_AG(T280, T278, T265, partc44_in_aaaa(T280, T265, T270, T279))
U37_AG(T280, T278, T265, partc44_out_aaaa(T280, T265, T270, T279)) → U38_AG(T280, T278, T265, T279, qsc60_in_aa(T270, T277))
U38_AG(T280, T278, T265, T279, qsc60_out_aa(T270, T277)) → U39_AG(T280, T278, T265, qs39_in_aaa(T278, T279, X12))
U38_AG(T280, T278, T265, T279, qsc60_out_aa(T270, T277)) → QS39_IN_AAA(T278, T279, X12)
QS1_IN_AG(.(T347, .(T348, T349)), T329) → U40_AG(T347, T348, T349, T329, less13_in_aa(T347, T348))
QS1_IN_AG(.(T347, .(T348, T349)), T329) → LESS13_IN_AA(T347, T348)
QS1_IN_AG(.(T352, .(T354, T353)), T329) → U41_AG(T352, T354, T353, T329, lessc13_in_aa(T352, T354))
U41_AG(T352, T354, T353, T329, lessc13_out_aa(T352, T354)) → U42_AG(T352, T354, T353, T329, part23_in_gaaa(T352, T353, X581, X582))
U41_AG(T352, T354, T353, T329, lessc13_out_aa(T352, T354)) → PART23_IN_GAAA(T352, T353, X581, X582)
QS1_IN_AG(.(T352, .(T363, T353)), T329) → U43_AG(T352, T363, T353, T329, lessc13_in_aa(T352, T363))
U43_AG(T352, T363, T353, T329, lessc13_out_aa(T352, T363)) → U44_AG(T352, T363, T353, T329, partc23_in_gaaa(T352, T353, T361, T362))
U44_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T362)) → U45_AG(T352, T363, T353, T329, qs39_in_aaa(T363, T361, X543))
U44_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T362)) → QS39_IN_AAA(T363, T361, X543)
U43_AG(T352, T363, T353, T329, lessc13_out_aa(T352, T363)) → U46_AG(T352, T363, T353, T329, partc23_in_gaaa(T352, T353, T361, T371))
U46_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T371)) → U47_AG(T352, T363, T353, T329, T371, qsc39_in_aaa(T363, T361, T370))
U47_AG(T352, T363, T353, T329, T371, qsc39_out_aaa(T363, T361, T370)) → U48_AG(T352, T363, T353, T329, qs60_in_aa(T371, X544))
U47_AG(T352, T363, T353, T329, T371, qsc39_out_aaa(T363, T361, T370)) → QS60_IN_AA(T371, X544)
QS1_IN_AG(.(T404, .(T363, T353)), .(T402, T406)) → U49_AG(T404, T363, T353, T402, T406, lessc13_in_aa(T404, T363))
U49_AG(T404, T363, T353, T402, T406, lessc13_out_aa(T404, T363)) → U50_AG(T404, T363, T353, T402, T406, partc23_in_gaaa(T404, T353, T361, T371))
U50_AG(T404, T363, T353, T402, T406, partc23_out_gaaa(T404, T353, T361, T371)) → U51_AG(T404, T363, T353, T402, T406, T371, qsc39_in_aaa(T363, T361, .(T402, T407)))
U51_AG(T404, T363, T353, T402, T406, T371, qsc39_out_aaa(T363, T361, .(T402, T407))) → U52_AG(T404, T363, T353, T402, T406, T407, qsc60_in_aa(T371, T408))
U52_AG(T404, T363, T353, T402, T406, T407, qsc60_out_aa(T371, T408)) → U53_AG(T404, T363, T353, T402, T406, app131_in_agag(T407, T404, T408, T406))
U52_AG(T404, T363, T353, T402, T406, T407, qsc60_out_aa(T371, T408)) → APP131_IN_AGAG(T407, T404, T408, T406)
APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → U23_AGAG(T433, T438, T435, T439, T437, app131_in_agag(T438, T435, T439, T437))
APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → APP131_IN_AGAG(T438, T435, T439, T437)
QS1_IN_AG(.(T459, .(T461, T460)), T329) → U54_AG(T459, T461, T460, T329, part44_in_aaaa(T459, T460, X705, X706))
QS1_IN_AG(.(T459, .(T461, T460)), T329) → PART44_IN_AAAA(T459, T460, X705, X706)
QS1_IN_AG(.(T468, .(T467, T460)), T329) → U55_AG(T468, T467, T460, T329, partc44_in_aaaa(T468, T460, T465, T466))
U55_AG(T468, T467, T460, T329, partc44_out_aaaa(T468, T460, T465, T466)) → U56_AG(T468, T467, T460, T329, qs60_in_aa(T465, X543))
U55_AG(T468, T467, T460, T329, partc44_out_aaaa(T468, T460, T465, T466)) → QS60_IN_AA(T465, X543)
QS1_IN_AG(.(T476, .(T474, T460)), T329) → U57_AG(T476, T474, T460, T329, partc44_in_aaaa(T476, T460, T465, T475))
U57_AG(T476, T474, T460, T329, partc44_out_aaaa(T476, T460, T465, T475)) → U58_AG(T476, T474, T460, T329, T475, qsc60_in_aa(T465, T473))
U58_AG(T476, T474, T460, T329, T475, qsc60_out_aa(T465, T473)) → U59_AG(T476, T474, T460, T329, qs39_in_aaa(T474, T475, X544))
U58_AG(T476, T474, T460, T329, T475, qsc60_out_aa(T465, T473)) → QS39_IN_AAA(T474, T475, X544)
QS1_IN_AG(.(T516, .(T474, T460)), .(T510, T514)) → U60_AG(T516, T474, T460, T510, T514, partc44_in_aaaa(T516, T460, T465, T475))
U60_AG(T516, T474, T460, T510, T514, partc44_out_aaaa(T516, T460, T465, T475)) → U61_AG(T516, T474, T460, T510, T514, T475, qsc60_in_aa(T465, .(T510, T515)))
U61_AG(T516, T474, T460, T510, T514, T475, qsc60_out_aa(T465, .(T510, T515))) → U62_AG(T516, T474, T460, T510, T514, T515, qsc39_in_aaa(T474, T475, T517))
U62_AG(T516, T474, T460, T510, T514, T515, qsc39_out_aaa(T474, T475, T517)) → U63_AG(T516, T474, T460, T510, T514, app157_in_aaag(T515, T516, T517, T514))
U62_AG(T516, T474, T460, T510, T514, T515, qsc39_out_aaa(T474, T475, T517)) → APP157_IN_AAAG(T515, T516, T517, T514)
APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → U24_AAAG(T542, T547, T548, T549, T546, app157_in_aaag(T547, T548, T549, T546))
APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_AAAG(T547, T548, T549, T546)
QS1_IN_AG(.(T557, []), T329) → U64_AG(T557, T329, qsc100_in_a(T560))
U64_AG(T557, T329, qsc100_out_a(T560)) → U65_AG(T557, T329, T560, qsc100_in_a(T563))
U65_AG(T557, T329, T560, qsc100_out_a(T563)) → U66_AG(T557, T329, app157_in_gagg(T560, T557, T563, T329))
U65_AG(T557, T329, T560, qsc100_out_a(T563)) → APP157_IN_GAGG(T560, T557, T563, T329)
APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → U24_GAGG(T542, T547, T548, T549, T546, app157_in_gagg(T547, T548, T549, T546))
APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_GAGG(T547, T548, T549, T546)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
less13_in_aa(x1, x2) = less13_in_aa
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
part23_in_gaaa(x1, x2, x3, x4) = part23_in_gaaa(x1)
less13_in_ga(x1, x2) = less13_in_ga(x1)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
qs39_in_aaa(x1, x2, x3) = qs39_in_aaa
p43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = p43_in_aaaaaaa
part44_in_aaaa(x1, x2, x3, x4) = part44_in_aaaa
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qs60_in_aa(x1, x2) = qs60_in_aa
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
app71_in_aaaa(x1, x2, x3, x4) = app71_in_aaaa
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
app131_in_agag(x1, x2, x3, x4) = app131_in_agag(x2, x4)
app157_in_aaag(x1, x2, x3, x4) = app157_in_aaag(x4)
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
app157_in_gagg(x1, x2, x3, x4) = app157_in_gagg(x1, x3, x4)
QS1_IN_AG(x1, x2) = QS1_IN_AG(x2)
U25_AG(x1, x2, x3, x4) = U25_AG(x4)
LESS13_IN_AA(x1, x2) = LESS13_IN_AA
U1_AA(x1, x2, x3) = U1_AA(x3)
U26_AG(x1, x2, x3, x4) = U26_AG(x4)
U27_AG(x1, x2, x3, x4) = U27_AG(x4)
PART23_IN_GAAA(x1, x2, x3, x4) = PART23_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6)
LESS13_IN_GA(x1, x2) = LESS13_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)
U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
U28_AG(x1, x2, x3, x4) = U28_AG(x4)
U29_AG(x1, x2, x3, x4) = U29_AG(x4)
U30_AG(x1, x2, x3, x4) = U30_AG(x4)
QS39_IN_AAA(x1, x2, x3) = QS39_IN_AAA
U22_AAA(x1, x2, x3, x4) = U22_AAA(x4)
P43_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = P43_IN_AAAAAAA
U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_AAAAAAA(x8)
PART44_IN_AAAA(x1, x2, x3, x4) = PART44_IN_AAAA
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6)
U8_AAAA(x1, x2, x3, x4, x5, x6) = U8_AAAA(x1, x6)
U9_AAAA(x1, x2, x3, x4, x5, x6) = U9_AAAA(x6)
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8)
U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_AAAAAAA(x8)
QS60_IN_AA(x1, x2) = QS60_IN_AA
U20_AA(x1, x2, x3, x4) = U20_AA(x4)
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8)
U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_AAAAAAA(x8)
U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_AAAAAAA(x8)
U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U17_AAAAAAA(x8)
U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U18_AAAAAAA(x8)
U19_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U19_AAAAAAA(x8)
APP71_IN_AAAA(x1, x2, x3, x4) = APP71_IN_AAAA
U21_AAAA(x1, x2, x3, x4, x5, x6) = U21_AAAA(x6)
U31_AG(x1, x2, x3, x4) = U31_AG(x4)
U32_AG(x1, x2, x3, x4, x5) = U32_AG(x5)
U33_AG(x1, x2, x3, x4) = U33_AG(x4)
U34_AG(x1, x2, x3, x4) = U34_AG(x4)
U35_AG(x1, x2, x3, x4) = U35_AG(x4)
U36_AG(x1, x2, x3, x4) = U36_AG(x4)
U37_AG(x1, x2, x3, x4) = U37_AG(x4)
U38_AG(x1, x2, x3, x4, x5) = U38_AG(x5)
U39_AG(x1, x2, x3, x4) = U39_AG(x4)
U40_AG(x1, x2, x3, x4, x5) = U40_AG(x4, x5)
U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5)
U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5)
U43_AG(x1, x2, x3, x4, x5) = U43_AG(x4, x5)
U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5)
U45_AG(x1, x2, x3, x4, x5) = U45_AG(x4, x5)
U46_AG(x1, x2, x3, x4, x5) = U46_AG(x4, x5)
U47_AG(x1, x2, x3, x4, x5, x6) = U47_AG(x4, x6)
U48_AG(x1, x2, x3, x4, x5) = U48_AG(x4, x5)
U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x5, x6)
U50_AG(x1, x2, x3, x4, x5, x6) = U50_AG(x1, x4, x5, x6)
U51_AG(x1, x2, x3, x4, x5, x6, x7) = U51_AG(x1, x4, x5, x7)
U52_AG(x1, x2, x3, x4, x5, x6, x7) = U52_AG(x1, x4, x5, x7)
U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x5, x6)
APP131_IN_AGAG(x1, x2, x3, x4) = APP131_IN_AGAG(x2, x4)
U23_AGAG(x1, x2, x3, x4, x5, x6) = U23_AGAG(x1, x3, x5, x6)
U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5)
U55_AG(x1, x2, x3, x4, x5) = U55_AG(x4, x5)
U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5)
U57_AG(x1, x2, x3, x4, x5) = U57_AG(x4, x5)
U58_AG(x1, x2, x3, x4, x5, x6) = U58_AG(x4, x6)
U59_AG(x1, x2, x3, x4, x5) = U59_AG(x4, x5)
U60_AG(x1, x2, x3, x4, x5, x6) = U60_AG(x4, x5, x6)
U61_AG(x1, x2, x3, x4, x5, x6, x7) = U61_AG(x4, x5, x7)
U62_AG(x1, x2, x3, x4, x5, x6, x7) = U62_AG(x4, x5, x7)
U63_AG(x1, x2, x3, x4, x5, x6) = U63_AG(x4, x5, x6)
APP157_IN_AAAG(x1, x2, x3, x4) = APP157_IN_AAAG(x4)
U24_AAAG(x1, x2, x3, x4, x5, x6) = U24_AAAG(x1, x5, x6)
U64_AG(x1, x2, x3) = U64_AG(x2, x3)
U65_AG(x1, x2, x3, x4) = U65_AG(x2, x3, x4)
U66_AG(x1, x2, x3) = U66_AG(x2, x3)
APP157_IN_GAGG(x1, x2, x3, x4) = APP157_IN_GAGG(x1, x3, x4)
U24_GAGG(x1, x2, x3, x4, x5, x6) = U24_GAGG(x1, x2, x4, x5, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

QS1_IN_AG(.(T24, .(T25, T26)), []) → U25_AG(T24, T25, T26, less13_in_aa(T24, T25))
QS1_IN_AG(.(T24, .(T25, T26)), []) → LESS13_IN_AA(T24, T25)
LESS13_IN_AA(s(T45), s(T46)) → U1_AA(T45, T46, less13_in_aa(T45, T46))
LESS13_IN_AA(s(T45), s(T46)) → LESS13_IN_AA(T45, T46)
QS1_IN_AG(.(T29, .(T31, T30)), []) → U26_AG(T29, T31, T30, lessc13_in_aa(T29, T31))
U26_AG(T29, T31, T30, lessc13_out_aa(T29, T31)) → U27_AG(T29, T31, T30, part23_in_gaaa(T29, T30, X49, X50))
U26_AG(T29, T31, T30, lessc13_out_aa(T29, T31)) → PART23_IN_GAAA(T29, T30, X49, X50)
PART23_IN_GAAA(T70, .(T73, T74), .(T73, X123), X124) → U2_GAAA(T70, T73, T74, X123, X124, less13_in_ga(T70, T73))
PART23_IN_GAAA(T70, .(T73, T74), .(T73, X123), X124) → LESS13_IN_GA(T70, T73)
LESS13_IN_GA(s(T45), s(T46)) → U1_GA(T45, T46, less13_in_ga(T45, T46))
LESS13_IN_GA(s(T45), s(T46)) → LESS13_IN_GA(T45, T46)
PART23_IN_GAAA(T70, .(T73, T77), .(T73, X123), X124) → U3_GAAA(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U4_GAAA(T70, T73, T77, X123, X124, part23_in_gaaa(T70, T77, X123, X124))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → PART23_IN_GAAA(T70, T77, X123, X124)
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → U5_GAAA(T95, T96, T98, X174, X175, part23_in_gaaa(T95, T98, X174, X175))
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → PART23_IN_GAAA(T95, T98, X174, X175)
QS1_IN_AG(.(T29, .(T54, T30)), []) → U28_AG(T29, T54, T30, lessc13_in_aa(T29, T54))
U28_AG(T29, T54, T30, lessc13_out_aa(T29, T54)) → U29_AG(T29, T54, T30, partc23_in_gaaa(T29, T30, T52, T53))
U29_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T53)) → U30_AG(T29, T54, T30, qs39_in_aaa(T54, T52, X11))
U29_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T53)) → QS39_IN_AAA(T54, T52, X11)
QS39_IN_AAA(T122, T123, X234) → U22_AAA(T122, T123, X234, p43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
QS39_IN_AAA(T122, T123, X234) → P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234)
P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234) → U10_AAAAAAA(T122, T123, X230, X231, X232, X233, X234, part44_in_aaaa(T122, T123, X230, X231))
P43_IN_AAAAAAA(T122, T123, X230, X231, X232, X233, X234) → PART44_IN_AAAA(T122, T123, X230, X231)
PART44_IN_AAAA(T148, .(T149, T150), .(T149, X282), X283) → U6_AAAA(T148, T149, T150, X282, X283, less13_in_aa(T148, T149))
PART44_IN_AAAA(T148, .(T149, T150), .(T149, X282), X283) → LESS13_IN_AA(T148, T149)
PART44_IN_AAAA(T153, .(T149, T154), .(T149, X282), X283) → U7_AAAA(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U7_AAAA(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U8_AAAA(T153, T149, T154, X282, X283, part23_in_gaaa(T153, T154, X282, X283))
U7_AAAA(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → PART23_IN_GAAA(T153, T154, X282, X283)
PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → U9_AAAA(T175, T173, T176, X333, X334, part44_in_aaaa(T175, T176, X333, X334))
PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → PART44_IN_AAAA(T175, T176, X333, X334)
P43_IN_AAAAAAA(T129, T123, T127, T128, X232, X233, X234) → U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_in_aaaa(T129, T123, T127, T128))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → U12_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, qs60_in_aa(T127, X232))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → QS60_IN_AA(T127, X232)
QS60_IN_AA(.(T192, T193), X370) → U20_AA(T192, T193, X370, p43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
QS60_IN_AA(.(T192, T193), X370) → P43_IN_AAAAAAA(T192, T193, X366, X367, X368, X369, X370)
P43_IN_AAAAAAA(T185, T123, T127, T184, T183, X233, X234) → U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_in_aaaa(T185, T123, T127, T184))
U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_out_aaaa(T185, T123, T127, T184)) → U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_in_aa(T127, T183))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → U15_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qs60_in_aa(T184, X233))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → QS60_IN_AA(T184, X233)
P43_IN_AAAAAAA(T199, T123, T127, T184, T198, T197, X234) → U16_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U16_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U17_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U17_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U19_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, app71_in_aaaa(T198, T199, T197, X234))
U18_AAAAAAA(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → APP71_IN_AAAA(T198, T199, T197, X234)
APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → U21_AAAA(T222, T226, T227, T228, X403, app71_in_aaaa(T226, T227, T228, X403))
APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → APP71_IN_AAAA(T226, T227, T228, X403)
U28_AG(T29, T54, T30, lessc13_out_aa(T29, T54)) → U31_AG(T29, T54, T30, partc23_in_gaaa(T29, T30, T52, T108))
U31_AG(T29, T54, T30, partc23_out_gaaa(T29, T30, T52, T108)) → U32_AG(T29, T54, T30, T108, qsc39_in_aaa(T54, T52, T107))
U32_AG(T29, T54, T30, T108, qsc39_out_aaa(T54, T52, T107)) → U33_AG(T29, T54, T30, qs60_in_aa(T108, X12))
U32_AG(T29, T54, T30, T108, qsc39_out_aaa(T54, T52, T107)) → QS60_IN_AA(T108, X12)
QS1_IN_AG(.(T264, .(T266, T265)), []) → U34_AG(T264, T266, T265, part44_in_aaaa(T264, T265, X454, X455))
QS1_IN_AG(.(T264, .(T266, T265)), []) → PART44_IN_AAAA(T264, T265, X454, X455)
QS1_IN_AG(.(T273, .(T272, T265)), []) → U35_AG(T273, T272, T265, partc44_in_aaaa(T273, T265, T270, T271))
U35_AG(T273, T272, T265, partc44_out_aaaa(T273, T265, T270, T271)) → U36_AG(T273, T272, T265, qs60_in_aa(T270, X11))
U35_AG(T273, T272, T265, partc44_out_aaaa(T273, T265, T270, T271)) → QS60_IN_AA(T270, X11)
QS1_IN_AG(.(T280, .(T278, T265)), []) → U37_AG(T280, T278, T265, partc44_in_aaaa(T280, T265, T270, T279))
U37_AG(T280, T278, T265, partc44_out_aaaa(T280, T265, T270, T279)) → U38_AG(T280, T278, T265, T279, qsc60_in_aa(T270, T277))
U38_AG(T280, T278, T265, T279, qsc60_out_aa(T270, T277)) → U39_AG(T280, T278, T265, qs39_in_aaa(T278, T279, X12))
U38_AG(T280, T278, T265, T279, qsc60_out_aa(T270, T277)) → QS39_IN_AAA(T278, T279, X12)
QS1_IN_AG(.(T347, .(T348, T349)), T329) → U40_AG(T347, T348, T349, T329, less13_in_aa(T347, T348))
QS1_IN_AG(.(T347, .(T348, T349)), T329) → LESS13_IN_AA(T347, T348)
QS1_IN_AG(.(T352, .(T354, T353)), T329) → U41_AG(T352, T354, T353, T329, lessc13_in_aa(T352, T354))
U41_AG(T352, T354, T353, T329, lessc13_out_aa(T352, T354)) → U42_AG(T352, T354, T353, T329, part23_in_gaaa(T352, T353, X581, X582))
U41_AG(T352, T354, T353, T329, lessc13_out_aa(T352, T354)) → PART23_IN_GAAA(T352, T353, X581, X582)
QS1_IN_AG(.(T352, .(T363, T353)), T329) → U43_AG(T352, T363, T353, T329, lessc13_in_aa(T352, T363))
U43_AG(T352, T363, T353, T329, lessc13_out_aa(T352, T363)) → U44_AG(T352, T363, T353, T329, partc23_in_gaaa(T352, T353, T361, T362))
U44_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T362)) → U45_AG(T352, T363, T353, T329, qs39_in_aaa(T363, T361, X543))
U44_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T362)) → QS39_IN_AAA(T363, T361, X543)
U43_AG(T352, T363, T353, T329, lessc13_out_aa(T352, T363)) → U46_AG(T352, T363, T353, T329, partc23_in_gaaa(T352, T353, T361, T371))
U46_AG(T352, T363, T353, T329, partc23_out_gaaa(T352, T353, T361, T371)) → U47_AG(T352, T363, T353, T329, T371, qsc39_in_aaa(T363, T361, T370))
U47_AG(T352, T363, T353, T329, T371, qsc39_out_aaa(T363, T361, T370)) → U48_AG(T352, T363, T353, T329, qs60_in_aa(T371, X544))
U47_AG(T352, T363, T353, T329, T371, qsc39_out_aaa(T363, T361, T370)) → QS60_IN_AA(T371, X544)
QS1_IN_AG(.(T404, .(T363, T353)), .(T402, T406)) → U49_AG(T404, T363, T353, T402, T406, lessc13_in_aa(T404, T363))
U49_AG(T404, T363, T353, T402, T406, lessc13_out_aa(T404, T363)) → U50_AG(T404, T363, T353, T402, T406, partc23_in_gaaa(T404, T353, T361, T371))
U50_AG(T404, T363, T353, T402, T406, partc23_out_gaaa(T404, T353, T361, T371)) → U51_AG(T404, T363, T353, T402, T406, T371, qsc39_in_aaa(T363, T361, .(T402, T407)))
U51_AG(T404, T363, T353, T402, T406, T371, qsc39_out_aaa(T363, T361, .(T402, T407))) → U52_AG(T404, T363, T353, T402, T406, T407, qsc60_in_aa(T371, T408))
U52_AG(T404, T363, T353, T402, T406, T407, qsc60_out_aa(T371, T408)) → U53_AG(T404, T363, T353, T402, T406, app131_in_agag(T407, T404, T408, T406))
U52_AG(T404, T363, T353, T402, T406, T407, qsc60_out_aa(T371, T408)) → APP131_IN_AGAG(T407, T404, T408, T406)
APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → U23_AGAG(T433, T438, T435, T439, T437, app131_in_agag(T438, T435, T439, T437))
APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → APP131_IN_AGAG(T438, T435, T439, T437)
QS1_IN_AG(.(T459, .(T461, T460)), T329) → U54_AG(T459, T461, T460, T329, part44_in_aaaa(T459, T460, X705, X706))
QS1_IN_AG(.(T459, .(T461, T460)), T329) → PART44_IN_AAAA(T459, T460, X705, X706)
QS1_IN_AG(.(T468, .(T467, T460)), T329) → U55_AG(T468, T467, T460, T329, partc44_in_aaaa(T468, T460, T465, T466))
U55_AG(T468, T467, T460, T329, partc44_out_aaaa(T468, T460, T465, T466)) → U56_AG(T468, T467, T460, T329, qs60_in_aa(T465, X543))
U55_AG(T468, T467, T460, T329, partc44_out_aaaa(T468, T460, T465, T466)) → QS60_IN_AA(T465, X543)
QS1_IN_AG(.(T476, .(T474, T460)), T329) → U57_AG(T476, T474, T460, T329, partc44_in_aaaa(T476, T460, T465, T475))
U57_AG(T476, T474, T460, T329, partc44_out_aaaa(T476, T460, T465, T475)) → U58_AG(T476, T474, T460, T329, T475, qsc60_in_aa(T465, T473))
U58_AG(T476, T474, T460, T329, T475, qsc60_out_aa(T465, T473)) → U59_AG(T476, T474, T460, T329, qs39_in_aaa(T474, T475, X544))
U58_AG(T476, T474, T460, T329, T475, qsc60_out_aa(T465, T473)) → QS39_IN_AAA(T474, T475, X544)
QS1_IN_AG(.(T516, .(T474, T460)), .(T510, T514)) → U60_AG(T516, T474, T460, T510, T514, partc44_in_aaaa(T516, T460, T465, T475))
U60_AG(T516, T474, T460, T510, T514, partc44_out_aaaa(T516, T460, T465, T475)) → U61_AG(T516, T474, T460, T510, T514, T475, qsc60_in_aa(T465, .(T510, T515)))
U61_AG(T516, T474, T460, T510, T514, T475, qsc60_out_aa(T465, .(T510, T515))) → U62_AG(T516, T474, T460, T510, T514, T515, qsc39_in_aaa(T474, T475, T517))
U62_AG(T516, T474, T460, T510, T514, T515, qsc39_out_aaa(T474, T475, T517)) → U63_AG(T516, T474, T460, T510, T514, app157_in_aaag(T515, T516, T517, T514))
U62_AG(T516, T474, T460, T510, T514, T515, qsc39_out_aaa(T474, T475, T517)) → APP157_IN_AAAG(T515, T516, T517, T514)
APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → U24_AAAG(T542, T547, T548, T549, T546, app157_in_aaag(T547, T548, T549, T546))
APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_AAAG(T547, T548, T549, T546)
QS1_IN_AG(.(T557, []), T329) → U64_AG(T557, T329, qsc100_in_a(T560))
U64_AG(T557, T329, qsc100_out_a(T560)) → U65_AG(T557, T329, T560, qsc100_in_a(T563))
U65_AG(T557, T329, T560, qsc100_out_a(T563)) → U66_AG(T557, T329, app157_in_gagg(T560, T557, T563, T329))
U65_AG(T557, T329, T560, qsc100_out_a(T563)) → APP157_IN_GAGG(T560, T557, T563, T329)
APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → U24_GAGG(T542, T547, T548, T549, T546, app157_in_gagg(T547, T548, T549, T546))
APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_GAGG(T547, T548, T549, T546)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 9 SCCs with 87 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_GAGG(T547, T548, T549, T546)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
APP157_IN_GAGG(x1, x2, x3, x4) = APP157_IN_GAGG(x1, x3, x4)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP157_IN_GAGG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_GAGG(T547, T548, T549, T546)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP157_IN_GAGG(x1, x2, x3, x4) = APP157_IN_GAGG(x1, x3, x4)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP157_IN_GAGG(.(T542, T547), T549, .(T542, T546)) → APP157_IN_GAGG(T547, T549, T546)

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:

APP157_IN_GAGG(.(T542, T547), T549, .(T542, T546)) → APP157_IN_GAGG(T547, T549, T546)
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:

APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_AAAG(T547, T548, T549, T546)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
APP157_IN_AAAG(x1, x2, x3, x4) = APP157_IN_AAAG(x4)

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:

APP157_IN_AAAG(.(T542, T547), T548, T549, .(T542, T546)) → APP157_IN_AAAG(T547, T548, T549, T546)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP157_IN_AAAG(x1, x2, x3, x4) = APP157_IN_AAAG(x4)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP157_IN_AAAG(.(T542, T546)) → APP157_IN_AAAG(T546)

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:

APP157_IN_AAAG(.(T542, T546)) → APP157_IN_AAAG(T546)
The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → APP131_IN_AGAG(T438, T435, T439, T437)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
APP131_IN_AGAG(x1, x2, x3, x4) = APP131_IN_AGAG(x2, x4)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APP131_IN_AGAG(.(T433, T438), T435, T439, .(T433, T437)) → APP131_IN_AGAG(T438, T435, T439, T437)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP131_IN_AGAG(x1, x2, x3, x4) = APP131_IN_AGAG(x2, x4)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APP131_IN_AGAG(T435, .(T433, T437)) → APP131_IN_AGAG(T435, T437)

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:

APP131_IN_AGAG(T435, .(T433, T437)) → APP131_IN_AGAG(T435, T437)
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:

APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → APP71_IN_AAAA(T226, T227, T228, X403)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
APP71_IN_AAAA(x1, x2, x3, x4) = APP71_IN_AAAA

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:

APP71_IN_AAAA(.(T222, T226), T227, T228, .(T222, X403)) → APP71_IN_AAAA(T226, T227, T228, X403)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP71_IN_AAAA(x1, x2, x3, x4) = APP71_IN_AAAA

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP71_IN_AAAA → APP71_IN_AAAA

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

(35) NonTerminationProof (EQUIVALENT transformation)

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

s = APP71_IN_AAAA evaluates to t =APP71_IN_AAAA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP71_IN_AAAA to APP71_IN_AAAA.

(36) NO

(37) Obligation:

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

LESS13_IN_GA(s(T45), s(T46)) → LESS13_IN_GA(T45, T46)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
LESS13_IN_GA(x1, x2) = LESS13_IN_GA(x1)

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:

LESS13_IN_GA(s(T45), s(T46)) → LESS13_IN_GA(T45, T46)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESS13_IN_GA(x1, x2) = LESS13_IN_GA(x1)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

LESS13_IN_GA(s(T45)) → LESS13_IN_GA(T45)

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

(42) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

LESS13_IN_GA(s(T45)) → LESS13_IN_GA(T45)
The graph contains the following edges 1 > 1

(43) YES

(44) Obligation:

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

PART23_IN_GAAA(T70, .(T73, T77), .(T73, X123), X124) → U3_GAAA(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → PART23_IN_GAAA(T70, T77, X123, X124)
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → PART23_IN_GAAA(T95, T98, X174, X175)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
PART23_IN_GAAA(x1, x2, x3, x4) = PART23_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

PART23_IN_GAAA(T70, .(T73, T77), .(T73, X123), X124) → U3_GAAA(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U3_GAAA(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → PART23_IN_GAAA(T70, T77, X123, X124)
PART23_IN_GAAA(T95, .(T96, T98), X174, .(T96, X175)) → PART23_IN_GAAA(T95, T98, X174, X175)

The TRS R consists of the following rules:

lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
.(x1, x2) = .(x1, x2)
PART23_IN_GAAA(x1, x2, x3, x4) = PART23_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

PART23_IN_GAAA(T70) → U3_GAAA(T70, lessc13_in_ga(T70))
U3_GAAA(T70, lessc13_out_ga(T70)) → PART23_IN_GAAA(T70)
PART23_IN_GAAA(T95) → PART23_IN_GAAA(T95)

The TRS R consists of the following rules:

lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))

The set Q consists of the following terms:

lessc13_in_ga(x0)
U68_ga(x0, x1)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART23_IN_GAAA(T70) → U3_GAAA(T70, lessc13_in_ga(T70)) at position [1] we obtained the following new rules [LPAR04]:

PART23_IN_GAAA(0) → U3_GAAA(0, lessc13_out_ga(0))
PART23_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U68_ga(x0, lessc13_in_ga(x0)))

(50) Obligation:

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

U3_GAAA(T70, lessc13_out_ga(T70)) → PART23_IN_GAAA(T70)
PART23_IN_GAAA(T95) → PART23_IN_GAAA(T95)
PART23_IN_GAAA(0) → U3_GAAA(0, lessc13_out_ga(0))
PART23_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U68_ga(x0, lessc13_in_ga(x0)))

The TRS R consists of the following rules:

lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))

The set Q consists of the following terms:

lessc13_in_ga(x0)
U68_ga(x0, x1)

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

(51) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GAAA(T70, lessc13_out_ga(T70)) → PART23_IN_GAAA(T70) we obtained the following new rules [LPAR04]:

U3_GAAA(0, lessc13_out_ga(0)) → PART23_IN_GAAA(0)
U3_GAAA(s(z0), lessc13_out_ga(s(z0))) → PART23_IN_GAAA(s(z0))

(52) Obligation:

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

PART23_IN_GAAA(T95) → PART23_IN_GAAA(T95)
PART23_IN_GAAA(0) → U3_GAAA(0, lessc13_out_ga(0))
PART23_IN_GAAA(s(x0)) → U3_GAAA(s(x0), U68_ga(x0, lessc13_in_ga(x0)))
U3_GAAA(0, lessc13_out_ga(0)) → PART23_IN_GAAA(0)
U3_GAAA(s(z0), lessc13_out_ga(s(z0))) → PART23_IN_GAAA(s(z0))

The TRS R consists of the following rules:

lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))

The set Q consists of the following terms:

lessc13_in_ga(x0)
U68_ga(x0, x1)

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

(53) NonTerminationProof (EQUIVALENT transformation)

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

s = PART23_IN_GAAA(T95) evaluates to t =PART23_IN_GAAA(T95)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART23_IN_GAAA(T95) to PART23_IN_GAAA(T95).

(54) NO

(55) Obligation:

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

LESS13_IN_AA(s(T45), s(T46)) → LESS13_IN_AA(T45, T46)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
LESS13_IN_AA(x1, x2) = LESS13_IN_AA

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

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

LESS13_IN_AA(s(T45), s(T46)) → LESS13_IN_AA(T45, T46)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESS13_IN_AA(x1, x2) = LESS13_IN_AA

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

(58) PiDPToQDPProof (SOUND transformation)

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

(59) Obligation:

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

LESS13_IN_AA → LESS13_IN_AA

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

(60) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS13_IN_AA evaluates to t =LESS13_IN_AA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS13_IN_AA to LESS13_IN_AA.

(61) NO

(62) Obligation:

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

PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → PART44_IN_AAAA(T175, T176, X333, X334)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
PART44_IN_AAAA(x1, x2, x3, x4) = PART44_IN_AAAA

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

(63) UsableRulesProof (EQUIVALENT transformation)

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

(64) Obligation:

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

PART44_IN_AAAA(T175, .(T173, T176), X333, .(T173, X334)) → PART44_IN_AAAA(T175, T176, X333, X334)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
PART44_IN_AAAA(x1, x2, x3, x4) = PART44_IN_AAAA

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

(65) PiDPToQDPProof (SOUND transformation)

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

(66) Obligation:

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

PART44_IN_AAAA → PART44_IN_AAAA

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

(67) NonTerminationProof (EQUIVALENT transformation)

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

s = PART44_IN_AAAA evaluates to t =PART44_IN_AAAA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART44_IN_AAAA to PART44_IN_AAAA.

(68) NO

(69) Obligation:

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

P43_IN_AAAAAAA(T129, T123, T127, T128, X232, X233, X234) → U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_in_aaaa(T129, T123, T127, T128))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → QS60_IN_AA(T127, X232)
QS60_IN_AA(.(T192, T193), X370) → P43_IN_AAAAAAA(T192, T193, X366, X367, X368, X369, X370)
P43_IN_AAAAAAA(T185, T123, T127, T184, T183, X233, X234) → U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_in_aaaa(T185, T123, T127, T184))
U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_out_aaaa(T185, T123, T127, T184)) → U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_in_aa(T127, T183))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → QS60_IN_AA(T184, X233)

The TRS R consists of the following rules:

lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
qsc39_in_aaa(T122, T123, X234) → U81_aaa(T122, T123, X234, qc43_in_aaaaaaa(T122, T123, X230, X231, X232, X233, X234))
U81_aaa(T122, T123, X234, qc43_out_aaaaaaa(T122, T123, X230, X231, X232, X233, X234)) → qsc39_out_aaa(T122, T123, X234)
qsc100_in_a([]) → qsc100_out_a([])

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
qsc39_in_aaa(x1, x2, x3) = qsc39_in_aaa
U81_aaa(x1, x2, x3, x4) = U81_aaa(x4)
qsc39_out_aaa(x1, x2, x3) = qsc39_out_aaa
qsc100_in_a(x1) = qsc100_in_a
qsc100_out_a(x1) = qsc100_out_a(x1)
P43_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = P43_IN_AAAAAAA
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8)
QS60_IN_AA(x1, x2) = QS60_IN_AA
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8)

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) Obligation:

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

P43_IN_AAAAAAA(T129, T123, T127, T128, X232, X233, X234) → U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_in_aaaa(T129, T123, T127, T128))
U11_AAAAAAA(T129, T123, T127, T128, X232, X233, X234, partc44_out_aaaa(T129, T123, T127, T128)) → QS60_IN_AA(T127, X232)
QS60_IN_AA(.(T192, T193), X370) → P43_IN_AAAAAAA(T192, T193, X366, X367, X368, X369, X370)
P43_IN_AAAAAAA(T185, T123, T127, T184, T183, X233, X234) → U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_in_aaaa(T185, T123, T127, T184))
U13_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, partc44_out_aaaa(T185, T123, T127, T184)) → U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_in_aa(T127, T183))
U14_AAAAAAA(T185, T123, T127, T184, T183, X233, X234, qsc60_out_aa(T127, T183)) → QS60_IN_AA(T184, X233)

The TRS R consists of the following rules:

partc44_in_aaaa(T153, .(T149, T154), .(T149, X282), X283) → U72_aaaa(T153, T149, T154, X282, X283, lessc13_in_aa(T153, T149))
partc44_in_aaaa(T175, .(T173, T176), X333, .(T173, X334)) → U74_aaaa(T175, T173, T176, X333, X334, partc44_in_aaaa(T175, T176, X333, X334))
partc44_in_aaaa(T182, [], [], []) → partc44_out_aaaa(T182, [], [], [])
qsc60_in_aa([], []) → qsc60_out_aa([], [])
qsc60_in_aa(.(T192, T193), X370) → U79_aa(T192, T193, X370, qc43_in_aaaaaaa(T192, T193, X366, X367, X368, X369, X370))
U72_aaaa(T153, T149, T154, X282, X283, lessc13_out_aa(T153, T149)) → U73_aaaa(T153, T149, T154, X282, X283, partc23_in_gaaa(T153, T154, X282, X283))
U74_aaaa(T175, T173, T176, X333, X334, partc44_out_aaaa(T175, T176, X333, X334)) → partc44_out_aaaa(T175, .(T173, T176), X333, .(T173, X334))
U79_aa(T192, T193, X370, qc43_out_aaaaaaa(T192, T193, X366, X367, X368, X369, X370)) → qsc60_out_aa(.(T192, T193), X370)
lessc13_in_aa(0, s(T38)) → lessc13_out_aa(0, s(T38))
lessc13_in_aa(s(T45), s(T46)) → U68_aa(T45, T46, lessc13_in_aa(T45, T46))
U73_aaaa(T153, T149, T154, X282, X283, partc23_out_gaaa(T153, T154, X282, X283)) → partc44_out_aaaa(T153, .(T149, T154), .(T149, X282), X283)
qc43_in_aaaaaaa(T199, T123, T127, T184, T198, T197, X234) → U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_in_aaaa(T199, T123, T127, T184))
U68_aa(T45, T46, lessc13_out_aa(T45, T46)) → lessc13_out_aa(s(T45), s(T46))
partc23_in_gaaa(T70, .(T73, T77), .(T73, X123), X124) → U69_gaaa(T70, T73, T77, X123, X124, lessc13_in_ga(T70, T73))
partc23_in_gaaa(T95, .(T96, T98), X174, .(T96, X175)) → U71_gaaa(T95, T96, T98, X174, X175, partc23_in_gaaa(T95, T98, X174, X175))
partc23_in_gaaa(T104, [], [], []) → partc23_out_gaaa(T104, [], [], [])
U75_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, partc44_out_aaaa(T199, T123, T127, T184)) → U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T127, T198))
U69_gaaa(T70, T73, T77, X123, X124, lessc13_out_ga(T70, T73)) → U70_gaaa(T70, T73, T77, X123, X124, partc23_in_gaaa(T70, T77, X123, X124))
U71_gaaa(T95, T96, T98, X174, X175, partc23_out_gaaa(T95, T98, X174, X175)) → partc23_out_gaaa(T95, .(T96, T98), X174, .(T96, X175))
U76_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T127, T198)) → U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_in_aa(T184, T197))
lessc13_in_ga(0, s(T38)) → lessc13_out_ga(0, s(T38))
lessc13_in_ga(s(T45), s(T46)) → U68_ga(T45, T46, lessc13_in_ga(T45, T46))
U70_gaaa(T70, T73, T77, X123, X124, partc23_out_gaaa(T70, T77, X123, X124)) → partc23_out_gaaa(T70, .(T73, T77), .(T73, X123), X124)
U77_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, qsc60_out_aa(T184, T197)) → U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_in_aaaa(T198, T199, T197, X234))
U68_ga(T45, T46, lessc13_out_ga(T45, T46)) → lessc13_out_ga(s(T45), s(T46))
U78_aaaaaaa(T199, T123, T127, T184, T198, T197, X234, appc71_out_aaaa(T198, T199, T197, X234)) → qc43_out_aaaaaaa(T199, T123, T127, T184, T198, T197, X234)
appc71_in_aaaa([], T212, T213, .(T212, T213)) → appc71_out_aaaa([], T212, T213, .(T212, T213))
appc71_in_aaaa(.(T222, T226), T227, T228, .(T222, X403)) → U80_aaaa(T222, T226, T227, T228, X403, appc71_in_aaaa(T226, T227, T228, X403))
U80_aaaa(T222, T226, T227, T228, X403, appc71_out_aaaa(T226, T227, T228, X403)) → appc71_out_aaaa(.(T222, T226), T227, T228, .(T222, X403))

The argument filtering Pi contains the following mapping:
[] = []
lessc13_in_aa(x1, x2) = lessc13_in_aa
lessc13_out_aa(x1, x2) = lessc13_out_aa(x1)
U68_aa(x1, x2, x3) = U68_aa(x3)
s(x1) = s(x1)
lessc13_in_ga(x1, x2) = lessc13_in_ga(x1)
0 = 0
lessc13_out_ga(x1, x2) = lessc13_out_ga(x1)
U68_ga(x1, x2, x3) = U68_ga(x1, x3)
partc23_in_gaaa(x1, x2, x3, x4) = partc23_in_gaaa(x1)
U69_gaaa(x1, x2, x3, x4, x5, x6) = U69_gaaa(x1, x6)
U70_gaaa(x1, x2, x3, x4, x5, x6) = U70_gaaa(x1, x6)
U71_gaaa(x1, x2, x3, x4, x5, x6) = U71_gaaa(x1, x6)
partc23_out_gaaa(x1, x2, x3, x4) = partc23_out_gaaa(x1)
partc44_in_aaaa(x1, x2, x3, x4) = partc44_in_aaaa
U72_aaaa(x1, x2, x3, x4, x5, x6) = U72_aaaa(x6)
U73_aaaa(x1, x2, x3, x4, x5, x6) = U73_aaaa(x6)
partc44_out_aaaa(x1, x2, x3, x4) = partc44_out_aaaa
U74_aaaa(x1, x2, x3, x4, x5, x6) = U74_aaaa(x6)
qsc60_in_aa(x1, x2) = qsc60_in_aa
qsc60_out_aa(x1, x2) = qsc60_out_aa
U79_aa(x1, x2, x3, x4) = U79_aa(x4)
qc43_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_in_aaaaaaa
U75_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U75_aaaaaaa(x8)
U76_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U76_aaaaaaa(x8)
U77_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U77_aaaaaaa(x8)
U78_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U78_aaaaaaa(x8)
.(x1, x2) = .(x1, x2)
qc43_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qc43_out_aaaaaaa
appc71_in_aaaa(x1, x2, x3, x4) = appc71_in_aaaa
appc71_out_aaaa(x1, x2, x3, x4) = appc71_out_aaaa
U80_aaaa(x1, x2, x3, x4, x5, x6) = U80_aaaa(x6)
P43_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = P43_IN_AAAAAAA
U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8)
QS60_IN_AA(x1, x2) = QS60_IN_AA
U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8)
U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8)

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

(72) PiDPToQDPProof (SOUND transformation)

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

(73) Obligation:

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

P43_IN_AAAAAAA → U11_AAAAAAA(partc44_in_aaaa)
U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA
QS60_IN_AA → P43_IN_AAAAAAA
P43_IN_AAAAAAA → U13_AAAAAAA(partc44_in_aaaa)
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_in_aa)
U14_AAAAAAA(qsc60_out_aa) → QS60_IN_AA

The TRS R consists of the following rules:

partc44_in_aaaa → U72_aaaa(lessc13_in_aa)
partc44_in_aaaa → U74_aaaa(partc44_in_aaaa)
partc44_in_aaaa → partc44_out_aaaa
qsc60_in_aa → qsc60_out_aa
qsc60_in_aa → U79_aa(qc43_in_aaaaaaa)
U72_aaaa(lessc13_out_aa(T153)) → U73_aaaa(partc23_in_gaaa(T153))
U74_aaaa(partc44_out_aaaa) → partc44_out_aaaa
U79_aa(qc43_out_aaaaaaa) → qsc60_out_aa
lessc13_in_aa → lessc13_out_aa(0)
lessc13_in_aa → U68_aa(lessc13_in_aa)
U73_aaaa(partc23_out_gaaa(T153)) → partc44_out_aaaa
qc43_in_aaaaaaa → U75_aaaaaaa(partc44_in_aaaa)
U68_aa(lessc13_out_aa(T45)) → lessc13_out_aa(s(T45))
partc23_in_gaaa(T70) → U69_gaaa(T70, lessc13_in_ga(T70))
partc23_in_gaaa(T95) → U71_gaaa(T95, partc23_in_gaaa(T95))
partc23_in_gaaa(T104) → partc23_out_gaaa(T104)
U75_aaaaaaa(partc44_out_aaaa) → U76_aaaaaaa(qsc60_in_aa)
U69_gaaa(T70, lessc13_out_ga(T70)) → U70_gaaa(T70, partc23_in_gaaa(T70))
U71_gaaa(T95, partc23_out_gaaa(T95)) → partc23_out_gaaa(T95)
U76_aaaaaaa(qsc60_out_aa) → U77_aaaaaaa(qsc60_in_aa)
lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U70_gaaa(T70, partc23_out_gaaa(T70)) → partc23_out_gaaa(T70)
U77_aaaaaaa(qsc60_out_aa) → U78_aaaaaaa(appc71_in_aaaa)
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))
U78_aaaaaaa(appc71_out_aaaa) → qc43_out_aaaaaaa
appc71_in_aaaa → appc71_out_aaaa
appc71_in_aaaa → U80_aaaa(appc71_in_aaaa)
U80_aaaa(appc71_out_aaaa) → appc71_out_aaaa

The set Q consists of the following terms:

partc44_in_aaaa
qsc60_in_aa
U72_aaaa(x0)
U74_aaaa(x0)
U79_aa(x0)
lessc13_in_aa
U73_aaaa(x0)
qc43_in_aaaaaaa
U68_aa(x0)
partc23_in_gaaa(x0)
U75_aaaaaaa(x0)
U69_gaaa(x0, x1)
U71_gaaa(x0, x1)
U76_aaaaaaa(x0)
lessc13_in_ga(x0)
U70_gaaa(x0, x1)
U77_aaaaaaa(x0)
U68_ga(x0, x1)
U78_aaaaaaa(x0)
appc71_in_aaaa
U80_aaaa(x0)

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

(74) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P43_IN_AAAAAAA → U11_AAAAAAA(partc44_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P43_IN_AAAAAAA → U11_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U11_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa)

(75) Obligation:

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

U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA
QS60_IN_AA → P43_IN_AAAAAAA
P43_IN_AAAAAAA → U13_AAAAAAA(partc44_in_aaaa)
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_in_aa)
U14_AAAAAAA(qsc60_out_aa) → QS60_IN_AA
P43_IN_AAAAAAA → U11_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U11_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa)

The TRS R consists of the following rules:

partc44_in_aaaa → U72_aaaa(lessc13_in_aa)
partc44_in_aaaa → U74_aaaa(partc44_in_aaaa)
partc44_in_aaaa → partc44_out_aaaa
qsc60_in_aa → qsc60_out_aa
qsc60_in_aa → U79_aa(qc43_in_aaaaaaa)
U72_aaaa(lessc13_out_aa(T153)) → U73_aaaa(partc23_in_gaaa(T153))
U74_aaaa(partc44_out_aaaa) → partc44_out_aaaa
U79_aa(qc43_out_aaaaaaa) → qsc60_out_aa
lessc13_in_aa → lessc13_out_aa(0)
lessc13_in_aa → U68_aa(lessc13_in_aa)
U73_aaaa(partc23_out_gaaa(T153)) → partc44_out_aaaa
qc43_in_aaaaaaa → U75_aaaaaaa(partc44_in_aaaa)
U68_aa(lessc13_out_aa(T45)) → lessc13_out_aa(s(T45))
partc23_in_gaaa(T70) → U69_gaaa(T70, lessc13_in_ga(T70))
partc23_in_gaaa(T95) → U71_gaaa(T95, partc23_in_gaaa(T95))
partc23_in_gaaa(T104) → partc23_out_gaaa(T104)
U75_aaaaaaa(partc44_out_aaaa) → U76_aaaaaaa(qsc60_in_aa)
U69_gaaa(T70, lessc13_out_ga(T70)) → U70_gaaa(T70, partc23_in_gaaa(T70))
U71_gaaa(T95, partc23_out_gaaa(T95)) → partc23_out_gaaa(T95)
U76_aaaaaaa(qsc60_out_aa) → U77_aaaaaaa(qsc60_in_aa)
lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U70_gaaa(T70, partc23_out_gaaa(T70)) → partc23_out_gaaa(T70)
U77_aaaaaaa(qsc60_out_aa) → U78_aaaaaaa(appc71_in_aaaa)
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))
U78_aaaaaaa(appc71_out_aaaa) → qc43_out_aaaaaaa
appc71_in_aaaa → appc71_out_aaaa
appc71_in_aaaa → U80_aaaa(appc71_in_aaaa)
U80_aaaa(appc71_out_aaaa) → appc71_out_aaaa

The set Q consists of the following terms:

partc44_in_aaaa
qsc60_in_aa
U72_aaaa(x0)
U74_aaaa(x0)
U79_aa(x0)
lessc13_in_aa
U73_aaaa(x0)
qc43_in_aaaaaaa
U68_aa(x0)
partc23_in_gaaa(x0)
U75_aaaaaaa(x0)
U69_gaaa(x0, x1)
U71_gaaa(x0, x1)
U76_aaaaaaa(x0)
lessc13_in_ga(x0)
U70_gaaa(x0, x1)
U77_aaaaaaa(x0)
U68_ga(x0, x1)
U78_aaaaaaa(x0)
appc71_in_aaaa
U80_aaaa(x0)

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

(76) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P43_IN_AAAAAAA → U13_AAAAAAA(partc44_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

P43_IN_AAAAAAA → U13_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U13_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U13_AAAAAAA(partc44_out_aaaa)

(77) Obligation:

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

U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA
QS60_IN_AA → P43_IN_AAAAAAA
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_in_aa)
U14_AAAAAAA(qsc60_out_aa) → QS60_IN_AA
P43_IN_AAAAAAA → U11_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U11_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa)
P43_IN_AAAAAAA → U13_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U13_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U13_AAAAAAA(partc44_out_aaaa)

The TRS R consists of the following rules:

partc44_in_aaaa → U72_aaaa(lessc13_in_aa)
partc44_in_aaaa → U74_aaaa(partc44_in_aaaa)
partc44_in_aaaa → partc44_out_aaaa
qsc60_in_aa → qsc60_out_aa
qsc60_in_aa → U79_aa(qc43_in_aaaaaaa)
U72_aaaa(lessc13_out_aa(T153)) → U73_aaaa(partc23_in_gaaa(T153))
U74_aaaa(partc44_out_aaaa) → partc44_out_aaaa
U79_aa(qc43_out_aaaaaaa) → qsc60_out_aa
lessc13_in_aa → lessc13_out_aa(0)
lessc13_in_aa → U68_aa(lessc13_in_aa)
U73_aaaa(partc23_out_gaaa(T153)) → partc44_out_aaaa
qc43_in_aaaaaaa → U75_aaaaaaa(partc44_in_aaaa)
U68_aa(lessc13_out_aa(T45)) → lessc13_out_aa(s(T45))
partc23_in_gaaa(T70) → U69_gaaa(T70, lessc13_in_ga(T70))
partc23_in_gaaa(T95) → U71_gaaa(T95, partc23_in_gaaa(T95))
partc23_in_gaaa(T104) → partc23_out_gaaa(T104)
U75_aaaaaaa(partc44_out_aaaa) → U76_aaaaaaa(qsc60_in_aa)
U69_gaaa(T70, lessc13_out_ga(T70)) → U70_gaaa(T70, partc23_in_gaaa(T70))
U71_gaaa(T95, partc23_out_gaaa(T95)) → partc23_out_gaaa(T95)
U76_aaaaaaa(qsc60_out_aa) → U77_aaaaaaa(qsc60_in_aa)
lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U70_gaaa(T70, partc23_out_gaaa(T70)) → partc23_out_gaaa(T70)
U77_aaaaaaa(qsc60_out_aa) → U78_aaaaaaa(appc71_in_aaaa)
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))
U78_aaaaaaa(appc71_out_aaaa) → qc43_out_aaaaaaa
appc71_in_aaaa → appc71_out_aaaa
appc71_in_aaaa → U80_aaaa(appc71_in_aaaa)
U80_aaaa(appc71_out_aaaa) → appc71_out_aaaa

The set Q consists of the following terms:

partc44_in_aaaa
qsc60_in_aa
U72_aaaa(x0)
U74_aaaa(x0)
U79_aa(x0)
lessc13_in_aa
U73_aaaa(x0)
qc43_in_aaaaaaa
U68_aa(x0)
partc23_in_gaaa(x0)
U75_aaaaaaa(x0)
U69_gaaa(x0, x1)
U71_gaaa(x0, x1)
U76_aaaaaaa(x0)
lessc13_in_ga(x0)
U70_gaaa(x0, x1)
U77_aaaaaaa(x0)
U68_ga(x0, x1)
U78_aaaaaaa(x0)
appc71_in_aaaa
U80_aaaa(x0)

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

(78) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_out_aa)
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(U79_aa(qc43_in_aaaaaaa))

(79) Obligation:

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

U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA
QS60_IN_AA → P43_IN_AAAAAAA
U14_AAAAAAA(qsc60_out_aa) → QS60_IN_AA
P43_IN_AAAAAAA → U11_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U11_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa)
P43_IN_AAAAAAA → U13_AAAAAAA(U72_aaaa(lessc13_in_aa))
P43_IN_AAAAAAA → U13_AAAAAAA(U74_aaaa(partc44_in_aaaa))
P43_IN_AAAAAAA → U13_AAAAAAA(partc44_out_aaaa)
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(qsc60_out_aa)
U13_AAAAAAA(partc44_out_aaaa) → U14_AAAAAAA(U79_aa(qc43_in_aaaaaaa))

The TRS R consists of the following rules:

partc44_in_aaaa → U72_aaaa(lessc13_in_aa)
partc44_in_aaaa → U74_aaaa(partc44_in_aaaa)
partc44_in_aaaa → partc44_out_aaaa
qsc60_in_aa → qsc60_out_aa
qsc60_in_aa → U79_aa(qc43_in_aaaaaaa)
U72_aaaa(lessc13_out_aa(T153)) → U73_aaaa(partc23_in_gaaa(T153))
U74_aaaa(partc44_out_aaaa) → partc44_out_aaaa
U79_aa(qc43_out_aaaaaaa) → qsc60_out_aa
lessc13_in_aa → lessc13_out_aa(0)
lessc13_in_aa → U68_aa(lessc13_in_aa)
U73_aaaa(partc23_out_gaaa(T153)) → partc44_out_aaaa
qc43_in_aaaaaaa → U75_aaaaaaa(partc44_in_aaaa)
U68_aa(lessc13_out_aa(T45)) → lessc13_out_aa(s(T45))
partc23_in_gaaa(T70) → U69_gaaa(T70, lessc13_in_ga(T70))
partc23_in_gaaa(T95) → U71_gaaa(T95, partc23_in_gaaa(T95))
partc23_in_gaaa(T104) → partc23_out_gaaa(T104)
U75_aaaaaaa(partc44_out_aaaa) → U76_aaaaaaa(qsc60_in_aa)
U69_gaaa(T70, lessc13_out_ga(T70)) → U70_gaaa(T70, partc23_in_gaaa(T70))
U71_gaaa(T95, partc23_out_gaaa(T95)) → partc23_out_gaaa(T95)
U76_aaaaaaa(qsc60_out_aa) → U77_aaaaaaa(qsc60_in_aa)
lessc13_in_ga(0) → lessc13_out_ga(0)
lessc13_in_ga(s(T45)) → U68_ga(T45, lessc13_in_ga(T45))
U70_gaaa(T70, partc23_out_gaaa(T70)) → partc23_out_gaaa(T70)
U77_aaaaaaa(qsc60_out_aa) → U78_aaaaaaa(appc71_in_aaaa)
U68_ga(T45, lessc13_out_ga(T45)) → lessc13_out_ga(s(T45))
U78_aaaaaaa(appc71_out_aaaa) → qc43_out_aaaaaaa
appc71_in_aaaa → appc71_out_aaaa
appc71_in_aaaa → U80_aaaa(appc71_in_aaaa)
U80_aaaa(appc71_out_aaaa) → appc71_out_aaaa

The set Q consists of the following terms:

partc44_in_aaaa
qsc60_in_aa
U72_aaaa(x0)
U74_aaaa(x0)
U79_aa(x0)
lessc13_in_aa
U73_aaaa(x0)
qc43_in_aaaaaaa
U68_aa(x0)
partc23_in_gaaa(x0)
U75_aaaaaaa(x0)
U69_gaaa(x0, x1)
U71_gaaa(x0, x1)
U76_aaaaaaa(x0)
lessc13_in_ga(x0)
U70_gaaa(x0, x1)
U77_aaaaaaa(x0)
U68_ga(x0, x1)
U78_aaaaaaa(x0)
appc71_in_aaaa
U80_aaaa(x0)

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

(80) NonTerminationProof (EQUIVALENT transformation)

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

s = QS60_IN_AA evaluates to t =QS60_IN_AA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

QS60_IN_AA → P43_IN_AAAAAAA
with rule QS60_IN_AA → P43_IN_AAAAAAA at position [] and matcher [ ]

P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa)
with rule P43_IN_AAAAAAA → U11_AAAAAAA(partc44_out_aaaa) at position [] and matcher [ ]

U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA
with rule U11_AAAAAAA(partc44_out_aaaa) → QS60_IN_AA

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

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