Left Termination of the query pattern maxsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(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))).

Queries:

maxsort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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(T22, T18)\n\nT22 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(T31, T31)\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: eq(T31, T31), SCOPE: 4, CLAUSE: 0 | eq(T31, T31), SCOPE: 4, CLAUSE: 1\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: eq(T31, T31), SCOPE: 4, CLAUSE: 0\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: eq(T31, T31), SCOPE: 4, CLAUSE: 1\n\nT31 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(T35, T35), del(T35, [], X66))\n\nT35 is ground\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(neq(T35, T35), del(T35, [], X66)), SCOPE: 5, CLAUSE: 2 | ','(neq(T35, T35), del(T35, [], X66)), SCOPE: 5, CLAUSE: 3\n\nT35 is ground\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(neq(T35, T35), del(T35, [], X66)), SCOPE: 5, CLAUSE: 3\n\nT35 is ground\nX66 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(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58))))\n\nT53 is ground\nT54 is ground\nT55 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(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 7 | ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 8 | ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 9\n\nT53 is ground\nT54 is ground\nT55 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 7\n\nT53 is ground\nT54 is ground\nT55 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 8 | ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 9\n\nT53 is ground\nT54 is ground\nT55 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(max(.(t, T55), T57), ','(del(T57, .(t, .(t, T55)), X14), maxsort(X14, T58)))\n\nT55 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, T55), T57)\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(del(T61, .(t, .(t, T55)), X14), maxsort(X14, T62))\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: max(.(t, T55), T57), SCOPE: 7, CLAUSE: 11 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 12 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 13 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: max(.(t, T55), T57), SCOPE: 7, CLAUSE: 12 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 13 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: max(.(t, T55), T57), SCOPE: 7, CLAUSE: 12\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: max(.(t, T55), T57), SCOPE: 7, CLAUSE: 13 | max(.(t, T55), T57), SCOPE: 7, CLAUSE: 14\n\nT55 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, T55), T57), SCOPE: 7, CLAUSE: 13\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: max(.(t, T55), T57), SCOPE: 7, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(ge(t, T75), max(.(t, T76), T78))\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 7 | ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 9\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 7\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 8 | ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 9\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: max(.(t, T76), T78)\n\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 8\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(ge(t, T75), max(.(t, T76), T78)), SCOPE: 8, CLAUSE: 9\n\nT75 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: max(.(t, T76), T78)\n\nT76 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(T85, t), max(.(T85, T86), T88))\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(gt(T85, t), max(.(T85, T86), T88)), SCOPE: 9, CLAUSE: 10\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: del(T61, .(t, .(t, T55)), X14)\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: maxsort(T95, T62)\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 4 | del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 5 | del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 6\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 5 | del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 6\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 5\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: del(T61, .(t, .(t, T55)), X14), SCOPE: 10, CLAUSE: 6\n\nT55 is ground\nT61 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: eq(T109, t)\n\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: eq(T109, t), SCOPE: 11, CLAUSE: 0 | eq(T109, t), SCOPE: 11, CLAUSE: 1\n\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: eq(T109, t), SCOPE: 11, CLAUSE: 0\n\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: eq(T109, t), SCOPE: 11, CLAUSE: 1\n\nT109 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(T119, t), del(T119, .(t, T120), X197))\n\nT119 is ground\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(neq(T119, t), del(T119, .(t, T120), X197)), SCOPE: 12, CLAUSE: 2 | ','(neq(T119, t), del(T119, .(t, T120), X197)), SCOPE: 12, CLAUSE: 3\n\nT119 is ground\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(neq(T119, t), del(T119, .(t, T120), X197)), SCOPE: 12, CLAUSE: 3\n\nT119 is ground\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: del(f, .(t, T120), X197)\n\nT120 is ground\nX197 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, T120), X197), SCOPE: 13, CLAUSE: 4 | del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 6\n\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 5 | del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 6\n\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 5\n\nT120 is ground\nX197 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: del(f, .(t, T120), X197), SCOPE: 13, CLAUSE: 6\n\nT120 is ground\nX197 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, T133, X240))\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(neq(f, t), del(f, T133, X240)), SCOPE: 15, CLAUSE: 2 | ','(neq(f, t), del(f, T133, X240)), SCOPE: 15, CLAUSE: 3\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(neq(f, t), del(f, T133, X240)), SCOPE: 15, CLAUSE: 3\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: del(f, T133, X240)\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: del(f, T133, X240), SCOPE: 16, CLAUSE: 4 | del(f, T133, X240), SCOPE: 16, CLAUSE: 5 | del(f, T133, X240), SCOPE: 16, CLAUSE: 6\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: del(f, T133, X240), SCOPE: 16, CLAUSE: 4\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: del(f, T133, X240), SCOPE: 16, CLAUSE: 5 | del(f, T133, X240), SCOPE: 16, CLAUSE: 6\n\nT133 is ground\nX240 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, T133, X240), SCOPE: 16, CLAUSE: 5\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: del(f, T133, X240), SCOPE: 16, CLAUSE: 6\n\nT133 is ground\nX240 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: eq(f, T142)\n\nT142 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: eq(f, T142), SCOPE: 17, CLAUSE: 0 | eq(f, T142), SCOPE: 17, CLAUSE: 1\n\nT142 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: eq(f, T142), SCOPE: 17, CLAUSE: 1\n\nT142 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, T148), del(f, T149, X285))\n\nT148 is ground\nT149 is ground\nX285 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(neq(f, T148), del(f, T149, X285)), SCOPE: 18, CLAUSE: 2 | ','(neq(f, T148), del(f, T149, X285)), SCOPE: 18, CLAUSE: 3\n\nT148 is ground\nT149 is ground\nX285 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(neq(f, T148), del(f, T149, X285)), SCOPE: 18, CLAUSE: 3\n\nT148 is ground\nT149 is ground\nX285 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: del(f, T149, X285)\n\nT149 is ground\nX285 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 8\n\nT53 is ground\nT54 is ground\nT55 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(ge(T53, T54), ','(max(.(T53, T55), T57), ','(del(T57, .(T53, .(T54, T55)), X14), maxsort(X14, T58)))), SCOPE: 6, CLAUSE: 9\n\nT53 is ground\nT54 is ground\nT55 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(max(.(t, T55), T57), ','(del(T57, .(t, .(f, T55)), X14), maxsort(X14, T58)))\n\nT55 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, T55), T57)\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: ','(del(T150, .(t, .(f, T55)), X14), maxsort(X14, T151))\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: del(T150, .(t, .(f, T55)), X14)\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: maxsort(T156, T151)\n\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 4 | del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 5 | del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 6\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 5 | del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 6\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 5\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: del(T150, .(t, .(f, T55)), X14), SCOPE: 19, CLAUSE: 6\n\nT55 is ground\nT150 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: eq(T170, t)\n\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: eq(T170, t), SCOPE: 20, CLAUSE: 0 | eq(T170, t), SCOPE: 20, CLAUSE: 1\n\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: eq(T170, t), SCOPE: 20, CLAUSE: 0\n\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: eq(T170, t), SCOPE: 20, CLAUSE: 1\n\nT170 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(T180, t), del(T180, .(f, T181), X343))\n\nT180 is ground\nT181 is ground\nX343 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: ','(neq(T180, t), del(T180, .(f, T181), X343)), SCOPE: 21, CLAUSE: 2 | ','(neq(T180, t), del(T180, .(f, T181), X343)), SCOPE: 21, CLAUSE: 3\n\nT180 is ground\nT181 is ground\nX343 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: ','(neq(T180, t), del(T180, .(f, T181), X343)), SCOPE: 21, CLAUSE: 3\n\nT180 is ground\nT181 is ground\nX343 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: del(f, .(f, T181), X343)\n\nT181 is ground\nX343 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: ','(max(.(f, T55), T57), ','(del(T57, .(f, .(f, T55)), X14), maxsort(X14, T58)))\n\nT55 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, T55), T57)\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: ','(del(T187, .(f, .(f, T55)), X14), maxsort(X14, T188))\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: max(.(f, T55), T57), SCOPE: 22, CLAUSE: 11 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 12 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 13 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: max(.(f, T55), T57), SCOPE: 22, CLAUSE: 12 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 13 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: max(.(f, T55), T57), SCOPE: 22, CLAUSE: 12\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: max(.(f, T55), T57), SCOPE: 22, CLAUSE: 13 | max(.(f, T55), T57), SCOPE: 22, CLAUSE: 14\n\nT55 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, T55), T57), SCOPE: 22, CLAUSE: 13\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: max(.(f, T55), T57), SCOPE: 22, CLAUSE: 14\n\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: ','(ge(f, T201), max(.(f, T202), T204))\n\nT201 is ground\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 7 | ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 9\n\nT201 is ground\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 8 | ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 9\n\nT201 is ground\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: ','(ge(f, T201), max(.(f, T202), T204)), SCOPE: 23, CLAUSE: 9\n\nT201 is ground\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: max(.(f, T202), T204)\n\nT202 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: ','(gt(T211, f), max(.(T211, T212), T214))\n\nT211 is ground\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: ','(gt(T211, f), max(.(T211, T212), T214)), SCOPE: 24, CLAUSE: 10\n\nT211 is ground\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: max(.(t, T212), T214)\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: del(T187, .(f, .(f, T55)), X14)\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: maxsort(T221, T188)\n\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 4 | del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 5 | del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 6\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 5 | del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 6\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 5\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: del(T187, .(f, .(f, T55)), X14), SCOPE: 25, CLAUSE: 6\n\nT55 is ground\nT187 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="179: eq(T235, f)\n\nT235 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: eq(T235, f), SCOPE: 26, CLAUSE: 0 | eq(T235, f), SCOPE: 26, CLAUSE: 1\n\nT235 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
181 [label="181: eq(T235, f), SCOPE: 26, CLAUSE: 1\n\nT235 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(T245, f), del(T245, .(f, T246), X457))\n\nT245 is ground\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: ','(neq(T245, f), del(T245, .(f, T246), X457)), SCOPE: 27, CLAUSE: 2 | ','(neq(T245, f), del(T245, .(f, T246), X457)), SCOPE: 27, CLAUSE: 3\n\nT245 is ground\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: ','(neq(T245, f), del(T245, .(f, T246), X457)), SCOPE: 27, CLAUSE: 2\n\nT245 is ground\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: ','(neq(T245, f), del(T245, .(f, T246), X457)), SCOPE: 27, CLAUSE: 3\n\nT245 is ground\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: del(t, .(f, T246), X457)\n\nT246 is ground\nX457 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, T246), X457), SCOPE: 28, CLAUSE: 4 | del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 6\n\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 5 | del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 6\n\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 5\n\nT246 is ground\nX457 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: del(t, .(f, T246), X457), SCOPE: 28, CLAUSE: 6\n\nT246 is ground\nX457 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, T259, X500))\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: ','(neq(t, f), del(t, T259, X500)), SCOPE: 30, CLAUSE: 2 | ','(neq(t, f), del(t, T259, X500)), SCOPE: 30, CLAUSE: 3\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: ','(neq(t, f), del(t, T259, X500)), SCOPE: 30, CLAUSE: 2\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: ','(neq(t, f), del(t, T259, X500)), SCOPE: 30, CLAUSE: 3\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: del(t, T259, X500)\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: del(t, T259, X500), SCOPE: 31, CLAUSE: 4 | del(t, T259, X500), SCOPE: 31, CLAUSE: 5 | del(t, T259, X500), SCOPE: 31, CLAUSE: 6\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: del(t, T259, X500), SCOPE: 31, CLAUSE: 4\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: del(t, T259, X500), SCOPE: 31, CLAUSE: 5 | del(t, T259, X500), SCOPE: 31, CLAUSE: 6\n\nT259 is ground\nX500 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, T259, X500), SCOPE: 31, CLAUSE: 5\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: del(t, T259, X500), SCOPE: 31, CLAUSE: 6\n\nT259 is ground\nX500 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: eq(t, T268)\n\nT268 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: eq(t, T268), SCOPE: 32, CLAUSE: 0 | eq(t, T268), SCOPE: 32, CLAUSE: 1\n\nT268 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: eq(t, T268), SCOPE: 32, CLAUSE: 0\n\nT268 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: eq(t, T268), SCOPE: 32, CLAUSE: 1\n\nT268 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, T274), del(t, T275, X547))\n\nT274 is ground\nT275 is ground\nX547 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: ','(neq(t, T274), del(t, T275, X547)), SCOPE: 33, CLAUSE: 2 | ','(neq(t, T274), del(t, T275, X547)), SCOPE: 33, CLAUSE: 3\n\nT274 is ground\nT275 is ground\nX547 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: ','(neq(t, T274), del(t, T275, X547)), SCOPE: 33, CLAUSE: 2\n\nT274 is ground\nT275 is ground\nX547 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: ','(neq(t, T274), del(t, T275, X547)), SCOPE: 33, CLAUSE: 3\n\nT274 is ground\nT275 is ground\nX547 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: del(t, T275, X547)\n\nT275 is ground\nX547 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(T285, T284), ','(max(.(T285, T286), T288), ','(del(T288, .(T284, .(T285, T286)), X14), maxsort(X14, T289))))\n\nT284 is ground\nT285 is ground\nT286 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(T285, T284), ','(max(.(T285, T286), T288), ','(del(T288, .(T284, .(T285, T286)), X14), maxsort(X14, T289)))), SCOPE: 34, CLAUSE: 10\n\nT284 is ground\nT285 is ground\nT286 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: ','(max(.(t, T286), T288), ','(del(T288, .(f, .(t, T286)), X14), maxsort(X14, T289)))\n\nT286 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, T286), T288)\n\nT286 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: ','(del(T292, .(f, .(t, T286)), X14), maxsort(X14, T293))\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: del(T292, .(f, .(t, T286)), X14)\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: maxsort(T298, T293)\n\nT298 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 4 | del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 5 | del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 6\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 5 | del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 6\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 5\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: del(T292, .(f, .(t, T286)), X14), SCOPE: 35, CLAUSE: 6\n\nT286 is ground\nT292 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: eq(T312, f)\n\nT312 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="245: eq(T312, f), SCOPE: 36, CLAUSE: 0 | eq(T312, f), SCOPE: 36, CLAUSE: 1\n\nT312 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: eq(T312, f), SCOPE: 36, CLAUSE: 1\n\nT312 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(T322, f), del(T322, .(t, T323), X623))\n\nT322 is ground\nT323 is ground\nX623 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: ','(neq(T322, f), del(T322, .(t, T323), X623)), SCOPE: 37, CLAUSE: 2 | ','(neq(T322, f), del(T322, .(t, T323), X623)), SCOPE: 37, CLAUSE: 3\n\nT322 is ground\nT323 is ground\nX623 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="252: ','(neq(T322, f), del(T322, .(t, T323), X623)), SCOPE: 37, CLAUSE: 2\n\nT322 is ground\nT323 is ground\nX623 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
253 [label="253: ','(neq(T322, f), del(T322, .(t, T323), X623)), SCOPE: 37, CLAUSE: 3\n\nT322 is ground\nT323 is ground\nX623 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
254 [label="254: del(t, .(t, T323), X623)\n\nT323 is ground\nX623 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 substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 258 [arrowhead = none ];
258 -> 3;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nX10 -> 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 = lightblue];
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 = white];
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 = white];
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(.(X23, []), X23).\nand substitution\nT7 -> T17\nX23 -> 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 = lightblue];
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:\nT22 is ground\nreplacements:\nX14 -> T22", 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 -> T22\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 = white];
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(X48, .(X49, X50), X50) :- eq(X48, X49).\nand substitution\nT17 -> T31\nX48 -> T31\nX49 -> T31\nX50 -> []\nX14 -> []", fontsize=14, style = filled, fillcolor = white];
19 -> 279 [arrowhead = none ];
279 -> 21;

280 [label="ONLY EVAL with clause\ndel(X62, .(X63, X64), .(X63, X65)) :- ','(neq(X62, X63), del(X62, X64, X65)).\nand substitution\nT17 -> T35\nX62 -> T35\nX63 -> T35\nX64 -> []\nX65 -> X66\nX14 -> .(T35, X66)", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT31 -> t", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 284 [arrowhead = none ];
284 -> 25;

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

286 [label="EVAL with clause\neq(f, f).\nand substitution\nT31 -> f", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 286 [arrowhead = none ];
286 -> 28;

287 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
32 -> 291 [arrowhead = none ];
291 -> 33;

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

293 [label="EVAL with clause\nmax(.(X84, .(X85, X86)), X87) :- ','(ge(X84, X85), max(.(X84, X86), X87)).\nand substitution\nT7 -> T53\nX84 -> T53\nX85 -> T54\nX86 -> T55\nT8 -> .(T54, T55)\nT11 -> T57\nX87 -> T57\nT56 -> T57\nT12 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 293 [arrowhead = none ];
293 -> 37;

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

295 [label="EVAL with clause\nmax(.(X564, .(X565, X566)), X567) :- ','(gt(X565, X564), max(.(X565, X566), X567)).\nand substitution\nT7 -> T284\nX564 -> T284\nX565 -> T285\nX566 -> T286\nT8 -> .(T285, T286)\nT11 -> T288\nX567 -> T288\nT287 -> T288\nT12 -> T289", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 295 [arrowhead = none ];
295 -> 231;

296 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT53 -> t\nT54 -> t", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 300 [arrowhead = none ];
300 -> 42;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT61 is ground\nreplacements:\nT57 -> T61\nT58 -> T62", 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:\nT95 is ground\nreplacements:\nX14 -> T95", 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 = white];
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(.(X102, []), X102).\nand substitution\nX102 -> t\nT55 -> []\nT57 -> t", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 312 [arrowhead = none ];
312 -> 50;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X120, .(X121, X122)), X123) :- ','(ge(X120, X121), max(.(X120, X122), X123)).\nand substitution\nX120 -> t\nX121 -> T75\nX122 -> T76\nT55 -> .(T75, T76)\nT57 -> T78\nX123 -> T78\nT77 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 317 [arrowhead = none ];
317 -> 55;

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

319 [label="EVAL with clause\nmax(.(X142, .(X143, X144)), X145) :- ','(gt(X143, X142), max(.(X143, X144), X145)).\nand substitution\nX142 -> t\nX143 -> T85\nX144 -> T86\nT55 -> .(T85, T86)\nT57 -> T88\nX145 -> T88\nT87 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 319 [arrowhead = none ];
319 -> 67;

320 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT75 -> t", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 324 [arrowhead = none ];
324 -> 60;

325 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT55 -> T76\nT57 -> T78", 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 substitution\nT75 -> f", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 329 [arrowhead = none ];
329 -> 64;

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

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

332 [label="INSTANCE with matching:\nT55 -> T76\nT57 -> T78", 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 = white];
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 -> T95\nT2 -> T62", 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 = white];
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(X175, .(X176, X177), X177) :- eq(X175, X176).\nand substitution\nT61 -> T109\nX175 -> T109\nX176 -> t\nT55 -> T110\nX177 -> .(t, T110)\nX14 -> .(t, T110)", fontsize=14, style = filled, fillcolor = white];
75 -> 340 [arrowhead = none ];
340 -> 77;

341 [label="ONLY EVAL with clause\ndel(X193, .(X194, X195), .(X194, X196)) :- ','(neq(X193, X194), del(X193, X195, X196)).\nand substitution\nT61 -> T119\nX193 -> T119\nX194 -> t\nT55 -> T120\nX195 -> .(t, T120)\nX196 -> X197\nX14 -> .(t, X197)", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT109 -> t", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 345 [arrowhead = none ];
345 -> 81;

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

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

351 [label="EVAL with clause\nneq(f, t).\nand substitution\nT119 -> f", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 351 [arrowhead = none ];
351 -> 88;

352 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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(X221, .(X222, X223), X223) :- eq(X221, X222).\nand substitution\nX221 -> f\nX222 -> t\nT120 -> T130\nX223 -> T130\nX197 -> T130", fontsize=14, style = filled, fillcolor = white];
92 -> 357 [arrowhead = none ];
357 -> 94;

358 [label="ONLY EVAL with clause\ndel(X236, .(X237, X238), .(X237, X239)) :- ','(neq(X236, X237), del(X236, X238, X239)).\nand substitution\nX236 -> f\nX237 -> t\nT120 -> T133\nX238 -> T133\nX239 -> X240\nX197 -> .(t, X240)", fontsize=14, style = filled, fillcolor = white];
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 = white];
95 -> 360 [arrowhead = none ];
360 -> 96;

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

364 [label="ONLY EVAL with clause\nneq(f, t).\nand substitution", fontsize=14, style = filled, fillcolor = white];
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(X251, [], []).\nand substitution\nX251 -> f\nT133 -> []\nX240 -> []", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 368 [arrowhead = none ];
368 -> 105;

369 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X265, .(X266, X267), X267) :- eq(X265, X266).\nand substitution\nX265 -> f\nX266 -> T142\nX267 -> T143\nT133 -> .(T142, T143)\nX240 -> T143", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 373 [arrowhead = none ];
373 -> 110;

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

375 [label="EVAL with clause\ndel(X281, .(X282, X283), .(X282, X284)) :- ','(neq(X281, X282), del(X281, X283, X284)).\nand substitution\nX281 -> f\nX282 -> T148\nX283 -> T149\nT133 -> .(T148, T149)\nX284 -> X285\nX240 -> .(T148, X285)", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 375 [arrowhead = none ];
375 -> 117;

376 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
112 -> 378 [arrowhead = none ];
378 -> 113;

379 [label="EVAL with clause\neq(f, f).\nand substitution\nT142 -> f", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 379 [arrowhead = none ];
379 -> 114;

380 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
119 -> 383 [arrowhead = none ];
383 -> 120;

384 [label="EVAL with clause\nneq(f, t).\nand substitution\nT148 -> t", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 384 [arrowhead = none ];
384 -> 121;

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

386 [label="INSTANCE with matching:\nT133 -> T149\nX240 -> X285", 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 substitution\nT53 -> t\nT54 -> f", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 387 [arrowhead = none ];
387 -> 125;

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

389 [label="EVAL with clause\nge(f, f).\nand substitution\nT53 -> f\nT54 -> f", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 389 [arrowhead = none ];
389 -> 148;

390 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT150 is ground\nreplacements:\nT57 -> T150\nT58 -> T151", 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:\nT156 is ground\nreplacements:\nX14 -> T156", 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 -> T156\nT2 -> T151", 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 = white];
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(X321, .(X322, X323), X323) :- eq(X321, X322).\nand substitution\nT150 -> T170\nX321 -> T170\nX322 -> t\nT55 -> T171\nX323 -> .(f, T171)\nX14 -> .(f, T171)", fontsize=14, style = filled, fillcolor = white];
133 -> 401 [arrowhead = none ];
401 -> 135;

402 [label="ONLY EVAL with clause\ndel(X339, .(X340, X341), .(X340, X342)) :- ','(neq(X339, X340), del(X339, X341, X342)).\nand substitution\nT150 -> T180\nX339 -> T180\nX340 -> t\nT55 -> T181\nX341 -> .(f, T181)\nX342 -> X343\nX14 -> .(t, X343)", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT170 -> t", fontsize=14, style = filled, fillcolor = lightblue];
137 -> 406 [arrowhead = none ];
406 -> 139;

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

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

412 [label="EVAL with clause\nneq(f, t).\nand substitution\nT180 -> f", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 412 [arrowhead = none ];
412 -> 146;

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

414 [label="INSTANCE with matching:\nT133 -> .(f, T181)\nX240 -> X343", 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:\nT187 is ground\nreplacements:\nT57 -> T187\nT58 -> T188", 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:\nT221 is ground\nreplacements:\nX14 -> T221", 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 = white];
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(.(X362, []), X362).\nand substitution\nX362 -> f\nT55 -> []\nT57 -> f", fontsize=14, style = filled, fillcolor = lightblue];
154 -> 423 [arrowhead = none ];
423 -> 156;

424 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X380, .(X381, X382)), X383) :- ','(ge(X380, X381), max(.(X380, X382), X383)).\nand substitution\nX380 -> f\nX381 -> T201\nX382 -> T202\nT55 -> .(T201, T202)\nT57 -> T204\nX383 -> T204\nT203 -> T204", fontsize=14, style = filled, fillcolor = lightblue];
159 -> 428 [arrowhead = none ];
428 -> 161;

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

430 [label="EVAL with clause\nmax(.(X400, .(X401, X402)), X403) :- ','(gt(X401, X400), max(.(X401, X402), X403)).\nand substitution\nX400 -> f\nX401 -> T211\nX402 -> T212\nT55 -> .(T211, T212)\nT57 -> T214\nX403 -> T214\nT213 -> T214", fontsize=14, style = filled, fillcolor = lightblue];
160 -> 430 [arrowhead = none ];
430 -> 168;

431 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
163 -> 433 [arrowhead = none ];
433 -> 164;

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

435 [label="EVAL with clause\nge(f, f).\nand substitution\nT201 -> f", fontsize=14, style = filled, fillcolor = lightblue];
165 -> 435 [arrowhead = none ];
435 -> 166;

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

437 [label="INSTANCE with matching:\nT55 -> T202\nT57 -> T204", 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 substitution\nT211 -> t", fontsize=14, style = filled, fillcolor = lightblue];
170 -> 439 [arrowhead = none ];
439 -> 171;

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

441 [label="INSTANCE with matching:\nT55 -> T212\nT57 -> T214", 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 -> T221\nT2 -> T188", 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 = white];
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(X435, .(X436, X437), X437) :- eq(X435, X436).\nand substitution\nT187 -> T235\nX435 -> T235\nX436 -> f\nT55 -> T236\nX437 -> .(f, T236)\nX14 -> .(f, T236)", fontsize=14, style = filled, fillcolor = white];
177 -> 447 [arrowhead = none ];
447 -> 179;

448 [label="ONLY EVAL with clause\ndel(X453, .(X454, X455), .(X454, X456)) :- ','(neq(X453, X454), del(X453, X455, X456)).\nand substitution\nT187 -> T245\nX453 -> T245\nX454 -> f\nT55 -> T246\nX455 -> .(f, T246)\nX456 -> X457\nX14 -> .(f, X457)", fontsize=14, style = filled, fillcolor = white];
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 = white];
180 -> 450 [arrowhead = none ];
450 -> 181;

451 [label="EVAL with clause\neq(f, f).\nand substitution\nT235 -> f", fontsize=14, style = filled, fillcolor = lightblue];
181 -> 451 [arrowhead = none ];
451 -> 182;

452 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT245 -> t", fontsize=14, style = filled, fillcolor = lightblue];
187 -> 457 [arrowhead = none ];
457 -> 189;

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

459 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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 = white];
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(X481, .(X482, X483), X483) :- eq(X481, X482).\nand substitution\nX481 -> t\nX482 -> f\nT246 -> T256\nX483 -> T256\nX457 -> T256", fontsize=14, style = filled, fillcolor = white];
193 -> 464 [arrowhead = none ];
464 -> 195;

465 [label="ONLY EVAL with clause\ndel(X496, .(X497, X498), .(X497, X499)) :- ','(neq(X496, X497), del(X496, X498, X499)).\nand substitution\nX496 -> t\nX497 -> f\nT246 -> T259\nX498 -> T259\nX499 -> X500\nX457 -> .(f, X500)", fontsize=14, style = filled, fillcolor = white];
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 = white];
196 -> 467 [arrowhead = none ];
467 -> 197;

468 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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 = white];
201 -> 472 [arrowhead = none ];
472 -> 203;

473 [label="BACKTRACK\nfor clause: neq(f, t)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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(X513, [], []).\nand substitution\nX513 -> t\nT259 -> []\nX500 -> []", fontsize=14, style = filled, fillcolor = lightblue];
205 -> 477 [arrowhead = none ];
477 -> 207;

478 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X527, .(X528, X529), X529) :- eq(X527, X528).\nand substitution\nX527 -> t\nX528 -> T268\nX529 -> T269\nT259 -> .(T268, T269)\nX500 -> T269", fontsize=14, style = filled, fillcolor = lightblue];
210 -> 482 [arrowhead = none ];
482 -> 212;

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

484 [label="EVAL with clause\ndel(X543, .(X544, X545), .(X544, X546)) :- ','(neq(X543, X544), del(X543, X545, X546)).\nand substitution\nX543 -> t\nX544 -> T274\nX545 -> T275\nT259 -> .(T274, T275)\nX546 -> X547\nX500 -> .(T274, X547)", fontsize=14, style = filled, fillcolor = lightblue];
211 -> 484 [arrowhead = none ];
484 -> 221;

485 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT268 -> t", fontsize=14, style = filled, fillcolor = lightblue];
215 -> 489 [arrowhead = none ];
489 -> 217;

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

491 [label="BACKTRACK\nfor clause: eq(f, f)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT274 -> f", fontsize=14, style = filled, fillcolor = lightblue];
224 -> 496 [arrowhead = none ];
496 -> 226;

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

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

499 [label="INSTANCE with matching:\nT259 -> T275\nX500 -> X547", 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 substitution\nT285 -> t\nT284 -> f", fontsize=14, style = filled, fillcolor = lightblue];
233 -> 501 [arrowhead = none ];
501 -> 234;

502 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT292 is ground\nreplacements:\nT288 -> T292\nT289 -> T293", fontsize=14, style = filled, fillcolor = violet];
234 -> 504 [arrowhead = none ];
504 -> 237;

505 [label="INSTANCE with matching:\nT55 -> T286\nT57 -> T288", 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:\nT298 is ground\nreplacements:\nX14 -> T298", 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 -> T298\nT2 -> T293", 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 = white];
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(X601, .(X602, X603), X603) :- eq(X601, X602).\nand substitution\nT292 -> T312\nX601 -> T312\nX602 -> f\nT286 -> T313\nX603 -> .(t, T313)\nX14 -> .(t, T313)", fontsize=14, style = filled, fillcolor = white];
242 -> 513 [arrowhead = none ];
513 -> 244;

514 [label="ONLY EVAL with clause\ndel(X619, .(X620, X621), .(X620, X622)) :- ','(neq(X619, X620), del(X619, X621, X622)).\nand substitution\nT292 -> T322\nX619 -> T322\nX620 -> f\nT286 -> T323\nX621 -> .(t, T323)\nX622 -> X623\nX14 -> .(f, X623)", fontsize=14, style = filled, fillcolor = white];
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 = white];
245 -> 516 [arrowhead = none ];
516 -> 246;

517 [label="EVAL with clause\neq(f, f).\nand substitution\nT312 -> f", fontsize=14, style = filled, fillcolor = lightblue];
246 -> 517 [arrowhead = none ];
517 -> 247;

518 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT322 -> t", fontsize=14, style = filled, fillcolor = lightblue];
252 -> 523 [arrowhead = none ];
523 -> 254;

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

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

526 [label="INSTANCE with matching:\nT259 -> .(t, T323)\nX500 -> X623", fontsize=14, style = filled, fillcolor = lightgrey];
254 -> 526 [arrowhead = none , style=dashed];
526 -> 203[style=dashed];

}

(2) Obligation:

Clauses:

max44([], t).
max44(.(t, T76), T78) :- max44(T76, T78).
max44(.(f, T76), T78) :- max44(T76, T78).
del101([], []).
del101(.(f, T143), T143).
del101(.(t, T149), .(t, X285)) :- del101(T149, X285).
max150([], f).
max150(.(f, T202), T204) :- max150(T202, T204).
max150(.(t, T212), T214) :- max44(T212, T214).
del203([], []).
del203(.(t, T269), T269).
del203(.(f, T275), .(f, X547)) :- del203(T275, X547).
del15(t, []).
del15(f, []).
del71(t, T110, .(t, T110)).
del71(f, T133, .(t, .(t, X240))) :- del101(T133, X240).
del129(t, T171, .(f, T171)).
del129(f, T181, .(t, X343)) :- del101(.(f, T181), X343).
del173(f, T236, .(f, T236)).
del173(t, T259, .(f, .(f, X500))) :- del203(T259, X500).
del238(f, T313, .(t, T313)).
del238(t, T323, .(f, X623)) :- del203(.(t, T323), X623).
maxsort1([], []).
maxsort1(.(T17, []), .(T17, T18)) :- del15(T17, X14).
maxsort1(.(T17, []), .(T17, T18)) :- ','(del15(T17, T22), maxsort1(T22, T18)).
maxsort1(.(t, .(t, T55)), .(T57, T58)) :- max44(T55, T57).
maxsort1(.(t, .(t, T55)), .(T61, T62)) :- ','(max44(T55, T61), del71(T61, T55, X14)).
maxsort1(.(t, .(t, T55)), .(T61, T62)) :- ','(max44(T55, T61), ','(del71(T61, T55, T95), maxsort1(T95, T62))).
maxsort1(.(t, .(f, T55)), .(T57, T58)) :- max44(T55, T57).
maxsort1(.(t, .(f, T55)), .(T150, T151)) :- ','(max44(T55, T150), del129(T150, T55, X14)).
maxsort1(.(t, .(f, T55)), .(T150, T151)) :- ','(max44(T55, T150), ','(del129(T150, T55, T156), maxsort1(T156, T151))).
maxsort1(.(f, .(f, T55)), .(T57, T58)) :- max150(T55, T57).
maxsort1(.(f, .(f, T55)), .(T187, T188)) :- ','(max150(T55, T187), del173(T187, T55, X14)).
maxsort1(.(f, .(f, T55)), .(T187, T188)) :- ','(max150(T55, T187), ','(del173(T187, T55, T221), maxsort1(T221, T188))).
maxsort1(.(f, .(t, T286)), .(T288, T289)) :- max44(T286, T288).
maxsort1(.(f, .(t, T286)), .(T292, T293)) :- ','(max44(T286, T292), del238(T292, T286, X14)).
maxsort1(.(f, .(t, T286)), .(T292, T293)) :- ','(max44(T286, T292), ','(del238(T292, T286, T298), maxsort1(T298, T293))).

Queries:

maxsort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
maxsort1_in: (b,f)
max44_in: (b,f)
del71_in: (b,b,f)
del101_in: (b,f)
del129_in: (b,b,f)
max150_in: (b,f)
del173_in: (b,b,f)
del203_in: (b,f)
del238_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)

(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:

MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U11_GA(T17, T18, del15_in_ga(T17, X14))
MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → DEL15_IN_GA(T17, X14)
MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, del15_in_ga(T17, T22))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → U13_GA(T17, T18, maxsort1_in_ga(T22, T18))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → MAXSORT1_IN_GA(T22, T18)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T57, T58)) → U14_GA(T55, T57, T58, max44_in_ga(T55, T57))
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T57, T58)) → MAX44_IN_GA(T55, T57)
MAX44_IN_GA(.(t, T76), T78) → U1_GA(T76, T78, max44_in_ga(T76, T78))
MAX44_IN_GA(.(t, T76), T78) → MAX44_IN_GA(T76, T78)
MAX44_IN_GA(.(f, T76), T78) → U2_GA(T76, T78, max44_in_ga(T76, T78))
MAX44_IN_GA(.(f, T76), T78) → MAX44_IN_GA(T76, T78)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T61, T62)) → U15_GA(T55, T61, T62, max44_in_ga(T55, T61))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U16_GA(T55, T61, T62, del71_in_gga(T61, T55, X14))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → DEL71_IN_GGA(T61, T55, X14)
DEL71_IN_GGA(f, T133, .(t, .(t, X240))) → U7_GGA(T133, X240, del101_in_ga(T133, X240))
DEL71_IN_GGA(f, T133, .(t, .(t, X240))) → DEL101_IN_GA(T133, X240)
DEL101_IN_GA(.(t, T149), .(t, X285)) → U3_GA(T149, X285, del101_in_ga(T149, X285))
DEL101_IN_GA(.(t, T149), .(t, X285)) → DEL101_IN_GA(T149, X285)
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U17_GA(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_GA(T55, T61, T62, maxsort1_in_ga(T95, T62))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → MAXSORT1_IN_GA(T95, T62)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T57, T58)) → U19_GA(T55, T57, T58, max44_in_ga(T55, T57))
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T57, T58)) → MAX44_IN_GA(T55, T57)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T150, T151)) → U20_GA(T55, T150, T151, max44_in_ga(T55, T150))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U21_GA(T55, T150, T151, del129_in_gga(T150, T55, X14))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → DEL129_IN_GGA(T150, T55, X14)
DEL129_IN_GGA(f, T181, .(t, X343)) → U8_GGA(T181, X343, del101_in_ga(.(f, T181), X343))
DEL129_IN_GGA(f, T181, .(t, X343)) → DEL101_IN_GA(.(f, T181), X343)
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U22_GA(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_GA(T55, T150, T151, maxsort1_in_ga(T156, T151))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → MAXSORT1_IN_GA(T156, T151)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T57, T58)) → U24_GA(T55, T57, T58, max150_in_ga(T55, T57))
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T57, T58)) → MAX150_IN_GA(T55, T57)
MAX150_IN_GA(.(f, T202), T204) → U4_GA(T202, T204, max150_in_ga(T202, T204))
MAX150_IN_GA(.(f, T202), T204) → MAX150_IN_GA(T202, T204)
MAX150_IN_GA(.(t, T212), T214) → U5_GA(T212, T214, max44_in_ga(T212, T214))
MAX150_IN_GA(.(t, T212), T214) → MAX44_IN_GA(T212, T214)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T187, T188)) → U25_GA(T55, T187, T188, max150_in_ga(T55, T187))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U26_GA(T55, T187, T188, del173_in_gga(T187, T55, X14))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → DEL173_IN_GGA(T187, T55, X14)
DEL173_IN_GGA(t, T259, .(f, .(f, X500))) → U9_GGA(T259, X500, del203_in_ga(T259, X500))
DEL173_IN_GGA(t, T259, .(f, .(f, X500))) → DEL203_IN_GA(T259, X500)
DEL203_IN_GA(.(f, T275), .(f, X547)) → U6_GA(T275, X547, del203_in_ga(T275, X547))
DEL203_IN_GA(.(f, T275), .(f, X547)) → DEL203_IN_GA(T275, X547)
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U27_GA(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_GA(T55, T187, T188, maxsort1_in_ga(T221, T188))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → MAXSORT1_IN_GA(T221, T188)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T288, T289)) → U29_GA(T286, T288, T289, max44_in_ga(T286, T288))
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T288, T289)) → MAX44_IN_GA(T286, T288)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T292, T293)) → U30_GA(T286, T292, T293, max44_in_ga(T286, T292))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U31_GA(T286, T292, T293, del238_in_gga(T292, T286, X14))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → DEL238_IN_GGA(T292, T286, X14)
DEL238_IN_GGA(t, T323, .(f, X623)) → U10_GGA(T323, X623, del203_in_ga(.(t, T323), X623))
DEL238_IN_GGA(t, T323, .(f, X623)) → DEL203_IN_GA(.(t, T323), X623)
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U32_GA(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_GA(T286, T292, T293, maxsort1_in_ga(T298, T293))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → MAXSORT1_IN_GA(T298, T293)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
MAXSORT1_IN_GA(x1, x2)  =  MAXSORT1_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x3)
DEL15_IN_GA(x1, x2)  =  DEL15_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)
MAX44_IN_GA(x1, x2)  =  MAX44_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)
DEL71_IN_GGA(x1, x2, x3)  =  DEL71_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
DEL101_IN_GA(x1, x2)  =  DEL101_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)
DEL129_IN_GGA(x1, x2, x3)  =  DEL129_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)
MAX150_IN_GA(x1, x2)  =  MAX150_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)
DEL173_IN_GGA(x1, x2, x3)  =  DEL173_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3)  =  U9_GGA(x3)
DEL203_IN_GA(x1, x2)  =  DEL203_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)
DEL238_IN_GGA(x1, x2, x3)  =  DEL238_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:

MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U11_GA(T17, T18, del15_in_ga(T17, X14))
MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → DEL15_IN_GA(T17, X14)
MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, del15_in_ga(T17, T22))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → U13_GA(T17, T18, maxsort1_in_ga(T22, T18))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → MAXSORT1_IN_GA(T22, T18)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T57, T58)) → U14_GA(T55, T57, T58, max44_in_ga(T55, T57))
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T57, T58)) → MAX44_IN_GA(T55, T57)
MAX44_IN_GA(.(t, T76), T78) → U1_GA(T76, T78, max44_in_ga(T76, T78))
MAX44_IN_GA(.(t, T76), T78) → MAX44_IN_GA(T76, T78)
MAX44_IN_GA(.(f, T76), T78) → U2_GA(T76, T78, max44_in_ga(T76, T78))
MAX44_IN_GA(.(f, T76), T78) → MAX44_IN_GA(T76, T78)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T61, T62)) → U15_GA(T55, T61, T62, max44_in_ga(T55, T61))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U16_GA(T55, T61, T62, del71_in_gga(T61, T55, X14))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → DEL71_IN_GGA(T61, T55, X14)
DEL71_IN_GGA(f, T133, .(t, .(t, X240))) → U7_GGA(T133, X240, del101_in_ga(T133, X240))
DEL71_IN_GGA(f, T133, .(t, .(t, X240))) → DEL101_IN_GA(T133, X240)
DEL101_IN_GA(.(t, T149), .(t, X285)) → U3_GA(T149, X285, del101_in_ga(T149, X285))
DEL101_IN_GA(.(t, T149), .(t, X285)) → DEL101_IN_GA(T149, X285)
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U17_GA(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_GA(T55, T61, T62, maxsort1_in_ga(T95, T62))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → MAXSORT1_IN_GA(T95, T62)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T57, T58)) → U19_GA(T55, T57, T58, max44_in_ga(T55, T57))
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T57, T58)) → MAX44_IN_GA(T55, T57)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T150, T151)) → U20_GA(T55, T150, T151, max44_in_ga(T55, T150))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U21_GA(T55, T150, T151, del129_in_gga(T150, T55, X14))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → DEL129_IN_GGA(T150, T55, X14)
DEL129_IN_GGA(f, T181, .(t, X343)) → U8_GGA(T181, X343, del101_in_ga(.(f, T181), X343))
DEL129_IN_GGA(f, T181, .(t, X343)) → DEL101_IN_GA(.(f, T181), X343)
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U22_GA(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_GA(T55, T150, T151, maxsort1_in_ga(T156, T151))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → MAXSORT1_IN_GA(T156, T151)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T57, T58)) → U24_GA(T55, T57, T58, max150_in_ga(T55, T57))
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T57, T58)) → MAX150_IN_GA(T55, T57)
MAX150_IN_GA(.(f, T202), T204) → U4_GA(T202, T204, max150_in_ga(T202, T204))
MAX150_IN_GA(.(f, T202), T204) → MAX150_IN_GA(T202, T204)
MAX150_IN_GA(.(t, T212), T214) → U5_GA(T212, T214, max44_in_ga(T212, T214))
MAX150_IN_GA(.(t, T212), T214) → MAX44_IN_GA(T212, T214)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T187, T188)) → U25_GA(T55, T187, T188, max150_in_ga(T55, T187))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U26_GA(T55, T187, T188, del173_in_gga(T187, T55, X14))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → DEL173_IN_GGA(T187, T55, X14)
DEL173_IN_GGA(t, T259, .(f, .(f, X500))) → U9_GGA(T259, X500, del203_in_ga(T259, X500))
DEL173_IN_GGA(t, T259, .(f, .(f, X500))) → DEL203_IN_GA(T259, X500)
DEL203_IN_GA(.(f, T275), .(f, X547)) → U6_GA(T275, X547, del203_in_ga(T275, X547))
DEL203_IN_GA(.(f, T275), .(f, X547)) → DEL203_IN_GA(T275, X547)
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U27_GA(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_GA(T55, T187, T188, maxsort1_in_ga(T221, T188))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → MAXSORT1_IN_GA(T221, T188)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T288, T289)) → U29_GA(T286, T288, T289, max44_in_ga(T286, T288))
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T288, T289)) → MAX44_IN_GA(T286, T288)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T292, T293)) → U30_GA(T286, T292, T293, max44_in_ga(T286, T292))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U31_GA(T286, T292, T293, del238_in_gga(T292, T286, X14))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → DEL238_IN_GGA(T292, T286, X14)
DEL238_IN_GGA(t, T323, .(f, X623)) → U10_GGA(T323, X623, del203_in_ga(.(t, T323), X623))
DEL238_IN_GGA(t, T323, .(f, X623)) → DEL203_IN_GA(.(t, T323), X623)
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U32_GA(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_GA(T286, T292, T293, maxsort1_in_ga(T298, T293))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → MAXSORT1_IN_GA(T298, T293)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
MAXSORT1_IN_GA(x1, x2)  =  MAXSORT1_IN_GA(x1)
U11_GA(x1, x2, x3)  =  U11_GA(x3)
DEL15_IN_GA(x1, x2)  =  DEL15_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)
MAX44_IN_GA(x1, x2)  =  MAX44_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)
DEL71_IN_GGA(x1, x2, x3)  =  DEL71_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
DEL101_IN_GA(x1, x2)  =  DEL101_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)
DEL129_IN_GGA(x1, x2, x3)  =  DEL129_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)
MAX150_IN_GA(x1, x2)  =  MAX150_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)
DEL173_IN_GGA(x1, x2, x3)  =  DEL173_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3)  =  U9_GGA(x3)
DEL203_IN_GA(x1, x2)  =  DEL203_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)
DEL238_IN_GGA(x1, x2, x3)  =  DEL238_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

(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:

DEL203_IN_GA(.(f, T275), .(f, X547)) → DEL203_IN_GA(T275, X547)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
DEL203_IN_GA(x1, x2)  =  DEL203_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

DEL203_IN_GA(.(f, T275), .(f, X547)) → DEL203_IN_GA(T275, X547)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
f  =  f
DEL203_IN_GA(x1, x2)  =  DEL203_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:

DEL203_IN_GA(.(f, T275)) → DEL203_IN_GA(T275)

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:

• DEL203_IN_GA(.(f, T275)) → DEL203_IN_GA(T275)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

DEL101_IN_GA(.(t, T149), .(t, X285)) → DEL101_IN_GA(T149, X285)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
DEL101_IN_GA(x1, x2)  =  DEL101_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

DEL101_IN_GA(.(t, T149), .(t, X285)) → DEL101_IN_GA(T149, X285)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
t  =  t
DEL101_IN_GA(x1, x2)  =  DEL101_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:

DEL101_IN_GA(.(t, T149)) → DEL101_IN_GA(T149)

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:

• DEL101_IN_GA(.(t, T149)) → DEL101_IN_GA(T149)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

MAX44_IN_GA(.(f, T76), T78) → MAX44_IN_GA(T76, T78)
MAX44_IN_GA(.(t, T76), T78) → MAX44_IN_GA(T76, T78)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
MAX44_IN_GA(x1, x2)  =  MAX44_IN_GA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

MAX44_IN_GA(.(f, T76), T78) → MAX44_IN_GA(T76, T78)
MAX44_IN_GA(.(t, T76), T78) → MAX44_IN_GA(T76, T78)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
t  =  t
f  =  f
MAX44_IN_GA(x1, x2)  =  MAX44_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:

MAX44_IN_GA(.(f, T76)) → MAX44_IN_GA(T76)
MAX44_IN_GA(.(t, T76)) → MAX44_IN_GA(T76)

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:

• MAX44_IN_GA(.(f, T76)) → MAX44_IN_GA(T76)
  The graph contains the following edges 1 > 1

• MAX44_IN_GA(.(t, T76)) → MAX44_IN_GA(T76)
  The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

MAX150_IN_GA(.(f, T202), T204) → MAX150_IN_GA(T202, T204)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
MAX150_IN_GA(x1, x2)  =  MAX150_IN_GA(x1)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MAX150_IN_GA(.(f, T202), T204) → MAX150_IN_GA(T202, T204)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
f  =  f
MAX150_IN_GA(x1, x2)  =  MAX150_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:

MAX150_IN_GA(.(f, T202)) → MAX150_IN_GA(T202)

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:

• MAX150_IN_GA(.(f, T202)) → MAX150_IN_GA(T202)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, del15_in_ga(T17, T22))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → MAXSORT1_IN_GA(T22, T18)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T61, T62)) → U15_GA(T55, T61, T62, max44_in_ga(T55, T61))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U17_GA(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → MAXSORT1_IN_GA(T95, T62)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T150, T151)) → U20_GA(T55, T150, T151, max44_in_ga(T55, T150))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U22_GA(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → MAXSORT1_IN_GA(T156, T151)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T187, T188)) → U25_GA(T55, T187, T188, max150_in_ga(T55, T187))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U27_GA(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → MAXSORT1_IN_GA(T221, T188)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T292, T293)) → U30_GA(T286, T292, T293, max44_in_ga(T286, T292))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U32_GA(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → MAXSORT1_IN_GA(T298, T293)

The TRS R consists of the following rules:

maxsort1_in_ga([], []) → maxsort1_out_ga([], [])
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U11_ga(T17, T18, del15_in_ga(T17, X14))
del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
U11_ga(T17, T18, del15_out_ga(T17, X14)) → maxsort1_out_ga(.(T17, []), .(T17, T18))
maxsort1_in_ga(.(T17, []), .(T17, T18)) → U12_ga(T17, T18, del15_in_ga(T17, T22))
U12_ga(T17, T18, del15_out_ga(T17, T22)) → U13_ga(T17, T18, maxsort1_in_ga(T22, T18))
maxsort1_in_ga(.(t, .(t, T55)), .(T57, T58)) → U14_ga(T55, T57, T58, max44_in_ga(T55, T57))
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U14_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(t, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(t, T55)), .(T61, T62)) → U15_ga(T55, T61, T62, max44_in_ga(T55, T61))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U16_ga(T55, T61, T62, del71_in_gga(T61, T55, X14))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U16_ga(T55, T61, T62, del71_out_gga(T61, T55, X14)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U15_ga(T55, T61, T62, max44_out_ga(T55, T61)) → U17_ga(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_ga(T55, T61, T62, del71_out_gga(T61, T55, T95)) → U18_ga(T55, T61, T62, maxsort1_in_ga(T95, T62))
maxsort1_in_ga(.(t, .(f, T55)), .(T57, T58)) → U19_ga(T55, T57, T58, max44_in_ga(T55, T57))
U19_ga(T55, T57, T58, max44_out_ga(T55, T57)) → maxsort1_out_ga(.(t, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(t, .(f, T55)), .(T150, T151)) → U20_ga(T55, T150, T151, max44_in_ga(T55, T150))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U21_ga(T55, T150, T151, del129_in_gga(T150, T55, X14))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U21_ga(T55, T150, T151, del129_out_gga(T150, T55, X14)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U20_ga(T55, T150, T151, max44_out_ga(T55, T150)) → U22_ga(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_ga(T55, T150, T151, del129_out_gga(T150, T55, T156)) → U23_ga(T55, T150, T151, maxsort1_in_ga(T156, T151))
maxsort1_in_ga(.(f, .(f, T55)), .(T57, T58)) → U24_ga(T55, T57, T58, max150_in_ga(T55, T57))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U24_ga(T55, T57, T58, max150_out_ga(T55, T57)) → maxsort1_out_ga(.(f, .(f, T55)), .(T57, T58))
maxsort1_in_ga(.(f, .(f, T55)), .(T187, T188)) → U25_ga(T55, T187, T188, max150_in_ga(T55, T187))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U26_ga(T55, T187, T188, del173_in_gga(T187, T55, X14))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U26_ga(T55, T187, T188, del173_out_gga(T187, T55, X14)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U25_ga(T55, T187, T188, max150_out_ga(T55, T187)) → U27_ga(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_ga(T55, T187, T188, del173_out_gga(T187, T55, T221)) → U28_ga(T55, T187, T188, maxsort1_in_ga(T221, T188))
maxsort1_in_ga(.(f, .(t, T286)), .(T288, T289)) → U29_ga(T286, T288, T289, max44_in_ga(T286, T288))
U29_ga(T286, T288, T289, max44_out_ga(T286, T288)) → maxsort1_out_ga(.(f, .(t, T286)), .(T288, T289))
maxsort1_in_ga(.(f, .(t, T286)), .(T292, T293)) → U30_ga(T286, T292, T293, max44_in_ga(T286, T292))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U31_ga(T286, T292, T293, del238_in_gga(T292, T286, X14))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
U31_ga(T286, T292, T293, del238_out_gga(T292, T286, X14)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U30_ga(T286, T292, T293, max44_out_ga(T286, T292)) → U32_ga(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_ga(T286, T292, T293, del238_out_gga(T292, T286, T298)) → U33_ga(T286, T292, T293, maxsort1_in_ga(T298, T293))
U33_ga(T286, T292, T293, maxsort1_out_ga(T298, T293)) → maxsort1_out_ga(.(f, .(t, T286)), .(T292, T293))
U28_ga(T55, T187, T188, maxsort1_out_ga(T221, T188)) → maxsort1_out_ga(.(f, .(f, T55)), .(T187, T188))
U23_ga(T55, T150, T151, maxsort1_out_ga(T156, T151)) → maxsort1_out_ga(.(t, .(f, T55)), .(T150, T151))
U18_ga(T55, T61, T62, maxsort1_out_ga(T95, T62)) → maxsort1_out_ga(.(t, .(t, T55)), .(T61, T62))
U13_ga(T17, T18, maxsort1_out_ga(T22, T18)) → maxsort1_out_ga(.(T17, []), .(T17, T18))

The argument filtering Pi contains the following mapping:
maxsort1_in_ga(x1, x2)  =  maxsort1_in_ga(x1)
[]  =  []
maxsort1_out_ga(x1, x2)  =  maxsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_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)
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_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)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_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)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_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)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_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)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_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)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_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)
MAXSORT1_IN_GA(x1, x2)  =  MAXSORT1_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

(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:

MAXSORT1_IN_GA(.(T17, []), .(T17, T18)) → U12_GA(T17, T18, del15_in_ga(T17, T22))
U12_GA(T17, T18, del15_out_ga(T17, T22)) → MAXSORT1_IN_GA(T22, T18)
MAXSORT1_IN_GA(.(t, .(t, T55)), .(T61, T62)) → U15_GA(T55, T61, T62, max44_in_ga(T55, T61))
U15_GA(T55, T61, T62, max44_out_ga(T55, T61)) → U17_GA(T55, T61, T62, del71_in_gga(T61, T55, T95))
U17_GA(T55, T61, T62, del71_out_gga(T61, T55, T95)) → MAXSORT1_IN_GA(T95, T62)
MAXSORT1_IN_GA(.(t, .(f, T55)), .(T150, T151)) → U20_GA(T55, T150, T151, max44_in_ga(T55, T150))
U20_GA(T55, T150, T151, max44_out_ga(T55, T150)) → U22_GA(T55, T150, T151, del129_in_gga(T150, T55, T156))
U22_GA(T55, T150, T151, del129_out_gga(T150, T55, T156)) → MAXSORT1_IN_GA(T156, T151)
MAXSORT1_IN_GA(.(f, .(f, T55)), .(T187, T188)) → U25_GA(T55, T187, T188, max150_in_ga(T55, T187))
U25_GA(T55, T187, T188, max150_out_ga(T55, T187)) → U27_GA(T55, T187, T188, del173_in_gga(T187, T55, T221))
U27_GA(T55, T187, T188, del173_out_gga(T187, T55, T221)) → MAXSORT1_IN_GA(T221, T188)
MAXSORT1_IN_GA(.(f, .(t, T286)), .(T292, T293)) → U30_GA(T286, T292, T293, max44_in_ga(T286, T292))
U30_GA(T286, T292, T293, max44_out_ga(T286, T292)) → U32_GA(T286, T292, T293, del238_in_gga(T292, T286, T298))
U32_GA(T286, T292, T293, del238_out_gga(T292, T286, T298)) → MAXSORT1_IN_GA(T298, T293)

The TRS R consists of the following rules:

del15_in_ga(t, []) → del15_out_ga(t, [])
del15_in_ga(f, []) → del15_out_ga(f, [])
max44_in_ga([], t) → max44_out_ga([], t)
max44_in_ga(.(t, T76), T78) → U1_ga(T76, T78, max44_in_ga(T76, T78))
max44_in_ga(.(f, T76), T78) → U2_ga(T76, T78, max44_in_ga(T76, T78))
del71_in_gga(t, T110, .(t, T110)) → del71_out_gga(t, T110, .(t, T110))
del71_in_gga(f, T133, .(t, .(t, X240))) → U7_gga(T133, X240, del101_in_ga(T133, X240))
del129_in_gga(t, T171, .(f, T171)) → del129_out_gga(t, T171, .(f, T171))
del129_in_gga(f, T181, .(t, X343)) → U8_gga(T181, X343, del101_in_ga(.(f, T181), X343))
max150_in_ga([], f) → max150_out_ga([], f)
max150_in_ga(.(f, T202), T204) → U4_ga(T202, T204, max150_in_ga(T202, T204))
max150_in_ga(.(t, T212), T214) → U5_ga(T212, T214, max44_in_ga(T212, T214))
del173_in_gga(f, T236, .(f, T236)) → del173_out_gga(f, T236, .(f, T236))
del173_in_gga(t, T259, .(f, .(f, X500))) → U9_gga(T259, X500, del203_in_ga(T259, X500))
del238_in_gga(f, T313, .(t, T313)) → del238_out_gga(f, T313, .(t, T313))
del238_in_gga(t, T323, .(f, X623)) → U10_gga(T323, X623, del203_in_ga(.(t, T323), X623))
U1_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(t, T76), T78)
U2_ga(T76, T78, max44_out_ga(T76, T78)) → max44_out_ga(.(f, T76), T78)
U7_gga(T133, X240, del101_out_ga(T133, X240)) → del71_out_gga(f, T133, .(t, .(t, X240)))
U8_gga(T181, X343, del101_out_ga(.(f, T181), X343)) → del129_out_gga(f, T181, .(t, X343))
U4_ga(T202, T204, max150_out_ga(T202, T204)) → max150_out_ga(.(f, T202), T204)
U5_ga(T212, T214, max44_out_ga(T212, T214)) → max150_out_ga(.(t, T212), T214)
U9_gga(T259, X500, del203_out_ga(T259, X500)) → del173_out_gga(t, T259, .(f, .(f, X500)))
U10_gga(T323, X623, del203_out_ga(.(t, T323), X623)) → del238_out_gga(t, T323, .(f, X623))
del101_in_ga([], []) → del101_out_ga([], [])
del101_in_ga(.(f, T143), T143) → del101_out_ga(.(f, T143), T143)
del101_in_ga(.(t, T149), .(t, X285)) → U3_ga(T149, X285, del101_in_ga(T149, X285))
del203_in_ga([], []) → del203_out_ga([], [])
del203_in_ga(.(t, T269), T269) → del203_out_ga(.(t, T269), T269)
del203_in_ga(.(f, T275), .(f, X547)) → U6_ga(T275, X547, del203_in_ga(T275, X547))
U3_ga(T149, X285, del101_out_ga(T149, X285)) → del101_out_ga(.(t, T149), .(t, X285))
U6_ga(T275, X547, del203_out_ga(T275, X547)) → del203_out_ga(.(f, T275), .(f, X547))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
del15_in_ga(x1, x2)  =  del15_in_ga(x1)
t  =  t
del15_out_ga(x1, x2)  =  del15_out_ga(x2)
f  =  f
max44_in_ga(x1, x2)  =  max44_in_ga(x1)
max44_out_ga(x1, x2)  =  max44_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
del71_in_gga(x1, x2, x3)  =  del71_in_gga(x1, x2)
del71_out_gga(x1, x2, x3)  =  del71_out_gga(x3)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
del101_in_ga(x1, x2)  =  del101_in_ga(x1)
del101_out_ga(x1, x2)  =  del101_out_ga(x2)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
del129_in_gga(x1, x2, x3)  =  del129_in_gga(x1, x2)
del129_out_gga(x1, x2, x3)  =  del129_out_gga(x3)
U8_gga(x1, x2, x3)  =  U8_gga(x3)
max150_in_ga(x1, x2)  =  max150_in_ga(x1)
max150_out_ga(x1, x2)  =  max150_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
del173_in_gga(x1, x2, x3)  =  del173_in_gga(x1, x2)
del173_out_gga(x1, x2, x3)  =  del173_out_gga(x3)
U9_gga(x1, x2, x3)  =  U9_gga(x3)
del203_in_ga(x1, x2)  =  del203_in_ga(x1)
del203_out_ga(x1, x2)  =  del203_out_ga(x2)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del238_in_gga(x1, x2, x3)  =  del238_in_gga(x1, x2)
del238_out_gga(x1, x2, x3)  =  del238_out_gga(x3)
U10_gga(x1, x2, x3)  =  U10_gga(x3)
MAXSORT1_IN_GA(x1, x2)  =  MAXSORT1_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:

MAXSORT1_IN_GA(.(T17, [])) → U12_GA(del15_in_ga(T17))
U12_GA(del15_out_ga(T22)) → MAXSORT1_IN_GA(T22)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del15_in_ga(t) → del15_out_ga([])
del15_in_ga(f) → del15_out_ga([])
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

del15_in_ga(x0)
max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_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 MAXSORT1_IN_GA(.(T17, [])) → U12_GA(del15_in_ga(T17)) at position [0] we obtained the following new rules [LPAR04]:

MAXSORT1_IN_GA(.(t, [])) → U12_GA(del15_out_ga([]))
MAXSORT1_IN_GA(.(f, [])) → U12_GA(del15_out_ga([]))

(43) Obligation:

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

U12_GA(del15_out_ga(T22)) → MAXSORT1_IN_GA(T22)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)
MAXSORT1_IN_GA(.(t, [])) → U12_GA(del15_out_ga([]))
MAXSORT1_IN_GA(.(f, [])) → U12_GA(del15_out_ga([]))

The TRS R consists of the following rules:

del15_in_ga(t) → del15_out_ga([])
del15_in_ga(f) → del15_out_ga([])
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

del15_in_ga(x0)
max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_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(del15_out_ga(T22)) → MAXSORT1_IN_GA(T22)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)
MAXSORT1_IN_GA(.(t, [])) → U12_GA(del15_out_ga([]))
MAXSORT1_IN_GA(.(f, [])) → U12_GA(del15_out_ga([]))

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

del15_in_ga(x0)
max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_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].

del15_in_ga(x0)

(47) Obligation:

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

U12_GA(del15_out_ga(T22)) → MAXSORT1_IN_GA(T22)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)
MAXSORT1_IN_GA(.(t, [])) → U12_GA(del15_out_ga([]))
MAXSORT1_IN_GA(.(f, [])) → U12_GA(del15_out_ga([]))

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_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(del15_out_ga(T22)) → MAXSORT1_IN_GA(T22) we obtained the following new rules [LPAR04]:

U12_GA(del15_out_ga([])) → MAXSORT1_IN_GA([])

(49) Obligation:

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

MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)
MAXSORT1_IN_GA(.(t, [])) → U12_GA(del15_out_ga([]))
MAXSORT1_IN_GA(.(f, [])) → U12_GA(del15_out_ga([]))
U12_GA(del15_out_ga([])) → MAXSORT1_IN_GA([])

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_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(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U20_GA(T55, max44_out_ga(T150)) → U22_GA(del129_in_gga(T150, T55))

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

POL(U15_GA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(max44_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

POL(U17_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(del71_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(del71_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(MAXSORT1_IN_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(.(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(t) =

/ 0 \ \ 1 /

POL(max44_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(f) =

/ 1 \ \ 0 /

POL(U20_GA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U22_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(del129_in_gga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(del129_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U25_GA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(max150_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(max150_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U27_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(del173_in_gga(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(del173_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U30_GA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U32_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(del238_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(del238_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U7_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(del101_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL([]) =

/ 0 \ \ 0 /

POL(U1_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 1 /

·

x1

POL(U2_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U8_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U4_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U5_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_gga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(del203_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U10_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(del203_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U6_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(del101_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U3_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

(53) Obligation:

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

U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
MAXSORT1_IN_GA(.(t, .(f, T55))) → U20_GA(T55, max44_in_ga(T55))
U22_GA(del129_out_gga(T156)) → MAXSORT1_IN_GA(T156)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del129_in_gga(t, T171) → del129_out_gga(.(f, T171))
del129_in_gga(f, T181) → U8_gga(del101_in_ga(.(f, T181)))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
U8_gga(del101_out_ga(X343)) → del129_out_gga(.(t, X343))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(56) 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.

(57) Obligation:

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

U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
del129_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_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)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(58) 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].

del129_in_gga(x0, x1)
U8_gga(x0)

(59) Obligation:

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

U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U30_GA(T286, max44_out_ga(T292)) → U32_GA(del238_in_gga(T292, T286))

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

POL(U17_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(del71_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(MAXSORT1_IN_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(.(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(t) =

/ 0 \ \ 1 /

POL(U15_GA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(max44_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(max44_out_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(del71_in_gga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 1 0 /

·

x2

POL(f) =

/ 0 \ \ 0 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(max150_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(max150_out_ga(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(del173_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(del173_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U30_GA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(U32_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(del238_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(del238_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL([]) =

/ 0 \ \ 0 /

POL(U1_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U2_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U7_gga(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(del101_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U4_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U5_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(del203_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U10_gga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(del203_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U6_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(del101_out_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U3_ga(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

The following usable rules [FROCOS05] were oriented:

del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

(61) Obligation:

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

U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(t, T286))) → U30_GA(T286, max44_in_ga(T286))
U32_GA(del238_out_gga(T298)) → MAXSORT1_IN_GA(T298)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(62) DependencyGraphProof (EQUIVALENT transformation)

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

(63) Obligation:

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

MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)

The TRS R consists of the following rules:

del238_in_gga(f, T313) → del238_out_gga(.(t, T313))
del238_in_gga(t, T323) → U10_gga(del203_in_ga(.(t, T323)))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U10_gga(del203_out_ga(X623)) → del238_out_gga(.(f, X623))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(64) 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.

(65) Obligation:

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

MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)

The TRS R consists of the following rules:

del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
del238_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
U10_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(66) 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].

del238_in_gga(x0, x1)
U10_gga(x0)

(67) Obligation:

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

MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))
U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)

The TRS R consists of the following rules:

del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MAXSORT1_IN_GA(.(t, .(t, T55))) → U15_GA(T55, max44_in_ga(T55))

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

POL(MAXSORT1_IN_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(.(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(t) =

/ 0 \ \ 1 /

POL(U15_GA(x1, x2)) =

1

+

[ 0, 1 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(max44_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(max44_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U17_GA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(del71_in_gga(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(del71_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(f) =

/ 1 \ \ 0 /

POL(U25_GA(x1, x2)) =

0

+

[ 0, 1 ]

·

x1

+

[ 0, 0 ]

·

x2

POL(max150_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(max150_out_ga(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U27_GA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(del173_in_gga(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(del173_out_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL([]) =

/ 0 \ \ 0 /

POL(U1_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U2_ga(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U7_gga(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(del101_in_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U4_ga(x1)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U5_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U9_gga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(del203_in_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(del203_out_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U6_ga(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(del101_out_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U3_ga(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

The following usable rules [FROCOS05] were oriented:

max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

(69) Obligation:

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

U15_GA(T55, max44_out_ga(T61)) → U17_GA(del71_in_gga(T61, T55))
U17_GA(del71_out_gga(T95)) → MAXSORT1_IN_GA(T95)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))
U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)

The TRS R consists of the following rules:

del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_ga(x0)
U6_ga(x0)

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

(70) DependencyGraphProof (EQUIVALENT transformation)

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

(71) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))
max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del71_in_gga(t, T110) → del71_out_gga(.(t, T110))
del71_in_gga(f, T133) → U7_gga(del101_in_ga(T133))
del101_in_ga(.(f, T143)) → del101_out_ga(T143)
del101_in_ga([]) → del101_out_ga([])
del101_in_ga(.(t, T149)) → U3_ga(del101_in_ga(T149))
U7_gga(del101_out_ga(X240)) → del71_out_gga(.(t, .(t, X240)))
U3_ga(del101_out_ga(X285)) → del101_out_ga(.(t, X285))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_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:

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max44_in_ga(x0)
del71_in_gga(x0, x1)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U7_gga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del101_in_ga(x0)
del203_in_ga(x0)
U3_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].

del71_in_gga(x0, x1)
U7_gga(x0)
del101_in_ga(x0)
U3_ga(x0)

(75) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max44_in_ga(x0)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del203_in_ga(x0)
U6_ga(x0)

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

(76) 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 rules of the TRS R:

max150_in_ga(.(t, T212)) → U5_ga(max44_in_ga(T212))
max44_in_ga([]) → max44_out_ga(t)
max44_in_ga(.(t, T76)) → U1_ga(max44_in_ga(T76))
U5_ga(max44_out_ga(T214)) → max150_out_ga(T214)
U1_ga(max44_out_ga(T78)) → max44_out_ga(T78)
del203_in_ga(.(t, T269)) → del203_out_ga(T269)
U9_gga(del203_out_ga(X500)) → del173_out_gga(.(f, .(f, X500)))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(MAXSORT1_IN_GA(x₁)) = x₁   
POL(U1_ga(x₁)) = 1 + x₁   
POL(U25_GA(x₁, x₂)) = 2·x₁ + x₂   
POL(U27_GA(x₁)) = x₁   
POL(U2_ga(x₁)) = x₁   
POL(U4_ga(x₁)) = x₁   
POL(U5_ga(x₁)) = 1 + x₁   
POL(U6_ga(x₁)) = 2·x₁   
POL(U9_gga(x₁)) = 1 + 2·x₁   
POL([]) = 0   
POL(del173_in_gga(x₁, x₂)) = x₁ + 2·x₂   
POL(del173_out_gga(x₁)) = x₁   
POL(del203_in_ga(x₁)) = x₁   
POL(del203_out_ga(x₁)) = 2·x₁   
POL(f) = 0   
POL(max150_in_ga(x₁)) = 2·x₁   
POL(max150_out_ga(x₁)) = x₁   
POL(max44_in_ga(x₁)) = 2 + x₁   
POL(max44_out_ga(x₁)) = x₁   
POL(t) = 1   

(77) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
max44_in_ga(.(f, T76)) → U2_ga(max44_in_ga(T76))
U2_ga(max44_out_ga(T78)) → max44_out_ga(T78)
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max44_in_ga(x0)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del203_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:

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max44_in_ga(x0)
max150_in_ga(x0)
del173_in_gga(x0, x1)
U1_ga(x0)
U2_ga(x0)
U4_ga(x0)
U5_ga(x0)
U9_gga(x0)
del203_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].

max44_in_ga(x0)
U1_ga(x0)
U2_ga(x0)
U5_ga(x0)

(81) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max150_in_ga(x0)
del173_in_gga(x0, x1)
U4_ga(x0)
U9_gga(x0)
del203_in_ga(x0)
U6_ga(x0)

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

(82) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

del173_in_gga(t, T259) → U9_gga(del203_in_ga(T259))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(MAXSORT1_IN_GA(x₁)) = x₁   
POL(U25_GA(x₁, x₂)) = 2·x₁ + x₂   
POL(U27_GA(x₁)) = x₁   
POL(U4_ga(x₁)) = x₁   
POL(U6_ga(x₁)) = 2·x₁   
POL(U9_gga(x₁)) = 2 + x₁   
POL([]) = 0   
POL(del173_in_gga(x₁, x₂)) = x₁ + 2·x₂   
POL(del173_out_gga(x₁)) = x₁   
POL(del203_in_ga(x₁)) = x₁   
POL(del203_out_ga(x₁)) = x₁   
POL(f) = 0   
POL(max150_in_ga(x₁)) = 2·x₁   
POL(max150_out_ga(x₁)) = 2·x₁   
POL(t) = 2   

(83) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))
del203_in_ga([]) → del203_out_ga([])
del203_in_ga(.(f, T275)) → U6_ga(del203_in_ga(T275))
U6_ga(del203_out_ga(X547)) → del203_out_ga(.(f, X547))

The set Q consists of the following terms:

max150_in_ga(x0)
del173_in_gga(x0, x1)
U4_ga(x0)
U9_gga(x0)
del203_in_ga(x0)
U6_ga(x0)

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

(84) 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.

(85) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))

The set Q consists of the following terms:

max150_in_ga(x0)
del173_in_gga(x0, x1)
U4_ga(x0)
U9_gga(x0)
del203_in_ga(x0)
U6_ga(x0)

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

(86) 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].

U9_gga(x0)
del203_in_ga(x0)
U6_ga(x0)

(87) Obligation:

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

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)
MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))
U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))

The set Q consists of the following terms:

max150_in_ga(x0)
del173_in_gga(x0, x1)
U4_ga(x0)

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

(88) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U25_GA(T55, max150_out_ga(T187)) → U27_GA(del173_in_gga(T187, T55))
U27_GA(del173_out_gga(T221)) → MAXSORT1_IN_GA(T221)

The following rules are removed from R:

max150_in_ga([]) → max150_out_ga(f)
max150_in_ga(.(f, T202)) → U4_ga(max150_in_ga(T202))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + 2·x₂   
POL(MAXSORT1_IN_GA(x₁)) = x₁   
POL(U25_GA(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(U27_GA(x₁)) = 1 + x₁   
POL(U4_ga(x₁)) = x₁   
POL([]) = 2   
POL(del173_in_gga(x₁, x₂)) = x₁ + 2·x₂   
POL(del173_out_gga(x₁)) = x₁   
POL(f) = 1   
POL(max150_in_ga(x₁)) = 1 + x₁   
POL(max150_out_ga(x₁)) = 2·x₁   

(89) Obligation:

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

MAXSORT1_IN_GA(.(f, .(f, T55))) → U25_GA(T55, max150_in_ga(T55))

The TRS R consists of the following rules:

U4_ga(max150_out_ga(T204)) → max150_out_ga(T204)
del173_in_gga(f, T236) → del173_out_gga(.(f, T236))

The set Q consists of the following terms:

max150_in_ga(x0)
del173_in_gga(x0, x1)
U4_ga(x0)

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

(90) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(91) TRUE