Left Termination proof of /tmp/tmp9DY2t3/psk09-maxsort-bool-nocut.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 383 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 358 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇒, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇒, 0 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 0 ms)

        ↳39 QDP

        ↳40 Narrowing (⇒, 0 ms)

        ↳41 QDP

        ↳42 UsableRulesProof (⇔, 0 ms)

        ↳43 QDP

        ↳44 QReductionProof (⇔, 0 ms)

        ↳45 QDP

        ↳46 Instantiation (⇔, 0 ms)

        ↳47 QDP

        ↳48 DependencyGraphProof (⇔, 0 ms)

        ↳49 QDP

        ↳50 QDPQMonotonicMRRProof (⇔, 1118 ms)

        ↳51 QDP

        ↳52 QDPQMonotonicMRRProof (⇔, 1200 ms)

        ↳53 QDP

        ↳54 QDPOrderProof (⇔, 160 ms)

        ↳55 QDP

        ↳56 DependencyGraphProof (⇔, 0 ms)

        ↳57 QDP

        ↳58 UsableRulesProof (⇔, 0 ms)

        ↳59 QDP

        ↳60 QReductionProof (⇔, 0 ms)

        ↳61 QDP

        ↳62 QDPQMonotonicMRRProof (⇔, 528 ms)

        ↳63 QDP

        ↳64 QDPOrderProof (⇔, 222 ms)

        ↳65 QDP

        ↳66 DependencyGraphProof (⇔, 0 ms)

        ↳67 QDP

        ↳68 UsableRulesProof (⇔, 0 ms)

        ↳69 QDP

        ↳70 QReductionProof (⇔, 0 ms)

        ↳71 QDP

        ↳72 QDPQMonotonicMRRProof (⇔, 1805 ms)

        ↳73 QDP

        ↳74 DependencyGraphProof (⇔, 0 ms)

        ↳75 QDP

        ↳76 UsableRulesProof (⇔, 0 ms)

        ↳77 QDP

        ↳78 QReductionProof (⇔, 2 ms)

        ↳79 QDP

        ↳80 QDPQMonotonicMRRProof (⇔, 407 ms)

        ↳81 QDP

        ↳82 UsableRulesProof (⇔, 0 ms)

        ↳83 QDP

        ↳84 QReductionProof (⇔, 0 ms)

        ↳85 QDP

        ↳86 Narrowing (⇒, 0 ms)

        ↳87 QDP

        ↳88 UsableRulesProof (⇔, 0 ms)

        ↳89 QDP

        ↳90 QReductionProof (⇔, 0 ms)

        ↳91 QDP

        ↳92 Instantiation (⇔, 0 ms)

        ↳93 QDP

        ↳94 MRRProof (⇔, 23 ms)

        ↳95 QDP

        ↳96 PisEmptyProof (⇔, 0 ms)

        ↳97 YES

(0) Obligation:

Clauses:

eq(t, t).
eq(f, f).
neq(t, f).
neq(f, t).
del(X1, [], []).
del(X, .(Y, YS), YS) :- eq(X, Y).
del(X, .(Y, YS), .(Y, ZS)) :- ','(neq(X, Y), del(X, YS, ZS)).
ge(t, t).
ge(t, f).
ge(f, f).
gt(t, f).
max([], f).
max(.(X, []), X).
max(.(X, .(Y, XS)), Z) :- ','(ge(X, Y), max(.(X, XS), Z)).
max(.(X, .(Y, XS)), Z) :- ','(gt(Y, X), max(.(Y, XS), Z)).
maxsort([], []).
maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS))).

Query: maxsort(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: maxsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: maxsort(T1, T2), SCOPE: 1, CLAUSE: 15 | maxsort(T1, T2), SCOPE: 1, CLAUSE: 16\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | maxsort([], T2), SCOPE: 1, CLAUSE: 16\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: maxsort(T1, T2), SCOPE: 1, CLAUSE: 16\n\nT1 is ground\nmaxsort(T1, T2) !=? maxsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: maxsort([], T2), SCOPE: 1, CLAUSE: 16\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12)))\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 11 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 12 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 13 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 14\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 12 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 13 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 14\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 12\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 13 | ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 14\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(del(T17, .(T17, []), X14), maxsort(X14, T18))\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="B: del(T17, .(T17, []), X14)\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: maxsort(T21, T18)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 4 | del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 5 | del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 6\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 5 | del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 6\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 5\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: del(T17, .(T17, []), X14), SCOPE: 3, CLAUSE: 6\n\nT17 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(T29, T29)\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: eq(T29, T29), SCOPE: 4, CLAUSE: 0 | eq(T29, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: eq(T29, T29), SCOPE: 4, CLAUSE: 0\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: eq(T29, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(neq(T33, T33), del(T33, [], X55))\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 2 | ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 3\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(neq(T33, T33), del(T33, [], X55)), SCOPE: 5, CLAUSE: 3\n\nT33 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 13\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(max(.(T7, T8), T11), ','(del(T11, .(T7, T8), X14), maxsort(X14, T12))), SCOPE: 2, CLAUSE: 14\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56)))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 7 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 7\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8 | ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(max(.(t, T53), T55), ','(del(T55, .(t, .(t, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="C: max(.(t, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(del(T57, .(t, .(t, T53)), X14), maxsort(X14, T58))\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 11 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 12\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13 | max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 13\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: max(.(t, T53), T55), SCOPE: 7, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(ge(t, T71), max(.(t, T72), T74))\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 7 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 7\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: max(.(t, T72), T74)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 8\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(ge(t, T71), max(.(t, T72), T74)), SCOPE: 8, CLAUSE: 9\n\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: max(.(t, T72), T74)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(gt(T81, t), max(.(T81, T82), T84))\n\nT81 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(gt(T81, t), max(.(T81, T82), T84)), SCOPE: 9, CLAUSE: 10\n\nT81 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="D: del(T57, .(t, .(t, T53)), X14)\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: maxsort(T87, T58)\n\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 4 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5 | del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 5\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: del(T57, .(t, .(t, T53)), X14), SCOPE: 10, CLAUSE: 6\n\nT53 is ground\nT57 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: eq(T99, t)\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: eq(T99, t), SCOPE: 11, CLAUSE: 0 | eq(T99, t), SCOPE: 11, CLAUSE: 1\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: eq(T99, t), SCOPE: 11, CLAUSE: 0\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: eq(T99, t), SCOPE: 11, CLAUSE: 1\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(neq(T107, t), del(T107, .(t, T108), X150))\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 2 | ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 3\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(neq(T107, t), del(T107, .(t, T108), X150)), SCOPE: 12, CLAUSE: 3\n\nT107 is ground\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: del(f, .(t, T108), X150)\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 4 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 5\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: del(f, .(t, T108), X150), SCOPE: 13, CLAUSE: 6\n\nT108 is ground\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: eq(f, t)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: eq(f, t), SCOPE: 14, CLAUSE: 0 | eq(f, t), SCOPE: 14, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: eq(f, t), SCOPE: 14, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(neq(f, t), del(f, T118, X187))\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 2 | ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 3\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(neq(f, t), del(f, T118, X187)), SCOPE: 15, CLAUSE: 3\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="I: del(f, T118, X187)\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: del(f, T118, X187), SCOPE: 16, CLAUSE: 4 | del(f, T118, X187), SCOPE: 16, CLAUSE: 5 | del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: del(f, T118, X187), SCOPE: 16, CLAUSE: 4\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: del(f, T118, X187), SCOPE: 16, CLAUSE: 5 | del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 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: del(f, T118, X187), SCOPE: 16, CLAUSE: 5\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: del(f, T118, X187), SCOPE: 16, CLAUSE: 6\n\nT118 is ground\nX187 is free", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: eq(f, T127)\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: eq(f, T127), SCOPE: 17, CLAUSE: 0 | eq(f, T127), SCOPE: 17, CLAUSE: 1\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: eq(f, T127), SCOPE: 17, CLAUSE: 1\n\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: ','(neq(f, T133), del(f, T134, X228))\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 2 | ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 3\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(neq(f, T133), del(f, T134, X228)), SCOPE: 18, CLAUSE: 3\n\nT133 is ground\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: del(f, T134, X228)\n\nT134 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 8\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(','(ge(T51, T52), max(.(T51, T53), T55)), ','(del(T55, .(T51, .(T52, T53)), X14), maxsort(X14, T56))), SCOPE: 6, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(max(.(t, T53), T55), ','(del(T55, .(t, .(f, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: max(.(t, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: ','(del(T135, .(t, .(f, T53)), X14), maxsort(X14, T136))\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="E: del(T135, .(t, .(f, T53)), X14)\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: maxsort(T139, T136)\n\nT139 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 4 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5 | del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 5\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: del(T135, .(t, .(f, T53)), X14), SCOPE: 19, CLAUSE: 6\n\nT53 is ground\nT135 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: eq(T151, t)\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: eq(T151, t), SCOPE: 20, CLAUSE: 0 | eq(T151, t), SCOPE: 20, CLAUSE: 1\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: eq(T151, t), SCOPE: 20, CLAUSE: 0\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: eq(T151, t), SCOPE: 20, CLAUSE: 1\n\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: ','(neq(T159, t), del(T159, .(f, T160), X269))\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 2 | ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 3\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: ','(neq(T159, t), del(T159, .(f, T160), X269)), SCOPE: 21, CLAUSE: 3\n\nT159 is ground\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: del(f, .(f, T160), X269)\n\nT160 is ground\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: ','(max(.(f, T53), T55), ','(del(T55, .(f, .(f, T53)), X14), maxsort(X14, T56)))\n\nT53 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="F: max(.(f, T53), T55)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(del(T163, .(f, .(f, T53)), X14), maxsort(X14, T164))\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 11 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 12\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13 | max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 13\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: max(.(f, T53), T55), SCOPE: 22, CLAUSE: 14\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: ','(ge(f, T177), max(.(f, T178), T180))\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 7 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: ','(ge(f, T177), max(.(f, T178), T180)), SCOPE: 23, CLAUSE: 9\n\nT177 is ground\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: max(.(f, T178), T180)\n\nT178 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: ','(gt(T187, f), max(.(T187, T188), T190))\n\nT187 is ground\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: ','(gt(T187, f), max(.(T187, T188), T190)), SCOPE: 24, CLAUSE: 10\n\nT187 is ground\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: max(.(t, T188), T190)\n\nT188 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="G: del(T163, .(f, .(f, T53)), X14)\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: maxsort(T193, T164)\n\nT193 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 4 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5 | del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 5\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: del(T163, .(f, .(f, T53)), X14), SCOPE: 25, CLAUSE: 6\n\nT53 is ground\nT163 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="179: eq(T205, f)\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: eq(T205, f), SCOPE: 26, CLAUSE: 0 | eq(T205, f), SCOPE: 26, CLAUSE: 1\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
181 [label="181: eq(T205, f), SCOPE: 26, CLAUSE: 1\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="182: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: ','(neq(T213, f), del(T213, .(f, T214), X347))\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 2 | ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 2\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: ','(neq(T213, f), del(T213, .(f, T214), X347)), SCOPE: 27, CLAUSE: 3\n\nT213 is ground\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: del(t, .(f, T214), X347)\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 4 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 5\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: del(t, .(f, T214), X347), SCOPE: 28, CLAUSE: 6\n\nT214 is ground\nX347 is free", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: eq(t, f)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: eq(t, f), SCOPE: 29, CLAUSE: 0 | eq(t, f), SCOPE: 29, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: eq(t, f), SCOPE: 29, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: ','(neq(t, f), del(t, T224, X384))\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 2 | ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 3\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 2\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: ','(neq(t, f), del(t, T224, X384)), SCOPE: 30, CLAUSE: 3\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="J: del(t, T224, X384)\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: del(t, T224, X384), SCOPE: 31, CLAUSE: 4 | del(t, T224, X384), SCOPE: 31, CLAUSE: 5 | del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: del(t, T224, X384), SCOPE: 31, CLAUSE: 4\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: del(t, T224, X384), SCOPE: 31, CLAUSE: 5 | del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: del(t, T224, X384), SCOPE: 31, CLAUSE: 5\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: del(t, T224, X384), SCOPE: 31, CLAUSE: 6\n\nT224 is ground\nX384 is free", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: eq(t, T233)\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: eq(t, T233), SCOPE: 32, CLAUSE: 0 | eq(t, T233), SCOPE: 32, CLAUSE: 1\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: eq(t, T233), SCOPE: 32, CLAUSE: 0\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: eq(t, T233), SCOPE: 32, CLAUSE: 1\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: ','(neq(t, T239), del(t, T240, X427))\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 2 | ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 3\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 2\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: ','(neq(t, T239), del(t, T240, X427)), SCOPE: 33, CLAUSE: 3\n\nT239 is ground\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: del(t, T240, X427)\n\nT240 is ground\nX427 is free", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: ','(','(gt(T250, T249), max(.(T250, T251), T253)), ','(del(T253, .(T249, .(T250, T251)), X14), maxsort(X14, T254)))\n\nT249 is ground\nT250 is ground\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: ','(','(gt(T250, T249), max(.(T250, T251), T253)), ','(del(T253, .(T249, .(T250, T251)), X14), maxsort(X14, T254))), SCOPE: 34, CLAUSE: 10\n\nT249 is ground\nT250 is ground\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: ','(max(.(t, T251), T253), ','(del(T253, .(f, .(t, T251)), X14), maxsort(X14, T254)))\n\nT251 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="236: max(.(t, T251), T253)\n\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: ','(del(T255, .(f, .(t, T251)), X14), maxsort(X14, T256))\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="H: del(T255, .(f, .(t, T251)), X14)\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: maxsort(T259, T256)\n\nT259 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 4 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5 | del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 5\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: del(T255, .(f, .(t, T251)), X14), SCOPE: 35, CLAUSE: 6\n\nT251 is ground\nT255 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: eq(T271, f)\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="245: eq(T271, f), SCOPE: 36, CLAUSE: 0 | eq(T271, f), SCOPE: 36, CLAUSE: 1\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: eq(T271, f), SCOPE: 36, CLAUSE: 1\n\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
247 [label="247: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
248 [label="248: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
249 [label="249: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: ','(neq(T279, f), del(T279, .(t, T280), X480))\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 2 | ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 3\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="252: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 2\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
253 [label="253: ','(neq(T279, f), del(T279, .(t, T280), X480)), SCOPE: 37, CLAUSE: 3\n\nT279 is ground\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
254 [label="254: del(t, .(t, T280), X480)\n\nT280 is ground\nX480 is free", fontsize=16, style=filled, fillcolor=lightyellow];
255 [label="255: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
256 [label="256: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
257 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 257 [arrowhead = none ];
257 -> 2;

258 [label="EVAL with clause\nmaxsort([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 258 [arrowhead = none ];
258 -> 3;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 259 [arrowhead = none ];
259 -> 4;

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 260 [arrowhead = none ];
260 -> 5;

261 [label="EVAL with clause\nmaxsort(.(X10, X11), .(X12, X13)) :- ','(max(.(X10, X11), X12), ','(del(X12, .(X10, X11), X14), maxsort(X14, X13))).\nand substitutionX10 -> T7,\nX11 -> T8,\nT1 -> .(T7, T8),\nX12 -> T11,\nX13 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 261 [arrowhead = none ];
261 -> 7;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 262 [arrowhead = none ];
262 -> 8;

263 [label="BACKTRACK\nfor clause: maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 263 [arrowhead = none ];
263 -> 6;

264 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 264 [arrowhead = none ];
264 -> 9;

265 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 265 [arrowhead = none ];
265 -> 10;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 266 [arrowhead = none ];
266 -> 11;

267 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 267 [arrowhead = none ];
267 -> 12;

268 [label="EVAL with clause\nmax(.(X19, []), X19).\nand substitutionT7 -> T17,\nX19 -> T17,\nT8 -> [],\nT11 -> T17,\nT12 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 268 [arrowhead = none ];
268 -> 13;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 269 [arrowhead = none ];
269 -> 14;

270 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 270 [arrowhead = none ];
270 -> 35;

271 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 271 [arrowhead = none ];
271 -> 36;

272 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 272 [arrowhead = none ];
272 -> 15;

273 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nT21 is ground\nreplacements:X14 -> T21", fontsize=14, style = filled, fillcolor = violet];
13 -> 273 [arrowhead = none ];
273 -> 16;

274 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 274 [arrowhead = none ];
274 -> 17;

275 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 275 [arrowhead = none , style=dashed];
275 -> 1[style=dashed];

276 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 276 [arrowhead = none ];
276 -> 18;

277 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 277 [arrowhead = none ];
277 -> 19;

278 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 278 [arrowhead = none ];
278 -> 20;

279 [label="ONLY EVAL with clause\ndel(X37, .(X38, X39), X39) :- eq(X37, X38).\nand substitutionT17 -> T29,\nX37 -> T29,\nX38 -> T29,\nX39 -> [],\nX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 279 [arrowhead = none ];
279 -> 21;

280 [label="ONLY EVAL with clause\ndel(X51, .(X52, X53), .(X52, X54)) :- ','(neq(X51, X52), del(X51, X53, X54)).\nand substitutionT17 -> T33,\nX51 -> T33,\nX52 -> T33,\nX53 -> [],\nX54 -> X55,\nX14 -> .(T33, X55)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 280 [arrowhead = none ];
280 -> 31;

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 281 [arrowhead = none ];
281 -> 22;

282 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 282 [arrowhead = none ];
282 -> 23;

283 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 283 [arrowhead = none ];
283 -> 24;

284 [label="EVAL with clause\neq(t, t).\nand substitutionT29 -> t", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 284 [arrowhead = none ];
284 -> 25;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 285 [arrowhead = none ];
285 -> 26;

286 [label="EVAL with clause\neq(f, f).\nand substitutionT29 -> f", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 286 [arrowhead = none ];
286 -> 28;

287 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 287 [arrowhead = none ];
287 -> 29;

288 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 288 [arrowhead = none ];
288 -> 27;

289 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 289 [arrowhead = none ];
289 -> 30;

290 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 290 [arrowhead = none ];
290 -> 32;

291 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 291 [arrowhead = none ];
291 -> 33;

292 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 292 [arrowhead = none ];
292 -> 34;

293 [label="EVAL with clause\nmax(.(X73, .(X74, X75)), X76) :- ','(ge(X73, X74), max(.(X73, X75), X76)).\nand substitutionT7 -> T51,\nX73 -> T51,\nX74 -> T52,\nX75 -> T53,\nT8 -> .(T52, T53),\nT11 -> T55,\nX76 -> T55,\nT54 -> T55,\nT12 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 293 [arrowhead = none ];
293 -> 37;

294 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 294 [arrowhead = none ];
294 -> 38;

295 [label="EVAL with clause\nmax(.(X440, .(X441, X442)), X443) :- ','(gt(X441, X440), max(.(X441, X442), X443)).\nand substitutionT7 -> T249,\nX440 -> T249,\nX441 -> T250,\nX442 -> T251,\nT8 -> .(T250, T251),\nT11 -> T253,\nX443 -> T253,\nT252 -> T253,\nT12 -> T254", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 295 [arrowhead = none ];
295 -> 231;

296 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 296 [arrowhead = none ];
296 -> 232;

297 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 297 [arrowhead = none ];
297 -> 39;

298 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 298 [arrowhead = none ];
298 -> 40;

299 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 299 [arrowhead = none ];
299 -> 41;

300 [label="EVAL with clause\nge(t, t).\nand substitutionT51 -> t,\nT52 -> t", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 300 [arrowhead = none ];
300 -> 42;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 301 [arrowhead = none ];
301 -> 43;

302 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 302 [arrowhead = none ];
302 -> 123;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 303 [arrowhead = none ];
303 -> 124;

304 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
42 -> 304 [arrowhead = none ];
304 -> 44;

305 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT57 is ground\nreplacements:T55 -> T57,\nT56 -> T58", fontsize=14, style = filled, fillcolor = violet];
42 -> 305 [arrowhead = none ];
305 -> 45;

306 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 306 [arrowhead = none ];
306 -> 46;

307 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 307 [arrowhead = none ];
307 -> 71;

308 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nT53 is ground\nT87 is ground\nreplacements:X14 -> T87", fontsize=14, style = filled, fillcolor = violet];
45 -> 308 [arrowhead = none ];
308 -> 72;

309 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 309 [arrowhead = none ];
309 -> 47;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 310 [arrowhead = none ];
310 -> 48;

311 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 311 [arrowhead = none ];
311 -> 49;

312 [label="EVAL with clause\nmax(.(X81, []), X81).\nand substitutionX81 -> t,\nT53 -> [],\nT55 -> t", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 312 [arrowhead = none ];
312 -> 50;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 313 [arrowhead = none ];
313 -> 51;

314 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 314 [arrowhead = none ];
314 -> 53;

315 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 315 [arrowhead = none ];
315 -> 54;

316 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 316 [arrowhead = none ];
316 -> 52;

317 [label="EVAL with clause\nmax(.(X98, .(X99, X100)), X101) :- ','(ge(X98, X99), max(.(X98, X100), X101)).\nand substitutionX98 -> t,\nX99 -> T71,\nX100 -> T72,\nT53 -> .(T71, T72),\nT55 -> T74,\nX101 -> T74,\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 317 [arrowhead = none ];
317 -> 55;

318 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 318 [arrowhead = none ];
318 -> 56;

319 [label="EVAL with clause\nmax(.(X110, .(X111, X112)), X113) :- ','(gt(X111, X110), max(.(X111, X112), X113)).\nand substitutionX110 -> t,\nX111 -> T81,\nX112 -> T82,\nT53 -> .(T81, T82),\nT55 -> T84,\nX113 -> T84,\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 319 [arrowhead = none ];
319 -> 67;

320 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 320 [arrowhead = none ];
320 -> 68;

321 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 321 [arrowhead = none ];
321 -> 57;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 322 [arrowhead = none ];
322 -> 58;

323 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 323 [arrowhead = none ];
323 -> 59;

324 [label="EVAL with clause\nge(t, t).\nand substitutionT71 -> t", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 324 [arrowhead = none ];
324 -> 60;

325 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 325 [arrowhead = none ];
325 -> 61;

326 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 326 [arrowhead = none ];
326 -> 62;

327 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 327 [arrowhead = none ];
327 -> 63;

328 [label="INSTANCE with matching:\nT53 -> T72\nT55 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 328 [arrowhead = none , style=dashed];
328 -> 44[style=dashed];

329 [label="EVAL with clause\nge(t, f).\nand substitutionT71 -> f", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 329 [arrowhead = none ];
329 -> 64;

330 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 330 [arrowhead = none ];
330 -> 65;

331 [label="BACKTRACK\nfor clause: ge(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 331 [arrowhead = none ];
331 -> 66;

332 [label="INSTANCE with matching:\nT53 -> T72\nT55 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 332 [arrowhead = none , style=dashed];
332 -> 44[style=dashed];

333 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 333 [arrowhead = none ];
333 -> 69;

334 [label="BACKTRACK\nfor clause: gt(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 334 [arrowhead = none ];
334 -> 70;

335 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 335 [arrowhead = none ];
335 -> 73;

336 [label="INSTANCE with matching:\nT1 -> T87\nT2 -> T58", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 336 [arrowhead = none , style=dashed];
336 -> 1[style=dashed];

337 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 337 [arrowhead = none ];
337 -> 74;

338 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 338 [arrowhead = none ];
338 -> 75;

339 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 339 [arrowhead = none ];
339 -> 76;

340 [label="ONLY EVAL with clause\ndel(X131, .(X132, X133), X133) :- eq(X131, X132).\nand substitutionT57 -> T99,\nX131 -> T99,\nX132 -> t,\nT53 -> T100,\nX133 -> .(t, T100),\nX14 -> .(t, T100)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 340 [arrowhead = none ];
340 -> 77;

341 [label="ONLY EVAL with clause\ndel(X146, .(X147, X148), .(X147, X149)) :- ','(neq(X146, X147), del(X146, X148, X149)).\nand substitutionT57 -> T107,\nX146 -> T107,\nX147 -> t,\nT53 -> T108,\nX148 -> .(t, T108),\nX149 -> X150,\nX14 -> .(t, X150)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 341 [arrowhead = none ];
341 -> 85;

342 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 342 [arrowhead = none ];
342 -> 78;

343 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 343 [arrowhead = none ];
343 -> 79;

344 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 344 [arrowhead = none ];
344 -> 80;

345 [label="EVAL with clause\neq(t, t).\nand substitutionT99 -> t", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 345 [arrowhead = none ];
345 -> 81;

346 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 346 [arrowhead = none ];
346 -> 82;

347 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 347 [arrowhead = none ];
347 -> 84;

348 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 348 [arrowhead = none ];
348 -> 83;

349 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 349 [arrowhead = none ];
349 -> 86;

350 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 350 [arrowhead = none ];
350 -> 87;

351 [label="EVAL with clause\nneq(f, t).\nand substitutionT107 -> f", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 351 [arrowhead = none ];
351 -> 88;

352 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 352 [arrowhead = none ];
352 -> 89;

353 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 353 [arrowhead = none ];
353 -> 90;

354 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 354 [arrowhead = none ];
354 -> 91;

355 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 355 [arrowhead = none ];
355 -> 92;

356 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 356 [arrowhead = none ];
356 -> 93;

357 [label="ONLY EVAL with clause\ndel(X168, .(X169, X170), X170) :- eq(X168, X169).\nand substitutionX168 -> f,\nX169 -> t,\nT108 -> T115,\nX170 -> T115,\nX150 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 357 [arrowhead = none ];
357 -> 94;

358 [label="ONLY EVAL with clause\ndel(X183, .(X184, X185), .(X184, X186)) :- ','(neq(X183, X184), del(X183, X185, X186)).\nand substitutionX183 -> f,\nX184 -> t,\nT108 -> T118,\nX185 -> T118,\nX186 -> X187,\nX150 -> .(t, X187)", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 358 [arrowhead = none ];
358 -> 98;

359 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 359 [arrowhead = none ];
359 -> 95;

360 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 360 [arrowhead = none ];
360 -> 96;

361 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 361 [arrowhead = none ];
361 -> 97;

362 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 362 [arrowhead = none ];
362 -> 99;

363 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 363 [arrowhead = none ];
363 -> 100;

364 [label="ONLY EVAL with clause\nneq(f, t).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 364 [arrowhead = none ];
364 -> 101;

365 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
101 -> 365 [arrowhead = none ];
365 -> 102;

366 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 366 [arrowhead = none ];
366 -> 103;

367 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
102 -> 367 [arrowhead = none ];
367 -> 104;

368 [label="EVAL with clause\ndel(X198, [], []).\nand substitutionX198 -> f,\nT118 -> [],\nX187 -> []", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 368 [arrowhead = none ];
368 -> 105;

369 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 369 [arrowhead = none ];
369 -> 106;

370 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 370 [arrowhead = none ];
370 -> 108;

371 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 371 [arrowhead = none ];
371 -> 109;

372 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 372 [arrowhead = none ];
372 -> 107;

373 [label="EVAL with clause\ndel(X211, .(X212, X213), X213) :- eq(X211, X212).\nand substitutionX211 -> f,\nX212 -> T127,\nX213 -> T128,\nT118 -> .(T127, T128),\nX187 -> T128", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 373 [arrowhead = none ];
373 -> 110;

374 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 374 [arrowhead = none ];
374 -> 111;

375 [label="EVAL with clause\ndel(X224, .(X225, X226), .(X225, X227)) :- ','(neq(X224, X225), del(X224, X226, X227)).\nand substitutionX224 -> f,\nX225 -> T133,\nX226 -> T134,\nT118 -> .(T133, T134),\nX227 -> X228,\nX187 -> .(T133, X228)", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 375 [arrowhead = none ];
375 -> 117;

376 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 376 [arrowhead = none ];
376 -> 118;

377 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
110 -> 377 [arrowhead = none ];
377 -> 112;

378 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 378 [arrowhead = none ];
378 -> 113;

379 [label="EVAL with clause\neq(f, f).\nand substitutionT127 -> f", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 379 [arrowhead = none ];
379 -> 114;

380 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 380 [arrowhead = none ];
380 -> 115;

381 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 381 [arrowhead = none ];
381 -> 116;

382 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
117 -> 382 [arrowhead = none ];
382 -> 119;

383 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 383 [arrowhead = none ];
383 -> 120;

384 [label="EVAL with clause\nneq(f, t).\nand substitutionT133 -> t", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 384 [arrowhead = none ];
384 -> 121;

385 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 385 [arrowhead = none ];
385 -> 122;

386 [label="INSTANCE with matching:\nT118 -> T134\nX187 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 386 [arrowhead = none , style=dashed];
386 -> 101[style=dashed];

387 [label="EVAL with clause\nge(t, f).\nand substitutionT51 -> t,\nT52 -> f", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 387 [arrowhead = none ];
387 -> 125;

388 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 388 [arrowhead = none ];
388 -> 126;

389 [label="EVAL with clause\nge(f, f).\nand substitutionT51 -> f,\nT52 -> f", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 389 [arrowhead = none ];
389 -> 148;

390 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 390 [arrowhead = none ];
390 -> 149;

391 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
125 -> 391 [arrowhead = none ];
391 -> 127;

392 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT135 is ground\nreplacements:T55 -> T135,\nT56 -> T136", fontsize=14, style = filled, fillcolor = violet];
125 -> 392 [arrowhead = none ];
392 -> 128;

393 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
127 -> 393 [arrowhead = none , style=dashed];
393 -> 44[style=dashed];

394 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
128 -> 394 [arrowhead = none ];
394 -> 129;

395 [label="SPLIT 2\nnew knowledge:\nT135 is ground\nT53 is ground\nT139 is ground\nreplacements:X14 -> T139", fontsize=14, style = filled, fillcolor = violet];
128 -> 395 [arrowhead = none ];
395 -> 130;

396 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
129 -> 396 [arrowhead = none ];
396 -> 131;

397 [label="INSTANCE with matching:\nT1 -> T139\nT2 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
130 -> 397 [arrowhead = none , style=dashed];
397 -> 1[style=dashed];

398 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
131 -> 398 [arrowhead = none ];
398 -> 132;

399 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
132 -> 399 [arrowhead = none ];
399 -> 133;

400 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
132 -> 400 [arrowhead = none ];
400 -> 134;

401 [label="ONLY EVAL with clause\ndel(X250, .(X251, X252), X252) :- eq(X250, X251).\nand substitutionT135 -> T151,\nX250 -> T151,\nX251 -> t,\nT53 -> T152,\nX252 -> .(f, T152),\nX14 -> .(f, T152)", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 401 [arrowhead = none ];
401 -> 135;

402 [label="ONLY EVAL with clause\ndel(X265, .(X266, X267), .(X266, X268)) :- ','(neq(X265, X266), del(X265, X267, X268)).\nand substitutionT135 -> T159,\nX265 -> T159,\nX266 -> t,\nT53 -> T160,\nX267 -> .(f, T160),\nX268 -> X269,\nX14 -> .(t, X269)", fontsize=14, style = filled, fillcolor = lightblue];
134 -> 402 [arrowhead = none ];
402 -> 143;

403 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
135 -> 403 [arrowhead = none ];
403 -> 136;

404 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 404 [arrowhead = none ];
404 -> 137;

405 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 405 [arrowhead = none ];
405 -> 138;

406 [label="EVAL with clause\neq(t, t).\nand substitutionT151 -> t", fontsize=14, style = filled, fillcolor = lightblue];
137 -> 406 [arrowhead = none ];
406 -> 139;

407 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 407 [arrowhead = none ];
407 -> 140;

408 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 408 [arrowhead = none ];
408 -> 142;

409 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
139 -> 409 [arrowhead = none ];
409 -> 141;

410 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
143 -> 410 [arrowhead = none ];
410 -> 144;

411 [label="BACKTRACK\nfor clause: neq(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
144 -> 411 [arrowhead = none ];
411 -> 145;

412 [label="EVAL with clause\nneq(f, t).\nand substitutionT159 -> f", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 412 [arrowhead = none ];
412 -> 146;

413 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
145 -> 413 [arrowhead = none ];
413 -> 147;

414 [label="INSTANCE with matching:\nT118 -> .(f, T160)\nX187 -> X269", fontsize=14, style = filled, fillcolor = lightgrey];
146 -> 414 [arrowhead = none , style=dashed];
414 -> 101[style=dashed];

415 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
148 -> 415 [arrowhead = none ];
415 -> 150;

416 [label="SPLIT 2\nnew knowledge:\nT53 is ground\nT163 is ground\nreplacements:T55 -> T163,\nT56 -> T164", fontsize=14, style = filled, fillcolor = violet];
148 -> 416 [arrowhead = none ];
416 -> 151;

417 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
150 -> 417 [arrowhead = none ];
417 -> 152;

418 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
151 -> 418 [arrowhead = none ];
418 -> 173;

419 [label="SPLIT 2\nnew knowledge:\nT163 is ground\nT53 is ground\nT193 is ground\nreplacements:X14 -> T193", fontsize=14, style = filled, fillcolor = violet];
151 -> 419 [arrowhead = none ];
419 -> 174;

420 [label="BACKTRACK\nfor clause: max([], f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 420 [arrowhead = none ];
420 -> 153;

421 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 421 [arrowhead = none ];
421 -> 154;

422 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 422 [arrowhead = none ];
422 -> 155;

423 [label="EVAL with clause\nmax(.(X278, []), X278).\nand substitutionX278 -> f,\nT53 -> [],\nT55 -> f", fontsize=14, style = filled, fillcolor = lightblue];
154 -> 423 [arrowhead = none ];
423 -> 156;

424 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
154 -> 424 [arrowhead = none ];
424 -> 157;

425 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
155 -> 425 [arrowhead = none ];
425 -> 159;

426 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
155 -> 426 [arrowhead = none ];
426 -> 160;

427 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
156 -> 427 [arrowhead = none ];
427 -> 158;

428 [label="EVAL with clause\nmax(.(X295, .(X296, X297)), X298) :- ','(ge(X295, X296), max(.(X295, X297), X298)).\nand substitutionX295 -> f,\nX296 -> T177,\nX297 -> T178,\nT53 -> .(T177, T178),\nT55 -> T180,\nX298 -> T180,\nT179 -> T180", fontsize=14, style = filled, fillcolor = lightblue];
159 -> 428 [arrowhead = none ];
428 -> 161;

429 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
159 -> 429 [arrowhead = none ];
429 -> 162;

430 [label="EVAL with clause\nmax(.(X307, .(X308, X309)), X310) :- ','(gt(X308, X307), max(.(X308, X309), X310)).\nand substitutionX307 -> f,\nX308 -> T187,\nX309 -> T188,\nT53 -> .(T187, T188),\nT55 -> T190,\nX310 -> T190,\nT189 -> T190", fontsize=14, style = filled, fillcolor = lightblue];
160 -> 430 [arrowhead = none ];
430 -> 168;

431 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
160 -> 431 [arrowhead = none ];
431 -> 169;

432 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
161 -> 432 [arrowhead = none ];
432 -> 163;

433 [label="BACKTRACK\nfor clause: ge(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
163 -> 433 [arrowhead = none ];
433 -> 164;

434 [label="BACKTRACK\nfor clause: ge(t, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
164 -> 434 [arrowhead = none ];
434 -> 165;

435 [label="EVAL with clause\nge(f, f).\nand substitutionT177 -> f", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 435 [arrowhead = none ];
435 -> 166;

436 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
165 -> 436 [arrowhead = none ];
436 -> 167;

437 [label="INSTANCE with matching:\nT53 -> T178\nT55 -> T180", fontsize=14, style = filled, fillcolor = lightgrey];
166 -> 437 [arrowhead = none , style=dashed];
437 -> 150[style=dashed];

438 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
168 -> 438 [arrowhead = none ];
438 -> 170;

439 [label="EVAL with clause\ngt(t, f).\nand substitutionT187 -> t", fontsize=14, style = filled, fillcolor = lightblue];
170 -> 439 [arrowhead = none ];
439 -> 171;

440 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
170 -> 440 [arrowhead = none ];
440 -> 172;

441 [label="INSTANCE with matching:\nT53 -> T188\nT55 -> T190", fontsize=14, style = filled, fillcolor = lightgrey];
171 -> 441 [arrowhead = none , style=dashed];
441 -> 44[style=dashed];

442 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
173 -> 442 [arrowhead = none ];
442 -> 175;

443 [label="INSTANCE with matching:\nT1 -> T193\nT2 -> T164", fontsize=14, style = filled, fillcolor = lightgrey];
174 -> 443 [arrowhead = none , style=dashed];
443 -> 1[style=dashed];

444 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
175 -> 444 [arrowhead = none ];
444 -> 176;

445 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
176 -> 445 [arrowhead = none ];
445 -> 177;

446 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
176 -> 446 [arrowhead = none ];
446 -> 178;

447 [label="ONLY EVAL with clause\ndel(X328, .(X329, X330), X330) :- eq(X328, X329).\nand substitutionT163 -> T205,\nX328 -> T205,\nX329 -> f,\nT53 -> T206,\nX330 -> .(f, T206),\nX14 -> .(f, T206)", fontsize=14, style = filled, fillcolor = lightblue];
177 -> 447 [arrowhead = none ];
447 -> 179;

448 [label="ONLY EVAL with clause\ndel(X343, .(X344, X345), .(X344, X346)) :- ','(neq(X343, X344), del(X343, X345, X346)).\nand substitutionT163 -> T213,\nX343 -> T213,\nX344 -> f,\nT53 -> T214,\nX345 -> .(f, T214),\nX346 -> X347,\nX14 -> .(f, X347)", fontsize=14, style = filled, fillcolor = lightblue];
178 -> 448 [arrowhead = none ];
448 -> 185;

449 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
179 -> 449 [arrowhead = none ];
449 -> 180;

450 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
180 -> 450 [arrowhead = none ];
450 -> 181;

451 [label="EVAL with clause\neq(f, f).\nand substitutionT205 -> f", fontsize=14, style = filled, fillcolor = lightblue];
181 -> 451 [arrowhead = none ];
451 -> 182;

452 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
181 -> 452 [arrowhead = none ];
452 -> 183;

453 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
182 -> 453 [arrowhead = none ];
453 -> 184;

454 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
185 -> 454 [arrowhead = none ];
454 -> 186;

455 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
186 -> 455 [arrowhead = none ];
455 -> 187;

456 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
186 -> 456 [arrowhead = none ];
456 -> 188;

457 [label="EVAL with clause\nneq(t, f).\nand substitutionT213 -> t", fontsize=14, style = filled, fillcolor = lightblue];
187 -> 457 [arrowhead = none ];
457 -> 189;

458 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
187 -> 458 [arrowhead = none ];
458 -> 190;

459 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
188 -> 459 [arrowhead = none ];
459 -> 230;

460 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
189 -> 460 [arrowhead = none ];
460 -> 191;

461 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
191 -> 461 [arrowhead = none ];
461 -> 192;

462 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
192 -> 462 [arrowhead = none ];
462 -> 193;

463 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
192 -> 463 [arrowhead = none ];
463 -> 194;

464 [label="ONLY EVAL with clause\ndel(X365, .(X366, X367), X367) :- eq(X365, X366).\nand substitutionX365 -> t,\nX366 -> f,\nT214 -> T221,\nX367 -> T221,\nX347 -> T221", fontsize=14, style = filled, fillcolor = lightblue];
193 -> 464 [arrowhead = none ];
464 -> 195;

465 [label="ONLY EVAL with clause\ndel(X380, .(X381, X382), .(X381, X383)) :- ','(neq(X380, X381), del(X380, X382, X383)).\nand substitutionX380 -> t,\nX381 -> f,\nT214 -> T224,\nX382 -> T224,\nX383 -> X384,\nX347 -> .(f, X384)", fontsize=14, style = filled, fillcolor = lightblue];
194 -> 465 [arrowhead = none ];
465 -> 199;

466 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
195 -> 466 [arrowhead = none ];
466 -> 196;

467 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
196 -> 467 [arrowhead = none ];
467 -> 197;

468 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
197 -> 468 [arrowhead = none ];
468 -> 198;

469 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
199 -> 469 [arrowhead = none ];
469 -> 200;

470 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
200 -> 470 [arrowhead = none ];
470 -> 201;

471 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
200 -> 471 [arrowhead = none ];
471 -> 202;

472 [label="ONLY EVAL with clause\nneq(t, f).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
201 -> 472 [arrowhead = none ];
472 -> 203;

473 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
202 -> 473 [arrowhead = none ];
473 -> 229;

474 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
203 -> 474 [arrowhead = none ];
474 -> 204;

475 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
204 -> 475 [arrowhead = none ];
475 -> 205;

476 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
204 -> 476 [arrowhead = none ];
476 -> 206;

477 [label="EVAL with clause\ndel(X397, [], []).\nand substitutionX397 -> t,\nT224 -> [],\nX384 -> []", fontsize=14, style = filled, fillcolor = lightblue];
205 -> 477 [arrowhead = none ];
477 -> 207;

478 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
205 -> 478 [arrowhead = none ];
478 -> 208;

479 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
206 -> 479 [arrowhead = none ];
479 -> 210;

480 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
206 -> 480 [arrowhead = none ];
480 -> 211;

481 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
207 -> 481 [arrowhead = none ];
481 -> 209;

482 [label="EVAL with clause\ndel(X410, .(X411, X412), X412) :- eq(X410, X411).\nand substitutionX410 -> t,\nX411 -> T233,\nX412 -> T234,\nT224 -> .(T233, T234),\nX384 -> T234", fontsize=14, style = filled, fillcolor = lightblue];
210 -> 482 [arrowhead = none ];
482 -> 212;

483 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
210 -> 483 [arrowhead = none ];
483 -> 213;

484 [label="EVAL with clause\ndel(X423, .(X424, X425), .(X424, X426)) :- ','(neq(X423, X424), del(X423, X425, X426)).\nand substitutionX423 -> t,\nX424 -> T239,\nX425 -> T240,\nT224 -> .(T239, T240),\nX426 -> X427,\nX384 -> .(T239, X427)", fontsize=14, style = filled, fillcolor = lightblue];
211 -> 484 [arrowhead = none ];
484 -> 221;

485 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
211 -> 485 [arrowhead = none ];
485 -> 222;

486 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
212 -> 486 [arrowhead = none ];
486 -> 214;

487 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
214 -> 487 [arrowhead = none ];
487 -> 215;

488 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
214 -> 488 [arrowhead = none ];
488 -> 216;

489 [label="EVAL with clause\neq(t, t).\nand substitutionT233 -> t", fontsize=14, style = filled, fillcolor = lightblue];
215 -> 489 [arrowhead = none ];
489 -> 217;

490 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
215 -> 490 [arrowhead = none ];
490 -> 218;

491 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
216 -> 491 [arrowhead = none ];
491 -> 220;

492 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
217 -> 492 [arrowhead = none ];
492 -> 219;

493 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
221 -> 493 [arrowhead = none ];
493 -> 223;

494 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
223 -> 494 [arrowhead = none ];
494 -> 224;

495 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
223 -> 495 [arrowhead = none ];
495 -> 225;

496 [label="EVAL with clause\nneq(t, f).\nand substitutionT239 -> f", fontsize=14, style = filled, fillcolor = lightblue];
224 -> 496 [arrowhead = none ];
496 -> 226;

497 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 497 [arrowhead = none ];
497 -> 227;

498 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
225 -> 498 [arrowhead = none ];
498 -> 228;

499 [label="INSTANCE with matching:\nT224 -> T240\nX384 -> X427", fontsize=14, style = filled, fillcolor = lightgrey];
226 -> 499 [arrowhead = none , style=dashed];
499 -> 203[style=dashed];

500 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
231 -> 500 [arrowhead = none ];
500 -> 233;

501 [label="EVAL with clause\ngt(t, f).\nand substitutionT250 -> t,\nT249 -> f", fontsize=14, style = filled, fillcolor = lightblue];
233 -> 501 [arrowhead = none ];
501 -> 234;

502 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
233 -> 502 [arrowhead = none ];
502 -> 235;

503 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
234 -> 503 [arrowhead = none ];
503 -> 236;

504 [label="SPLIT 2\nnew knowledge:\nT251 is ground\nT255 is ground\nreplacements:T253 -> T255,\nT254 -> T256", fontsize=14, style = filled, fillcolor = violet];
234 -> 504 [arrowhead = none ];
504 -> 237;

505 [label="INSTANCE with matching:\nT53 -> T251\nT55 -> T253", fontsize=14, style = filled, fillcolor = lightgrey];
236 -> 505 [arrowhead = none , style=dashed];
505 -> 44[style=dashed];

506 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
237 -> 506 [arrowhead = none ];
506 -> 238;

507 [label="SPLIT 2\nnew knowledge:\nT255 is ground\nT251 is ground\nT259 is ground\nreplacements:X14 -> T259", fontsize=14, style = filled, fillcolor = violet];
237 -> 507 [arrowhead = none ];
507 -> 239;

508 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
238 -> 508 [arrowhead = none ];
508 -> 240;

509 [label="INSTANCE with matching:\nT1 -> T259\nT2 -> T256", fontsize=14, style = filled, fillcolor = lightgrey];
239 -> 509 [arrowhead = none , style=dashed];
509 -> 1[style=dashed];

510 [label="BACKTRACK\nfor clause: del(X1, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
240 -> 510 [arrowhead = none ];
510 -> 241;

511 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
241 -> 511 [arrowhead = none ];
511 -> 242;

512 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
241 -> 512 [arrowhead = none ];
512 -> 243;

513 [label="ONLY EVAL with clause\ndel(X461, .(X462, X463), X463) :- eq(X461, X462).\nand substitutionT255 -> T271,\nX461 -> T271,\nX462 -> f,\nT251 -> T272,\nX463 -> .(t, T272),\nX14 -> .(t, T272)", fontsize=14, style = filled, fillcolor = lightblue];
242 -> 513 [arrowhead = none ];
513 -> 244;

514 [label="ONLY EVAL with clause\ndel(X476, .(X477, X478), .(X477, X479)) :- ','(neq(X476, X477), del(X476, X478, X479)).\nand substitutionT255 -> T279,\nX476 -> T279,\nX477 -> f,\nT251 -> T280,\nX478 -> .(t, T280),\nX479 -> X480,\nX14 -> .(f, X480)", fontsize=14, style = filled, fillcolor = lightblue];
243 -> 514 [arrowhead = none ];
514 -> 250;

515 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
244 -> 515 [arrowhead = none ];
515 -> 245;

516 [label="BACKTRACK\nfor clause: eq(t, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
245 -> 516 [arrowhead = none ];
516 -> 246;

517 [label="EVAL with clause\neq(f, f).\nand substitutionT271 -> f", fontsize=14, style = filled, fillcolor = lightblue];
246 -> 517 [arrowhead = none ];
517 -> 247;

518 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
246 -> 518 [arrowhead = none ];
518 -> 248;

519 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
247 -> 519 [arrowhead = none ];
519 -> 249;

520 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
250 -> 520 [arrowhead = none ];
520 -> 251;

521 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
251 -> 521 [arrowhead = none ];
521 -> 252;

522 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
251 -> 522 [arrowhead = none ];
522 -> 253;

523 [label="EVAL with clause\nneq(t, f).\nand substitutionT279 -> t", fontsize=14, style = filled, fillcolor = lightblue];
252 -> 523 [arrowhead = none ];
523 -> 254;

524 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
252 -> 524 [arrowhead = none ];
524 -> 255;

525 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
253 -> 525 [arrowhead = none ];
525 -> 256;

526 [label="INSTANCE with matching:\nT224 -> .(t, T280)\nX384 -> X480", fontsize=14, style = filled, fillcolor = lightgrey];
254 -> 526 [arrowhead = none , style=dashed];
526 -> 203[style=dashed];

}

(2) Obligation:

Triples:

maxC(.(t, X1), X2) :- maxC(X1, X2).
maxC(.(f, X1), X2) :- maxC(X1, X2).
delI(.(t, X1), .(t, X2)) :- delI(X1, X2).
maxF(.(f, X1), X2) :- maxF(X1, X2).
maxF(.(t, X1), X2) :- maxC(X1, X2).
delJ(.(f, X1), .(f, X2)) :- delJ(X1, X2).
maxsortA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortA(X3, X2)).
maxsortA(.(t, .(t, X1)), .(X2, X3)) :- maxC(X1, X2).
maxsortA(.(t, .(t, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(X1, X3)).
maxsortA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortA(X4, X3))).
maxsortA(.(t, .(f, X1)), .(X2, X3)) :- maxC(X1, X2).
maxsortA(.(t, .(f, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(.(f, X1), X3)).
maxsortA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortA(X4, X3))).
maxsortA(.(f, .(f, X1)), .(X2, X3)) :- maxF(X1, X2).
maxsortA(.(f, .(f, X1)), .(t, X2)) :- ','(maxcF(X1, t), delJ(X1, X3)).
maxsortA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortA(X4, X3))).
maxsortA(.(f, .(t, X1)), .(X2, X3)) :- maxC(X1, X2).
maxsortA(.(f, .(t, X1)), .(t, X2)) :- ','(maxcC(X1, t), delJ(.(t, X1), X3)).
maxsortA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortA(X4, X3))).

Clauses:

maxsortcA([], []).
maxsortcA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortcA(X3, X2)).
maxsortcA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortcA(X4, X3))).
maxsortcA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortcA(X4, X3))).
maxsortcA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortcA(X4, X3))).
maxsortcA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortcA(X4, X3))).
maxcC([], t).
maxcC(.(t, X1), X2) :- maxcC(X1, X2).
maxcC(.(f, X1), X2) :- maxcC(X1, X2).
delcI([], []).
delcI(.(f, X1), X1).
delcI(.(t, X1), .(t, X2)) :- delcI(X1, X2).
maxcF([], f).
maxcF(.(f, X1), X2) :- maxcF(X1, X2).
maxcF(.(t, X1), X2) :- maxcC(X1, X2).
delcJ([], []).
delcJ(.(t, X1), X1).
delcJ(.(f, X1), .(f, X2)) :- delcJ(X1, X2).
delcB(t, []).
delcB(f, []).
delcD(t, X1, .(t, X1)).
delcD(f, X1, .(t, .(t, X2))) :- delcI(X1, X2).
delcE(t, X1, .(f, X1)).
delcE(f, X1, .(t, X2)) :- delcI(.(f, X1), X2).
delcG(f, X1, .(f, X1)).
delcG(t, X1, .(f, .(f, X2))) :- delcJ(X1, X2).
delcH(f, X1, .(t, X1)).
delcH(t, X1, .(f, X2)) :- delcJ(.(t, X1), X2).

Afs:

maxsortA(x1, x2) = maxsortA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
maxsortA_in: (b,f)
maxC_in: (b,f)
maxcC_in: (b,b) (b,f)
delI_in: (b,f)
delcD_in: (b,b,f)
delcI_in: (b,f)
delcE_in: (b,b,f)
maxF_in: (b,f)
maxcF_in: (b,b) (b,f)
delJ_in: (b,f)
delcG_in: (b,b,f)
delcJ_in: (b,f)
delcH_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) → U7_GA(X1, X2, delcB_in_ga(X1, X3))
U7_GA(X1, X2, delcB_out_ga(X1, X3)) → U8_GA(X1, X2, maxsortA_in_ga(X3, X2))
U7_GA(X1, X2, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U9_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(t, X1), X2) → U1_GA(X1, X2, maxC_in_ga(X1, X2))
MAXC_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(f, X1), X2) → U2_GA(X1, X2, maxC_in_ga(X1, X2))
MAXC_IN_GA(.(f, X1), X2) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) → U10_GA(X1, X2, maxcC_in_gg(X1, f))
U10_GA(X1, X2, maxcC_out_gg(X1, f)) → U11_GA(X1, X2, delI_in_ga(X1, X3))
U10_GA(X1, X2, maxcC_out_gg(X1, f)) → DELI_IN_GA(X1, X3)
DELI_IN_GA(.(t, X1), .(t, X2)) → U3_GA(X1, X2, delI_in_ga(X1, X2))
DELI_IN_GA(.(t, X1), .(t, X2)) → DELI_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U15_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) → U16_GA(X1, X2, maxcC_in_gg(X1, f))
U16_GA(X1, X2, maxcC_out_gg(X1, f)) → U17_GA(X1, X2, delI_in_ga(.(f, X1), X3))
U16_GA(X1, X2, maxcC_out_gg(X1, f)) → DELI_IN_GA(.(f, X1), X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U21_GA(X1, X2, X3, maxF_in_ga(X1, X2))
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → MAXF_IN_GA(X1, X2)
MAXF_IN_GA(.(f, X1), X2) → U4_GA(X1, X2, maxF_in_ga(X1, X2))
MAXF_IN_GA(.(f, X1), X2) → MAXF_IN_GA(X1, X2)
MAXF_IN_GA(.(t, X1), X2) → U5_GA(X1, X2, maxC_in_ga(X1, X2))
MAXF_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) → U22_GA(X1, X2, maxcF_in_gg(X1, t))
U22_GA(X1, X2, maxcF_out_gg(X1, t)) → U23_GA(X1, X2, delJ_in_ga(X1, X3))
U22_GA(X1, X2, maxcF_out_gg(X1, t)) → DELJ_IN_GA(X1, X3)
DELJ_IN_GA(.(f, X1), .(f, X2)) → U6_GA(X1, X2, delJ_in_ga(X1, X2))
DELJ_IN_GA(.(f, X1), .(f, X2)) → DELJ_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2))
U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) → U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U27_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) → U28_GA(X1, X2, maxcC_in_gg(X1, t))
U28_GA(X1, X2, maxcC_out_gg(X1, t)) → U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3))
U28_GA(X1, X2, maxcC_out_gg(X1, t)) → DELJ_IN_GA(.(t, X1), X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
maxsortA_in_ga(x1, x2) = maxsortA_in_ga(x1)
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxC_in_ga(x1, x2) = maxC_in_ga(x1)
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
delI_in_ga(x1, x2) = delI_in_ga(x1)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxF_in_ga(x1, x2) = maxF_in_ga(x1)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
delJ_in_ga(x1, x2) = delJ_in_ga(x1)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1)
U7_GA(x1, x2, x3) = U7_GA(x1, x3)
U8_GA(x1, x2, x3) = U8_GA(x1, x3)
U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4)
MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U10_GA(x1, x2, x3) = U10_GA(x1, x3)
U11_GA(x1, x2, x3) = U11_GA(x1, x3)
DELI_IN_GA(x1, x2) = DELI_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4)
U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4) = U14_GA(x1, x4)
U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4)
U16_GA(x1, x2, x3) = U16_GA(x1, x3)
U17_GA(x1, x2, x3) = U17_GA(x1, x3)
U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4)
U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4)
U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4)
U21_GA(x1, x2, x3, x4) = U21_GA(x1, x4)
MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
U5_GA(x1, x2, x3) = U5_GA(x1, x3)
U22_GA(x1, x2, x3) = U22_GA(x1, x3)
U23_GA(x1, x2, x3) = U23_GA(x1, x3)
DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1)
U6_GA(x1, x2, x3) = U6_GA(x1, x3)
U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4)
U26_GA(x1, x2, x3, x4) = U26_GA(x1, x4)
U27_GA(x1, x2, x3, x4) = U27_GA(x1, x4)
U28_GA(x1, x2, x3) = U28_GA(x1, x3)
U29_GA(x1, x2, x3) = U29_GA(x1, x3)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4)
U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4)
U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) → U7_GA(X1, X2, delcB_in_ga(X1, X3))
U7_GA(X1, X2, delcB_out_ga(X1, X3)) → U8_GA(X1, X2, maxsortA_in_ga(X3, X2))
U7_GA(X1, X2, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U9_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(t, X1), X2) → U1_GA(X1, X2, maxC_in_ga(X1, X2))
MAXC_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(f, X1), X2) → U2_GA(X1, X2, maxC_in_ga(X1, X2))
MAXC_IN_GA(.(f, X1), X2) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) → U10_GA(X1, X2, maxcC_in_gg(X1, f))
U10_GA(X1, X2, maxcC_out_gg(X1, f)) → U11_GA(X1, X2, delI_in_ga(X1, X3))
U10_GA(X1, X2, maxcC_out_gg(X1, f)) → DELI_IN_GA(X1, X3)
DELI_IN_GA(.(t, X1), .(t, X2)) → U3_GA(X1, X2, delI_in_ga(X1, X2))
DELI_IN_GA(.(t, X1), .(t, X2)) → DELI_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U15_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) → U16_GA(X1, X2, maxcC_in_gg(X1, f))
U16_GA(X1, X2, maxcC_out_gg(X1, f)) → U17_GA(X1, X2, delI_in_ga(.(f, X1), X3))
U16_GA(X1, X2, maxcC_out_gg(X1, f)) → DELI_IN_GA(.(f, X1), X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U21_GA(X1, X2, X3, maxF_in_ga(X1, X2))
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → MAXF_IN_GA(X1, X2)
MAXF_IN_GA(.(f, X1), X2) → U4_GA(X1, X2, maxF_in_ga(X1, X2))
MAXF_IN_GA(.(f, X1), X2) → MAXF_IN_GA(X1, X2)
MAXF_IN_GA(.(t, X1), X2) → U5_GA(X1, X2, maxC_in_ga(X1, X2))
MAXF_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) → U22_GA(X1, X2, maxcF_in_gg(X1, t))
U22_GA(X1, X2, maxcF_out_gg(X1, t)) → U23_GA(X1, X2, delJ_in_ga(X1, X3))
U22_GA(X1, X2, maxcF_out_gg(X1, t)) → DELJ_IN_GA(X1, X3)
DELJ_IN_GA(.(f, X1), .(f, X2)) → U6_GA(X1, X2, delJ_in_ga(X1, X2))
DELJ_IN_GA(.(f, X1), .(f, X2)) → DELJ_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2))
U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) → U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U27_GA(X1, X2, X3, maxC_in_ga(X1, X2))
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → MAXC_IN_GA(X1, X2)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) → U28_GA(X1, X2, maxcC_in_gg(X1, t))
U28_GA(X1, X2, maxcC_out_gg(X1, t)) → U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3))
U28_GA(X1, X2, maxcC_out_gg(X1, t)) → DELJ_IN_GA(.(t, X1), X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 32 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELJ_IN_GA(.(f, X1), .(f, X2)) → DELJ_IN_GA(X1, X2)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

DELJ_IN_GA(.(f, X1), .(f, X2)) → DELJ_IN_GA(X1, X2)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

DELJ_IN_GA(.(f, X1)) → DELJ_IN_GA(X1)

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

(12) 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:

DELJ_IN_GA(.(f, X1)) → DELJ_IN_GA(X1)
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

DELI_IN_GA(.(t, X1), .(t, X2)) → DELI_IN_GA(X1, X2)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
DELI_IN_GA(x1, x2) = DELI_IN_GA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

DELI_IN_GA(.(t, X1), .(t, X2)) → DELI_IN_GA(X1, X2)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

DELI_IN_GA(.(t, X1)) → DELI_IN_GA(X1)

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

(19) 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:

DELI_IN_GA(.(t, X1)) → DELI_IN_GA(X1)
The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

MAXC_IN_GA(.(f, X1), X2) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MAXC_IN_GA(.(f, X1), X2) → MAXC_IN_GA(X1, X2)
MAXC_IN_GA(.(t, X1), X2) → MAXC_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
t = t
f = f
MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MAXC_IN_GA(.(f, X1)) → MAXC_IN_GA(X1)
MAXC_IN_GA(.(t, X1)) → MAXC_IN_GA(X1)

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

(26) 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:

MAXC_IN_GA(.(f, X1)) → MAXC_IN_GA(X1)
The graph contains the following edges 1 > 1
MAXC_IN_GA(.(t, X1)) → MAXC_IN_GA(X1)
The graph contains the following edges 1 > 1

(27) YES

(28) Obligation:

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

MAXF_IN_GA(.(f, X1), X2) → MAXF_IN_GA(X1, X2)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

MAXF_IN_GA(.(f, X1), X2) → MAXF_IN_GA(X1, X2)

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

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

MAXF_IN_GA(.(f, X1)) → MAXF_IN_GA(X1)

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

(33) 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:

MAXF_IN_GA(.(f, X1)) → MAXF_IN_GA(X1)
The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

U7_GA(X1, X2, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3, X2)
MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) → U7_GA(X1, X2, delcB_in_ga(X1, X3))
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2))
U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) → U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_gg([], t) → maxcC_out_gg([], t)
maxcC_in_gg(.(t, X1), X2) → U48_gg(X1, X2, maxcC_in_gg(X1, X2))
maxcC_in_gg(.(f, X1), X2) → U49_gg(X1, X2, maxcC_in_gg(X1, X2))
U49_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(f, X1), X2)
U48_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcC_out_gg(.(t, X1), X2)
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
maxcF_in_gg([], f) → maxcF_out_gg([], f)
maxcF_in_gg(.(f, X1), X2) → U51_gg(X1, X2, maxcF_in_gg(X1, X2))
maxcF_in_gg(.(t, X1), X2) → U52_gg(X1, X2, maxcC_in_gg(X1, X2))
U52_gg(X1, X2, maxcC_out_gg(X1, X2)) → maxcF_out_gg(.(t, X1), X2)
U51_gg(X1, X2, maxcF_out_gg(X1, X2)) → maxcF_out_gg(.(f, X1), X2)
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2)
maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2)
U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3)
U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3)
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2)
maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2)
U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3)
U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1)
U7_GA(x1, x2, x3) = U7_GA(x1, x3)
U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4)
U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4)
U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4)
U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4)
U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4)
U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

U7_GA(X1, X2, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3, X2)
MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) → U7_GA(X1, X2, delcB_in_ga(X1, X3))
MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) → U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4))
U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) → U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4))
U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) → U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2))
U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) → U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4))
U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)
MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) → U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2))
U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) → U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4))
U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4, X3)

The TRS R consists of the following rules:

delcB_in_ga(t, []) → delcB_out_ga(t, [])
delcB_in_ga(f, []) → delcB_out_ga(f, [])
maxcC_in_ga([], t) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1), X2) → U48_ga(X1, X2, maxcC_in_ga(X1, X2))
maxcC_in_ga(.(f, X1), X2) → U49_ga(X1, X2, maxcC_in_ga(X1, X2))
delcD_in_gga(t, X1, .(t, X1)) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1, .(t, .(t, X2))) → U54_gga(X1, X2, delcI_in_ga(X1, X2))
delcE_in_gga(t, X1, .(f, X1)) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1, .(t, X2)) → U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2))
maxcF_in_ga([], f) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1), X2) → U51_ga(X1, X2, maxcF_in_ga(X1, X2))
maxcF_in_ga(.(t, X1), X2) → U52_ga(X1, X2, maxcC_in_ga(X1, X2))
delcG_in_gga(f, X1, .(f, X1)) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1, .(f, .(f, X2))) → U56_gga(X1, X2, delcJ_in_ga(X1, X2))
delcH_in_gga(f, X1, .(t, X1)) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1, .(f, X2)) → U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2))
U48_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U49_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U54_gga(X1, X2, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
U51_ga(X1, X2, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
U52_ga(X1, X2, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U56_gga(X1, X2, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
delcI_in_ga([], []) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1), X1) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1), .(t, X2)) → U50_ga(X1, X2, delcI_in_ga(X1, X2))
delcJ_in_ga([], []) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1), X1) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1), .(f, X2)) → U53_ga(X1, X2, delcJ_in_ga(X1, X2))
U50_ga(X1, X2, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U53_ga(X1, X2, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
delcB_in_ga(x1, x2) = delcB_in_ga(x1)
t = t
delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2)
f = f
maxcC_in_ga(x1, x2) = maxcC_in_ga(x1)
maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2)
U48_ga(x1, x2, x3) = U48_ga(x1, x3)
U49_ga(x1, x2, x3) = U49_ga(x1, x3)
delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2)
delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x3)
delcI_in_ga(x1, x2) = delcI_in_ga(x1)
delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2)
U50_ga(x1, x2, x3) = U50_ga(x1, x3)
delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2)
delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x3)
maxcF_in_ga(x1, x2) = maxcF_in_ga(x1)
maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2)
U51_ga(x1, x2, x3) = U51_ga(x1, x3)
U52_ga(x1, x2, x3) = U52_ga(x1, x3)
delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2)
delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x3)
delcJ_in_ga(x1, x2) = delcJ_in_ga(x1)
delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2)
U53_ga(x1, x2, x3) = U53_ga(x1, x3)
delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2)
delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x3)
MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1)
U7_GA(x1, x2, x3) = U7_GA(x1, x3)
U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4)
U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4)
U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4)
U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4)
U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4)
U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

U7_GA(X1, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3)
MAXSORTA_IN_GA(.(X1, [])) → U7_GA(X1, delcB_in_ga(X1))
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcB_in_ga(t) → delcB_out_ga(t, [])
delcB_in_ga(f) → delcB_out_ga(f, [])
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

delcB_in_ga(x0)
maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(40) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MAXSORTA_IN_GA(.(X1, [])) → U7_GA(X1, delcB_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]:

MAXSORTA_IN_GA(.(t, [])) → U7_GA(t, delcB_out_ga(t, []))
MAXSORTA_IN_GA(.(f, [])) → U7_GA(f, delcB_out_ga(f, []))

(41) Obligation:

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

U7_GA(X1, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, [])) → U7_GA(t, delcB_out_ga(t, []))
MAXSORTA_IN_GA(.(f, [])) → U7_GA(f, delcB_out_ga(f, []))

The TRS R consists of the following rules:

delcB_in_ga(t) → delcB_out_ga(t, [])
delcB_in_ga(f) → delcB_out_ga(f, [])
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

delcB_in_ga(x0)
maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) Obligation:

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

U7_GA(X1, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, [])) → U7_GA(t, delcB_out_ga(t, []))
MAXSORTA_IN_GA(.(f, [])) → U7_GA(f, delcB_out_ga(f, []))

The TRS R consists of the following rules:

delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

delcB_in_ga(x0)
maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(44) QReductionProof (EQUIVALENT transformation)

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

delcB_in_ga(x0)

(45) Obligation:

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

U7_GA(X1, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, [])) → U7_GA(t, delcB_out_ga(t, []))
MAXSORTA_IN_GA(.(f, [])) → U7_GA(f, delcB_out_ga(f, []))

The TRS R consists of the following rules:

delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GA(X1, delcB_out_ga(X1, X3)) → MAXSORTA_IN_GA(X3) we obtained the following new rules [LPAR04]:

U7_GA(t, delcB_out_ga(t, [])) → MAXSORTA_IN_GA([])
U7_GA(f, delcB_out_ga(f, [])) → MAXSORTA_IN_GA([])

(47) Obligation:

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

MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, [])) → U7_GA(t, delcB_out_ga(t, []))
MAXSORTA_IN_GA(.(f, [])) → U7_GA(f, delcB_out_ga(f, []))
U7_GA(t, delcB_out_ga(t, [])) → MAXSORTA_IN_GA([])
U7_GA(f, delcB_out_ga(f, [])) → MAXSORTA_IN_GA([])

The TRS R consists of the following rules:

delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(49) Obligation:

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

U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(50) QDPQMonotonicMRRProof (EQUIVALENT transformation)

By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented rules of the TRS R:

delcH_in_gga(f, X1) → delcH_out_gga(f, X1, .(t, X1))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0
POL(MAXSORTA_IN_GA(x₁)) = 0
POL(U12_GA(x₁, x₂)) = 2·x₂
POL(U13_GA(x₁, x₂)) = 0
POL(U18_GA(x₁, x₂)) = 0
POL(U19_GA(x₁, x₂)) = 0
POL(U24_GA(x₁, x₂)) = 0
POL(U25_GA(x₁, x₂)) = 0
POL(U30_GA(x₁, x₂)) = 2·x₂
POL(U31_GA(x₁, x₂)) = x₂
POL(U48_ga(x₁, x₂)) = x₂
POL(U49_ga(x₁, x₂)) = 2·x₂
POL(U50_ga(x₁, x₂)) = 0
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = 2·x₂
POL(U54_gga(x₁, x₂)) = 0
POL(U55_gga(x₁, x₂)) = x₁
POL(U56_gga(x₁, x₂)) = 1 + x₁
POL(U57_gga(x₁, x₂)) = 0
POL([]) = 0
POL(delcD_in_gga(x₁, x₂)) = 2·x₁
POL(delcD_out_gga(x₁, x₂, x₃)) = 0
POL(delcE_in_gga(x₁, x₂)) = x₂
POL(delcE_out_gga(x₁, x₂, x₃)) = x₂
POL(delcG_in_gga(x₁, x₂)) = 2 + x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = 1
POL(delcH_in_gga(x₁, x₂)) = x₁
POL(delcH_out_gga(x₁, x₂, x₃)) = 0
POL(delcI_in_ga(x₁)) = 0
POL(delcI_out_ga(x₁, x₂)) = 0
POL(delcJ_in_ga(x₁)) = 0
POL(delcJ_out_ga(x₁, x₂)) = 0
POL(f) = 2
POL(maxcC_in_ga(x₁)) = 0
POL(maxcC_out_ga(x₁, x₂)) = 2·x₁ + 2·x₂
POL(maxcF_in_ga(x₁)) = 0
POL(maxcF_out_ga(x₁, x₂)) = 0
POL(t) = 0

(51) Obligation:

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

U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(52) QDPQMonotonicMRRProof (EQUIVALENT transformation)

U55_gga(X1, delcI_out_ga(.(f, X1), X2)) → delcE_out_gga(f, X1, .(t, X2))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0
POL(MAXSORTA_IN_GA(x₁)) = 0
POL(U12_GA(x₁, x₂)) = 0
POL(U13_GA(x₁, x₂)) = 0
POL(U18_GA(x₁, x₂)) = 2·x₂
POL(U19_GA(x₁, x₂)) = x₂
POL(U24_GA(x₁, x₂)) = 0
POL(U25_GA(x₁, x₂)) = 0
POL(U30_GA(x₁, x₂)) = 0
POL(U31_GA(x₁, x₂)) = 0
POL(U48_ga(x₁, x₂)) = x₂
POL(U49_ga(x₁, x₂)) = x₂
POL(U50_ga(x₁, x₂)) = 2
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = 0
POL(U54_gga(x₁, x₂)) = x₁
POL(U55_gga(x₁, x₂)) = 2·x₂
POL(U56_gga(x₁, x₂)) = x₁
POL(U57_gga(x₁, x₂)) = 1
POL([]) = 0
POL(delcD_in_gga(x₁, x₂)) = 2·x₂
POL(delcD_out_gga(x₁, x₂, x₃)) = x₃
POL(delcE_in_gga(x₁, x₂)) = 2·x₁
POL(delcE_out_gga(x₁, x₂, x₃)) = 2·x₃
POL(delcG_in_gga(x₁, x₂)) = 2 + 2·x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = x₃
POL(delcH_in_gga(x₁, x₂)) = 1
POL(delcH_out_gga(x₁, x₂, x₃)) = 0
POL(delcI_in_ga(x₁)) = 2
POL(delcI_out_ga(x₁, x₂)) = 1
POL(delcJ_in_ga(x₁)) = 1
POL(delcJ_out_ga(x₁, x₂)) = 0
POL(f) = 2
POL(maxcC_in_ga(x₁)) = 0
POL(maxcC_out_ga(x₁, x₂)) = x₂
POL(maxcF_in_ga(x₁)) = 0
POL(maxcF_out_ga(x₁, x₂)) = 0
POL(t) = 0

(53) Obligation:

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

U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MAXSORTA_IN_GA(.(t, .(f, X1))) → U18_GA(X1, maxcC_in_ga(X1))

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

POL(.(x₁, x₂)) = x₁
POL(MAXSORTA_IN_GA(x₁)) = x₁
POL(U12_GA(x₁, x₂)) = 1
POL(U13_GA(x₁, x₂)) = x₂
POL(U18_GA(x₁, x₂)) = 0
POL(U19_GA(x₁, x₂)) = x₂
POL(U24_GA(x₁, x₂)) = 0
POL(U25_GA(x₁, x₂)) = x₂
POL(U30_GA(x₁, x₂)) = 0
POL(U31_GA(x₁, x₂)) = x₂
POL(U48_ga(x₁, x₂)) = 0
POL(U49_ga(x₁, x₂)) = 0
POL(U50_ga(x₁, x₂)) = 0
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = 0
POL(U54_gga(x₁, x₂)) = 1
POL(U55_gga(x₁, x₂)) = 0
POL(U56_gga(x₁, x₂)) = 0
POL(U57_gga(x₁, x₂)) = 0
POL([]) = 0
POL(delcD_in_gga(x₁, x₂)) = 1
POL(delcD_out_gga(x₁, x₂, x₃)) = x₃
POL(delcE_in_gga(x₁, x₂)) = 0
POL(delcE_out_gga(x₁, x₂, x₃)) = x₃
POL(delcG_in_gga(x₁, x₂)) = 0
POL(delcG_out_gga(x₁, x₂, x₃)) = x₃
POL(delcH_in_gga(x₁, x₂)) = 0
POL(delcH_out_gga(x₁, x₂, x₃)) = x₃
POL(delcI_in_ga(x₁)) = 0
POL(delcI_out_ga(x₁, x₂)) = 0
POL(delcJ_in_ga(x₁)) = 0
POL(delcJ_out_ga(x₁, x₂)) = 0
POL(f) = 0
POL(maxcC_in_ga(x₁)) = 0
POL(maxcC_out_ga(x₁, x₂)) = 0
POL(maxcF_in_ga(x₁)) = 0
POL(maxcF_out_ga(x₁, x₂)) = 0
POL(t) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))

(55) Obligation:

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

U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U18_GA(X1, maxcC_out_ga(X1, X2)) → U19_GA(X1, delcE_in_gga(X2, X1))
U19_GA(X1, delcE_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(56) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(57) Obligation:

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

U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcE_in_gga(t, X1) → delcE_out_gga(t, X1, .(f, X1))
delcE_in_gga(f, X1) → U55_gga(X1, delcI_in_ga(.(f, X1)))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
delcE_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U55_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(60) QReductionProof (EQUIVALENT transformation)

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

delcE_in_gga(x0, x1)
U55_gga(x0, x1)

(61) Obligation:

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

U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(62) QDPQMonotonicMRRProof (EQUIVALENT transformation)

U54_gga(X1, delcI_out_ga(X1, X2)) → delcD_out_gga(f, X1, .(t, .(t, X2)))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0
POL(MAXSORTA_IN_GA(x₁)) = 0
POL(U12_GA(x₁, x₂)) = 2·x₂
POL(U13_GA(x₁, x₂)) = x₂
POL(U24_GA(x₁, x₂)) = 0
POL(U25_GA(x₁, x₂)) = 0
POL(U30_GA(x₁, x₂)) = 0
POL(U31_GA(x₁, x₂)) = 0
POL(U48_ga(x₁, x₂)) = 2·x₂
POL(U49_ga(x₁, x₂)) = x₂
POL(U50_ga(x₁, x₂)) = 0
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = 1
POL(U54_gga(x₁, x₂)) = 2
POL(U56_gga(x₁, x₂)) = 2·x₁
POL(U57_gga(x₁, x₂)) = 0
POL([]) = 0
POL(delcD_in_gga(x₁, x₂)) = x₁
POL(delcD_out_gga(x₁, x₂, x₃)) = 0
POL(delcG_in_gga(x₁, x₂)) = 2·x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = x₂
POL(delcH_in_gga(x₁, x₂)) = 1
POL(delcH_out_gga(x₁, x₂, x₃)) = x₁
POL(delcI_in_ga(x₁)) = 0
POL(delcI_out_ga(x₁, x₂)) = 2·x₁
POL(delcJ_in_ga(x₁)) = 2
POL(delcJ_out_ga(x₁, x₂)) = 1
POL(f) = 2
POL(maxcC_in_ga(x₁)) = 0
POL(maxcC_out_ga(x₁, x₂)) = 2·x₂
POL(maxcF_in_ga(x₁)) = 0
POL(maxcF_out_ga(x₁, x₂)) = 2·x₁
POL(t) = 0

(63) Obligation:

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

U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MAXSORTA_IN_GA(.(t, .(t, X1))) → U12_GA(X1, maxcC_in_ga(X1))

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

POL(.(x₁, x₂)) = x₁ + x₂
POL(MAXSORTA_IN_GA(x₁)) = x₁
POL(U12_GA(x₁, x₂)) = 1 + x₁
POL(U13_GA(x₁, x₂)) = x₂
POL(U24_GA(x₁, x₂)) = x₁
POL(U25_GA(x₁, x₂)) = x₂
POL(U30_GA(x₁, x₂)) = 1 + x₁
POL(U31_GA(x₁, x₂)) = x₂
POL(U48_ga(x₁, x₂)) = 0
POL(U49_ga(x₁, x₂)) = 0
POL(U50_ga(x₁, x₂)) = 0
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = x₂
POL(U54_gga(x₁, x₂)) = 0
POL(U56_gga(x₁, x₂)) = x₂
POL(U57_gga(x₁, x₂)) = x₂
POL([]) = 0
POL(delcD_in_gga(x₁, x₂)) = 1 + x₂
POL(delcD_out_gga(x₁, x₂, x₃)) = x₃
POL(delcG_in_gga(x₁, x₂)) = x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = x₃
POL(delcH_in_gga(x₁, x₂)) = 1 + x₂
POL(delcH_out_gga(x₁, x₂, x₃)) = x₃
POL(delcI_in_ga(x₁)) = 0
POL(delcI_out_ga(x₁, x₂)) = 0
POL(delcJ_in_ga(x₁)) = x₁
POL(delcJ_out_ga(x₁, x₂)) = x₂
POL(f) = 0
POL(maxcC_in_ga(x₁)) = 0
POL(maxcC_out_ga(x₁, x₂)) = 0
POL(maxcF_in_ga(x₁)) = 0
POL(maxcF_out_ga(x₁, x₂)) = 0
POL(t) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

(65) Obligation:

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

U13_GA(X1, delcD_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
U12_GA(X1, maxcC_out_ga(X1, X2)) → U13_GA(X1, delcD_in_gga(X2, X1))
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(66) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(67) Obligation:

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

U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcD_in_gga(t, X1) → delcD_out_gga(t, X1, .(t, X1))
delcD_in_gga(f, X1) → U54_gga(X1, delcI_in_ga(X1))
delcI_in_ga(.(f, X1)) → delcI_out_ga(.(f, X1), X1)
delcI_in_ga([]) → delcI_out_ga([], [])
delcI_in_ga(.(t, X1)) → U50_ga(X1, delcI_in_ga(X1))
U50_ga(X1, delcI_out_ga(X1, X2)) → delcI_out_ga(.(t, X1), .(t, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(68) UsableRulesProof (EQUIVALENT transformation)

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

(69) Obligation:

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

U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
delcD_in_gga(x0, x1)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U54_gga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcI_in_ga(x0)
delcJ_in_ga(x0)
U50_ga(x0, x1)
U53_ga(x0, x1)

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

(70) QReductionProof (EQUIVALENT transformation)

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

delcD_in_gga(x0, x1)
U54_gga(x0, x1)
delcI_in_ga(x0)
U50_ga(x0, x1)

(71) Obligation:

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

U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(72) QDPQMonotonicMRRProof (EQUIVALENT transformation)

delcJ_in_ga(.(t, X1)) → delcJ_out_ga(.(t, X1), X1)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂
POL(MAXSORTA_IN_GA(x₁)) = x₁
POL(U24_GA(x₁, x₂)) = x₁
POL(U25_GA(x₁, x₂)) = x₂
POL(U30_GA(x₁, x₂)) = x₁ + 2·x₂
POL(U31_GA(x₁, x₂)) = x₂
POL(U48_ga(x₁, x₂)) = x₂
POL(U49_ga(x₁, x₂)) = x₂
POL(U51_ga(x₁, x₂)) = 0
POL(U52_ga(x₁, x₂)) = 0
POL(U53_ga(x₁, x₂)) = x₂
POL(U56_gga(x₁, x₂)) = x₂
POL(U57_gga(x₁, x₂)) = x₂
POL([]) = 0
POL(delcG_in_gga(x₁, x₂)) = x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = x₃
POL(delcH_in_gga(x₁, x₂)) = 2·x₁ + x₂
POL(delcH_out_gga(x₁, x₂, x₃)) = x₃
POL(delcJ_in_ga(x₁)) = x₁
POL(delcJ_out_ga(x₁, x₂)) = x₂
POL(f) = 0
POL(maxcC_in_ga(x₁)) = 2
POL(maxcC_out_ga(x₁, x₂)) = x₂
POL(maxcF_in_ga(x₁)) = 1
POL(maxcF_out_ga(x₁, x₂)) = 0
POL(t) = 2

(73) Obligation:

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

U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))
U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
MAXSORTA_IN_GA(.(f, .(t, X1))) → U30_GA(X1, maxcC_in_ga(X1))
U30_GA(X1, maxcC_out_ga(X1, X2)) → U31_GA(X1, delcH_in_gga(X2, X1))
U31_GA(X1, delcH_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(74) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(75) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcH_in_gga(t, X1) → U57_gga(X1, delcJ_in_ga(.(t, X1)))
U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) → delcH_out_gga(t, X1, .(f, X2))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)
delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(76) UsableRulesProof (EQUIVALENT transformation)

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

(77) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
delcH_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
U57_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(78) QReductionProof (EQUIVALENT transformation)

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

delcH_in_gga(x0, x1)
U57_gga(x0, x1)

(79) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(80) QDPQMonotonicMRRProof (EQUIVALENT transformation)

delcG_in_gga(t, X1) → U56_gga(X1, delcJ_in_ga(X1))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(MAXSORTA_IN_GA(x₁)) = x₁
POL(U24_GA(x₁, x₂)) = 2·x₁ + x₂
POL(U25_GA(x₁, x₂)) = x₂
POL(U48_ga(x₁, x₂)) = x₂
POL(U49_ga(x₁, x₂)) = x₂
POL(U51_ga(x₁, x₂)) = 2·x₂
POL(U52_ga(x₁, x₂)) = 2·x₂
POL(U53_ga(x₁, x₂)) = 2·x₂
POL(U56_gga(x₁, x₂)) = 2·x₂
POL([]) = 0
POL(delcG_in_gga(x₁, x₂)) = 2·x₁ + 2·x₂
POL(delcG_out_gga(x₁, x₂, x₃)) = x₃
POL(delcJ_in_ga(x₁)) = 0
POL(delcJ_out_ga(x₁, x₂)) = 2·x₂
POL(f) = 0
POL(maxcC_in_ga(x₁)) = 2
POL(maxcC_out_ga(x₁, x₂)) = x₂
POL(maxcF_in_ga(x₁)) = x₁
POL(maxcF_out_ga(x₁, x₂)) = 2·x₂
POL(t) = 2

(81) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
delcJ_in_ga([]) → delcJ_out_ga([], [])
delcJ_in_ga(.(f, X1)) → U53_ga(X1, delcJ_in_ga(X1))
U56_gga(X1, delcJ_out_ga(X1, X2)) → delcG_out_gga(t, X1, .(f, .(f, X2)))
U53_ga(X1, delcJ_out_ga(X1, X2)) → delcJ_out_ga(.(f, X1), .(f, X2))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(82) UsableRulesProof (EQUIVALENT transformation)

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

(83) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)
U56_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

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

(84) QReductionProof (EQUIVALENT transformation)

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

U56_gga(x0, x1)
delcJ_in_ga(x0)
U53_ga(x0, x1)

(85) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(86) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_GA(X1, maxcF_out_ga(X1, X2)) → U25_GA(X1, delcG_in_gga(X2, X1)) at position [1] we obtained the following new rules [LPAR04]:

U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))

(87) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))

The TRS R consists of the following rules:

delcG_in_gga(f, X1) → delcG_out_gga(f, X1, .(f, X1))
maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(88) UsableRulesProof (EQUIVALENT transformation)

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

(89) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))

The TRS R consists of the following rules:

maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
delcG_in_gga(x0, x1)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(90) QReductionProof (EQUIVALENT transformation)

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

delcG_in_gga(x0, x1)

(91) Obligation:

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

U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4)
MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))

The TRS R consists of the following rules:

maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(92) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_GA(X1, delcG_out_gga(X2, X1, X4)) → MAXSORTA_IN_GA(X4) we obtained the following new rules [LPAR04]:

U25_GA(z0, delcG_out_gga(f, z0, .(f, z0))) → MAXSORTA_IN_GA(.(f, z0))

(93) Obligation:

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

MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))
U25_GA(z0, delcG_out_gga(f, z0, .(f, z0))) → MAXSORTA_IN_GA(.(f, z0))

The TRS R consists of the following rules:

maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(94) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MAXSORTA_IN_GA(.(f, .(f, X1))) → U24_GA(X1, maxcF_in_ga(X1))
U24_GA(x0, maxcF_out_ga(x0, f)) → U25_GA(x0, delcG_out_gga(f, x0, .(f, x0)))
U25_GA(z0, delcG_out_gga(f, z0, .(f, z0))) → MAXSORTA_IN_GA(.(f, z0))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(MAXSORTA_IN_GA(x₁)) = 2·x₁
POL(U24_GA(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U25_GA(x₁, x₂)) = 2 + x₁ + x₂
POL(U48_ga(x₁, x₂)) = 1 + x₁ + x₂
POL(U49_ga(x₁, x₂)) = 1 + x₁ + x₂
POL(U51_ga(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(U52_ga(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL([]) = 0
POL(delcG_out_gga(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(f) = 0
POL(maxcC_in_ga(x₁)) = 1 + x₁
POL(maxcC_out_ga(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(maxcF_in_ga(x₁)) = 2 + 2·x₁
POL(maxcF_out_ga(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(t) = 0

(95) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

maxcF_in_ga([]) → maxcF_out_ga([], f)
maxcF_in_ga(.(f, X1)) → U51_ga(X1, maxcF_in_ga(X1))
maxcF_in_ga(.(t, X1)) → U52_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga([]) → maxcC_out_ga([], t)
maxcC_in_ga(.(t, X1)) → U48_ga(X1, maxcC_in_ga(X1))
maxcC_in_ga(.(f, X1)) → U49_ga(X1, maxcC_in_ga(X1))
U52_ga(X1, maxcC_out_ga(X1, X2)) → maxcF_out_ga(.(t, X1), X2)
U49_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(f, X1), X2)
U48_ga(X1, maxcC_out_ga(X1, X2)) → maxcC_out_ga(.(t, X1), X2)
U51_ga(X1, maxcF_out_ga(X1, X2)) → maxcF_out_ga(.(f, X1), X2)

The set Q consists of the following terms:

maxcC_in_ga(x0)
maxcF_in_ga(x0)
U48_ga(x0, x1)
U49_ga(x0, x1)
U51_ga(x0, x1)
U52_ga(x0, x1)

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

(96) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.