Left Termination proof of /tmp/tmp3BfGN8/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 PrologToPrologProblemTransformerProof (⇒, 386 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 80 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 201 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 39 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 4 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇒, 0 ms)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔, 0 ms)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔, 0 ms)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇒, 0 ms)

        ↳34 QDP

        ↳35 QDPSizeChangeProof (⇔, 0 ms)

        ↳36 YES

    ↳37 PiDP

        ↳38 UsableRulesProof (⇔, 0 ms)

        ↳39 PiDP

        ↳40 PiDPToQDPProof (⇒, 0 ms)

        ↳41 QDP

        ↳42 Narrowing (⇒, 0 ms)

        ↳43 QDP

        ↳44 UsableRulesProof (⇔, 0 ms)

        ↳45 QDP

        ↳46 QReductionProof (⇔, 0 ms)

        ↳47 QDP

        ↳48 Instantiation (⇔, 0 ms)

        ↳49 QDP

        ↳50 DependencyGraphProof (⇔, 0 ms)

        ↳51 QDP

        ↳52 QDPQMonotonicMRRProof (⇔, 906 ms)

        ↳53 QDP

        ↳54 UsableRulesProof (⇔, 0 ms)

        ↳55 QDP

        ↳56 QReductionProof (⇔, 2 ms)

        ↳57 QDP

        ↳58 Narrowing (⇒, 0 ms)

        ↳59 QDP

        ↳60 UsableRulesProof (⇔, 0 ms)

        ↳61 QDP

        ↳62 QReductionProof (⇔, 2 ms)

        ↳63 QDP

        ↳64 Narrowing (⇒, 0 ms)

        ↳65 QDP

        ↳66 UsableRulesProof (⇔, 0 ms)

        ↳67 QDP

        ↳68 QReductionProof (⇔, 8 ms)

        ↳69 QDP

        ↳70 Rewriting (⇔, 21 ms)

        ↳71 QDP

        ↳72 UsableRulesProof (⇔, 0 ms)

        ↳73 QDP

        ↳74 QReductionProof (⇔, 33 ms)

        ↳75 QDP

        ↳76 Rewriting (⇔, 0 ms)

        ↳77 QDP

        ↳78 UsableRulesProof (⇔, 0 ms)

        ↳79 QDP

        ↳80 QReductionProof (⇔, 31 ms)

        ↳81 QDP

        ↳82 QDPOrderProof (⇔, 226 ms)

        ↳83 QDP

        ↳84 MRRProof (⇔, 361 ms)

        ↳85 QDP

        ↳86 DependencyGraphProof (⇔, 0 ms)

        ↳87 QDP

        ↳88 QReductionProof (⇔, 0 ms)

        ↳89 QDP

        ↳90 MRRProof (⇔, 160 ms)

        ↳91 QDP

        ↳92 DependencyGraphProof (⇔, 0 ms)

        ↳93 TRUE

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="15: 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="44: 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="71: 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="101: 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="129: 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="150: 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="173: 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="203: 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="238: 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:

Clauses:

maxA([], t).
maxA(.(t, T72), T74) :- maxA(T72, T74).
maxA(.(f, T72), T74) :- maxA(T72, T74).
delB([], []).
delB(.(f, T128), T128).
delB(.(t, T134), .(t, X228)) :- delB(T134, X228).
maxC([], f).
maxC(.(f, T178), T180) :- maxC(T178, T180).
maxC(.(t, T188), T190) :- maxA(T188, T190).
delD([], []).
delD(.(t, T234), T234).
delD(.(f, T240), .(f, X427)) :- delD(T240, X427).
delE(t, []).
delE(f, []).
delF(t, T100, .(t, T100)).
delF(f, T118, .(t, .(t, X187))) :- delB(T118, X187).
delG(t, T152, .(f, T152)).
delG(f, T160, .(t, X269)) :- delB(.(f, T160), X269).
delH(f, T206, .(f, T206)).
delH(t, T224, .(f, .(f, X384))) :- delD(T224, X384).
delI(f, T272, .(t, T272)).
delI(t, T280, .(f, X480)) :- delD(.(t, T280), X480).
maxsortJ([], []).
maxsortJ(.(T17, []), .(T17, T18)) :- delE(T17, X14).
maxsortJ(.(T17, []), .(T17, T18)) :- ','(delE(T17, T21), maxsortJ(T21, T18)).
maxsortJ(.(t, .(t, T53)), .(T55, T56)) :- maxA(T53, T55).
maxsortJ(.(t, .(t, T53)), .(T57, T58)) :- ','(maxA(T53, T57), delF(T57, T53, X14)).
maxsortJ(.(t, .(t, T53)), .(T57, T58)) :- ','(maxA(T53, T57), ','(delF(T57, T53, T87), maxsortJ(T87, T58))).
maxsortJ(.(t, .(f, T53)), .(T55, T56)) :- maxA(T53, T55).
maxsortJ(.(t, .(f, T53)), .(T135, T136)) :- ','(maxA(T53, T135), delG(T135, T53, X14)).
maxsortJ(.(t, .(f, T53)), .(T135, T136)) :- ','(maxA(T53, T135), ','(delG(T135, T53, T139), maxsortJ(T139, T136))).
maxsortJ(.(f, .(f, T53)), .(T55, T56)) :- maxC(T53, T55).
maxsortJ(.(f, .(f, T53)), .(T163, T164)) :- ','(maxC(T53, T163), delH(T163, T53, X14)).
maxsortJ(.(f, .(f, T53)), .(T163, T164)) :- ','(maxC(T53, T163), ','(delH(T163, T53, T193), maxsortJ(T193, T164))).
maxsortJ(.(f, .(t, T251)), .(T253, T254)) :- maxA(T251, T253).
maxsortJ(.(f, .(t, T251)), .(T255, T256)) :- ','(maxA(T251, T255), delI(T255, T251, X14)).
maxsortJ(.(f, .(t, T251)), .(T255, T256)) :- ','(maxA(T251, T255), ','(delI(T255, T251, T259), maxsortJ(T259, T256))).

Query: maxsortJ(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
maxsortJ_in: (b,f)
maxA_in: (b,f)
delF_in: (b,b,f)
delB_in: (b,f)
delG_in: (b,b,f)
maxC_in: (b,f)
delH_in: (b,b,f)
delD_in: (b,f)
delI_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U11_GA(T17, T18, delE_in_ga(T17, X14))
MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → DELE_IN_GA(T17, X14)
MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, delE_in_ga(T17, T21))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → U13_GA(T17, T18, maxsortJ_in_ga(T21, T18))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → MAXSORTJ_IN_GA(T21, T18)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T55, T56)) → U14_GA(T53, T55, T56, maxA_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T55, T56)) → MAXA_IN_GA(T53, T55)
MAXA_IN_GA(.(t, T72), T74) → U1_GA(T72, T74, maxA_in_ga(T72, T74))
MAXA_IN_GA(.(t, T72), T74) → MAXA_IN_GA(T72, T74)
MAXA_IN_GA(.(f, T72), T74) → U2_GA(T72, T74, maxA_in_ga(T72, T74))
MAXA_IN_GA(.(f, T72), T74) → MAXA_IN_GA(T72, T74)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T57, T58)) → U15_GA(T53, T57, T58, maxA_in_ga(T53, T57))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_GA(T53, T57, T58, delF_in_gga(T57, T53, X14))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → DELF_IN_GGA(T57, T53, X14)
DELF_IN_GGA(f, T118, .(t, .(t, X187))) → U7_GGA(T118, X187, delB_in_ga(T118, X187))
DELF_IN_GGA(f, T118, .(t, .(t, X187))) → DELB_IN_GA(T118, X187)
DELB_IN_GA(.(t, T134), .(t, X228)) → U3_GA(T134, X228, delB_in_ga(T134, X228))
DELB_IN_GA(.(t, T134), .(t, X228)) → DELB_IN_GA(T134, X228)
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_GA(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_GA(T53, T57, T58, maxsortJ_in_ga(T87, T58))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → MAXSORTJ_IN_GA(T87, T58)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T55, T56)) → U19_GA(T53, T55, T56, maxA_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T55, T56)) → MAXA_IN_GA(T53, T55)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T135, T136)) → U20_GA(T53, T135, T136, maxA_in_ga(T53, T135))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_GA(T53, T135, T136, delG_in_gga(T135, T53, X14))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → DELG_IN_GGA(T135, T53, X14)
DELG_IN_GGA(f, T160, .(t, X269)) → U8_GGA(T160, X269, delB_in_ga(.(f, T160), X269))
DELG_IN_GGA(f, T160, .(t, X269)) → DELB_IN_GA(.(f, T160), X269)
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_GA(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_GA(T53, T135, T136, maxsortJ_in_ga(T139, T136))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → MAXSORTJ_IN_GA(T139, T136)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T55, T56)) → U24_GA(T53, T55, T56, maxC_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T55, T56)) → MAXC_IN_GA(T53, T55)
MAXC_IN_GA(.(f, T178), T180) → U4_GA(T178, T180, maxC_in_ga(T178, T180))
MAXC_IN_GA(.(f, T178), T180) → MAXC_IN_GA(T178, T180)
MAXC_IN_GA(.(t, T188), T190) → U5_GA(T188, T190, maxA_in_ga(T188, T190))
MAXC_IN_GA(.(t, T188), T190) → MAXA_IN_GA(T188, T190)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T163, T164)) → U25_GA(T53, T163, T164, maxC_in_ga(T53, T163))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_GA(T53, T163, T164, delH_in_gga(T163, T53, X14))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → DELH_IN_GGA(T163, T53, X14)
DELH_IN_GGA(t, T224, .(f, .(f, X384))) → U9_GGA(T224, X384, delD_in_ga(T224, X384))
DELH_IN_GGA(t, T224, .(f, .(f, X384))) → DELD_IN_GA(T224, X384)
DELD_IN_GA(.(f, T240), .(f, X427)) → U6_GA(T240, X427, delD_in_ga(T240, X427))
DELD_IN_GA(.(f, T240), .(f, X427)) → DELD_IN_GA(T240, X427)
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_GA(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_GA(T53, T163, T164, maxsortJ_in_ga(T193, T164))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → MAXSORTJ_IN_GA(T193, T164)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T253, T254)) → U29_GA(T251, T253, T254, maxA_in_ga(T251, T253))
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T253, T254)) → MAXA_IN_GA(T251, T253)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T255, T256)) → U30_GA(T251, T255, T256, maxA_in_ga(T251, T255))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_GA(T251, T255, T256, delI_in_gga(T255, T251, X14))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → DELI_IN_GGA(T255, T251, X14)
DELI_IN_GGA(t, T280, .(f, X480)) → U10_GGA(T280, X480, delD_in_ga(.(t, T280), X480))
DELI_IN_GGA(t, T280, .(f, X480)) → DELD_IN_GA(.(t, T280), X480)
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_GA(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_GA(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → MAXSORTJ_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsortJ_in_ga(x1, x2) = maxsortJ_in_ga(x1)
[] = []
maxsortJ_out_ga(x1, x2) = maxsortJ_out_ga
.(x1, x2) = .(x1, x2)
U11_ga(x1, x2, x3) = U11_ga(x3)
delE_in_ga(x1, x2) = delE_in_ga(x1)
t = t
delE_out_ga(x1, x2) = delE_out_ga(x2)
f = f
U12_ga(x1, x2, x3) = U12_ga(x3)
U13_ga(x1, x2, x3) = U13_ga(x3)
U14_ga(x1, x2, x3, x4) = U14_ga(x4)
maxA_in_ga(x1, x2) = maxA_in_ga(x1)
maxA_out_ga(x1, x2) = maxA_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x3)
U15_ga(x1, x2, x3, x4) = U15_ga(x1, x4)
U16_ga(x1, x2, x3, x4) = U16_ga(x4)
delF_in_gga(x1, x2, x3) = delF_in_gga(x1, x2)
delF_out_gga(x1, x2, x3) = delF_out_gga(x3)
U7_gga(x1, x2, x3) = U7_gga(x3)
delB_in_ga(x1, x2) = delB_in_ga(x1)
delB_out_ga(x1, x2) = delB_out_ga(x2)
U3_ga(x1, x2, x3) = U3_ga(x3)
U17_ga(x1, x2, x3, x4) = U17_ga(x4)
U18_ga(x1, x2, x3, x4) = U18_ga(x4)
U19_ga(x1, x2, x3, x4) = U19_ga(x4)
U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4) = U21_ga(x4)
delG_in_gga(x1, x2, x3) = delG_in_gga(x1, x2)
delG_out_gga(x1, x2, x3) = delG_out_gga(x3)
U8_gga(x1, x2, x3) = U8_gga(x3)
U22_ga(x1, x2, x3, x4) = U22_ga(x4)
U23_ga(x1, x2, x3, x4) = U23_ga(x4)
U24_ga(x1, x2, x3, x4) = U24_ga(x4)
maxC_in_ga(x1, x2) = maxC_in_ga(x1)
maxC_out_ga(x1, x2) = maxC_out_ga(x2)
U4_ga(x1, x2, x3) = U4_ga(x3)
U5_ga(x1, x2, x3) = U5_ga(x3)
U25_ga(x1, x2, x3, x4) = U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4) = U26_ga(x4)
delH_in_gga(x1, x2, x3) = delH_in_gga(x1, x2)
delH_out_gga(x1, x2, x3) = delH_out_gga(x3)
U9_gga(x1, x2, x3) = U9_gga(x3)
delD_in_ga(x1, x2) = delD_in_ga(x1)
delD_out_ga(x1, x2) = delD_out_ga(x2)
U6_ga(x1, x2, x3) = U6_ga(x3)
U27_ga(x1, x2, x3, x4) = U27_ga(x4)
U28_ga(x1, x2, x3, x4) = U28_ga(x4)
U29_ga(x1, x2, x3, x4) = U29_ga(x4)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x4)
U31_ga(x1, x2, x3, x4) = U31_ga(x4)
delI_in_gga(x1, x2, x3) = delI_in_gga(x1, x2)
delI_out_gga(x1, x2, x3) = delI_out_gga(x3)
U10_gga(x1, x2, x3) = U10_gga(x3)
U32_ga(x1, x2, x3, x4) = U32_ga(x4)
U33_ga(x1, x2, x3, x4) = U33_ga(x4)
MAXSORTJ_IN_GA(x1, x2) = MAXSORTJ_IN_GA(x1)
U11_GA(x1, x2, x3) = U11_GA(x3)
DELE_IN_GA(x1, x2) = DELE_IN_GA(x1)
U12_GA(x1, x2, x3) = U12_GA(x3)
U13_GA(x1, x2, x3) = U13_GA(x3)
U14_GA(x1, x2, x3, x4) = U14_GA(x4)
MAXA_IN_GA(x1, x2) = MAXA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
U2_GA(x1, x2, x3) = U2_GA(x3)
U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4)
U16_GA(x1, x2, x3, x4) = U16_GA(x4)
DELF_IN_GGA(x1, x2, x3) = DELF_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3) = U7_GGA(x3)
DELB_IN_GA(x1, x2) = DELB_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x3)
U17_GA(x1, x2, x3, x4) = U17_GA(x4)
U18_GA(x1, x2, x3, x4) = U18_GA(x4)
U19_GA(x1, x2, x3, x4) = U19_GA(x4)
U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4)
U21_GA(x1, x2, x3, x4) = U21_GA(x4)
DELG_IN_GGA(x1, x2, x3) = DELG_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3) = U8_GGA(x3)
U22_GA(x1, x2, x3, x4) = U22_GA(x4)
U23_GA(x1, x2, x3, x4) = U23_GA(x4)
U24_GA(x1, x2, x3, x4) = U24_GA(x4)
MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1)
U4_GA(x1, x2, x3) = U4_GA(x3)
U5_GA(x1, x2, x3) = U5_GA(x3)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4)
U26_GA(x1, x2, x3, x4) = U26_GA(x4)
DELH_IN_GGA(x1, x2, x3) = DELH_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3) = U9_GGA(x3)
DELD_IN_GA(x1, x2) = DELD_IN_GA(x1)
U6_GA(x1, x2, x3) = U6_GA(x3)
U27_GA(x1, x2, x3, x4) = U27_GA(x4)
U28_GA(x1, x2, x3, x4) = U28_GA(x4)
U29_GA(x1, x2, x3, x4) = U29_GA(x4)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4)
U31_GA(x1, x2, x3, x4) = U31_GA(x4)
DELI_IN_GGA(x1, x2, x3) = DELI_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3) = U10_GGA(x3)
U32_GA(x1, x2, x3, x4) = U32_GA(x4)
U33_GA(x1, x2, x3, x4) = U33_GA(x4)

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

(6) Obligation:

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

MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U11_GA(T17, T18, delE_in_ga(T17, X14))
MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → DELE_IN_GA(T17, X14)
MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, delE_in_ga(T17, T21))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → U13_GA(T17, T18, maxsortJ_in_ga(T21, T18))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → MAXSORTJ_IN_GA(T21, T18)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T55, T56)) → U14_GA(T53, T55, T56, maxA_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T55, T56)) → MAXA_IN_GA(T53, T55)
MAXA_IN_GA(.(t, T72), T74) → U1_GA(T72, T74, maxA_in_ga(T72, T74))
MAXA_IN_GA(.(t, T72), T74) → MAXA_IN_GA(T72, T74)
MAXA_IN_GA(.(f, T72), T74) → U2_GA(T72, T74, maxA_in_ga(T72, T74))
MAXA_IN_GA(.(f, T72), T74) → MAXA_IN_GA(T72, T74)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T57, T58)) → U15_GA(T53, T57, T58, maxA_in_ga(T53, T57))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_GA(T53, T57, T58, delF_in_gga(T57, T53, X14))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → DELF_IN_GGA(T57, T53, X14)
DELF_IN_GGA(f, T118, .(t, .(t, X187))) → U7_GGA(T118, X187, delB_in_ga(T118, X187))
DELF_IN_GGA(f, T118, .(t, .(t, X187))) → DELB_IN_GA(T118, X187)
DELB_IN_GA(.(t, T134), .(t, X228)) → U3_GA(T134, X228, delB_in_ga(T134, X228))
DELB_IN_GA(.(t, T134), .(t, X228)) → DELB_IN_GA(T134, X228)
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_GA(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_GA(T53, T57, T58, maxsortJ_in_ga(T87, T58))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → MAXSORTJ_IN_GA(T87, T58)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T55, T56)) → U19_GA(T53, T55, T56, maxA_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T55, T56)) → MAXA_IN_GA(T53, T55)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T135, T136)) → U20_GA(T53, T135, T136, maxA_in_ga(T53, T135))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_GA(T53, T135, T136, delG_in_gga(T135, T53, X14))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → DELG_IN_GGA(T135, T53, X14)
DELG_IN_GGA(f, T160, .(t, X269)) → U8_GGA(T160, X269, delB_in_ga(.(f, T160), X269))
DELG_IN_GGA(f, T160, .(t, X269)) → DELB_IN_GA(.(f, T160), X269)
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_GA(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_GA(T53, T135, T136, maxsortJ_in_ga(T139, T136))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → MAXSORTJ_IN_GA(T139, T136)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T55, T56)) → U24_GA(T53, T55, T56, maxC_in_ga(T53, T55))
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T55, T56)) → MAXC_IN_GA(T53, T55)
MAXC_IN_GA(.(f, T178), T180) → U4_GA(T178, T180, maxC_in_ga(T178, T180))
MAXC_IN_GA(.(f, T178), T180) → MAXC_IN_GA(T178, T180)
MAXC_IN_GA(.(t, T188), T190) → U5_GA(T188, T190, maxA_in_ga(T188, T190))
MAXC_IN_GA(.(t, T188), T190) → MAXA_IN_GA(T188, T190)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T163, T164)) → U25_GA(T53, T163, T164, maxC_in_ga(T53, T163))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_GA(T53, T163, T164, delH_in_gga(T163, T53, X14))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → DELH_IN_GGA(T163, T53, X14)
DELH_IN_GGA(t, T224, .(f, .(f, X384))) → U9_GGA(T224, X384, delD_in_ga(T224, X384))
DELH_IN_GGA(t, T224, .(f, .(f, X384))) → DELD_IN_GA(T224, X384)
DELD_IN_GA(.(f, T240), .(f, X427)) → U6_GA(T240, X427, delD_in_ga(T240, X427))
DELD_IN_GA(.(f, T240), .(f, X427)) → DELD_IN_GA(T240, X427)
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_GA(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_GA(T53, T163, T164, maxsortJ_in_ga(T193, T164))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → MAXSORTJ_IN_GA(T193, T164)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T253, T254)) → U29_GA(T251, T253, T254, maxA_in_ga(T251, T253))
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T253, T254)) → MAXA_IN_GA(T251, T253)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T255, T256)) → U30_GA(T251, T255, T256, maxA_in_ga(T251, T255))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_GA(T251, T255, T256, delI_in_gga(T255, T251, X14))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → DELI_IN_GGA(T255, T251, X14)
DELI_IN_GGA(t, T280, .(f, X480)) → U10_GGA(T280, X480, delD_in_ga(.(t, T280), X480))
DELI_IN_GGA(t, T280, .(f, X480)) → DELD_IN_GA(.(t, T280), X480)
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_GA(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_GA(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → MAXSORTJ_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 38 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DELD_IN_GA(.(f, T240), .(f, X427)) → DELD_IN_GA(T240, X427)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

DELD_IN_GA(.(f, T240), .(f, X427)) → DELD_IN_GA(T240, X427)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

DELD_IN_GA(.(f, T240)) → DELD_IN_GA(T240)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

DELD_IN_GA(.(f, T240)) → DELD_IN_GA(T240)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

DELB_IN_GA(.(t, T134), .(t, X228)) → DELB_IN_GA(T134, X228)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

DELB_IN_GA(.(t, T134), .(t, X228)) → DELB_IN_GA(T134, X228)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

DELB_IN_GA(.(t, T134)) → DELB_IN_GA(T134)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

DELB_IN_GA(.(t, T134)) → DELB_IN_GA(T134)
The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

MAXA_IN_GA(.(f, T72), T74) → MAXA_IN_GA(T72, T74)
MAXA_IN_GA(.(t, T72), T74) → MAXA_IN_GA(T72, T74)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

MAXA_IN_GA(.(f, T72), T74) → MAXA_IN_GA(T72, T74)
MAXA_IN_GA(.(t, T72), T74) → MAXA_IN_GA(T72, T74)

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

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

MAXA_IN_GA(.(f, T72)) → MAXA_IN_GA(T72)
MAXA_IN_GA(.(t, T72)) → MAXA_IN_GA(T72)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

MAXA_IN_GA(.(f, T72)) → MAXA_IN_GA(T72)
The graph contains the following edges 1 > 1
MAXA_IN_GA(.(t, T72)) → MAXA_IN_GA(T72)
The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

MAXC_IN_GA(.(f, T178), T180) → MAXC_IN_GA(T178, T180)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MAXC_IN_GA(.(f, T178), T180) → MAXC_IN_GA(T178, T180)

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

MAXC_IN_GA(.(f, T178)) → MAXC_IN_GA(T178)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

MAXC_IN_GA(.(f, T178)) → MAXC_IN_GA(T178)
The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, delE_in_ga(T17, T21))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → MAXSORTJ_IN_GA(T21, T18)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T57, T58)) → U15_GA(T53, T57, T58, maxA_in_ga(T53, T57))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_GA(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → MAXSORTJ_IN_GA(T87, T58)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T135, T136)) → U20_GA(T53, T135, T136, maxA_in_ga(T53, T135))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_GA(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → MAXSORTJ_IN_GA(T139, T136)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T163, T164)) → U25_GA(T53, T163, T164, maxC_in_ga(T53, T163))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_GA(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → MAXSORTJ_IN_GA(T193, T164)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T255, T256)) → U30_GA(T251, T255, T256, maxA_in_ga(T251, T255))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_GA(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → MAXSORTJ_IN_GA(T259, T256)

The TRS R consists of the following rules:

maxsortJ_in_ga([], []) → maxsortJ_out_ga([], [])
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, delE_in_ga(T17, X14))
delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
U11_ga(T17, T18, delE_out_ga(T17, X14)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))
maxsortJ_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, delE_in_ga(T17, T21))
U12_ga(T17, T18, delE_out_ga(T17, T21)) → U13_ga(T17, T18, maxsortJ_in_ga(T21, T18))
maxsortJ_in_ga(.(t, .(t, T53)), .(T55, T56)) → U14_ga(T53, T55, T56, maxA_in_ga(T53, T55))
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U14_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(t, T53)), .(T57, T58)) → U15_ga(T53, T57, T58, maxA_in_ga(T53, T57))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U16_ga(T53, T57, T58, delF_in_gga(T57, T53, X14))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U16_ga(T53, T57, T58, delF_out_gga(T57, T53, X14)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U15_ga(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_ga(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_ga(T53, T57, T58, delF_out_gga(T57, T53, T87)) → U18_ga(T53, T57, T58, maxsortJ_in_ga(T87, T58))
maxsortJ_in_ga(.(t, .(f, T53)), .(T55, T56)) → U19_ga(T53, T55, T56, maxA_in_ga(T53, T55))
U19_ga(T53, T55, T56, maxA_out_ga(T53, T55)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(t, .(f, T53)), .(T135, T136)) → U20_ga(T53, T135, T136, maxA_in_ga(T53, T135))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U21_ga(T53, T135, T136, delG_in_gga(T135, T53, X14))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U21_ga(T53, T135, T136, delG_out_gga(T135, T53, X14)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U20_ga(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_ga(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_ga(T53, T135, T136, delG_out_gga(T135, T53, T139)) → U23_ga(T53, T135, T136, maxsortJ_in_ga(T139, T136))
maxsortJ_in_ga(.(f, .(f, T53)), .(T55, T56)) → U24_ga(T53, T55, T56, maxC_in_ga(T53, T55))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U24_ga(T53, T55, T56, maxC_out_ga(T53, T55)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T55, T56))
maxsortJ_in_ga(.(f, .(f, T53)), .(T163, T164)) → U25_ga(T53, T163, T164, maxC_in_ga(T53, T163))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U26_ga(T53, T163, T164, delH_in_gga(T163, T53, X14))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U26_ga(T53, T163, T164, delH_out_gga(T163, T53, X14)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U25_ga(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_ga(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_ga(T53, T163, T164, delH_out_gga(T163, T53, T193)) → U28_ga(T53, T163, T164, maxsortJ_in_ga(T193, T164))
maxsortJ_in_ga(.(f, .(t, T251)), .(T253, T254)) → U29_ga(T251, T253, T254, maxA_in_ga(T251, T253))
U29_ga(T251, T253, T254, maxA_out_ga(T251, T253)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T253, T254))
maxsortJ_in_ga(.(f, .(t, T251)), .(T255, T256)) → U30_ga(T251, T255, T256, maxA_in_ga(T251, T255))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U31_ga(T251, T255, T256, delI_in_gga(T255, T251, X14))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
U31_ga(T251, T255, T256, delI_out_gga(T255, T251, X14)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U30_ga(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_ga(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_ga(T251, T255, T256, delI_out_gga(T255, T251, T259)) → U33_ga(T251, T255, T256, maxsortJ_in_ga(T259, T256))
U33_ga(T251, T255, T256, maxsortJ_out_ga(T259, T256)) → maxsortJ_out_ga(.(f, .(t, T251)), .(T255, T256))
U28_ga(T53, T163, T164, maxsortJ_out_ga(T193, T164)) → maxsortJ_out_ga(.(f, .(f, T53)), .(T163, T164))
U23_ga(T53, T135, T136, maxsortJ_out_ga(T139, T136)) → maxsortJ_out_ga(.(t, .(f, T53)), .(T135, T136))
U18_ga(T53, T57, T58, maxsortJ_out_ga(T87, T58)) → maxsortJ_out_ga(.(t, .(t, T53)), .(T57, T58))
U13_ga(T17, T18, maxsortJ_out_ga(T21, T18)) → maxsortJ_out_ga(.(T17, []), .(T17, T18))

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

MAXSORTJ_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, delE_in_ga(T17, T21))
U12_GA(T17, T18, delE_out_ga(T17, T21)) → MAXSORTJ_IN_GA(T21, T18)
MAXSORTJ_IN_GA(.(t, .(t, T53)), .(T57, T58)) → U15_GA(T53, T57, T58, maxA_in_ga(T53, T57))
U15_GA(T53, T57, T58, maxA_out_ga(T53, T57)) → U17_GA(T53, T57, T58, delF_in_gga(T57, T53, T87))
U17_GA(T53, T57, T58, delF_out_gga(T57, T53, T87)) → MAXSORTJ_IN_GA(T87, T58)
MAXSORTJ_IN_GA(.(t, .(f, T53)), .(T135, T136)) → U20_GA(T53, T135, T136, maxA_in_ga(T53, T135))
U20_GA(T53, T135, T136, maxA_out_ga(T53, T135)) → U22_GA(T53, T135, T136, delG_in_gga(T135, T53, T139))
U22_GA(T53, T135, T136, delG_out_gga(T135, T53, T139)) → MAXSORTJ_IN_GA(T139, T136)
MAXSORTJ_IN_GA(.(f, .(f, T53)), .(T163, T164)) → U25_GA(T53, T163, T164, maxC_in_ga(T53, T163))
U25_GA(T53, T163, T164, maxC_out_ga(T53, T163)) → U27_GA(T53, T163, T164, delH_in_gga(T163, T53, T193))
U27_GA(T53, T163, T164, delH_out_gga(T163, T53, T193)) → MAXSORTJ_IN_GA(T193, T164)
MAXSORTJ_IN_GA(.(f, .(t, T251)), .(T255, T256)) → U30_GA(T251, T255, T256, maxA_in_ga(T251, T255))
U30_GA(T251, T255, T256, maxA_out_ga(T251, T255)) → U32_GA(T251, T255, T256, delI_in_gga(T255, T251, T259))
U32_GA(T251, T255, T256, delI_out_gga(T255, T251, T259)) → MAXSORTJ_IN_GA(T259, T256)

The TRS R consists of the following rules:

delE_in_ga(t, []) → delE_out_ga(t, [])
delE_in_ga(f, []) → delE_out_ga(f, [])
maxA_in_ga([], t) → maxA_out_ga([], t)
maxA_in_ga(.(t, T72), T74) → U1_ga(T72, T74, maxA_in_ga(T72, T74))
maxA_in_ga(.(f, T72), T74) → U2_ga(T72, T74, maxA_in_ga(T72, T74))
delF_in_gga(t, T100, .(t, T100)) → delF_out_gga(t, T100, .(t, T100))
delF_in_gga(f, T118, .(t, .(t, X187))) → U7_gga(T118, X187, delB_in_ga(T118, X187))
delG_in_gga(t, T152, .(f, T152)) → delG_out_gga(t, T152, .(f, T152))
delG_in_gga(f, T160, .(t, X269)) → U8_gga(T160, X269, delB_in_ga(.(f, T160), X269))
maxC_in_ga([], f) → maxC_out_ga([], f)
maxC_in_ga(.(f, T178), T180) → U4_ga(T178, T180, maxC_in_ga(T178, T180))
maxC_in_ga(.(t, T188), T190) → U5_ga(T188, T190, maxA_in_ga(T188, T190))
delH_in_gga(f, T206, .(f, T206)) → delH_out_gga(f, T206, .(f, T206))
delH_in_gga(t, T224, .(f, .(f, X384))) → U9_gga(T224, X384, delD_in_ga(T224, X384))
delI_in_gga(f, T272, .(t, T272)) → delI_out_gga(f, T272, .(t, T272))
delI_in_gga(t, T280, .(f, X480)) → U10_gga(T280, X480, delD_in_ga(.(t, T280), X480))
U1_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(t, T72), T74)
U2_ga(T72, T74, maxA_out_ga(T72, T74)) → maxA_out_ga(.(f, T72), T74)
U7_gga(T118, X187, delB_out_ga(T118, X187)) → delF_out_gga(f, T118, .(t, .(t, X187)))
U8_gga(T160, X269, delB_out_ga(.(f, T160), X269)) → delG_out_gga(f, T160, .(t, X269))
U4_ga(T178, T180, maxC_out_ga(T178, T180)) → maxC_out_ga(.(f, T178), T180)
U5_ga(T188, T190, maxA_out_ga(T188, T190)) → maxC_out_ga(.(t, T188), T190)
U9_gga(T224, X384, delD_out_ga(T224, X384)) → delH_out_gga(t, T224, .(f, .(f, X384)))
U10_gga(T280, X480, delD_out_ga(.(t, T280), X480)) → delI_out_gga(t, T280, .(f, X480))
delB_in_ga([], []) → delB_out_ga([], [])
delB_in_ga(.(f, T128), T128) → delB_out_ga(.(f, T128), T128)
delB_in_ga(.(t, T134), .(t, X228)) → U3_ga(T134, X228, delB_in_ga(T134, X228))
delD_in_ga([], []) → delD_out_ga([], [])
delD_in_ga(.(t, T234), T234) → delD_out_ga(.(t, T234), T234)
delD_in_ga(.(f, T240), .(f, X427)) → U6_ga(T240, X427, delD_in_ga(T240, X427))
U3_ga(T134, X228, delB_out_ga(T134, X228)) → delB_out_ga(.(t, T134), .(t, X228))
U6_ga(T240, X427, delD_out_ga(T240, X427)) → delD_out_ga(.(f, T240), .(f, X427))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
delE_in_ga(x1, x2) = delE_in_ga(x1)
t = t
delE_out_ga(x1, x2) = delE_out_ga(x2)
f = f
maxA_in_ga(x1, x2) = maxA_in_ga(x1)
maxA_out_ga(x1, x2) = maxA_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x3)
delF_in_gga(x1, x2, x3) = delF_in_gga(x1, x2)
delF_out_gga(x1, x2, x3) = delF_out_gga(x3)
U7_gga(x1, x2, x3) = U7_gga(x3)
delB_in_ga(x1, x2) = delB_in_ga(x1)
delB_out_ga(x1, x2) = delB_out_ga(x2)
U3_ga(x1, x2, x3) = U3_ga(x3)
delG_in_gga(x1, x2, x3) = delG_in_gga(x1, x2)
delG_out_gga(x1, x2, x3) = delG_out_gga(x3)
U8_gga(x1, x2, x3) = U8_gga(x3)
maxC_in_ga(x1, x2) = maxC_in_ga(x1)
maxC_out_ga(x1, x2) = maxC_out_ga(x2)
U4_ga(x1, x2, x3) = U4_ga(x3)
U5_ga(x1, x2, x3) = U5_ga(x3)
delH_in_gga(x1, x2, x3) = delH_in_gga(x1, x2)
delH_out_gga(x1, x2, x3) = delH_out_gga(x3)
U9_gga(x1, x2, x3) = U9_gga(x3)
delD_in_ga(x1, x2) = delD_in_ga(x1)
delD_out_ga(x1, x2) = delD_out_ga(x2)
U6_ga(x1, x2, x3) = U6_ga(x3)
delI_in_gga(x1, x2, x3) = delI_in_gga(x1, x2)
delI_out_gga(x1, x2, x3) = delI_out_gga(x3)
U10_gga(x1, x2, x3) = U10_gga(x3)
MAXSORTJ_IN_GA(x1, x2) = MAXSORTJ_IN_GA(x1)
U12_GA(x1, x2, x3) = U12_GA(x3)
U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4)
U17_GA(x1, x2, x3, x4) = U17_GA(x4)
U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4)
U22_GA(x1, x2, x3, x4) = U22_GA(x4)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4)
U27_GA(x1, x2, x3, x4) = U27_GA(x4)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4)
U32_GA(x1, x2, x3, x4) = U32_GA(x4)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

MAXSORTJ_IN_GA(.(T17, [])) → U12_GA(delE_in_ga(T17))
U12_GA(delE_out_ga(T21)) → MAXSORTJ_IN_GA(T21)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delE_in_ga(t) → delE_out_ga([])
delE_in_ga(f) → delE_out_ga([])
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))

The set Q consists of the following terms:

delE_in_ga(x0)
maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(42) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MAXSORTJ_IN_GA(.(T17, [])) → U12_GA(delE_in_ga(T17)) at position [0] we obtained the following new rules [LPAR04]:

MAXSORTJ_IN_GA(.(t, [])) → U12_GA(delE_out_ga([]))
MAXSORTJ_IN_GA(.(f, [])) → U12_GA(delE_out_ga([]))

(43) Obligation:

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

U12_GA(delE_out_ga(T21)) → MAXSORTJ_IN_GA(T21)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
MAXSORTJ_IN_GA(.(t, [])) → U12_GA(delE_out_ga([]))
MAXSORTJ_IN_GA(.(f, [])) → U12_GA(delE_out_ga([]))

The TRS R consists of the following rules:

delE_in_ga(t) → delE_out_ga([])
delE_in_ga(f) → delE_out_ga([])
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))

The set Q consists of the following terms:

delE_in_ga(x0)
maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(44) 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.

(45) Obligation:

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

U12_GA(delE_out_ga(T21)) → MAXSORTJ_IN_GA(T21)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
MAXSORTJ_IN_GA(.(t, [])) → U12_GA(delE_out_ga([]))
MAXSORTJ_IN_GA(.(f, [])) → U12_GA(delE_out_ga([]))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))

The set Q consists of the following terms:

delE_in_ga(x0)
maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(46) 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].

delE_in_ga(x0)

(47) Obligation:

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

U12_GA(delE_out_ga(T21)) → MAXSORTJ_IN_GA(T21)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
MAXSORTJ_IN_GA(.(t, [])) → U12_GA(delE_out_ga([]))
MAXSORTJ_IN_GA(.(f, [])) → U12_GA(delE_out_ga([]))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(48) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GA(delE_out_ga(T21)) → MAXSORTJ_IN_GA(T21) we obtained the following new rules [LPAR04]:

U12_GA(delE_out_ga([])) → MAXSORTJ_IN_GA([])

(49) Obligation:

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

MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
MAXSORTJ_IN_GA(.(t, [])) → U12_GA(delE_out_ga([]))
MAXSORTJ_IN_GA(.(f, [])) → U12_GA(delE_out_ga([]))
U12_GA(delE_out_ga([])) → MAXSORTJ_IN_GA([])

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(50) DependencyGraphProof (EQUIVALENT transformation)

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

(51) Obligation:

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

U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(52) 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:

delF_in_gga(f, T118) → U7_gga(delB_in_ga(T118))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0
POL(MAXSORTJ_IN_GA(x₁)) = 0
POL(U10_gga(x₁)) = 1
POL(U15_GA(x₁, x₂)) = 2·x₂
POL(U17_GA(x₁)) = 2·x₁
POL(U1_ga(x₁)) = 2·x₁
POL(U20_GA(x₁, x₂)) = 0
POL(U22_GA(x₁)) = 0
POL(U25_GA(x₁, x₂)) = 0
POL(U27_GA(x₁)) = 0
POL(U2_ga(x₁)) = 2·x₁
POL(U30_GA(x₁, x₂)) = 0
POL(U32_GA(x₁)) = 0
POL(U3_ga(x₁)) = 0
POL(U4_ga(x₁)) = 1
POL(U5_ga(x₁)) = x₁
POL(U6_ga(x₁)) = 2
POL(U7_gga(x₁)) = x₁
POL(U8_gga(x₁)) = 0
POL(U9_gga(x₁)) = 0
POL([]) = 1
POL(delB_in_ga(x₁)) = 0
POL(delB_out_ga(x₁)) = 0
POL(delD_in_ga(x₁)) = 2
POL(delD_out_ga(x₁)) = 0
POL(delF_in_gga(x₁, x₂)) = x₁
POL(delF_out_gga(x₁)) = 2·x₁
POL(delG_in_gga(x₁, x₂)) = 2 + x₁
POL(delG_out_gga(x₁)) = 0
POL(delH_in_gga(x₁, x₂)) = 2 + 2·x₂
POL(delH_out_gga(x₁)) = x₁
POL(delI_in_gga(x₁, x₂)) = 2
POL(delI_out_gga(x₁)) = 0
POL(f) = 1
POL(maxA_in_ga(x₁)) = 0
POL(maxA_out_ga(x₁)) = x₁
POL(maxC_in_ga(x₁)) = 2
POL(maxC_out_ga(x₁)) = 0
POL(t) = 0

(53) Obligation:

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

U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))
delB_in_ga([]) → delB_out_ga([])
delB_in_ga(.(t, T134)) → U3_ga(delB_in_ga(T134))
U7_gga(delB_out_ga(X187)) → delF_out_gga(.(t, .(t, X187)))
U3_ga(delB_out_ga(X228)) → delB_out_ga(.(t, X228))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(54) 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.

(55) Obligation:

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

U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(56) 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].

U7_gga(x0)
U3_ga(x0)

(57) Obligation:

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

U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53))
U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(58) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_GA(T53, maxA_out_ga(T57)) → U17_GA(delF_in_gga(T57, T53)) at position [0] we obtained the following new rules [LPAR04]:

U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))

(59) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delF_in_gga(t, T100) → delF_out_gga(.(t, T100))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(60) 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.

(61) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))

The set Q consists of the following terms:

maxA_in_ga(x0)
delF_in_gga(x0, x1)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(62) 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].

delF_in_gga(x0, x1)

(63) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))

The set Q consists of the following terms:

maxA_in_ga(x0)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(64) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U20_GA(T53, maxA_out_ga(T135)) → U22_GA(delG_in_gga(T135, T53)) at position [0] we obtained the following new rules [LPAR04]:

U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_in_ga(.(f, x0))))

(65) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_in_ga(.(f, x0))))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)
delG_in_gga(t, T152) → delG_out_gga(.(f, T152))
delG_in_gga(f, T160) → U8_gga(delB_in_ga(.(f, T160)))
delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))

The set Q consists of the following terms:

maxA_in_ga(x0)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(66) 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.

(67) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_in_ga(.(f, x0))))

The TRS R consists of the following rules:

delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
delG_in_gga(x0, x1)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(68) 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].

delG_in_gga(x0, x1)

(69) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_in_ga(.(f, x0))))

The TRS R consists of the following rules:

delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(70) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_in_ga(.(f, x0)))) at position [0,0] we obtained the following new rules [LPAR04]:

U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_out_ga(x0)))

(71) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_out_ga(x0)))

The TRS R consists of the following rules:

delB_in_ga(.(f, T128)) → delB_out_ga(T128)
U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(72) 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.

(73) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_out_ga(x0)))

The TRS R consists of the following rules:

U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delB_in_ga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(74) 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].

delB_in_ga(x0)

(75) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_out_ga(x0)))

The TRS R consists of the following rules:

U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(76) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U20_GA(x0, maxA_out_ga(f)) → U22_GA(U8_gga(delB_out_ga(x0))) at position [0] we obtained the following new rules [LPAR04]:

U20_GA(x0, maxA_out_ga(f)) → U22_GA(delG_out_gga(.(t, x0)))

(77) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(delG_out_gga(.(t, x0)))

The TRS R consists of the following rules:

U8_gga(delB_out_ga(X269)) → delG_out_gga(.(t, X269))
delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(78) 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.

(79) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(delG_out_gga(.(t, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U8_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(80) 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].

U8_gga(x0)

(81) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))
U20_GA(x0, maxA_out_ga(f)) → U22_GA(delG_out_gga(.(t, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(82) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U20_GA(x0, maxA_out_ga(f)) → U22_GA(delG_out_gga(.(t, x0)))

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

POL(.(x₁, x₂)) = x₂
POL(MAXSORTJ_IN_GA(x₁)) = 0
POL(U10_gga(x₁)) = 0
POL(U15_GA(x₁, x₂)) = 0
POL(U17_GA(x₁)) = 0
POL(U1_ga(x₁)) = x₁
POL(U20_GA(x₁, x₂)) = x₂
POL(U22_GA(x₁)) = 0
POL(U25_GA(x₁, x₂)) = 0
POL(U27_GA(x₁)) = 0
POL(U2_ga(x₁)) = x₁
POL(U30_GA(x₁, x₂)) = 0
POL(U32_GA(x₁)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_ga(x₁)) = 0
POL(U6_ga(x₁)) = 0
POL(U9_gga(x₁)) = 0
POL([]) = 0
POL(delD_in_ga(x₁)) = 1 + x₁
POL(delD_out_ga(x₁)) = 0
POL(delF_out_gga(x₁)) = 0
POL(delG_out_gga(x₁)) = 0
POL(delH_in_gga(x₁, x₂)) = 0
POL(delH_out_gga(x₁)) = 0
POL(delI_in_gga(x₁, x₂)) = 0
POL(delI_out_gga(x₁)) = 0
POL(f) = 1
POL(maxA_in_ga(x₁)) = 0
POL(maxA_out_ga(x₁)) = x₁
POL(maxC_in_ga(x₁)) = 0
POL(maxC_out_ga(x₁)) = 0
POL(t) = 0

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

maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)

(83) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(84) 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:

MAXSORTJ_IN_GA(.(t, .(t, T53))) → U15_GA(T53, maxA_in_ga(T53))
MAXSORTJ_IN_GA(.(t, .(f, T53))) → U20_GA(T53, maxA_in_ga(T53))

Strictly oriented rules of the TRS R:

delI_in_gga(t, T280) → U10_gga(delD_in_ga(.(t, T280)))
delD_in_ga(.(t, T234)) → delD_out_ga(T234)
U10_gga(delD_out_ga(X480)) → delI_out_gga(.(f, X480))
maxA_in_ga(.(t, T72)) → U1_ga(maxA_in_ga(T72))
U1_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
U9_gga(delD_out_ga(X384)) → delH_out_gga(.(f, .(f, X384)))
maxC_in_ga(.(t, T188)) → U5_ga(maxA_in_ga(T188))
U5_ga(maxA_out_ga(T190)) → maxC_out_ga(T190)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(MAXSORTJ_IN_GA(x₁)) = 2 + x₁
POL(U10_gga(x₁)) = 1 + x₁
POL(U15_GA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U17_GA(x₁)) = 1 + x₁
POL(U1_ga(x₁)) = 1 + x₁
POL(U20_GA(x₁, x₂)) = 2·x₁ + x₂
POL(U22_GA(x₁)) = 1 + x₁
POL(U25_GA(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U27_GA(x₁)) = 1 + x₁
POL(U2_ga(x₁)) = x₁
POL(U30_GA(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U32_GA(x₁)) = 2 + x₁
POL(U4_ga(x₁)) = x₁
POL(U5_ga(x₁)) = 2 + x₁
POL(U6_ga(x₁)) = 2·x₁
POL(U9_gga(x₁)) = 2 + 2·x₁
POL([]) = 0
POL(delD_in_ga(x₁)) = x₁
POL(delD_out_ga(x₁)) = 2·x₁
POL(delF_out_gga(x₁)) = 1 + x₁
POL(delG_out_gga(x₁)) = 1 + x₁
POL(delH_in_gga(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(delH_out_gga(x₁)) = 1 + x₁
POL(delI_in_gga(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(delI_out_gga(x₁)) = x₁
POL(f) = 0
POL(maxA_in_ga(x₁)) = 2 + x₁
POL(maxA_out_ga(x₁)) = 1 + x₁
POL(maxC_in_ga(x₁)) = 1 + 2·x₁
POL(maxC_out_ga(x₁)) = 1 + x₁
POL(t) = 1

(85) Obligation:

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

U17_GA(delF_out_gga(T87)) → MAXSORTJ_IN_GA(T87)
U22_GA(delG_out_gga(T139)) → MAXSORTJ_IN_GA(T139)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)
U15_GA(x0, maxA_out_ga(t)) → U17_GA(delF_out_gga(.(t, x0)))
U20_GA(x0, maxA_out_ga(t)) → U22_GA(delG_out_gga(.(f, x0)))

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(86) DependencyGraphProof (EQUIVALENT transformation)

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

(87) Obligation:

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

U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(88) 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].

U1_ga(x0)
U5_ga(x0)
U10_gga(x0)

(89) Obligation:

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

U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

The TRS R consists of the following rules:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
maxA_in_ga([]) → maxA_out_ga(t)
maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga([]) → delD_out_ga([])
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U2_ga(x0)
U4_ga(x0)
U9_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(90) MRRProof (EQUIVALENT transformation)

MAXSORTJ_IN_GA(.(f, .(f, T53))) → U25_GA(T53, maxC_in_ga(T53))
U32_GA(delI_out_gga(T259)) → MAXSORTJ_IN_GA(T259)

Strictly oriented rules of the TRS R:

delI_in_gga(f, T272) → delI_out_gga(.(t, T272))
maxA_in_ga([]) → maxA_out_ga(t)
U2_ga(maxA_out_ga(T74)) → maxA_out_ga(T74)
delH_in_gga(t, T224) → U9_gga(delD_in_ga(T224))
delD_in_ga(.(f, T240)) → U6_ga(delD_in_ga(T240))
U6_ga(delD_out_ga(X427)) → delD_out_ga(.(f, X427))
U4_ga(maxC_out_ga(T180)) → maxC_out_ga(T180)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(MAXSORTJ_IN_GA(x₁)) = 1 + x₁
POL(U25_GA(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U27_GA(x₁)) = 1 + x₁
POL(U2_ga(x₁)) = 2·x₁
POL(U30_GA(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U32_GA(x₁)) = 1 + x₁
POL(U4_ga(x₁)) = 2 + 2·x₁
POL(U6_ga(x₁)) = 2 + 2·x₁
POL(U9_gga(x₁)) = x₁
POL([]) = 1
POL(delD_in_ga(x₁)) = 2·x₁
POL(delD_out_ga(x₁)) = 1 + x₁
POL(delH_in_gga(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(delH_out_gga(x₁)) = x₁
POL(delI_in_gga(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(delI_out_gga(x₁)) = 2 + x₁
POL(f) = 1
POL(maxA_in_ga(x₁)) = 2 + x₁
POL(maxA_out_ga(x₁)) = 2 + 2·x₁
POL(maxC_in_ga(x₁)) = x₁
POL(maxC_out_ga(x₁)) = x₁
POL(t) = 0

(91) Obligation:

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

U25_GA(T53, maxC_out_ga(T163)) → U27_GA(delH_in_gga(T163, T53))
U27_GA(delH_out_gga(T193)) → MAXSORTJ_IN_GA(T193)
MAXSORTJ_IN_GA(.(f, .(t, T251))) → U30_GA(T251, maxA_in_ga(T251))
U30_GA(T251, maxA_out_ga(T255)) → U32_GA(delI_in_gga(T255, T251))

The TRS R consists of the following rules:

maxA_in_ga(.(f, T72)) → U2_ga(maxA_in_ga(T72))
delH_in_gga(f, T206) → delH_out_gga(.(f, T206))
delD_in_ga([]) → delD_out_ga([])
maxC_in_ga([]) → maxC_out_ga(f)
maxC_in_ga(.(f, T178)) → U4_ga(maxC_in_ga(T178))

The set Q consists of the following terms:

maxA_in_ga(x0)
maxC_in_ga(x0)
delH_in_gga(x0, x1)
delI_in_gga(x0, x1)
U2_ga(x0)
U4_ga(x0)
U9_gga(x0)
delD_in_ga(x0)
U6_ga(x0)

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

(92) DependencyGraphProof (EQUIVALENT transformation)

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